CHAPTER_NN1 8_MC Polymers_NN2 in_II Solution_NN1 8.1_MC Thermodynamics_NN1 of_IO polymer_NN1 solutions_NN2 The_AT interaction_NN1 of_IO long_JJ chain_NN1 molecules_NN2 with_IW liquids_NN2 is_VBZ of_IO considerable_JJ interest_NN1 from_II both_RR a_AT1 practical_JJ and_CC theoretical_JJ viewpoint_NN1 ._. 
For_IF linear_JJ and_CC branched_JJ polymers_NN2 ,_, liquids_NN2 can_VM usually_RR be_VBI found_VVN which_DDQ will_VM dissolve_VVI the_AT polymer_NN1 completely_RR to_TO form_VVI a_AT1 homogeneous_JJ solution_NN1 ,_, whereas_CS cross-linked_JJ networks_NN2 will_VM only_RR swell_VVI when_RRQ in_II31 contact_II32 with_II33 compatible_JJ liquids_NN2 ._. 
In_II this_DD1 chapter_NN1 we_PPIS2 shall_VM deal_VVI with_IW linear_JJ or_CC branched_JJ polymers_NN2 and_CC treat_VV0 the_AT swelling_JJ of_IO networks_NN2 in_II chapter_NN1 14_MC ._. 
When_CS an_AT1 amorphous_JJ polymer_NN1 is_VBZ mixed_VVN with_IW a_AT1 suitable_JJ solvent_NN1 ,_, it_PPH1 disperses_VVZ in_II the_AT solvent_NN1 and_CC behaves_VVZ as_CS21 though_CS22 it_PPH1 too_RR is_VBZ a_AT1 liquid_NN1 ._. 
In_II a_AT1 good_JJ solvent_NN1 ,_, classed_VVN as_CSA one_PN1 which_DDQ is_VBZ highly_RR compatible_JJ with_IW the_AT polymer_NN1 ,_, the_AT liquid-polymer_JJ interactions_NN2 expand_VV0 the_AT polymer_NN1 coil_NN1 ,_, from_II its_APPGE unperturbed_JJ dimensions_NN2 ,_, in_II31 proportion_II32 to_II33 the_AT extent_NN1 of_IO these_DD2 interactions_NN2 ._. 
In_II a_AT1 '_GE poor_JJ '_GE solvent_NN1 ,_, the_AT interactions_NN2 are_VBR fewer_DAR and_CC coil_VV0 expansion_NN1 or_CC perturbation_NN1 is_VBZ restricted_VVN ._. 
The_AT fundamental_JJ thermodynamic_JJ equation_NN1 used_VMK to_TO describe_VVI these_DD2 systems_NN2 relates_VVZ the_AT Gibbs_NP1 free_JJ energy_NN1 function_NN1 G_ZZ1 to_II the_AT enthalpy_NN1 H_ZZ1 and_CC entropy_NN1 S_ZZ1 ,_, i.e_REX .._... 
A_AT1 homogeneous_JJ solution_NN1 is_VBZ obtained_VVN when_CS the_AT Gibbs_NP1 free_JJ energy_NN1 of_IO mixing_NN1 ,_, i.e._REX when_CS the_AT Gibbs_NP1 free_JJ energy_NN1 of_IO the_AT solution_NN1 G_ZZ1 12_MC is_VBZ lower_JJR than_CSN the_AT Gibbs_NP1 functions_NN2 of_IO the_AT components_NN2 of_IO the_AT mixture_NN1 G_ZZ1 1_MC1 and_CC G_ZZ1 2._MC 8.2_MC Ideal_JJ mixtures_NN2 of_IO small_JJ molecules_NN2 To_TO understand_VVI the_AT behaviour_NN1 of_IO polymers_NN2 in_II solution_NN1 more_RGR fully_RR ,_, a_AT1 knowledge_NN1 of_IO the_AT enthalpic_JJ and_CC entropic_JJ contributions_NN2 to_II G_ZZ1 M_ZZ1 is_VBZ essential_JJ ,_, and_CC it_PPH1 is_VBZ instructive_JJ to_TO consider_VVI first_MD mixtures_NN2 of_IO small_JJ molecules_NN2 ,_, to_TO establish_VVI some_DD fundamental_JJ rules_NN2 concerning_II ideal_JJ and_CC non-ideal_JJ behaviour_NN1 ._. 
Raoult_NP1 's_GE law_NN1 is_VBZ a_AT1 useful_JJ starting_NN1 point_NN1 and_CC defines_VVZ an_AT1 ideal_JJ solution_NN1 as_CSA one_PN1 in_II which_DDQ the_AT activity_NN1 of_IO each_DD1 component_NN1 in_II a_AT1 mixture_NN1 a_AT1 i_ZZ1 is_VBZ equal_JJ to_II its_APPGE mole_NN1 fraction_NN1 x_ZZ1 i_ZZ1 ._. 
This_DD1 is_VBZ valid_JJ only_RR for_IF components_NN2 of_IO comparable_JJ size_NN1 ,_, and_CC where_CS the_AT intermolecular_JJ forces_NN2 acting_VVG between_II both_DB2 like_VV0 and_CC unlike_JJ molecules_NN2 are_VBR equal_JJ ._. 
The_AT latter_DA requirement_NN1 means_VVZ that_DD1 component_NN1 molecules_NN2 of_IO each_DD1 species_NN can_VM interchange_NN1 positions_NN2 without_IW altering_VVG the_AT total_JJ energy_NN1 of_IO the_AT system_NN1 ,_, i.e._REX and_CC consequently_RR it_PPH1 only_RR remains_VVZ for_IF the_AT entropy_NN1 contribution_NN1 S_ZZ1 M_ZZ1 to_TO be_VBI calculated_VVN ._. 
For_IF a_AT1 system_NN1 in_II a_AT1 given_JJ state_NN1 ,_, the_AT entropy_NN1 is_VBZ related_VVN to_II the_AT number_NN1 of_IO distinguishable_JJ arrangements_NN2 the_AT components_NN2 in_II that_DD1 state_NN1 can_VM adopt_VVI ,_, and_CC can_VM be_VBI calculated_VVN from_II the_AT Boltzmann_NP1 law_NN1 ,_, where_CS W_ZZ1 is_VBZ the_AT number_NN1 of_IO statistical_JJ microstates_NN2 available_JJ to_II the_AT system_NN1 ._. 
We_PPIS2 can_VM begin_VVI by_II considering_VVG the_AT mixing_NN1 of_IO N_ZZ1 1_MC1 molecules_NN2 of_IO component_NN1 (_( 1_MC1 )_) with_IW N_ZZ1 2_MC molecules_NN2 of_IO component_NN1 (_( 2_MC )_) and_CC this_DD1 can_VM be_VBI assumed_VVN to_TO take_VVI place_NN1 on_II a_AT1 hypothetical_JJ lattice_NN1 containing_VVG cells_NN2 of_IO equal_JJ size_NN1 ._. 
Although_CS this_DD1 formalism_NN1 is_VBZ not_XX strictly_RR necessary_JJ for_IF the_AT analysis_NN1 ,_, the_AT arrangement_NN1 of_IO spherical_JJ molecules_NN2 of_IO equal_JJ size_NN1 in_II the_AT liquid_JJ state_NN1 will_VM ,_, to_II the_AT first_MD near_II neighbour_NN1 approximation_NN1 ,_, be_VBI similar_JJ to_II a_AT1 regular_JJ lattice_NN1 structure_NN1 and_CC so_RR it_PPH1 is_VBZ a_AT1 useful_JJ structure_NN1 to_TO use_VVI as_II a_AT1 framework_NN1 for_IF the_AT mixing_NN1 process_NN1 ._. 
The_AT total_JJ number_NN1 of_IO possible_JJ ways_NN2 in_II which_DDQ the_AT component_NN1 molecules_NN2 can_VM be_VBI arranged_VVN on_II the_AT lattice_NN1 increases_VVZ when_RRQ mixing_NN1 takes_VVZ place_NN1 and_CC is_VBZ equal_JJ to_TO ,_, but_CCB as_CSA the_AT interchanging_NN1 of_IO a_AT1 molecule_NN1 of_IO (_( 1_MC1 )_) with_IW another_DD1 molecule_NN1 (_( 1_MC1 )_) ,_, or_CC (_( 2_MC )_) with_IW (_( 2_MC )_) will_VM be_VBI an_AT1 indistinguishable_JJ process_NN1 ,_, the_AT net_JJ number_NN1 of_IO distinguishable_JJ arrangements_NN2 will_VM be_VBI The_AT configurational_JJ (_( or_CC combinatorial_JJ )_) entropy_NN1 S_ZZ1 c_ZZ1 can_VM then_RT be_VBI derived_VVN from_II the_AT Boltzmann_NP1 law_NN1 and_CC For_IF large_JJ values_NN2 of_IO N_ZZ1 i_ZZ1 ,_, Stirling_NP1 's_GE approximation_NN1 can_VM be_VBI used_VVN to_TO deal_VVI with_IW the_AT factorials_NN2 viz._REX In_II ,_, and_CC equation_NN1 (_( 8.3_MC )_) becomes_VVZ which_DDQ on_II dividing_VVG by_II N_ZZ1 o_ZZ1 gives_VVZ If_CS ,_, the_AT mole_NN1 fraction_NN1 of_IO component_NN1 i_ZZ1 ,_, then_RT For_IF the_AT pure_JJ components_NN2 ,_, ,_, and_CC as_CSA S_ZZ1 M_ZZ1 ,_, the_AT change_NN1 in_II entropy_NN1 on_II mixing_NN1 ,_, is_VBZ given_VVN by_II then_RT we_PPIS2 can_VM write_VVI so_RR for_IF a_AT1 two_MC component_JJ mixture_NN1 This_DD1 expression_NN1 is_VBZ derived_VVN assuming_VVG (_( a_ZZ1 )_) the_AT volume_NN1 change_NN1 on_II mixing_NN1 ,_, (_( b_ZZ1 )_) the_AT molecules_NN2 are_VBR all_DB of_IO equal_JJ size_NN1 ,_, (_( c_ZZ1 )_) all_DB possible_JJ arrangements_NN2 have_VH0 the_AT same_DA energy_NN1 ,_, ,_, and_CC (_( d_ZZ1 )_) the_AT motion_NN1 of_IO the_AT components_NN2 about_II their_APPGE equilibrium_NN1 positions_NN2 remains_VVZ unchanged_JJ on_II mixing_NN1 ._. 
Thus_RR the_AT free_JJ energy_NN1 of_IO mixing_NN1 ,_, G_ZZ1 M_ZZ1 is_VBZ which_DDQ shows_VVZ that_CST mixing_VVG in_II ideal_JJ systems_NN2 is_VBZ an_AT1 entropically_RR driven_VVN ,_, spontaneous_JJ process._NNU 8.3_MC Non-ideal_JJ solutions_NN2 Any_DD deviations_NN2 from_II assumptions_NN2 (_( a_ZZ1 )_) to_II (_( d_ZZ1 )_) will_VM constitute_VVI a_AT1 deviation_NN1 from_II ideality_NN1 (_( an_AT1 ideal_JJ solution_NN1 is_VBZ a_AT1 rare_JJ occurrence_NN1 )_) and_CC several_DA2 more_DAR realistic_JJ types_NN2 of_IO solution_NN1 can_VM be_VBI identified_VVN :_: (_( i_ZZ1 )_) Athermal_JJ solutions_NN2 ;_; where_RRQ but_CCB S_ZZ1 M_ZZ1 is_VBZ not_XX ideal_JJ (_( ii_MC )_) Regular_JJ solutions_NN2 ;_; where_RRQ S_ZZ1 M_ZZ1 is_VBZ ideal_JJ but_CCB ,_, (_( iii_MC )_) Irregular_JJ solutions_NN2 ;_; in_II which_DDQ both_RR S_ZZ1 M_ZZ1 and_CC H_ZZ1 M_ZZ1 deviate_VV0 from_II their_APPGE ideal_JJ values_NN2 ._. 
Polymer_NN1 solutions_NN2 tend_VV0 to_TO fall_VVI into_II category_NN1 (_( iii_MC )_) and_CC the_AT non-ideal_JJ behaviour_NN1 can_VM be_VBI attributed_VVN not_XX only_RR to_II the_AT existence_NN1 of_IO a_AT1 finite_JJ heat_NN1 of_IO mixing_NN1 but_CCB also_RR to_II the_AT large_JJ difference_NN1 in_II size_NN1 between_II the_AT polymer_NN1 and_CC solvent_NN1 molecules_NN2 ._. 
The_AT polymer_NN1 chain_NN1 can_VM be_VBI regarded_VVN as_II a_AT1 series_NN of_IO small_JJ segments_NN2 covalently_RR bonded_VVD together_RL and_CC it_PPH1 is_VBZ the_AT effect_NN1 of_IO this_DD1 chain_NN1 connectivity_NN1 which_DDQ leads_VVZ to_II deviations_NN2 from_II an_AT1 ideal_JJ entropy_NN1 of_IO mixing_NN1 ._. 
The_AT effect_NN1 of_IO connectivity_NN1 can_VM be_VBI assessed_VVN by_II calculating_VVG the_AT entropy_NN1 change_NN1 associated_VVN with_IW the_AT different_JJ number_NN1 of_IO ways_NN2 of_IO arranging_VVG polymer_NN1 chains_NN2 and_CC solvent_NN1 molecules_NN2 on_II a_AT1 lattice_NN1 and_CC ,_, as_CSA it_PPH1 will_VM be_VBI demonstrated_VVN ,_, this_DD1 differs_VVZ from_II that_DD1 calculated_VVD for_IF the_AT ideal_JJ solution_NN1 ._. 
This_DD1 is_VBZ embodied_VVN in_II the_AT theory_NN1 developed_VVN by_II Flory_NP1 and_CC Huggins_NP1 ,_, but_CCB still_RR represents_VVZ only_RR the_AT combinatorial_JJ contribution_NN1 ,_, whereas_CS there_EX are_VBR other_JJ (_( non-combinatorial_JJ )_) contributions_NN2 to_II the_AT entropy_NN1 which_DDQ come_VV0 from_II the_AT interaction_NN1 of_IO the_AT polymer_NN1 with_IW the_AT solvent_NN1 and_CC are_VBR much_RR harder_JJR to_TO quantify_VVI ._. 
Nevertheless_RR ,_, the_AT Flory-Huggins_NP1 theory_NN1 forms_VVZ the_AT cornerstone_NN1 of_IO polymer_NN1 solution_NN1 thermodynamics_NN1 and_CC is_VBZ worth_II considering_VVG further._NNU 8.4_MC Flory-Huggins_NP1 theory_NN1 The_AT dissolution_NN1 of_IO a_AT1 polymer_NN1 in_II a_AT1 solvent_NN1 can_VM be_VBI regarded_VVN as_II a_AT1 two_MC stage_NN1 process_NN1 ._. 
The_AT polymer_NN1 exists_VVZ initially_RR in_II the_AT solid_JJ state_NN1 where_CS it_PPH1 is_VBZ restricted_VVN to_II only_RR one_MC1 of_IO the_AT many_DA2 conformations_NN2 which_DDQ are_VBR available_JJ to_II it_PPH1 as_II a_AT1 free_JJ isolated_JJ molecule_NN1 ._. 
On_II passing_VVG into_II the_AT liquid_JJ solution_NN1 the_AT chain_NN1 achieves_VVZ relative_JJ freedom_NN1 and_CC can_VM now_RT change_VVI rapidly_RR among_II a_AT1 multitude_NN1 of_IO possible_JJ equi-energetic_JJ conformations_NN2 ,_, dictated_VVD partly_RR by_II the_AT chain_NN1 flexibility_NN1 and_CC partly_RR by_II the_AT interactions_NN2 with_IW the_AT solvent_NN1 ._. 
Flory_NN1 and_CC Huggins_NP1 considered_VVD that_DD1 formation_NN1 of_IO the_AT solution_NN1 depends_VVZ on_II (_( a_ZZ1 )_) the_AT transfer_NN1 of_IO the_AT polymer_NN1 chain_NN1 from_II a_AT1 pure_JJ ,_, perfectly_RR ordered_VVN state_NN1 to_II a_AT1 state_NN1 of_IO disorder_NN1 which_DDQ has_VHZ the_AT necessary_JJ freedom_NN1 to_TO allow_VVI the_AT chain_NN1 to_TO be_VBI placed_VVN randomly_RR on_II a_AT1 lattice_NN1 ,_, and_CC (_( b_ZZ1 )_) the_AT mixing_NN1 process_NN1 of_IO the_AT flexible_JJ chains_NN2 with_IW solvent_NN1 molecules_NN2 ._. 
The_AT formalism_NN1 of_IO the_AT lattice_NN1 was_VBDZ used_VVN ,_, for_IF convenience_NN1 ,_, to_TO calculate_VVI the_AT combinatorial_JJ entropy_NN1 of_IO mixing_VVG following_II the_AT method_NN1 outlined_VVN in_II section_NN1 8.2_MC for_IF small_JJ molecules_NN2 ,_, including_II the_AT same_DA starting_NN1 assumptions_NN2 and_CC restrictions_NN2 ._. 
ENTROPY_NN1 OF_IO MIXING_VVG FOR_IF ATHERMAL_JJ POLYMER_NN1 SOLUTIONS_NN2 Consider_VV0 a_AT1 polymer_NN1 chain_NN1 consisting_VVG of_IO r_ZZ1 covalently_RR bonded_VVD segments_NN2 whose_DDQGE size_NN1 is_VBZ the_AT same_DA as_CSA the_AT solvent_NN1 molecules_NN2 ,_, i.e._REX where_CS V_ZZ1 1_MC1 is_VBZ the_AT molar_JJ volume_NN1 of_IO component_NN1 i_ZZ1 ._. 
To_TO calculate_VVI the_AT number_NN1 of_IO ways_NN2 this_DD1 chain_NN1 can_VM be_VBI added_VVN to_II a_AT1 lattice_NN1 ,_, the_AT necessary_JJ restriction_NN1 imposed_VVN is_VBZ that_CST the_AT segments_NN2 must_VM occupy_VVI r_ZZ1 contiguous_JJ sites_NN2 on_II the_AT lattice_NN1 because_II21 of_II22 the_AT connectivity_NN1 ._. 
The_AT problem_NN1 is_VBZ to_TO examine_VVI the_AT mixing_NN1 of_IO N_ZZ1 1_MC1 solvent_NN1 molecules_NN2 with_IW N_ZZ1 2_MC monodisperse_NN1 polymer_NN1 molecules_NN2 comprising_VVG r_ZZ1 segments_NN2 and_CC we_PPIS2 can_VM begin_VVI by_II adding_VVG i_MC1 polymer_NN1 molecules_NN2 to_II an_AT1 empty_JJ lattice_NN1 with_IW a_AT1 total_JJ number_NN1 of_IO cells_NN2 N_ZZ1 Thus_RR the_AT number_NN1 of_IO vacant_JJ cells_NN2 left_VVD which_DDQ can_VM accommodate_VVI the_AT next_MD molecule_NN1 will_VM be_VBI The_AT molecule_NN1 can_VM now_RT be_VBI placed_VVN on_II the_AT lattice_NN1 ,_, segment_NN1 by_II segment_NN1 ,_, bearing_VVG in_II mind_NN1 the_AT restrictions_NN2 imposed_VVD ,_, viz._REX the_AT connectivity_NN1 of_IO the_AT segments_NN2 ,_, which_DDQ requires_VVZ the_AT placing_NN1 of_IO each_DD1 segment_NN1 in_II a_AT1 cell_NN1 adjoining_VVG the_AT preceding_JJ one_PN1 ._. 
This_DD1 in_II turn_NN1 will_VM depend_VVI on_II the_AT availability_NN1 of_IO a_AT1 suitable_JJ vacancy_NN1 ._. 
The_AT first_MD segment_NN1 can_VM be_VBI placed_VVN in_II any_DD empty_JJ cell_NN1 but_CCB the_AT second_MD segment_NN1 is_VBZ restricted_VVN to_II the_AT immediate_JJ near_II neighbours_NN2 surrounding_VVG the_AT first_MD ._. 
This_DD1 can_VM be_VBI given_VVN by_II the_AT co-ordination_NN1 number_NN1 of_IO the_AT lattice_NN1 z_ZZ1 but_CCB we_PPIS2 must_VM also_RR know_VVI if_CSW a_AT1 cell_NN1 in_II the_AT co-ordination_NN1 shell_NN1 is_VBZ empty_JJ ._. 
If_CS we_PPIS2 let_VV0 p_ZZ1 i_ZZ1 be_VBI the_AT probability_NN1 that_CST an_AT1 adjacent_JJ cell_NN1 is_VBZ vacant_JJ ,_, then_RT to_II a_AT1 reasonable_JJ approximation_NN1 this_DD1 can_VM be_VBI equated_VVN with_IW the_AT fraction_NN1 of_IO cells_NN2 occupied_VVN by_II i_MC1 polymer_NN1 chains_NN2 on_II the_AT lattice_NN1 i.e._REX which_DDQ is_VBZ valid_JJ for_IF large_JJ values_NN2 of_IO z_ZZ1 ._. 
So_RR the_AT expected_JJ number_NN1 of_IO empty_JJ cells_NN2 available_JJ for_IF the_AT second_MD segment_NN1 is_VBZ zp_NNU i_ZZ1 ,_, and_CC having_VHG removed_VVN one_PN1 more_RGR vacant_JJ cell_NN1 from_II the_AT immediate_JJ vicinity_NN1 ,_, the_AT third_MD and_CC each_DD1 succeeding_JJ segment_NN1 will_VM have_VHI empty_JJ cells_NN2 to_TO choose_VVI from_II ._. 
The_AT total_JJ number_NN1 of_IO ways_NN2 in_II which_DDQ the_AT molecule_NN1 can_VM be_VBI placed_VVN on_II the_AT lattice_NN1 is_VBZ then_RT This_DD1 gives_VVZ the_AT set_NN1 of_IO possible_JJ ways_NN2 in_II which_DDQ the_AT molecule_NN1 can_VM be_VBI accommodated_VVN on_II the_AT lattice_NN1 ._. 
The_AT total_JJ number_NN1 of_IO ways_NN2 for_IF all_DB N_ZZ1 2_MC molecules_NN2 to_TO be_VBI placed_VVN can_VM then_RT be_VBI obtained_VVN from_II the_AT product_NN1 of_IO all_DB possible_JJ ways_NN2 ,_, i.e._REX The_AT polymer_NN1 molecules_NN2 are_VBR all_DB identical_JJ and_CC so_RR by_II analogy_NN1 with_IW equation_NN1 (_( 8.2_MC )_) the_AT total_JJ number_NN1 of_IO distinguishable_JJ ways_NN2 of_IO adding_VVG N_ZZ1 2_MC polymer_NN1 molecules_NN2 is_VBZ Substituting_VVG for_IF W_ZZ1 i_ZZ1 gives_VVZ To_TO evaluate_VVI the_AT product_NN1 term_NN1 we_PPIS2 can_VM multiply_VVI and_CC divide_VVI by_II r_ZZ1 This_DD1 can_VM be_VBI converted_VVN into_II the_AT more_RGR convenient_JJ factorial_JJ form_NN1 by_II remembering_VVG that_CST the_AT product_NN1 is_VBZ equivalent_JJ to_II and_CC so_RR equation_NN1 (_( 8.14_MC )_) can_VM be_VBI written_VVN as_II The_AT remaining_JJ empty_JJ cells_NN2 on_II the_AT lattice_NN1 can_VM now_RT be_VBI filled_VVN by_II solvent_NN1 molecules_NN2 ,_, but_CCB as_CSA there_EX is_VBZ only_RR one_MC1 distinguishable_JJ way_NN1 in_II which_DDQ this_DD1 can_VM be_VBI done_VDN ,_, ,_, there_EX is_VBZ no_AT further_JJR contribution_NN1 to_II W_ZZ1 p_ZZ1 and_CC the_AT entropy_NN1 of_IO the_AT system_NN1 ._. 
The_AT latter_DA can_VM now_RT be_VBI calculated_VVN from_II the_AT Boltzmann_NP1 equation_NN1 ._. 
The_AT factorials_NN2 can_VM again_RT be_VBI approximated_VVN using_VVG Stirling_NP1 's_GE relation_NN1 and_CC while_CS this_DD1 requires_VVZ considerable_JJ manipulation_NN1 ,_, which_DDQ will_VM be_VBI omitted_VVN here_RL ,_, it_PPH1 can_VM eventually_RR be_VBI shown_VVN that_CST To_TO convert_VVI this_DD1 into_II a_AT1 form_NN1 which_DDQ will_VM allow_VVI us_PPIO2 to_TO express_VVI this_DD1 in_II the_AT correct_JJ site_NN1 fraction_NN1 form_VV0 we_PPIS2 can_VM add_VVI and_CC subtract_VVI on_II the_AT r.h.s._NNU of_IO equation_NN1 (_( 8.19_MC )_) to_TO give_VVI For_IF the_AT pure_JJ solvent_NN1 and_CC the_AT entropy_NN1 ._. 
Similarly_RR the_AT entropy_NN1 of_IO the_AT pure_JJ polymer_NN1 S_ZZ1 2_MC can_VM be_VBI obtained_VVN for_IF ,_, which_DDQ gives_VVZ Equation_NN1 8.21_MC then_RT represents_VVZ the_AT entropy_NN1 associated_VVN with_IW the_AT disordered_JJ or_CC amorphous_JJ polymer_NN1 on_II the_AT lattice_NN1 in_II the_AT absence_NN1 of_IO solvent._NNU and_CC so_RR It_PPH1 follows_VVZ that_CST the_AT entropy_NN1 change_NN1 on_II mixing_VVG disordered_JJ polymer_NN1 and_CC solvent_NN1 and_CC so_RR where_CS i_ZZ1 the_AT volume_NN1 fraction_NN1 can_VM replace_VVI the_AT site_NN1 fraction_NN1 if_CS it_PPH1 is_VBZ considered_VVN that_CST the_AT number_NN1 of_IO sites_NN2 occupied_VVN by_II the_AT polymer_NN1 and_CC solvent_NN1 is_VBZ proportional_JJ to_II their_APPGE respective_JJ volumes_NN2 ._. 
Equation_NN1 (_( 8.22_MC )_) is_VBZ the_AT expression_NN1 for_IF the_AT combinatorial_JJ entropy_NN1 of_IO mixing_NN1 of_IO an_AT1 athermal_JJ polymer_NN1 solution_NN1 and_CC comparison_NN1 with_IW equation_NN1 (_( 8.7_MC )_) shows_VVZ that_CST they_PPHS2 are_VBR similar_JJ in_II form_NN1 except_II21 for_II22 the_AT fact_NN1 that_CST now_RT the_AT volume_NN1 fraction_NN1 is_VBZ found_VVN to_TO be_VBI the_AT most_RGT convenient_JJ way_NN1 of_IO expressing_VVG the_AT entropy_NN1 change_NN1 ,_, rather_II21 than_II22 the_AT mole_NN1 fraction_NN1 used_VVN for_IF small_JJ molecules_NN2 ._. 
This_DD1 change_NN1 arises_VVZ from_II the_AT differences_NN2 in_II size_NN1 between_II the_AT components_NN2 which_DDQ would_VM normally_RR mean_VVI mole_NN1 fractions_NN2 close_RR to_II unity_NN1 for_IF the_AT solvent_NN1 especially_RR when_CS dilute_JJ solutions_NN2 are_VBR being_VBG studied_VVN ._. 
We_PPIS2 can_VM gain_VVI a_AT1 further_JJR understanding_NN1 of_IO how_RRQ the_AT size_NN1 of_IO the_AT polymer_NN1 chain_NN1 affects_VVZ the_AT magnitude_NN1 of_IO S_ZZ1 M_ZZ1 and_CC why_RRQ it_PPH1 differs_VVZ from_II (_( equation_NN1 8.7_MC )_) ,_, by_II recasting_VVG equation_NN1 (_( 8.22_MC )_) in_II the_AT following_JJ way_NN1 ._. 
The_AT volume_NN1 fraction_NN1 i_ZZ1 can_VM be_VBI expressed_VVN in_II31 terms_II32 of_II33 the_AT number_NN1 of_IO moles_NN2 n_ZZ1 i_ZZ1 ,_, and_CC the_AT volume_NN1 V_ZZ1 i_MC1 of_IO component_NN1 i_ZZ1 ,_, as_CSA where_CS V_ZZ1 is_VBZ the_AT total_JJ volume_NN1 and_CC ._. 
If_CS n_ZZ1 i_ZZ1 is_VBZ converted_VVN to_II molar_JJ quantities_NN2 then_RT As_CSA V_ZZ1 i_ZZ1 can_VM conveniently_RR be_VBI expressed_VVN as_II a_AT1 function_NN1 of_IO a_AT1 reference_NN1 volume_NN1 V_ZZ1 o_ZZ1 such_CS21 that_CS22 and_CC assuming_VVG that_CST ,_, without_IW introducing_VVG significant_JJ error_NN1 ,_, r_ZZ1 can_VM be_VBI equated_VVN with_IW the_AT degree_NN1 of_IO polymerization_NN1 for_IF the_AT polymer_NN1 then_RT If_CS the_AT volume_NN1 fraction_NN1 form_NN1 is_VBZ retained_VVN ,_, then_RT for_IF a_AT1 simple_JJ liquid_JJ mixture_NN1 ,_, but_II21 for_II22 a_AT1 polymer_NN1 solution_NN1 and_CC the_AT last_MD term_NN1 in_II equation_NN1 (_( 8.24_MC )_) will_VM be_VBI smaller_JJR than_CSN the_AT equivalent_JJ term_NN1 calculated_VVN for_IF small_JJ molecules_NN2 ._. 
Consequently_RR S_ZZ1 M_ZZ1 per_II mole_NN1 of_IO lattice_NN1 sites_NN2 (_( or_CC equivalent_JJ volume_NN1 )_) will_VM be_VBI very_RG much_DA1 less_DAR than_CSN and_CC the_AT contribution_NN1 of_IO the_AT combinatorial_JJ entropy_NN1 to_II the_AT mixing_NN1 process_NN1 in_II a_AT1 polymer_NN1 solution_NN1 is_VBZ not_XX as_RG large_JJ as_CSA that_DD1 for_IF solutions_NN2 of_IO small_JJ molecules_NN2 when_CS calculated_VVN in_II31 terms_II32 of_II33 volume_NN1 fractions_NN2 and_CC expressed_VVN as_CSA per_II mole_NN1 of_IO sites._NNU 8.5_MC Enthalpy_NN1 change_NN1 on_II mixing_VVG The_AT derivation_NN1 of_IO S_ZZ1 M_ZZ1 from_II the_AT lattice_NN1 theory_NN1 has_VHZ been_VBN made_VVN on_II the_AT assumption_NN1 that_CST no_AT heat_NN1 or_CC energy_NN1 change_NN1 occurs_VVZ on_II mixing_NN1 ._. 
This_DD1 is_VBZ an_AT1 uncommon_JJ situation_NN1 as_CSA experimental_JJ experience_NN1 suggests_VVZ that_CST the_AT energy_NN1 change_NN1 is_VBZ finite_JJ ._. 
We_PPIS2 can_VM make_VVI use_NN1 of_IO regular_JJ solution_NN1 theory_NN1 to_TO obtain_VVI an_AT1 expression_NN1 for_IF H_ZZ1 M_ZZ1 where_CS this_DD1 change_NN1 in_II energy_NN1 is_VBZ assumed_VVN to_TO arise_VVI from_II the_AT formation_NN1 of_IO new_JJ solvent-polymer_NN1 (_( 12_MC )_) contacts_NN2 on_II mixing_VVG which_DDQ replace_VV0 some_DD of_IO the_AT (_( 11_MC )_) and_CC (_( 22_MC )_) contacts_NN2 present_VV0 in_II the_AT pure_JJ solvent_NN1 ,_, and_CC the_AT pure_JJ polymer_NN1 components_NN2 respectively_RR ._. 
This_DD1 can_VM be_VBI represented_VVN by_II a_AT1 quasi-chemical_JJ process_NN1 where_CS the_AT formation_NN1 of_IO a_AT1 solvent-polymer_JJ contact_NN1 requires_VVZ first_MD the_AT breaking_NN1 of_IO (_( 11_MC )_) and_CC (_( 22_MC )_) contacts_NN2 ,_, and_CC can_VM be_VBI expressed_VVN as_II an_AT1 interchange_NN1 energy_NN1 A_ZZ1 12_MC per_II contact_NN1 ,_, given_VVN by_II Here_RL ii_MC and_CC ij_NN1 are_VBR the_AT contact_NN1 energies_NN2 for_IF each_DD1 species_NN ._. 
The_AT energy_NN1 of_IO mixing_VVG U_ZZ1 M_ZZ1 can_VM be_VBI replaced_VVN by_II H_ZZ1 M_ZZ1 if_CS no_AT volume_NN1 change_NN1 takes_VVZ place_NN1 on_II mixing_NN1 ,_, and_CC for_IF q_ZZ1 new_JJ contacts_NN2 formed_VVN in_II solution_NN1 The_AT number_NN1 of_IO contacts_NN2 can_VM be_VBI estimated_VVN from_II the_AT lattice_NN1 model_NN1 by_II assuming_VVG that_CST the_AT probability_NN1 of_IO having_VHG a_AT1 lattice_NN1 cell_NN1 occupied_VVN by_II a_AT1 solvent_NN1 molecule_NN1 is_VBZ simply_RR the_AT volume_NN1 fraction_NN1 1_MC1 ._. 
This_DD1 means_VVZ that_CST each_DD1 polymer_NN1 molecule_NN1 will_VM be_VBI surrounded_VVN by_II solvent_NN1 molecules_NN2 ,_, and_CC for_IF N_ZZ1 2_MC polymer_NN1 molecules_NN2 From_II the_AT definition_NN1 of_IO 2_MC we_PPIS2 obtain_VV0 ,_, hence_RR which_DDQ is_VBZ the_AT van_NP1 Laar_NP1 expression_NN1 derived_VVN for_IF regular_JJ solutions_NN2 and_CC shows_VVZ that_CST this_DD1 approach_NN1 can_VM be_VBI applied_VVN to_II polymer_NN1 systems_NN2 ._. 
To_TO eliminate_VVI z_ZZ1 ,_, a_AT1 dimensionless_JJ parameter_NN1 per_II solvent_NN1 molecule_NN1 ,_, is_VBZ defined_VVN as_II which_DDQ is_VBZ the_AT difference_NN1 in_II energy_NN1 between_II a_AT1 solvent_NN1 molecule_NN1 when_CS it_PPH1 is_VBZ immersed_VVN in_II pure_JJ polymer_NN1 and_CC when_CS in_II pure_JJ solvent_NN1 ._. 
It_PPH1 can_VM also_RR be_VBI expressed_VVN in_II the_AT alternative_JJ form_NN1 ,_, where_CS B_ZZ1 is_VBZ now_RT an_AT1 interaction_NN1 density_NN1 ._. 
The_AT final_JJ expression_NN1 is_VBZ and_CC the_AT interaction_NN1 parameter_NN1 1_MC1 is_VBZ an_AT1 important_JJ feature_NN1 of_IO polymer_NN1 solution_NN1 theory_NN1 which_DDQ will_VM be_VBI met_VVN with_IW frequently._NNU 8.6_MC Free_JJ energy_NN1 of_IO mixing_NN1 Having_VHG calculated_VVN the_AT entropy_NN1 and_CC enthalpy_NN1 contributions_NN2 to_II mixing_NN1 ,_, these_DD2 can_VM now_RT be_VBI combined_VVN to_TO give_VVI the_AT expression_NN1 for_IF the_AT free_JJ energy_NN1 of_IO mixing_NN1 ,_, as_CSA It_PPH1 is_VBZ more_RGR useful_JJ to_TO express_VVI equation_NN1 (_( 8.32_MC )_) in_II31 terms_II32 of_II33 the_AT chemical_JJ potentials_NN2 of_IO the_AT pure_JJ solvent_NN1 and_CC the_AT solvent_NN1 in_II solution_NN1 ,_, by_II differentiating_VVG the_AT expression_NN1 with_II31 respect_II32 to_II33 the_AT number_NN1 of_IO solvent_NN1 molecules_NN2 N_ZZ1 1_MC1 to_TO obtain_VVI the_AT partial_JJ molar_NN1 Gibbs_NP1 free_JJ energy_NN1 of_IO dilution_NN1 (_( after_II multiplying_VVG by_II Avogadro_NP1 's_GE number_NN1 )_) ,_, This_DD1 could_VM also_RR be_VBI carried_VVN out_RP for_IF the_AT polymer_NN1 ,_, but_CCB as_CSA it_PPH1 makes_VVZ no_AT difference_NN1 which_DDQ one_PN1 is_VBZ taken_VVN (_( both_RR having_VHG started_VVN from_II G_ZZ1 M_ZZ1 )_) ,_, equation_NN1 (_( 8.33_MC )_) is_VBZ more_RGR convenient_JJ to_TO use_VVI ._. 
While_CS this_DD1 expression_NN1 is_VBZ not_XX strictly_RR valid_JJ for_IF the_AT dilute_JJ solution_NN1 regime_NN1 it_PPH1 can_VM be_VBI converted_VVN into_II a_AT1 structure_NN1 which_DDQ is_VBZ extremely_RR informative_JJ about_II deviations_NN2 from_II ideal_JJ solution_NN1 behaviour_NN1 encountered_VVN when_CS measuring_VVG the_AT molar_JJ mass_NN1 by_II techniques_NN2 such_II21 as_II22 osmotic_JJ pressure_NN1 ._. 
If_CS the_AT logarithmic_JJ term_NN1 is_VBZ expanded_VVN using_VVG a_AT1 Taylor_NP1 series_NN :_: but_CCB truncated_VVN after_II the_AT squared_JJ term_NN1 ,_, assuming_VVG 2_MC is_VBZ small_JJ ,_, then_RT ,_, This_DD1 can_VM be_VBI modified_VVN by_II remembering_VVG that_DD1 and_CC ,_, where_CS v_ZZ1 2_MC is_VBZ the_AT partial_JJ specific_JJ volume_NN1 of_IO the_AT polymer_NN1 ._. 
This_DD1 can_VM be_VBI related_VVN to_II the_AT polymer_NN1 molecular_JJ weight_NN1 M_ZZ1 2_MC through_RP so_CS21 that_CS22 and_CC finally_RR Let_VV0 us_PPIO2 now_RT anticipate_VVI the_AT molar_JJ mass_JJ measurements_NN2 to_TO be_VBI described_VVN in_II chapter_NN1 9_MC and_CC examine_VV0 the_AT osmotic_JJ pressure_NN1 of_IO a_AT1 polymer_NN1 solution_NN1 in_II41 the_II42 light_II43 of_II44 equation_NN1 (_( 8.35_MC )_) ._. 
8.7_MC Osmotic_JJ pressure_NN1 The_AT osmotic_JJ pressure_NN1 of_IO a_AT1 solution_NN1 can_VM be_VBI regarded_VVN as_II the_AT pressure_NN1 which_DDQ must_VM be_VBI exerted_VVN on_II that_DD1 solution_NN1 to_TO raise_VVI the_AT chemical_JJ potential_NN1 of_IO the_AT solvent_NN1 in_II the_AT solution_NN1 back_RP up_II21 to_II22 that_DD1 of_IO the_AT pure_JJ solvent_NN1 at_II a_AT1 standard_JJ pressure_NN1 P_ZZ1 ,_, i.e._REX The_AT compressibility_NN1 of_IO the_AT solvent_NN1 is_VBZ equal_JJ to_II the_AT molar_JJ volume_NN1 of_IO the_AT solvent_NN1 in_II solution_NN1 ,_, V_ZZ1 1_MC1 ,_, and_CC can_VM be_VBI assumed_VVN to_TO be_VBI unchanged_JJ over_II a_AT1 small_JJ range_NN1 of_IO pressures_NN2 ,_, thus_RR giving_VVG Substitution_NN1 in_II (_( 8.35_MC )_) gives_VVZ or_CC This_DD1 is_VBZ a_AT1 limited_JJ virial_NN1 expansion_NN1 in_II which_DDQ the_AT first_MD term_NN1 is_VBZ the_AT classical_JJ va_FW n't_XX Hoff_VV0 expression_NN1 for_IF the_AT osmotic_JJ pressure_NN1 at_II infinite_JJ dilution_NN1 ._. 
The_AT second_MD term_NN1 is_VBZ related_VVN to_II the_AT deviation_NN1 from_II ideal_JJ behaviour_NN1 and_CC gives_VVZ a_AT1 relationship_NN1 between_II the_AT second_MD virial_NN1 coefficient_NN1 B_ZZ1 and_CC the_AT interaction_NN1 parameter_NN1 1_MC1 Thus_RR when_CS then_RT and_CC the_AT osmotic_JJ pressure_NN1 is_VBZ given_VVN by_II the_AT ideal_JJ solution_NN1 law._NNU 8.8_MC Limitations_NN2 of_IO the_AT Flory-Huggins_NP1 theory_NN1 The_AT simple_JJ lattice_NN1 theory_NN1 does_VDZ not_XX describe_VVI the_AT behaviour_NN1 of_IO dilute_JJ polymer_NN1 solutions_NN2 particularly_RR well_RR because_CS the_AT following_JJ simplifications_NN2 in_II the_AT theoretical_JJ treatment_NN1 are_VBR invalid_JJ :_: (_( 1_MC1 )_) it_PPH1 was_VBDZ assumed_VVN that_CST the_AT segment-locating_JJ process_NN1 is_VBZ purely_RR statistical_JJ ,_, but_CCB this_DD1 would_VM only_RR be_VBI true_JJ if_CS 12_MC was_VBDZ zero_MC ;_; (_( 2_MC )_) the_AT treatment_NN1 assumed_VVD that_CST the_AT flexibility_NN1 of_IO the_AT chain_NN1 is_VBZ unaltered_JJ on_II passing_VVG into_II the_AT solution_NN1 from_II the_AT solid_JJ state_NN1 this_DD1 limits_VVZ the_AT calculation_NN1 of_IO S_ZZ1 M_ZZ1 to_II the_AT combinatorial_JJ contribution_NN1 only_RR and_CC neglects_VVZ any_DD contribution_NN1 from_II continual_JJ flexing_NN1 of_IO the_AT chain_NN1 in_II solution_NN1 which_DDQ will_VM contribute_VVI to_II the_AT non-combinatorial_JJ or_CC excess_JJ entropy_NN1 of_IO mixing_NN1 ;_; (_( 3_MC )_) any_DD possible_JJ specific_JJ solvent-polymer_JJ interactions_NN2 which_DDQ might_VM lead_VVI to_II orientation_NN1 of_IO the_AT solvent_NN1 molecules_NN2 in_II the_AT vicinity_NN1 of_IO the_AT polymer_NN1 chain_NN1 are_VBR neglected_JJ i.e._REX polar_JJ solutions_NN2 may_VM be_VBI inadequately_RR catered_VVN for_IF by_II this_DD1 theory_NN1 ;_; (_( 4_MC )_) a_AT1 uniform_JJ density_NN1 of_IO lattice_NN1 site_NN1 occupation_NN1 is_VBZ assumed_VVN ,_, but_CCB this_DD1 will_VM only_RR apply_VVI to_II relatively_RR concentrated_JJ solutions_NN2 ;_; (_( 5_MC )_) the_AT parameter_NN1 1_MC1 is_VBZ often_RR concentration-dependent_JJ but_CCB this_DD1 is_VBZ ignored_VVN ._. 
It_PPH1 is_VBZ now_RT accepted_VVN that_CST a_AT1 non-combinatorial_JJ entropy_NN1 contribution_NN1 arises_VVZ from_II the_AT formation_NN1 of_IO new_JJ (_( 12_MC )_) contacts_NN2 in_II the_AT mixture_NN1 which_DDQ change_VV0 the_AT vibrational_JJ frequencies_NN2 of_IO the_AT two_MC components_NN2 ,_, i.e._REX assumption_NN1 (_( d_ZZ1 )_) in_II section_NN1 8.2_MC must_VM be_VBI relaxed_VVN ._. 
This_DD1 can_VM be_VBI allowed_VVN for_IF by_II recognizing_VVG that_CST 1_MC1 is_VBZ actually_RR a_AT1 free_JJ energy_NN1 parameter_NN1 comprising_VVG entropic_JJ H_ZZ1 and_CC enthalpic_JJ S_ZZ1 contributions_NN2 ,_, such_CS21 that_CS22 ._. 
These_DD2 are_VBR defined_VVN by_II Experiments_NN2 tend_VV0 to_TO show_VVI that_CST the_AT major_JJ contribution_NN1 comes_VVZ from_II the_AT s_ZZ1 component_NN1 ,_, indicating_VVG that_CST there_EX is_VBZ a_AT1 large_JJ decrease_NN1 in_II entropy_NN1 (_( non-combinatorial_JJ )_) which_DDQ is_VBZ acting_VVG against_II the_AT dissolution_NN1 process_NN1 of_IO a_AT1 polymer_NN1 in_II a_AT1 solvent_NN1 ._. 
In_II31 spite_II32 of_II33 much_RR justifiable_JJ criticism_NN1 ,_, the_AT Flory-Huggins_NP1 theory_NN1 can_VM still_RR generate_VVI considerable_JJ interest_NN1 because_II21 of_II22 the_AT limited_JJ amount_NN1 of_IO success_NN1 which_DDQ can_VM be_VBI claimed_VVN for_IF it_PPH1 in_II31 relation_II32 to_II33 phase_NN1 equilibria_NN2 studies._NNU 8.9_MC Phase_NN1 equilibria_NN2 Use_VV0 can_VM be_VBI made_VVN of_IO the_AT Flory-Huggins_NP1 theory_NN1 to_TO predict_VVI the_AT equilibrium_NN1 behaviour_NN1 of_IO two_MC liquid_JJ phases_NN2 when_CS both_DB2 contain_VV0 amorphous_JJ polymer_NN1 and_CC one_MC1 or_CC even_RR two_MC solvents_NN2 ._. 
Consider_VV0 a_AT1 two_MC component_JJ system_NN1 consisting_VVG of_IO a_AT1 liquid_NN1 (_( 1_MC1 )_) which_DDQ is_VBZ a_AT1 poor_JJ solvent_NN1 for_IF a_AT1 polymer_NN1 (_( 2_MC )_) ._. 
Complete_JJ miscibility_NN1 occurs_VVZ when_RRQ the_AT Gibbs_NP1 free_JJ energy_NN1 of_IO mixing_NN1 is_VBZ less_DAR than_CSN the_AT Gibbs_NP1 free_JJ energies_NN2 of_IO the_AT components_NN2 ,_, and_CC the_AT solution_NN1 maintains_VVZ its_APPGE homogeneity_NN1 only_RR as_CS31 long_CS32 as_CS33 G_ZZ1 M_ZZ1 remains_VVZ less_DAR than_CSN the_AT Gibbs_NP1 free_JJ energy_NN1 of_IO any_DD two_MC possible_JJ co-existing_JJ phases_NN2 ._. 
The_AT situation_NN1 is_VBZ represented_VVN by_II curve_NN1 T_ZZ1 5_MC in_II figure_NN1 8.2_MC ._. 
The_AT miscibility_NN1 of_IO this_DD1 type_NN1 of_IO system_NN1 is_VBZ observed_VVN to_TO be_VBI strongly_RR temperature_NN1 dependent_NN1 and_CC as_CSA T_ZZ1 decreases_VVZ the_AT solution_NN1 separates_VVZ into_II two_MC phases_NN2 ._. 
Thus_RR at_II any_DD temperature_NN1 ,_, say_VV0 T_ZZ1 1_MC1 ,_, ;_; the_AT Gibbs_NP1 free_JJ energy_NN1 of_IO any_DD mixture_NN1 ,_, composition_NN1 in_II the_AT composition_NN1 range_NN1 to_II ,_, is_VBZ higher_JJR than_CSN either_DD1 of_IO the_AT two_MC co-existing_JJ phases_NN2 whose_DDQGE compositions_NN2 are_VBR and_CC and_CC phase_NN1 separation_NN1 takes_VVZ place_NN1 ._. 
The_AT compositions_NN2 of_IO the_AT two_MC phases_NN2 and_CC do_VD0 not_XX correspond_VVI to_II the_AT two_MC minima_NN2 ,_, but_CCB are_VBR measured_VVN from_II the_AT points_NN2 of_IO contact_NN1 of_IO the_AT double_JJ tangent_NN1 AB_FO with_IW the_AT Gibbs_NP1 free_JJ energy_NN1 curve_NN1 ._. 
The_AT same_DA is_VBZ true_JJ for_IF other_JJ temperatures_NN2 lying_VVG below_II T_ZZ1 c_ZZ1 ,_, and_CC the_AT inflexion_NN1 points_NN2 can_VM be_VBI joined_VVN to_TO bound_VVI an_AT1 area_NN1 representing_VVG the_AT heterogeneous_JJ two_MC phase_NN1 system_NN1 ,_, where_CS there_EX is_VBZ limited_JJ solubility_NN1 of_IO component_NN1 2_MC in_II 1_MC1 and_CC vice-versa_RR ._. 
This_DD1 is_VBZ called_VVN a_AT1 cloud-point_JJ curve_NN1 ._. 
As_II the_AT temperature_NN1 is_VBZ increased_VVN the_AT limits_NN2 of_IO this_DD1 two_MC phase_NN1 co-existence_NN1 contract_NN1 ,_, until_CS eventually_RR they_PPHS2 coalesce_VV0 to_TO produce_VVI a_AT1 homogeneous_JJ ,_, one_MC1 phase_NN1 ,_, mixture_NN1 at_II T_ZZ1 c_ZZ1 ,_, the_AT critical_JJ solution_NN1 temperature_NN1 ._. 
This_DD1 is_VBZ sometimes_RT referred_VVN to_II as_II the_AT critical_JJ consolute_JJ point_NN1 ._. 
In_RR21 general_RR22 ,_, we_PPIS2 can_VM say_VVI that_CST if_CS the_AT free_JJ energy-composition_JJ curve_NN1 has_VHZ a_AT1 shape_NN1 which_DDQ allows_VVZ a_AT1 tangent_JJ to_TO touch_VVI it_PPH1 at_II two_MC points_NN2 ,_, phase_NN1 separation_NN1 will_VM occur_VVI ._. 
The_AT critical_JJ solution_NN1 temperature_NN1 is_VBZ an_AT1 important_JJ quantity_NN1 and_CC can_VM be_VBI accurately_RR defined_VVN in_II31 terms_II32 of_II33 the_AT chemical_JJ potential_NN1 ._. 
It_PPH1 represents_VVZ the_AT point_NN1 at_II which_DDQ the_AT inflexion_NN1 points_VVZ on_II the_AT curve_NN1 merge_VV0 ,_, and_CC so_RR it_PPH1 is_VBZ the_AT temperature_NN1 where_CS the_AT first_MD ,_, second_NNT1 ,_, and_CC third_MD derivatives_NN2 of_IO the_AT Gibbs_NP1 free_JJ energy_NN1 with_II31 respect_II32 to_II33 mole_NN1 fraction_NN1 are_VBR zero_MC ._. 
It_PPH1 is_VBZ also_RR true_JJ that_CST the_AT partial_JJ molar_NN1 Gibbs_NP1 free_JJ energies_NN2 of_IO each_DD1 component_NN1 are_VBR equal_JJ at_II this_DD1 point_NN1 and_CC it_PPH1 emerges_VVZ that_CST the_AT conditions_NN2 for_IF incipient_JJ phase_NN1 separation_NN1 are_VBR By_II remembering_VVG that_CST ,_, application_NN1 of_IO these_DD2 criteria_NN2 for_IF equilibrium_NN1 to_II equation_NN1 (_( 8.33_MC )_) leads_VVZ to_II the_AT first_MD derivative_NN1 of_IO that_DD1 equation_NN1 while_CS the_AT second_MD derivative_NN1 is_VBZ where_RRQ the_AT subscript_NN1 c_ZZ1 denotes_VVZ critical_JJ conditions_NN2 ._. 
The_AT critical_JJ composition_NN1 at_II which_DDQ phase_NN1 separation_NN1 is_VBZ first_MD detected_VVN is_VBZ then_RT and_CC which_DDQ indicates_VVZ that_CST at_II infinitely_RR large_JJ chain_NN1 length_NN1 ._. 
The_AT interaction_NN1 parameter_NN1 1_MC1 is_VBZ a_AT1 useful_JJ measure_NN1 of_IO the_AT solvent_NN1 power_NN1 ._. 
Poor_JJ solvents_NN2 have_VH0 values_NN2 of_IO 1_MC1 close_JJ to_II 0.5_MC while_CS an_AT1 improvement_NN1 in_II solvent_NN1 power_NN1 lowers_VVZ 1_MC1 ._. 
Generally_RR ,_, a_AT1 variation_NN1 from_II 0.5_MC to_II -1.0_MC can_VM be_VBI observed_VVN although_CS for_IF many_DA2 synthetic_JJ polymer_NN1 solutions_NN2 the_AT range_NN1 is_VBZ 0.6_MC to_II 0.3_MC ._. 
A_AT1 linear_JJ temperature_NN1 dependence_NN1 for_IF 1_MC1 is_VBZ also_RR predicted_VVN of_IO the_AT general_JJ form_NN1 ,_, which_DDQ suggests_VVZ that_CST as_II the_AT temperature_NN1 increases_VVZ the_AT solvating_JJ power_NN1 of_IO the_AT liquid_NN1 should_VM increase_VVI ._. 
This_DD1 has_VHZ implications_NN2 for_IF polymer_NN1 fractionation._NNU 8.10_MC Fractionation_NN1 The_AT relations_NN2 derived_VVN in_II this_DD1 and_CC other_JJ chapters_NN2 normally_RR assume_VV0 that_CST the_AT polymer_NN1 sample_NN1 has_VHZ a_AT1 unique_JJ molar_JJ mass_NN1 ._. 
This_DD1 situation_NN1 is_VBZ rarely_RR achieved_VVN in_II practice_NN1 and_CC it_PPH1 is_VBZ useful_JJ to_TO know_VVI the_AT form_NN1 of_IO the_AT molar_JJ mass_JJ distribution_NN1 in_II a_AT1 sample_NN1 ,_, as_CSA this_DD1 can_VM have_VHI a_AT1 significant_JJ bearing_NN1 on_II the_AT physical_JJ properties_NN2 ._. 
It_PPH1 is_VBZ also_RR advantageous_JJ to_TO be_VBI able_JK to_TO prepare_VVI sample_NN1 fractions_NN2 ,_, whose_DDQGE homogeneity_NN1 is_VBZ considerably_RR better_RRR than_CSN the_AT parent_NN1 polymer_NN1 ,_, especially_RR when_CS testing_VVG dilute_JJ solution_NN1 theory_NN1 ._. 
We_PPIS2 have_VH0 seen_VVN that_CST the_AT chain_NN1 length_NN1 can_VM be_VBI related_VVN to_II the_AT solvent_NN1 power_NN1 ,_, expressed_VVN as_CSA 1_MC1 ,_, by_II equation_NN1 (_( 8.46_MC )_) and_CC this_DD1 is_VBZ illustrated_VVN in_II figure_NN1 8.3_MC ._. 
The_AT implication_NN1 is_VBZ that_CST if_CS 1_MC1 can_VM be_VBI carefully_RR controlled_VVN ,_, conditions_NN2 could_VM be_VBI attained_VVN which_DDQ would_VM allow_VVI a_AT1 given_JJ molecular_JJ species_NN to_TO precipitate_VVI ,_, while_CS leaving_VVG larger_JJR or_CC smaller_JJR molecules_NN2 in_II solution_NN1 ._. 
This_DD1 process_NN1 is_VBZ known_VVN as_II fractionation_NN1 ._. 
Experimentally_RR ,_, a_AT1 polymer_NN1 sample_NN1 can_VM be_VBI fractionated_VVN in_II a_AT1 variety_NN1 of_IO ways_NN2 and_CC three_MC in_RR21 common_RR22 use_VV0 are_VBR :_: (_( 1_MC1 )_) addition_NN1 of_IO a_AT1 non-solvent_NN1 to_II a_AT1 polymer_NN1 solution_NN1 ;_; (_( 2_MC )_) lowering_VVG the_AT temperature_NN1 of_IO the_AT solution_NN1 ;_; and_CC (_( 3_MC )_) column_NN1 chromatography_NN1 ._. 
In_II the_AT first_MD method_NN1 the_AT control_NN1 of_IO 1_MC1 is_VBZ effected_VVN by_II adding_VVG a_AT1 non-solvent_NN1 to_II the_AT polymer_NN1 solution_NN1 ._. 
If_CS the_AT addition_NN1 is_VBZ slow_JJ ,_, 1_MC1 increases_VVZ gradually_RR until_CS the_AT critical_JJ value_NN1 for_IF large_JJ molecules_NN2 is_VBZ reached_VVN ._. 
This_DD1 causes_VVZ precipitation_NN1 of_IO the_AT longest_JJT chains_NN2 first_MD and_CC these_DD2 can_VM be_VBI separated_VVN from_II the_AT shorter_JJR chains_NN2 which_DDQ remain_VV0 in_II solution_NN1 ._. 
In_II practice_NN1 the_AT polymer_NN1 solution_NN1 is_VBZ held_VVN at_II a_AT1 constant_JJ temperature_NN1 while_CS precipitant_NN1 is_VBZ added_VVN to_II the_AT stirred_JJ solution_NN1 ._. 
When_CS the_AT solution_NN1 becomes_VVZ turbid_JJ the_AT mixture_NN1 is_VBZ warmed_VVN until_CS the_AT precipitate_NN1 dissolves_VVZ ._. 
The_AT solution_NN1 is_VBZ then_RT returned_VVN to_II the_AT original_JJ temperature_NN1 and_CC the_AT precipitate_NN1 which_DDQ reforms_VVZ is_VBZ allowed_VVN to_TO settle_VVI and_CC then_RT separated_VVN ._. 
This_DD1 ensures_VVZ that_CST the_AT precipitated_VVN fraction_NN1 is_VBZ not_XX broadened_VVN by_II local_JJ precipitation_NN1 during_II addition_NN1 of_IO the_AT non-solvent_NN1 ._. 
Successive_JJ additions_NN2 of_IO small_JJ quantities_NN2 of_IO non-solvent_NN1 to_II the_AT solution_NN1 allow_VV0 a_AT1 series_NN of_IO fractions_NN2 of_IO steadily_RR decreasing_VVG molar_JJ mass_NN1 to_TO be_VBI separated_VVN ._. 
In_II the_AT second_MD method_NN1 ,_, 1_MC1 is_VBZ varied_VVN by_II altering_VVG the_AT temperature_NN1 ,_, with_IW similar_JJ results_NN2 ._. 
For_IF both_DB2 techniques_NN2 ,_, it_PPH1 is_VBZ useful_JJ to_TO dissolve_VVI the_AT polymer_NN1 initially_RR in_II a_AT1 poor_JJ solvent_NN1 with_IW a_AT1 large_JJ 1_MC1 value_NN1 ._. 
This_DD1 ensures_VVZ that_CST only_RR small_JJ quantities_NN2 of_IO non-solvent_NN1 are_VBR required_VVN to_TO precipitate_VVI the_AT polymer_NN1 in_II method_NN1 1_MC1 ,_, and_CC that_CST the_AT temperature_NN1 changes_NN2 required_VVN in_II method_NN1 2_MC are_VBR small_JJ ._. 
In_II column_NN1 chromatography_NN1 the_AT polymer_NN1 is_VBZ precipitated_VVN on_II the_AT inert_JJ support_NN1 medium_NN1 at_II the_AT top_NN1 of_IO a_AT1 column_NN1 which_DDQ has_VHZ a_AT1 temperature_NN1 gradient_NN1 imposed_VVD along_RP its_APPGE length_NN1 ._. 
The_AT packing_NN1 is_VBZ usually_RR glass_NN1 beads_NN2 of_IO 0.1_MC to_II 0.3_MC mm_NNU diameter_NN1 ._. 
A_AT1 solvent_NN1 +_FO non-solvent_NN1 mixture_NN1 is_VBZ used_VVN to_TO elute_VVI the_AT sample_NN1 and_CC fractionation_NN1 is_VBZ achieved_VVN by_II using_VVG a_AT1 solvent_NN1 gradient_NN1 ._. 
This_DD1 is_VBZ generated_VVN in_II a_AT1 mixing_NN1 system_NN1 ,_, situated_VVN above_II the_AT column_NN1 ,_, by_II constantly_RR increasing_VVG the_AT solvent_NN1 to_II non-solvent_NN1 ratio_NN1 and_CC as_II the_AT mixture_NN1 is_VBZ initially_RR a_AT1 poor_JJ solvent_NN1 which_DDQ is_VBZ gradually_RR enriched_VVN by_II the_AT good_JJ solvent_NN1 the_AT low_JJ molar_JJ mass_JJ fractions_NN2 are_VBR eluted_VVN first_MD ._. 
Fractions_NN2 of_IO increasing_JJ molar_JJ mass_NN1 are_VBR collected_VVN from_II the_AT bottom_NN1 of_IO the_AT column_NN1 ._. 
In_II each_DD1 of_IO the_AT techniques_NN2 ,_, the_AT mass_NN1 and_CC molar_JJ mass_NN1 of_IO the_AT fractions_NN2 are_VBR recorded_VVN and_CC a_AT1 distribution_NN1 curve_NN1 for_IF the_AT sample_NN1 can_VM be_VBI constructed_VVN from_II the_AT results_NN2 ._. 
However_RR ,_, as_CSA the_AT fractions_NN2 themselves_PPX2 have_VH0 a_AT1 molar_JJ mass_JJ distribution_NN1 ,_, extensive_JJ overlapping_JJ of_IO the_AT fractions_NN2 will_VM occur_VVI as_CSA shown_VVN schematically_RR in_II figure_NN1 8.4_MC ._. 
Consequently_RR a_AT1 simple_JJ histogram_NN1 constructed_VVN from_II the_AT mass_NN1 and_CC molar_JJ mass_NN1 of_IO each_DD1 fraction_NN1 will_VM not_XX provide_VVI a_AT1 good_JJ representation_NN1 of_IO the_AT distribution_NN1 and_CC a_AT1 method_NN1 must_VM be_VBI used_VVN to_TO compensate_VVI for_IF the_AT overlapping_JJ ._. 
A_AT1 useful_JJ approach_NN1 was_VBDZ proposed_VVN by_II Schulz_NP1 who_PNQS suggested_VVD that_CST a_AT1 cumulative_JJ mass_JJ fraction_NN1 be_VBI plotted_VVN against_II the_AT molar_JJ mass_NN1 ._. 
The_AT cumulative_JJ mass_JJ fraction_NN1 can_VM be_VBI calculated_VVN by_II adding_VVG half_DB the_AT mass_JJ fraction_NN1 W_ZZ1 i_MC1 of_IO the_AT i_MC1 th_NNU fraction_NN1 to_II the_AT total_JJ mass_JJ fraction_NN1 of_IO those_DD2 fractions_NN2 preceding_VVG it_PPH1 ,_, i.e._REX The_AT values_NN2 of_IO are_VBR plotted_VVN against_II the_AT corresponding_JJ M_ZZ1 i_ZZ1 and_CC connected_VVN by_II a_AT1 smooth_JJ curve_NN1 as_CSA shown_VVN in_II figure_NN1 8.5_MC ,_, to_TO give_VVI the_AT integral_JJ distribution_NN1 curve_NN1 ._. 
The_AT differential_JJ curve_NN1 can_VM be_VBI obtained_VVN by_II determining_VVG the_AT slope_NN1 of_IO this_DD1 curve_NN1 at_II selected_JJ molar_JJ masses_NN2 and_CC plotting_VVG this_DD1 against_II the_AT appropriate_JJ molar_NN1 mass._NNU 8.11_MC Flory-Krigbaum_NP1 theory_NN1 To_TO overcome_VVI the_AT limitations_NN2 of_IO the_AT lattice_NN1 theory_NN1 resulting_VVG from_II the_AT discontinuous_JJ nature_NN1 of_IO a_AT1 dilute_JJ polymer_NN1 solution_NN1 ,_, Flory_NP1 and_CC Krigbaum_NP1 discarded_VVD the_AT idea_NN1 of_IO a_AT1 uniform_JJ distribution_NN1 of_IO chain_NN1 segments_NN2 in_II the_AT liquid_NN1 ._. 
Instead_RR they_PPHS2 considered_VVD the_AT solution_NN1 to_TO be_VBI composed_VVN of_IO areas_NN2 containing_VVG polymer_NN1 which_DDQ were_VBDR separated_VVN by_II the_AT solvent_NN1 ._. 
In_II these_DD2 areas_NN2 the_AT polymer_NN1 segments_NN2 were_VBDR assumed_VVN to_TO possess_VVI a_AT1 Gaussian_JJ distribution_NN1 about_II the_AT centre_NN1 of_IO mass_NN1 ,_, but_CCB even_RR with_IW this_DD1 distribution_NN1 the_AT chain_NN1 segments_NN2 still_RR occupy_VV0 a_AT1 finite_JJ volume_NN1 from_II which_DDQ all_DB other_JJ chain_NN1 segments_NN2 are_VBR excluded_VVN ._. 
It_PPH1 is_VBZ within_II this_DD1 excluded_JJ volume_NN1 that_CST the_AT long_JJ range_NN1 interactions_NN2 originate_VV0 which_DDQ are_VBR discussed_VVN more_RGR fully_RR in_II chapter_NN1 10_MC ._. 
Flory_NN1 and_CC Krigbaum_NP1 defined_VVD an_AT1 enthalpy_NN1 parameter_NN1 and_CC an_AT1 entropy_NN1 of_IO dilution_NN1 parameter_NN1 such_CS21 that_CS22 the_AT thermodynamic_JJ functions_NN2 used_VMK to_TO describe_VVI these_DD2 long_JJ range_NN1 effects_NN2 are_VBR given_VVN in_II31 terms_II32 of_II33 the_AT excess_JJ partial_JJ molar_JJ quantities_NN2 From_II equation_NN1 (_( 8.33_MC )_) it_PPH1 can_VM be_VBI seen_VVN that_CST the_AT excess_JJ free_JJ energy_NN1 of_IO dilution_NN1 is_VBZ Combination_NN1 of_IO these_DD2 non-ideal_JJ terms_NN2 then_RT yields_VVZ As_CSA we_PPIS2 saw_VVD from_II equation_NN1 (_( 8.40_MC )_) ,_, when_RRQ and_CC the_AT solution_NN1 appears_VVZ to_TO behave_VVI as_CS21 though_CS22 it_PPH1 were_VBDR ideal_JJ ._. 
The_AT point_NN1 at_II which_DDQ this_DD1 occurs_VVZ is_VBZ known_VVN as_II the_AT FLORY_NN1 or_CC THETA_NN1 point_NN1 and_CC is_VBZ in_II some_DD ways_NN2 analogous_JJ to_II the_AT Boyle_NP1 point_NN1 for_IF a_AT1 non-ideal_JJ gas_NN1 ._. 
Under_II these_DD2 conditions_NN2 The_AT temperature_NN1 at_II which_DDQ these_DD2 conditions_NN2 are_VBR obtained_VVN is_VBZ the_AT FLORY_NN1 or_CC THETA_NN1 temperature_NN1 ,_, conveniently_RR defined_VVN as_II ._. 
This_DD1 tells_VVZ us_PPIO2 that_CST will_VM only_RR have_VHI a_AT1 meaningful_JJ value_NN1 when_CS 1_MC1 and_CC 1_MC1 have_VH0 the_AT same_DA sign_NN1 ._. 
Substitution_NN1 in_II (_( 8.50_MC )_) followed_VVD by_II rearrangement_NN1 gives_VVZ and_CC shows_VVZ that_CST deviations_NN2 from_II ideal_JJ behaviour_NN1 vanish_VV0 when_RRQ ._. 
The_AT theta_NN1 temperature_NN1 is_VBZ a_AT1 well_NN1 defined_VVN state_NN1 of_IO the_AT polymer_NN1 solution_NN1 at_II which_DDQ the_AT excluded_JJ volume_NN1 effects_NN2 are_VBR eliminated_VVN and_CC the_AT polymer_NN1 coil_NN1 is_VBZ in_II an_AT1 unperturbed_JJ condition_NN1 (_( see_VV0 chapter_NN1 10_MC )_) ._. 
Above_II the_AT theta_NN1 temperature_NN1 expansion_NN1 of_IO the_AT coil_NN1 takes_VVZ place_NN1 ,_, caused_VVN by_II interactions_NN2 with_IW the_AT solvent_NN1 ,_, whereas_CS below_II *_FU the_AT polymer_NN1 segments_NN2 attract_VV0 one_PPX121 another_PPX122 ,_, the_AT excluded_JJ volume_NN1 is_VBZ negative_JJ ,_, the_AT coils_NN2 tend_VV0 to_TO collapse_VVI and_CC eventual_JJ phase_NN1 separation_NN1 occurs._NNU 8.12_MC Location_NN1 of_IO the_AT theta_NN1 temperature_NN1 The_AT theta_NN1 temperature_NN1 of_IO a_AT1 polymer-solvent_NN1 system_NN1 can_VM be_VBI measured_VVN from_II phase_NN1 separation_NN1 studies_NN2 ._. 
The_AT value_NN1 of_IO 1._MC c_ZZ1 ;_; at_II the_AT critical_JJ concentration_NN1 is_VBZ related_VVN to_II the_AT chain_NN1 length_NN1 of_IO the_AT polymer_NN1 by_II equation_NN1 (_( 8.46_MC )_) ,_, and_CC substitution_NN1 in_II (_( 8.52_MC )_) leads_VVZ to_II where_RRQ now_RT we_PPIS2 have_VH0 replaced_VVN r_ZZ1 with_IW the_AT equivalent_JJ degree_NN1 of_IO polymerization_NN1 n_ZZ1 ._. 
Rearrangement_NN1 gives_VVZ Remembering_VVG that_CST ,_, where_CS M_ZZ1 and_CC are_VBR the_AT molar_JJ mass_NN1 and_CC partial_JJ specific_JJ volume_NN1 of_IO the_AT polymer_NN1 ,_, and_CC V_ZZ1 1_MC1 is_VBZ the_AT molar_JJ volume_NN1 of_IO the_AT solvent_NN1 ,_, the_AT equation_NN1 states_VVZ that_CST the_AT critical_JJ temperature_NN1 is_VBZ a_AT1 function_NN1 of_IO M_ZZ1 and_CC the_AT value_NN1 of_IO T_ZZ1 c_ZZ1 at_II infinite_JJ M_NN1 is_VBZ the_AT theta_NN1 temperature_NN1 for_IF the_AT system_NN1 ._. 
Precipitation_NN1 data_NN for_IF several_DA2 systems_NN2 have_VH0 proved_VVN the_AT validity_NN1 of_IO equation_NN1 (_( 8.54_MC )_) ._. 
Linear_JJ plots_NN2 are_VBR obtained_VVN with_IW a_AT1 positive_JJ slope_NN1 from_II which_DDQ the_AT entropy_NN1 parameter_NN1 1_MC1 can_VM be_VBI calculated_VVN as_CSA shown_VVN in_II figure_NN1 8.6_MC ._. 
Typical_JJ values_NN2 are_VBR shown_VVN in_II table_NN1 8.1_MC ,_, but_CCB 1_MC1 values_NN2 measured_VVN for_IF systems_NN2 such_II21 as_II22 polystyrene_NN1 +_FO cyclohexane_NN1 have_VH0 been_VBN found_VVN to_TO be_VBI almost_RR ten_MC times_NNT2 larger_JJR than_CSN those_DD2 derived_VVN from_II other_JJ methods_NN2 of_IO measurement_NN1 ._. 
This_DD1 appears_VVZ to_TO arise_VVI from_II the_AT assumption_NN1 in_II the_AT Flory-Huggins_NP1 theory_NN1 that_CST 1_MC1 is_VBZ concentration_NN1 independent_JJ and_CC improved_JJ values_NN2 of_IO 1_MC1 are_VBR obtained_VVN when_CS this_DD1 is_VBZ rectified_VVN ._. 
The_AT theta_NN1 temperature_NN1 ,_, calculated_VVN from_II equation_NN1 (_( 8.54_MC )_) for_IF each_DD1 system_NN1 is_VBZ in_II good_JJ agreement_NN1 with_IW that_DD1 measured_VVD from_II the_AT temperature_NN1 variation_NN1 of_IO A_ZZ1 2_MC (_( =B/RT_NN1 see_VV0 chapter_NN1 9_MC )_) ._. 
Curves_NN2 of_IO A_ZZ1 2_MC ,_, measured_VVN at_II various_JJ temperatures_NN2 in_II the_AT vicinity_NN1 of_IO ,_, are_VBR constructed_VVN as_II a_AT1 function_NN1 of_IO temperature_NN1 for_IF one_MC1 or_CC more_DAR molar_JJ masses_NN2 as_CSA shown_VVN in_II figure_NN1 8.7_MC ._. 
Intersection_NN1 of_IO the_AT curves_NN2 with_IW the_AT T-axis_NP1 occurs_VVZ when_RRQ and_CC ._. 
The_AT curves_NN2 for_IF each_DD1 molar_JJ mass_NN1 of_IO the_AT same_DA polymer_NN1 should_VM all_DB intersect_VVI at._NNU 8.13_MC Lower_JJR critical_JJ solution_NN1 temperatures_NN2 So_RG far_RR we_PPIS2 have_VH0 been_VBN concerned_JJ with_IW non-polar_JJ solutions_NN2 of_IO amorphous_JJ polymers_NN2 ,_, whose_DDQGE solubility_NN1 is_VBZ increased_VVN with_IW rising_VVG temperature_NN1 ,_, because_CS the_AT additional_JJ thermal_JJ motion_NN1 helps_VVZ to_TO decrease_VVI attractive_JJ forces_NN2 between_II like_JJ molecules_NN2 ,_, and_CC encourages_VVZ energetically_RR less_RGR favourable_JJ contacts_NN2 ._. 
The_AT phase_NN1 diagram_NN1 for_IF such_DA systems_NN2 ,_, when_CS the_AT solvent_NN1 is_VBZ poor_JJ ,_, is_VBZ depicted_VVN by_II area_NN1 A_ZZ1 in_II figure_NN1 8.8_MC ,_, where_CS the_AT critical_JJ temperature_NN1 T_ZZ1 c_ZZ1 occurs_VVZ near_II the_AT maximum_NN1 of_IO the_AT cloud-point_JJ curve_NN1 and_CC is_VBZ often_RR referred_VVN to_II as_II the_AT upper_JJ critical_JJ solution_NN1 temperature_NN1 (_( UCST_NP1 )_) ._. 
This_DD1 behaviour_NN1 follows_VVZ from_II that_DD1 depicted_VVD in_II figure_NN1 8.2_MC ._. 
For_IF non-polar_JJ systems_NN2 S_ZZ1 M_ZZ1 is_VBZ normally_RR positive_JJ but_CCB weighted_VVD heavily_RR by_II T_ZZ1 and_CC so_RR solubility_NN1 depends_VVZ mainly_RR on_II the_AT magnitude_NN1 of_IO H_ZZ1 M_ZZ1 ,_, which_DDQ is_VBZ normally_RR endothermic_JJ (_( positive_JJ )_) ._. 
Consequently_RR as_CSA T_ZZ1 decreases_VVZ G_ZZ1 M_ZZ1 eventually_RR becomes_VVZ positive_JJ and_CC phase_NN1 separation_NN1 takes_VVZ place_NN1 ._. 
Values_NN2 of_IO and_CC 1_MC1 ,_, in_II table_NN1 8.1_MC ,_, show_VV0 that_CST for_IF systems_NN2 1_MC1 to_II 4_MC the_AT entropy_NN1 parameter_NN1 is_VBZ positive_JJ ,_, as_CSA expected_VVN ,_, but_II21 for_II22 poly_NN1 (_( acrylic_JJ acid_NN1 )_) in_II dioxan_NN1 and_CC polymethacrylonitrile_NN1 in_II butanone_NN1 ,_, is_VBZ negative_JJ at_II the_AT theta_NN1 temperature_NN1 ._. 
As_CSA ,_, when_RRQ ,_, the_AT enthalpy_NN1 is_VBZ also_RR negative_JJ for_IF these_DD2 systems_NN2 ._. 
This_DD1 means_VVZ that_CST systems_NN2 5_MC and_CC 6_MC exhibit_VV0 an_AT1 unusual_JJ decrease_NN1 in_II solubility_NN1 as_II the_AT temperature_NN1 rises_NN2 ,_, and_CC the_AT cloud-point_JJ curve_NN1 is_VBZ now_RT inverted_JJ as_CSA in_II area_NN1 B._NP1 The_AT corresponding_JJ critical_JJ temperature_NN1 is_VBZ located_VVN at_II the_AT minimum_NN1 of_IO the_AT miscibility_NN1 curve_NN1 and_CC is_VBZ known_VVN as_II the_AT lower_JJR critical_JJ solution_NN1 temperature_NN1 (_( LCST_NP1 )_) ._. 
In_II systems_NN2 5_MC and_CC 6_MC this_DD1 phenomenon_NN1 is_VBZ a_AT1 result_NN1 of_IO hydrogen-bond_JJ formation_NN1 between_II the_AT polymer_NN1 and_CC solvent_NN1 ,_, which_DDQ enhances_VVZ the_AT solubility_NN1 ._. 
As_CSA hydrogen_NN1 bonds_NN2 are_VBR thermally_RR labile_JJ a_AT1 rise_NN1 in_II T_ZZ1 reduces_VVZ the_AT number_NN1 of_IO bonds_NN2 and_CC causes_VVZ eventual_JJ phase_NN1 separation_NN1 ._. 
In_II solutions_NN2 ,_, which_DDQ are_VBR stabilized_VVN in_II this_DD1 way_NN1 by_II secondary_JJ bonding_NN1 ,_, the_AT LCST_NP1 usually_RR appears_VVZ below_II the_AT boiling_JJ temperature_NN1 of_IO the_AT solvent_NN1 but_CCB it_PPH1 has_VHZ been_VBN found_VVN experimentally_RR that_CST an_AT1 LCST_NP1 can_VM be_VBI detected_VVN in_II non-polar_JJ systems_NN2 when_CS these_DD2 are_VBR examined_VVN at_II temperatures_NN2 approaching_VVG the_AT critical_JJ temperature_NN1 of_IO the_AT solvent_NN1 ._. 
Polyisobutylene_NN1 in_II a_AT1 series_NN of_IO n-alkanes_NN2 ,_, polystyrene_NN1 in_II methyl_NN1 acetate_NN1 and_CC cyclohexane_NN1 ,_, and_CC cellulose_VV0 acetate_NN1 in_II acetone_NN1 all_DB exhibit_VV0 LCSTs_NP1 ._. 
The_AT LCST_NP1 is_VBZ located_VVN by_II heating_VVG the_AT solutions_NN2 ,_, in_II sealed_JJ tubes_NN2 ,_, up_II21 to_II22 temperatures_NN2 approaching_VVG the_AT gas-liquid_JJ critical_JJ point_NN1 of_IO the_AT solvent_NN1 ._. 
As_II the_AT temperature_NN1 rises_NN2 ,_, the_AT liquid_NN1 expands_VVZ much_RR more_RGR rapidly_RR than_CSN the_AT polymer_NN1 ,_, which_DDQ is_VBZ restrained_VVN by_II the_AT covalent_JJ bonding_NN1 between_II its_APPGE segments_NN2 ._. 
At_II high_JJ temperatures_NN2 ,_, the_AT spaces_NN2 between_II the_AT solvent_NN1 molecules_NN2 have_VH0 to_TO be_VBI reduced_VVN if_CS mixing_NN1 is_VBZ to_TO take_VVI place_NN1 and_CC when_CS this_DD1 eventually_RR results_VVZ in_II too_RG great_JJ a_AT1 loss_NN1 of_IO entropy_NN1 ,_, phase_NN1 separation_NN1 occurs_VVZ ._. 
The_AT separation_NN1 of_IO polymer/solvent_NN1 systems_NN2 into_II two_MC phases_NN2 as_II the_AT temperature_NN1 increases_NN2 is_VBZ now_RT recognized_VVN to_TO be_VBI a_AT1 characteristic_JJ feature_NN1 of_IO all_DB polymer_NN1 solutions_NN2 ._. 
This_DD1 presents_VVZ a_AT1 problem_NN1 of_IO interpretation_NN1 within_II the_AT framework_NN1 of_IO regular_JJ solution_NN1 theory_NN1 ,_, as_CSA the_AT accepted_JJ form_NN1 of_IO 1_MC1 predicts_VVZ a_AT1 monotonic_JJ change_NN1 with_IW temperature_NN1 and_CC is_VBZ incapable_JJ of_IO dealing_VVG with_IW two_MC critical_JJ consolute_JJ points_NN2 ._. 
The_AT problem_NN1 of_IO how_RRQ to_TO accommodate_VVI ,_, in_II a_AT1 theoretical_JJ framework_NN1 ,_, the_AT existence_NN1 of_IO two_MC miscibility_NN1 gaps_NN2 requires_VVZ a_AT1 new_JJ approach_NN1 ,_, and_CC a_AT1 more_RGR elaborate_JJ treatment_NN1 by_II Prigogine_NP1 and_CC co-workers_NN2 encompasses_VVZ the_AT difference_NN1 in_II size_NN1 between_II the_AT components_NN2 of_IO a_AT1 mixture_NN1 ,_, which_DDQ can_VM not_XX be_VBI ignored_VVN for_IF polymer_NN1 solutions_NN2 ._. 
They_PPHS2 replaced_VVD the_AT rigid_JJ lattice_NN1 model_NN1 used_VVN by_II Flory_NP1 and_CC Huggins_NP1 ,_, which_DDQ is_VBZ valid_JJ only_RR at_II absolute_JJ zero_NN1 ,_, with_IW a_AT1 flexible_JJ lattice_NN1 whose_DDQGE cells_NN2 change_VV0 volume_NN1 ,_, with_IW temperature_NN1 and_CC pressure_NN1 ._. 
This_DD1 allowed_VVD them_PPHO2 to_TO include_VVI in_II their_APPGE theory_NN1 dissimilarities_NN2 in_II free_JJ volume_NN1 between_II polymer_NN1 and_CC solvent_NN1 ,_, together_RL with_IW the_AT corresponding_JJ interactions_NN2 ._. 
The_AT same_DA approach_NN1 was_VBDZ extended_VVN by_II both_RR Patterson_NP1 and_CC Flory_JJ to_TO deal_VVI specifically_RR with_IW polymer_NN1 systems_NN2 ._. 
The_AT most_RGT important_JJ of_IO the_AT new_JJ parameters_NN2 is_VBZ the_AT so-called_JJ '_GE structural_JJ effect_NN1 '_GE which_DDQ is_VBZ related_VVN to_II the_AT number_NN1 of_IO degrees_NN2 of_IO freedom_NN1 '_GE 3_MC c_ZZ1 '_GE which_DDQ a_AT1 molecule_NN1 possesses_VVZ ,_, divided_VVN by_II the_AT number_NN1 of_IO external_JJ contacts_NN2 q_ZZ1 ._. 
This_DD1 structural_JJ factor_NN1 is_VBZ a_AT1 measure_NN1 of_IO the_AT number_NN1 of_IO external_JJ degrees_NN2 of_IO freedom_NN1 per_II segment_NN1 and_CC changes_NN2 with_IW the_AT length_NN1 of_IO the_AT component_NN1 ._. 
Thus_RR the_AT ratio_NN1 decreases_VVZ as_II a_AT1 liquid_NN1 becomes_VVZ increasingly_RR polymeric_JJ ._. 
The_AT expansion_NN1 and_CC free_JJ volume_NN1 can_VM then_RT be_VBI characterized_VVN by_II the_AT ratio_NN1 of_IO the_AT thermal_JJ energy_NN1 arising_VVG from_II the_AT external_JJ degrees_NN2 of_IO freedom_NN1 available_JJ to_II the_AT component_NN1 ,_, U_ZZ1 thermal_JJ ,_, and_CC the_AT interaction_NN1 energy_NN1 between_II neighbouring_JJ non-bonded_JJ segments_NN2 ,_, U_ZZ1 cohesive_JJ which_DDQ will_VM oppose_VVI the_AT thermal_JJ energy_NN1 effects_NN2 ,_, i.e._REX where_CS *_FU is_VBZ the_AT characteristic_JJ cohesive_JJ energy_NN1 per_II contact_NN1 ._. 
For_IF convenience_NN1 q_ZZ1 may_VM be_VBI replaced_VVN by_II r_ZZ1 ,_, the_AT number_NN1 of_IO chain_NN1 segments_NN2 ,_, although_CS q_ZZ1 will_VM actually_RR be_VBI less_DAR than_CSN r_ZZ1 because_CS some_DD of_IO the_AT external_JJ contacts_NN2 are_VBR used_VVN in_II forming_VVG the_AT covalent_JJ bonds_NN2 in_II the_AT chain_NN1 ._. 
Free_JJ volume_NN1 dissimilarities_NN2 become_VV0 increasingly_RR important_JJ as_II the_AT size_NN1 of_IO one_MC1 component_NN1 increases_VVZ with_II31 respect_II32 to_II33 the_AT second_NNT1 ,_, as_CSA in_II polymer_NN1 solutions_NN2 ,_, and_CC when_CS these_DD2 differences_NN2 are_VBR sufficiently_RR large_JJ ,_, phase_NN1 separation_NN1 can_VM be_VBI observed_VVN at_II the_AT LCST_NP1 ._. 
The_AT differences_NN2 in_II expansivity_NN1 can_VM be_VBI accounted_VVN for_IF if_CS the_AT interaction_NN1 parameter_NN1 is_VBZ now_RT expressed_VVN as_CSA where_CS the_AT first_MD term_NN1 reflects_VVZ the_AT interchange_NN1 energy_NN1 on_II forming_VVG contacts_NN2 of_IO unlike_JJ type_NN1 and_CC includes_VVZ segment_NN1 size_NN1 differences_NN2 ,_, while_CS the_AT second_MD term_NN1 is_VBZ the_AT new_JJ '_GE structural_JJ '_GE contribution_NN1 coming_VVG from_II free_JJ volume_NN1 changes_NN2 on_II mixing_VVG a_AT1 dense_JJ polymer_NN1 with_IW an_AT1 expanded_JJ solvent_NN1 ._. 
This_DD1 can_VM be_VBI represented_VVN schematically_RR in_II figure_NN1 8.9_MC ._. 
The_AT first_MD term_NN1 in_II (_( 8.56_MC )_) ,_, shown_VVN by_II curve_NN1 1_MC1 ,_, is_VBZ merely_RR an_AT1 expression_NN1 of_IO the_AT Flory-Huggins_NP1 theory_NN1 where_CS X_ZZ1 decreases_VVZ constantly_RR with_IW rising_VVG temperature_NN1 ,_, but_CCB now_RT inclusion_NN1 of_IO the_AT new_JJ free_JJ volume_NN1 term_NN1 ,_, shown_VVN by_II curve_NN1 2_MC ,_, modifies_VVZ the_AT behaviour_NN1 of_IO ._. 
The_AT second_MD term_NN1 gains_NN2 in_II importance_NN1 as_II the_AT expansivities_NN2 of_IO the_AT two_MC components_NN2 become_VV0 increasingly_RR divergent_JJ with_IW temperature_NN1 and_CC the_AT net_JJ effect_NN1 is_VBZ to_TO increase_VVI again_RT until_CS it_PPH1 once_RR21 more_RR22 attains_VVZ its_APPGE critical_JJ value_NN1 at_II high_JJ temperature_NN1 ._. 
The_AT LCST_NP1 which_DDQ results_VVZ ,_, is_VBZ then_RT a_AT1 consequence_NN1 of_IO these_DD2 free_JJ volume_NN1 differences_NN2 and_CC is_VBZ an_AT1 entropically_RR controlled_VVN phenomenon_NN1 ._. 
This_DD1 can_VM be_VBI illustrated_VVN in_II the_AT following_JJ ways_NN2 ._. 
In_II31 terms_II32 of_II33 the_AT flexible_JJ lattice_NN1 model_NN1 ,_, one_PN1 can_VM imagine_VVI the_AT polymer_NN1 and_CC liquid_JJ lattices_NN2 expanding_VVG at_II different_JJ rates_NN2 until_CS a_AT1 temperature_NN1 is_VBZ reached_VVN at_II which_DDQ the_AT highly_RR expanded_VVN liquid_JJ lattice_NN1 can_VM no_RR21 longer_RR22 be_VBI distorted_VVN sufficiently_RR to_TO accommodate_VVI the_AT less_RGR expanded_JJ polymer_NN1 lattice_NN1 and_CC form_VV0 a_AT1 solution_NN1 ,_, i.e._REX the_AT loss_NN1 in_II entropy_NN1 during_II the_AT distortion_NN1 becomes_VVZ so_RG large_JJ and_CC unfavourable_JJ that_CST phase_NN1 separation_NN1 (_( LCST_NP1 )_) takes_VVZ place_NN1 ._. 
Alternatively_RR ,_, a_AT1 polymer_NN1 solution_NN1 can_VM be_VBI thought_VVN of_IO as_II a_AT1 system_NN1 formed_VVN by_II the_AT condensation_NN1 of_IO solvent_NN1 into_II a_AT1 polymer_NN1 ._. 
As_II the_AT temperature_NN1 increases_NN2 ,_, the_AT entropy_NN1 loss_NN1 incurred_VVN during_II condensation_NN1 becomes_VVZ greater_JJR until_CS eventually_RR it_PPH1 is_VBZ so_RG unfavourable_JJ that_CST condensation_NN1 in_II the_AT polymer_NN1 is_VBZ impossible_JJ ,_, and_CC phase_NN1 separation_NN1 takes_VVZ place_NN1 ._. 
Neither_DD1 picture_NN1 is_VBZ particularly_RR rigorous_JJ but_CCB they_PPHS2 serve_VV0 to_TO emphasize_VVI the_AT fact_NN1 that_CST the_AT LCST_NP1 is_VBZ an_AT1 entropically_RR controlled_VVD phenomenon._NNU 8.14_MC Solubility_NP1 and_CC the_AT cohesive_JJ energy_NN1 density_NN1 Solvent-polymer_NN1 compatibility_NN1 problems_NN2 are_VBR repeatedly_RR encountered_VVN in_II industry_NN1 ._. 
For_REX21 example_REX22 ,_, in_II situations_NN2 requiring_VVG the_AT selection_NN1 of_IO elastomers_NN2 for_IF use_NN1 as_CSA hose-pipes_NN2 or_CC gaskets_NN2 ,_, the_AT correct_JJ choice_NN1 of_IO elastomer_NN1 is_VBZ of_IO prime_JJ importance_NN1 ,_, as_CSA contact_NN1 with_IW highly_RR compatible_JJ fluids_NN2 may_VM cause_VVI serious_JJ swelling_JJ and_CC impair_VVI the_AT operation_NN1 of_IO the_AT system_NN1 ._. 
The_AT wrong_JJ selection_NN1 can_VM have_VHI far_RR reaching_VVG consequences_NN2 ;_; the_AT initial_JJ choice_NN1 of_IO an_AT1 elastomer_NN1 for_IF the_AT seals_NN2 in_II the_AT landing_NN1 gear_NN1 of_IO the_AT DC-8_MC aircraft_NN resulted_VVN in_II serious_JJ jamming_NN1 because_CS the_AT seals_NN2 become_VV0 swollen_JJ when_CS in_II31 contact_II32 with_II33 the_AT hydraulic_JJ fluid_NN1 ._. 
This_DD1 almost_RR led_VVN to_II grounding_NN1 of_IO the_AT plane_NN1 but_CCB replacement_NN1 with_IW an_AT1 incompatible_JJ elastomer_NN1 made_VVN from_II ethylene-propylene_JJ copolymer_NN1 rectified_VVD the_AT fault_NN1 ._. 
To_TO avoid_VVI such_DA problems_NN2 a_AT1 technologist_NN1 may_VM wish_VVI to_TO have_VHI at_II his_APPGE disposal_NN1 a_AT1 rough_JJ guide_NN1 to_TO aid_VVI the_AT selection_NN1 of_IO solvents_NN2 for_IF a_AT1 polymer_NN1 or_CC to_TO assess_VVI the_AT extent_NN1 of_IO polymer-liquid_JJ interaction_NN1 other_II21 than_II22 those_DD2 already_RR described_VVN ._. 
Here_RL use_VV0 can_VM be_VBI made_VVN of_IO a_AT1 semi-empirical_JJ approach_NN1 suggested_VVN by_II Hildebrand_NP1 and_CC based_VVN on_II the_AT premise_NN1 that_DD1 '_VBZ like_NN1 dissolves_VVZ like_NN1 '_GE ._. 
The_AT treatment_NN1 involves_VVZ relating_VVG the_AT enthalpy_NN1 of_IO mixing_VVG to_II the_AT cohesive_JJ energy_NN1 density_NN1 and_CC defines_VVZ a_AT1 solubility_NN1 parameter_NN1 ,_, where_CS E_ZZ1 is_VBZ the_AT molar_JJ energy_NN1 of_IO vaporization_NN1 and_CC V_ZZ1 is_VBZ the_AT molar_JJ volume_NN1 of_IO the_AT component_NN1 ._. 
The_AT proposed_JJ relation_NN1 for_IF the_AT heat_NN1 of_IO mixing_NN1 of_IO two_MC non-polar_JJ components_NN2 shows_VVZ that_CST H_ZZ1 M_ZZ1 is_VBZ small_JJ for_IF mixtures_NN2 with_IW similar_JJ solubility_NN1 parameters_NN2 and_CC this_DD1 indicates_VVZ compatibility_NN1 ._. 
Values_NN2 of_IO the_AT solubility_NN1 parameter_NN1 for_IF simple_JJ liquids_NN2 can_VM be_VBI readily_RR calculated_VVN from_II the_AT enthalpy_NN1 of_IO vaporization_NN1 ._. 
The_AT same_DA method_NN1 can_VM not_XX be_VBI used_VVN for_IF a_AT1 polymer_NN1 and_CC one_PN1 must_VM resort_NN1 to_II comparative_JJ techniques_NN2 ._. 
Usually_RR for_IF a_AT1 polymer_NN1 is_VBZ established_VVN by_II finding_VVG the_AT solvent_NN1 which_DDQ will_VM produce_VVI maximum_JJ swelling_JJ of_IO a_AT1 network_NN1 or_CC the_AT largest_JJT value_NN1 of_IO the_AT limiting_JJ viscosity_NN1 number_NN1 ,_, as_CSA both_RR indicate_VV0 maximum_JJ compatibility_NN1 ._. 
The_AT polymer_NN1 is_VBZ then_RT assigned_VVN a_AT1 similar_JJ value_NN1 of_IO ._. 
Alternatively_RR ,_, Small_NP1 and_CC Hoy_UH have_VH0 tabulated_VVN a_AT1 series_NN of_IO group_NN1 molar_NN1 attraction_NN1 constants_NN2 from_II which_DDQ a_AT1 good_JJ estimate_NN1 of_IO for_IF most_DAT polymers_NN2 can_VM be_VBI made_VVN ._. 
The_AT suggested_JJ group_NN1 contributions_NN2 are_VBR shown_VVN in_II table_NN1 8.3_MC and_CC the_AT solubility_NN1 parameter_NN1 for_IF a_AT1 polymer_NN1 can_VM be_VBI estimated_VVN from_II the_AT sum_NN1 of_IO the_AT various_JJ molar_JJ attraction_NN1 constants_NN2 F_ZZ1 for_IF the_AT groups_NN2 which_DDQ make_VV0 up_RP the_AT repeat_NN1 unit_NN1 i.e._REX Here_RL V_ZZ1 is_VBZ the_AT molar_JJ volume_NN1 of_IO the_AT repeat_NN1 unit_NN1 whose_DDQGE molar_JJ mass_NN1 is_VBZ M_ZZ1 o_ZZ1 and_CC p_ZZ1 is_VBZ the_AT polymer_NN1 density_NN1 ._. 
Thus_RR for_IF poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) with_IW and_CC ,_, we_PPIS2 have_VH0 using_VVG the_AT Hoy_UH values_NN2 Therefore_RR For_IF a_AT1 more_RGR complex_JJ polyhydroxyether_VV0 of_IO Bisphenol_NP1 A_ZZ1 structure_NN1 and_CC with_IW Both_DB2 estimates_NN2 are_VBR within_II 10_MC per_NNU21 cent_NNU22 of_IO experimentally_RR determined_JJ values_NN2 ._. 
Attempts_NN2 to_TO correlate_VVI with_IW 1_MC1 from_II the_AT Flory-Huggins_NP1 equation_NN1 have_VH0 met_VVN with_IW limited_JJ success_NN1 because_II21 of_II22 the_AT unjustifiable_JJ assumptions_NN2 made_VVN in_II the_AT derivation_NN1 ._. 
It_PPH1 is_VBZ now_RT believed_VVN ,_, however_RR ,_, that_CST 1_MC1 is_VBZ not_XX an_AT1 enthalpy_NN1 parameter_NN1 but_CCB a_AT1 free_JJ energy_NN1 parameter_NN1 and_CC a_AT1 relation_NN1 of_IO the_AT form_NN1 c.f._VV0 section_NN1 8.8_MC has_VHZ improved_VVN the_AT correlation_NN1 ._. 
Here_RL is_VBZ supposed_JJ to_TO compensate_VVI for_IF the_AT lack_NN1 of_IO a_AT1 non-combinatorial_JJ entropy_NN1 contribution_NN1 in_II the_AT Flory-Huggins_NP1 treatment_NN1 ._. 
Unfortunately_RR ,_, solubility_NN1 is_VBZ not_XX a_AT1 simple_JJ process_NN1 and_CC secondary_JJ bonding_NN1 may_VM play_VVI an_AT1 important_JJ role_NN1 in_II determining_JJ component_NN1 interactions_NN2 ._. 
More_RGR detailed_JJ approaches_NN2 have_VH0 been_VBN suggested_VVN ,_, which_DDQ introduce_VV0 a_AT1 three-dimensional_JJ composed_JJ of_IO contributions_NN2 from_II van_NP1 der_NP1 Waals_NP1 dispersion_NN1 forces_NN2 ,_, dipole-dipole_JJ interaction_NN1 ,_, and_CC hydrogen_NN1 bonding_NN1 ._. 
The_AT overall_JJ solubility_NN1 parameter_NN1 is_VBZ then_RT the_AT sum_NN1 of_IO the_AT various_JJ contributions_NN2 Usually_RR two_MC dimensional_JJ plots_NN2 are_VBR constructed_VVN first_MD ,_, before_CS the_AT three_MC dimensional_JJ '_GE solubility_NN1 volume_NN1 '_GE is_VBZ established_VVN ,_, as_CSA shown_VVN in_II figure_NN1 8.10_MC ._. 
This_DD1 is_VBZ not_XX a_AT1 convenient_JJ construction_NN1 and_CC often_RR a_AT1 plot_NN1 of_IO versus_II H_ZZ1 is_VBZ considered_VVN to_TO be_VBI sufficiently_RR accurate_JJ as_CSA n_ZZ1 and_CC p_ZZ1 are_VBR usually_RR similar_JJ and_CC the_AT main_JJ polar_JJ contribution_NN1 comes_VVZ from_II the_AT hydrogen_NN1 bonding_NN1 factor_NN1 H._NP1 8.15_MC Polymer-polymer_JJ mixtures_NN2 In_II the_AT constant_JJ search_NN1 for_IF new_JJ materials_NN2 with_IW improved_JJ performance_NN1 ,_, the_AT idea_NN1 of_IO mixing_VVG two_MC or_CC more_RGR different_JJ polymers_NN2 to_TO form_VVI new_JJ substances_NN2 having_VHG a_AT1 combination_NN1 of_IO all_DB the_AT attributes_NN2 of_IO the_AT components_NN2 ,_, is_VBZ deceptively_RR attractive_JJ ._. 
Deceptively_RR ,_, because_CS in_II practice_NN1 it_PPH1 is_VBZ rarely_RR accomplished_VVN and_CC only_RR in_II a_AT1 few_DA2 cases_NN2 have_VH0 polymer_NN1 blends_NN2 or_CC mixtures_NN2 achieved_VVN industrial_JJ importance_NN1 ._. 
The_AT main_JJ reason_NN1 is_VBZ that_CST most_DAT common_JJ polymers_NN2 do_VD0 not_XX mix_VVI with_IW one_PPX121 another_PPX122 to_TO form_VVI homogeneous_JJ ,_, one_MC1 phase_NN1 solutions_NN2 or_CC blends_NN2 ,_, and_CC an_AT1 explanation_NN1 for_IF this_DD1 is_VBZ to_TO be_VBI found_VVN in_II the_AT thermodynamics_NN1 of_IO solutions_NN2 which_DDQ have_VH0 been_VBN outlined_VVN in_II the_AT previous_JJ sections_NN2 ._. 
As_CSA we_PPIS2 have_VH0 seen_VVN ,_, when_CS two_MC liquids_NN2 ,_, or_CC a_AT1 liquid_JJ and_CC a_AT1 polymer_NN1 are_VBR mixed_VVN ,_, the_AT formation_NN1 of_IO a_AT1 homogeneous_JJ ,_, one_MC1 phase_NN1 solution_NN1 is_VBZ assisted_VVN mainly_RR by_II the_AT large_JJ favourable_JJ gain_NN1 in_II combinatorial_JJ entropy_NN1 ._. 
This_DD1 entropic_JJ contribution_NN1 is_VBZ progressively_RR reduced_VVN when_CS one_PN1 or_CC both_DB2 components_NN2 increase_VV0 in_II size_NN1 ,_, and_CC the_AT reason_NN1 for_IF this_DD1 becomes_VVZ obvious_JJ on_II inspection_NN1 of_IO equation_NN1 (_( 8.24_MC )_) ._. 
When_CS r_ZZ1 1_MC1 and_CC r_ZZ1 2_MC both_RR increase_VV0 ,_, then_RT S_ZZ1 M_ZZ1 becomes_VVZ smaller_JJR ;_; consequently_RR attempts_VVZ to_TO mix_VVI two_MC high_JJ molar_JJ mass_JJ polymer_NN1 samples_NN2 will_VM receive_VVI little_DA1 assistance_NN1 from_II this_DD1 function_NN1 and_CC must_VM depend_VVI increasingly_RR on_II a_AT1 favourable_JJ (_( negative_JJ )_) heat_NN1 of_IO mixing_NN1 embodied_VVN in_II the_AT parameter_NN1 ._. 
This_DD1 loss_NN1 of_IO entropy_NN1 can_VM be_VBI conveniently_RR illustrated_VVN using_VVG the_AT simple_JJ lattice_NN1 model_NN1 shown_VVN in_II figure_NN1 8.11_MC ._. 
Here_RL a_AT1 10_MC x_ZZ1 10_MC lattice_NN1 ,_, containing_VVG 50_MC white_JJ and_CC 50_MC black_JJ units_NN2 randomly_RR mixed_VVN (_( a_ZZ1 )_) ,_, will_VM result_VVI in_II approximately_RR 10_MC 30_MC possible_JJ different_JJ arrangements_NN2 of_IO the_AT units_NN2 on_II mixing_NN1 ._. 
If_CS these_DD2 white_JJ units_NN2 are_VBR now_RT connected_VVN to_II other_JJ white_JJ units_NN2 ,_, and_CC black_JJ to_II black_JJ (_( b_ZZ1 )_) ,_, to_TO form_VVI five_MC equal_JJ chains_NN2 of_IO each_DD1 colour_NN1 with_IW ,_, then_RT the_AT number_NN1 of_IO possible_JJ arrangements_NN2 of_IO these_DD2 chains_NN2 decreases_VVZ to_II about_RG 10_MC 3_MC ._. 
Thus_RR as_CSA r_ZZ1 1_MC1 and_CC r_ZZ1 2_MC approach_NN1 infinity_NN1 S_ZZ1 M_ZZ1 will_VM become_VVI negligible_JJ and_CC the_AT free_JJ energy_NN1 of_IO mixing_NN1 will_VM become_VVI essentially_RR dependent_JJ on_II H_ZZ1 M_ZZ1 which_DDQ now_RT has_VHZ to_TO be_VBI either_RR very_RG small_JJ or_CC negative_JJ ._. 
The_AT heat_NN1 of_IO mixing_VVG for_IF the_AT majority_NN1 of_IO polymer_NN1 (_( 1_MC1 )_) polymer_NN1 (_( 2_MC )_) pairs_NN2 tends_VVZ to_TO be_VBI endothermic_JJ and_CC can_VM be_VBI approximated_VVN by_II reference_NN1 to_II the_AT solubility_NN1 parameters_NN2 using_VVG equation_NN1 (_( 8.57_MC )_) ._. 
This_DD1 can_VM be_VBI written_VVN as_CSA where_CS the_AT reference_NN1 volume_NN1 normally_RR assumes_VVZ a_AT1 value_NN1 of_IO ._. 
The_AT critical_JJ value_NN1 for_IF 12_MC can_VM be_VBI estimated_VVN from_II where_RRQ x_ZZ1 i_ZZ1 is_VBZ the_AT degree_NN1 of_IO polymerization_NN1 ,_, related_VVN to_II the_AT actual_JJ degree_NN1 of_IO polymerization_NN1 x_ZZ1 n_ZZ1 and_CC the_AT reference_NN1 volume_NN1 by_II with_IW V_ZZ1 R_ZZ1 the_AT molar_JJ volume_NN1 of_IO the_AT repeat_NN1 unit_NN1 ._. 
The_AT critical_JJ values_NN2 for_IF 12_MC above_II which_DDQ the_AT two_MC polymers_NN2 will_VM phase_NN1 separate_VV0 ,_, calculated_VVN for_IF various_JJ mixtures_NN2 with_IW ,_, are_VBR shown_VVN in_II table_NN1 8.4_MC along_II21 with_II22 the_AT corresponding_JJ differences_NN2 in_RP ._. 
This_DD1 shows_VVZ that_CST for_IF mixing_VVG to_TO take_VVI place_NN1 between_II high_JJ molecular_JJ weight_NN1 components_NN2 the_AT solubility_NN1 parameters_NN2 would_VM have_VHI to_TO be_VBI virtually_RR identical_JJ ._. 
This_DD1 limits_VVZ the_AT number_NN1 of_IO possible_JJ combinations_NN2 such_CS21 that_CS22 only_RR a_AT1 few_DA2 examples_NN2 exist_VV0 in_II this_DD1 category_NN1 ._. 
These_DD2 include_VV0 polystyrene/_NN1 poly_NN1 (_( -methyl_JJ styrene_NN1 )_) below_RL M_ZZ1 70_MC 000_MC ,_, and_CC the_AT polyacrylates_NN2 mixed_VVN with_IW the_AT corresponding_JJ poly_NN1 vinyl_NN1 esters_NN2 ,_, e.g._REX and_CC The_AT situation_NN1 changes_NN2 if_CS H_ZZ1 M_ZZ1 is_VBZ negative_JJ as_CSA this_DD1 will_VM encourage_VVI mixing_NN1 ,_, and_CC the_AT search_NN1 for_IF binary_JJ polymer_NN1 blends_NN2 which_DDQ are_VBR miscible_JJ has_VHZ focussed_VVN on_II combinations_NN2 in_II which_DDQ specific_JJ intermolecular_JJ interactions_NN2 ,_, such_II21 as_II22 hydrogen_NN1 bonds_NN2 ,_, dipole-dipole_JJ interactions_NN2 ,_, ion-dipole_JJ interactions_NN2 ,_, or_CC charge_NN1 transfer_NN1 complex_JJ formation_NN1 ,_, can_VM exist_VVI between_II the_AT component_NN1 polymers_NN2 ._. 
A_AT1 substantial_JJ number_NN1 of_IO miscible_JJ blends_NN2 have_VH0 now_RT been_VBN discovered_VVN using_VVG this_DD1 principle_NN1 and_CC it_PPH1 is_VBZ possible_JJ to_TO identify_VVI certain_JJ groups_NN2 or_CC repeat_VV0 units_NN2 ,_, which_DDQ when_CS incorporated_VVN in_II polymer_NN1 chains_NN2 tend_VV0 to_TO enter_VVI into_II these_DD2 intermolecular_JJ interactions_NN2 and_CC enhance_VVI the_AT miscibility_NN1 ._. 
A_AT1 short_JJ selection_NN1 of_IO some_DD of_IO these_DD2 complementary_JJ groups_NN2 is_VBZ given_VVN in_II table_NN1 8.5_MC ,_, where_CS a_AT1 polymer_NN1 containing_VVG groups_NN2 or_CC composed_VVD of_IO units_NN2 from_II column_NN1 1_MC1 will_VM tend_VVI to_TO form_VVI miscible_JJ blends_NN2 with_IW polymers_NN2 containing_VVG groups_NN2 or_CC composed_VVD of_IO units_NN2 from_II column_NN1 2_MC ._. 
Thus_RR it_PPH1 is_VBZ believed_VVN that_DD1 polystyrene_NN1 forms_VVZ miscible_JJ blends_NN2 with_IW poly_NN1 (_( vinyl_NN1 methylether_VV0 )_) ,_, and_CC with_IW poly_NN1 (_( phenylene_NN1 oxide_NN1 )_) s_ZZ1 (_( examples_NN2 1_MC1 and_CC 2_MC ,_, respectively_RR )_) because_II21 of_II22 interactions_NN2 between_II the_AT -electrons_NN2 of_IO the_AT phenyl_NN1 rings_NN2 and_CC the_AT lone_JJ pairs_NN2 of_IO the_AT ether_NN1 oxygens_NN2 ._. 
Similarly_RR it_PPH1 has_VHZ been_VBN suggested_VVN that_CST a_AT1 weak_JJ hydrogen_NN1 bond_NN1 ,_, which_DDQ is_VBZ strong_JJ enough_RR to_TO induce_VVI miscibility_NN1 can_VM form_VVI between_II the_AT carbonyl_NN1 unit_NN1 of_IO poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) and_CC the_AT -hydrogen_NN1 of_IO poly_NN1 (_( vinyl_NN1 chloride_NN1 )_) (_( example_NN1 3_MC where_RRQ )_) ._. 
i.e._REX Much_DA1 stronger_JJR hydrogen_NN1 bonding_NN1 interactions_NN2 can_VM be_VBI obtained_VVN if_CS units_NN2 such_II21 as_II22 or_CC sites_NN2 for_IF ion-dipole_JJ interactions_NN2 such_II21 as_II22 can_VM be_VBI built_VVN into_II chains_NN2 ,_, and_CC even_RR in_II relatively_RR small_JJ amounts_NN2 these_DD2 can_VM transform_VVI immiscible_JJ pairs_NN2 into_II totally_RR miscible_JJ blends_NN2 ._. 
Many_DA2 of_IO these_DD2 blends_NN2 undergo_VV0 quite_RG rapid_JJ demixing_NN1 as_II the_AT temperature_NN1 is_VBZ raised_VVN and_CC an_AT1 LCST_NP1 phase_NN1 boundary_NN1 can_VM be_VBI located_VVN above_II the_AT glass_NN1 transition_NN1 temperature_NN1 of_IO the_AT blend_NN1 ._. 
The_AT origins_NN2 of_IO the_AT lower_JJR critical_JJ phase_NN1 separation_NN1 phenomenon_NN1 in_II polymer_NN1 blends_NN2 are_VBR not_XX yet_RR clearly_RR understood_VVN and_CC three_MC possible_JJ causes_NN2 have_VH0 been_VBN proposed._NNU (_( i_ZZ1 )_) Free_JJ volume_NN1 dissimilarities_NN2 may_VM become_VVI unfavourable_JJ to_II mixing_VVG on_II increasing_VVG the_AT temperature._NNU (_( ii_MC )_) There_EX may_VM be_VBI unfavourable_JJ entropy_NN1 contributions_NN2 arising_VVG from_II non-random_JJ mixing._NNU (_( iii_MC )_) A_ZZ1 temperature_NN1 dependent_JJ heat_NN1 of_IO mixing_NN1 may_VM result_VVI if_CSW specific_JJ intermolecular_JJ interactions_NN2 ,_, which_DDQ dissociate_VV0 on_II heating_NN1 ,_, are_VBR responsible_JJ for_IF miscibility_NN1 at_II lower_JJR temperatures_NN2 ._. 
While_CS the_AT latter_DA seems_VVZ the_AT most_RGT likely_JJ cause_NN1 in_II many_DA2 blends_NN2 where_RRQ specific_JJ interactions_NN2 have_VH0 been_VBN identified_VVN ,_, miscible_JJ blends_NN2 can_VM also_RR be_VBI obtained_VVN when_CS certain_JJ statistical_JJ copolymers_NN2 are_VBR mixed_VVN with_IW either_RR a_AT1 homopolymer_NN1 ,_, or_CC another_DD1 copolymer_NN1 ,_, in_II which_DDQ no_AT such_DA interactions_NN2 have_VH0 been_VBN located_VVN ._. 
Thus_RR poly_NN1 (_( )_) will_VM form_VVI miscible_JJ blends_NN2 with_IW poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) if_CS the_AT composition_NN1 of_IO the_AT copolymer_NN1 lies_VVZ in_II the_AT range_NN1 1039_MC wt%_FO acrylonitrile_NN1 ._. 
This_DD1 range_NN1 of_IO compositions_NN2 is_VBZ called_VVN the_AT '_GE miscibility_NN1 window_NN1 '_GE and_CC has_VHZ been_VBN reported_VVN to_TO be_VBI present_JJ in_II other_JJ systems_NN2 ._. 
The_AT drive_NN1 towards_II formation_NN1 of_IO a_AT1 miscible_JJ solid_JJ solution_NN1 in_II these_DD2 cases_NN2 is_VBZ believed_VVN to_TO arise_VVI when_RRQ large_JJ repulsive_JJ interactions_NN2 exist_VV0 between_II the_AT monomer_NN1 units_NN2 (_( A_ZZ1 )_) and_CC (_( B_ZZ1 )_) comprising_VVG the_AT copolymer_NN1 ;_; on_II mixing_VVG with_IW a_AT1 polymer_NN1 (_( C_ZZ1 )_) ,_, the_AT number_NN1 of_IO these_DD2 unfavourable_JJ (_( A_ZZ1 B_ZZ1 )_) contacts_NN2 are_VBR reduced_VVN by_II forming_VVG less_RGR repulsive_JJ (_( A-C_NN1 )_) or_CC (_( B-C_RA )_) contacts_NN2 and_CC a_AT1 miscible_JJ blend_NN1 results_NN2 ._. 
Many_DA2 of_IO these_DD2 blends_NN2 also_RR exhibit_VV0 an_AT1 LCST_NP1 ._. 
Thus_RR the_AT driving_JJ force_NN1 towards_II lower_JJR critical_JJ phase_NN1 separation_NN1 in_II polymer-polymer_JJ solutions_NN2 may_VM depend_VVI on_II the_AT system_NN1 or_CC may_VM be_VBI a_AT1 combination_NN1 of_IO the_AT effects_NN2 (_( i_ZZ1 )_) (_( iii_MC )_) ._. 
CHAPTER_NN1 9_MC Polymer_NN1 Characterization_NN1 Molar_NN1 Masses_NN2 9.1_MC Introduction_NN1 Many_DA2 of_IO the_AT distinctive_JJ properties_NN2 of_IO polymers_NN2 are_VBR a_AT1 consequence_NN1 of_IO the_AT long_JJ chain_NN1 lengths_NN2 ,_, which_DDQ are_VBR reflected_VVN in_II the_AT large_JJ molar_JJ masses_NN2 of_IO these_DD2 substances_NN2 ._. 
While_CS such_DA large_JJ molar_JJ masses_NN2 are_VBR now_RT taken_VVN for_IF granted_VVN ,_, it_PPH1 was_VBDZ difficult_JJ in_II 1920_MC to_TO believe_VVI and_CC accept_VVI that_CST these_DD2 values_NN2 were_VBDR real_JJ and_CC not_XX just_RR caused_VVN by_II the_AT aggregation_NN1 of_IO much_DA1 smaller_JJR molecules_NN2 ._. 
Values_NN2 of_IO the_AT order_NN1 of_IO 10_MC 6_MC g_NNU mol_NN1 -1_MC are_VBR now_RT accepted_VVN without_IW question_NN1 ,_, but_CCB the_AT accuracy_NN1 of_IO the_AT measurements_NN2 is_VBZ much_RR lower_JJR than_CSN for_IF simple_JJ molecules_NN2 ._. 
This_DD1 is_VBZ not_XX surprising_JJ ,_, especially_RR when_CS polymer_NN1 samples_NN2 exhibit_VV0 polydispersity_NN1 ,_, and_CC the_AT molar_JJ mass_NN1 is_VBZ ,_, at_RR21 best_RR22 ,_, an_AT1 average_JJ dependent_JJ on_II the_AT particular_JJ method_NN1 of_IO measurement_NN1 used_VVD ._. 
Estimation_NN1 of_IO the_AT molar_JJ mass_NN1 of_IO a_AT1 polymer_NN1 is_VBZ of_IO considerable_JJ importance_NN1 ,_, as_CSA the_AT chain_NN1 length_NN1 can_VM be_VBI a_AT1 controlling_JJ factor_NN1 in_II determining_JJ solubility_NN1 ,_, elasticity_NN1 ,_, fibre_NN1 forming_VVG capacity_NN1 ,_, tear_NN1 strength_NN1 ,_, and_CC impact_NN1 strength_NN1 in_II many_DA2 polymers_NN2 ._. 
The_AT methods_NN2 used_VMK to_TO determine_VVI the_AT molar_JJ mass_NN1 M_ZZ1 are_VBR either_RR relative_JJ or_CC absolute_JJ ._. 
Relative_JJ methods_NN2 require_VV0 calibration_NN1 with_IW samples_NN2 of_IO known_JJ M_NN1 and_CC include_VV0 viscosity_NN1 and_CC vapour_NN1 pressure_NN1 osmometry_NN1 ._. 
The_AT absolute_JJ methods_NN2 are_VBR often_RR classified_VVN by_II the_AT type_NN1 of_IO average_NN1 they_PPHS2 yield_VV0 ,_, i.e._REX colligative_JJ techniques_NN2 yield_VV0 number_NN1 averages_NN2 ,_, light_JJ scattering_NN1 and_CC the_AT ultracentrifuge_NN1 yield_VV0 higher_JJR averages_NN2 ,_, e.g._REX weight_NN1 and_CC z-average._JJ 9.2_MC Molar_JJ masses_NN2 ,_, molecular_JJ weights_NN2 ,_, and_CC SI_FW units_NN2 The_AT dimensionless_JJ quantity_NN1 '_GE the_AT relative_JJ molecular_JJ mass_NN1 '_GE (_( molecular_JJ weight_NN1 )_) defined_VVD as_II the_AT average_JJ mass_NN1 of_IO the_AT molecule_NN1 divided_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) the_AT mass_NN1 of_IO an_AT1 atom_NN1 of_IO the_AT nuclide_NN1 C_ZZ1 12_MC ,_, is_VBZ often_RR used_VVN in_II polymer_NN1 chemistry_NN1 ,_, and_CC called_VVN the_AT molecular_JJ weight_NN1 ._. 
In_II this_DD1 book_NN1 the_AT quantity_NN1 molar_NN1 mass_NN1 is_VBZ used_VVN and_CC appropriate_JJ SI_FW units_NN2 are_VBR given._NNU 9.3_MC Number_NN1 average_JJ molar_JJ mass_NN1 M_ZZ1 n_ZZ1 Determination_NN1 of_IO the_AT number_NN1 average_JJ molar_JJ mass_NN1 M_ZZ1 n_ZZ1 involves_VVZ counting_VVG the_AT total_JJ number_NN1 of_IO molecules_NN2 ,_, regardless_RR of_IO their_APPGE shape_NN1 or_CC size_NN1 ,_, present_NN1 in_II a_AT1 unit_NN1 mass_NN1 of_IO the_AT polymer_NN1 ._. 
The_AT methods_NN2 are_VBR conveniently_RR grouped_VVN into_II three_MC categories_NN2 :_: end-group_JJ assay_NN1 ,_, thermodynamic_JJ ,_, and_CC transport_NN1 methods_NN2 ._. 
9.4_MC End-group_JJ assay_NN1 The_AT technique_NN1 is_VBZ of_IO limited_JJ value_NN1 and_CC can_VM only_RR be_VBI used_VVN when_CS the_AT polymer_NN1 has_VHZ an_AT1 end_NN1 group_NN1 amenable_JJ to_II analysis_NN1 ._. 
It_PPH1 can_VM be_VBI used_VVN to_TO follow_VVI the_AT progress_NN1 of_IO linear_JJ condensation_NN1 reactions_NN2 when_RRQ an_AT1 end_NN1 group_NN1 ,_, such_II21 as_II22 a_AT1 carboxyl_NN1 ,_, is_VBZ present_JJ which_DDQ can_VM be_VBI titrated_VVN ._. 
It_PPH1 is_VBZ used_VVN to_TO detect_VVI amino_NN1 end_NN1 groups_NN2 in_II nylons_NN2 dissolved_VVN in_II m-cresol_NN1 ,_, by_II titration_NN1 with_IW methanolic_JJ perchloric_JJ acid_NN1 solution_NN1 ,_, and_CC can_VM be_VBI applied_VVN to_II vinyl_NN1 polymers_NN2 if_CS an_AT1 initiator_NN1 fragment_NN1 ,_, perhaps_RR containing_VVG halogen_NN1 ,_, is_VBZ attached_VVN to_II the_AT end_NN1 of_IO the_AT chain_NN1 ._. 
The_AT sensitivity_NN1 of_IO the_AT method_NN1 decreases_VVZ rapidly_RR as_CSA the_AT chain_NN1 length_NN1 increases_NN2 and_CC the_AT number_NN1 of_IO end_NN1 groups_NN2 drops_VVZ ._. 
A_AT1 practical_JJ upper_JJ limit_NN1 might_VM reach_VVI an_AT1 M_ZZ1 n_ZZ1 of_IO about_RG 15000_MC g_NNU mol_NN1 -1._MC 9.5_MC Colligative_JJ properties_NN2 of_IO solutions_NN2 :_: thermodynamic_JJ considerations_NN2 Because_CS chemical_JJ methods_NN2 are_VBR rather_RG limited_JJ ,_, the_AT most_RGT widely_RR used_JJ techniques_NN2 for_IF measuring_VVG the_AT molar_JJ mass_NN1 of_IO a_AT1 polymer_NN1 are_VBR physical_JJ ._. 
Among_II the_AT more_RGR common_JJ methods_NN2 are_VBR those_DD2 which_DDQ depend_VV0 on_II the_AT colligative_JJ properties_NN2 of_IO dilute_JJ solutions_NN2 ._. 
These_DD2 are_VBR (_( a_ZZ1 )_) lowering_NN1 of_IO the_AT vapour_NN1 pressure_NN1 (_( b_ZZ1 )_) elevation_NN1 of_IO the_AT boiling_JJ point_NN1 (_( c_ZZ1 )_) depression_NN1 of_IO the_AT freezing_JJ point_NN1 (_( d_ZZ1 )_) osmotic_JJ pressure_NN1 ._. 
A_AT1 colligative_JJ property_NN1 is_VBZ defined_VVN as_CSA one_PN1 which_DDQ is_VBZ a_AT1 function_NN1 of_IO the_AT number_NN1 of_IO solute_NN1 molecules_NN2 present_VV0 per_II unit_NN1 volume_NN1 of_IO solution_NN1 and_CC is_VBZ unaffected_JJ by_II the_AT chemical_JJ nature_NN1 of_IO the_AT solute_NN1 ._. 
Thus_RR if_CS Y_ZZ1 represents_VVZ any_DD of_IO the_AT above_JJ colligative_JJ properties_NN2 then_RT where_CS N_ZZ1 i_ZZ1 is_VBZ the_AT number_NN1 of_IO particles_NN2 of_IO each_DD1 solute_NN1 component_NN1 i_ZZ1 ,_, and_CC K_ZZ1 is_VBZ a_AT1 proportionality_NN1 constant_NN1 ._. 
The_AT concentration_NN1 of_IO a_AT1 solution_NN1 per_II unit_NN1 volume_NN1 of_IO solution_NN1 V_ZZ1 is_VBZ where_RRQ w_ZZ1 i_ZZ1 is_VBZ the_AT mass_NN1 of_IO the_AT component_NN1 and_CC N_ZZ1 A_ZZ1 is_VBZ the_AT Avogadro_NP1 constant_JJ ._. 
The_AT colligative_JJ property_NN1 can_VM be_VBI expressed_VVN in_II the_AT reduced_JJ form_NN1 Y/c_ZZ1 so_CS21 that_CS22 Hence_RR any_DD colligative_JJ method_NN1 should_VM yield_VVI the_AT number_NN1 average_JJ molar_JJ mass_NN1 M_ZZ1 n_ZZ1 of_IO a_AT1 polydisperse_NN1 polymer_NN1 ._. 
When_CS a_AT1 solute_NN1 ,_, such_II21 as_II22 a_AT1 polymer_NN1 (_( component_NN1 2_MC )_) ,_, is_VBZ dissolved_VVN in_II a_AT1 solvent_NN1 (_( component_NN1 1_MC1 )_) to_TO form_VVI a_AT1 homogeneous_JJ solution_NN1 ,_, there_EX is_VBZ a_AT1 change_NN1 in_II the_AT chemical_JJ potential_NN1 which_DDQ can_VM be_VBI expressed_VVN in_II31 terms_II32 of_II33 the_AT activity_NN1 of_IO the_AT solvent_NN1 a_AT1 1_MC1 ._. 
During_II the_AT measurement_NN1 of_IO the_AT molar_JJ mass_NN1 of_IO the_AT polymer_NN1 using_VVG a_AT1 colligative_JJ method_NN1 ,_, an_AT1 equilibrium_NN1 is_VBZ established_VVN when_CS the_AT chemical_JJ potential_NN1 of_IO the_AT solvent_NN1 in_II the_AT solution_NN1 is_VBZ equal_JJ to_II that_DD1 of_IO the_AT pure_JJ solvent_NN1 ,_, where_CS the_AT pure_JJ solvent_NN1 is_VBZ either_RR in_II another_DD1 phase_NN1 or_CC separated_VVN from_II the_AT solution_NN1 by_II a_AT1 semi-permeable_JJ membrane_NN1 ._. 
The_AT equilibration_NN1 procedure_NN1 can_VM be_VBI achieved_VVN either_RR by_II changing_VVG the_AT temperature_NN1 or_CC the_AT pressure_NN1 of_IO the_AT system_NN1 ,_, and_CC the_AT amount_NN1 of_IO this_DD1 change_NN1 is_VBZ then_RT a_AT1 measure_NN1 of_IO the_AT activity_NN1 of_IO the_AT solvent_NN1 in_II solution_NN1 ._. 
This_DD1 tells_VVZ us_PPIO2 nothing_PN1 immediately_RR about_II the_AT solute_NN1 but_CCB ,_, if_CS very_RG dilute_JJ solutions_NN2 are_VBR used_VVN ,_, the_AT following_JJ useful_JJ approximations_NN2 can_VM be_VBI made_VVN ._. 
The_AT activity_NN1 of_IO the_AT solvent_NN1 can_VM be_VBI considered_VVN to_TO be_VBI equal_JJ to_II the_AT mole_NN1 fraction_NN1 of_IO the_AT solvent_NN1 x_ZZ1 1_MC1 ._. 
By_II expanding_VVG the_AT logarithmic_JJ term_NN1 and_CC assuming_VVG that_CST in_II dilute_JJ solutions_NN2 this_DD1 can_VM be_VBI restricted_VVN to_II the_AT first_MD expansion_NN1 term_NN1 ,_, ln_NNU a_AT1 1_MC1 can_VM be_VBI related_VVN to_II the_AT mole_NN1 fraction_NN1 of_IO the_AT solute_NN1 x_ZZ1 2_MC ._. 
We_PPIS2 can_VM now_RT make_VVI use_NN1 of_IO these_DD2 approximations_NN2 to_TO calculate_VVI M_ZZ1 n._NNU 9.6_MC Ebullioscopy_NP1 and_CC cryoscopy_NN1 In_II principle_NN1 these_DD2 two_MC methods_NN2 can_VM be_VBI treated_VVN together_RL and_CC the_AT relevant_JJ expressions_NN2 are_VBR derived_VVN from_II the_AT Clausius-Clapeyron_NP1 equation_NN1 describing_VVG the_AT temperature_NN1 dependence_NN1 of_IO the_AT vapour_NN1 pressure_NN1 of_IO a_AT1 liquid_NN1 where_CS H_ZZ1 1_MC1 is_VBZ the_AT latent_JJ heat_NN1 of_IO vapourization_NN1 ._. 
If_CS P_ZZ1 is_VBZ taken_VVN as_II the_AT vapour_NN1 pressure_NN1 of_IO a_AT1 solution_NN1 whose_DDQGE pure_JJ solvent_NN1 vapour_NN1 pressure_NN1 is_VBZ P_ZZ1 o_ZZ1 ,_, then_RT for_IF solutions_NN2 containing_VVG an_AT1 involatile_JJ solute_NN1 which_DDQ gives_VVZ If_CS the_AT solution_NN1 is_VBZ very_RG dilute_JJ the_AT change_NN1 in_II temperature_NN1 T_ZZ1 can_VM be_VBI related_VVN to_II the_AT solute_NN1 mole_NN1 fraction_NN1 by_II and_CC substituting_VVG for_IF gives_VVZ where_RRQ c_ZZ1 2_MC is_VBZ the_AT solute_NN1 concentration_NN1 (_( mass_NN1 per_II unit_NN1 volume_NN1 solution_NN1 )_) ._. 
Polymer_NN1 solutions_NN2 do_VD0 not_XX behave_VVI in_II this_DD1 ideal_JJ manner_NN1 even_RR in_II the_AT dilute_JJ solution_NN1 regime_NN1 and_CC for_IF accurate_JJ molar_JJ mass_JJ measurements_NN2 ,_, deviations_NN2 from_II ideality_NN1 must_VM be_VBI eliminated_VVN ._. 
A_AT1 more_RGR accurate_JJ representation_NN1 of_IO the_AT behaviour_NN1 of_IO the_AT polymer_NN1 solution_NN1 can_VM be_VBI obtained_VVN using_VVG equation_NN1 (_( 8.35_MC )_) where_RRQ Substitution_NN1 in_II equation_NN1 (_( 9.9_MC )_) yields_NN2 and_CC rearrangement_NN1 eliminating_VVG higher_JJR terms_NN2 gives_VVZ The_AT non-ideal_JJ behaviour_NN1 can_VM then_RT be_VBI eliminated_VVN by_II extrapolating_VVG the_AT experimental_JJ data_NN to_II c_ZZ1 2_MC 0_MC where_RRQ equation_NN1 (_( 9.14_MC )_) reduces_VVZ to_II equation_NN1 (_( 9.11_MC )_) and_CC can_VM be_VBI calculated_VVN ._. 
For_IF ebulliometry_NN1 ,_, T_ZZ1 ,_, H_ZZ1 ,_, and_CC T_ZZ1 are_VBR the_AT boiling_JJ temperature_NN1 of_IO the_AT solvent_NN1 ,_, the_AT enthalpy_NN1 of_IO vaporization_NN1 of_IO the_AT solvent_NN1 ,_, and_CC the_AT elevation_NN1 of_IO the_AT boiling_JJ temperature_NN1 respectively_RR ,_, while_CS for_IF cryoscopy_NN1 they_PPHS2 represent_VV0 the_AT freezing_JJ temperature_NN1 of_IO the_AT solvent_NN1 ,_, the_AT enthalpy_NN1 of_IO fusion_NN1 of_IO the_AT solvent_NN1 ,_, and_CC the_AT depression_NN1 of_IO the_AT freezing_JJ temperature_NN1 ._. 
The_AT equation_NN1 represents_VVZ the_AT limiting_JJ case_NN1 at_II infinite_JJ dilution_NN1 and_CC it_PPH1 is_VBZ necessary_JJ to_TO extrapolate_VVI for_IF a_AT1 series_NN of_IO solutions_NN2 to_II c_ZZ1 =_FO 0_MC in_BCL21 order_BCL22 to_TO calculate_VVI M_ZZ1 n_ZZ1 ._. 
The_AT measurements_NN2 are_VBR limited_VVN by_II the_AT sensitivity_NN1 of_IO the_AT thermometer_NN1 used_VMK to_TO obtain_VVI T._NP1 At_RR21 present_RR22 this_DD1 can_VM rarely_RR detect_VVI a_AT1 T_ZZ1 of_IO less_DAR than_CSN with_IW any_DD precision_NN1 ,_, and_CC the_AT limit_NN1 of_IO accurate_JJ measurement_NN1 of_IO M_ZZ1 n_ZZ1 is_VBZ in_II the_AT region_NN1 of_IO 25_MC 000_MC to_II 30_MC 000_MC g_NNU mol_NN1 -1._MC 9.7_MC Osmotic_JJ pressure_NN1 Measurement_NN1 of_IO the_AT osmotic_JJ pressure_NN1 of_IO a_AT1 polymer_NN1 solution_NN1 can_VM be_VBI carried_VVN out_RP in_II the_AT type_NN1 of_IO cell_NN1 represented_VVD schematically_RR in_II figure_NN1 9.1_MC ._. 
The_AT polymer_NN1 solution_NN1 is_VBZ separated_VVN from_II the_AT pure_JJ solvent_NN1 by_II a_AT1 membrane_NN1 ,_, permeable_JJ only_RR to_II solvent_NN1 molecules_NN2 ._. 
Initially_RR ,_, the_AT chemical_JJ potential_NN1 1_MC1 ,_, of_IO the_AT solvent_NN1 in_II the_AT solution_NN1 ,_, is_VBZ lower_JJR than_CSN that_DD1 of_IO the_AT pure_JJ solvent_NN1 ,_, 1_MC1 and_CC solvent_NN1 molecules_NN2 tend_VV0 to_TO pass_VVI through_II the_AT membrane_NN1 into_II the_AT solution_NN1 in_BCL21 order_BCL22 to_TO attain_VVI equilibrium_NN1 ._. 
This_DD1 causes_VVZ a_AT1 build_VV0 up_RP of_IO pressure_NN1 in_II the_AT solution_NN1 compartment_NN1 until_CS ,_, at_II equilibrium_NN1 ,_, the_AT pressure_NN1 exactly_RR counteracts_VVZ the_AT tendency_NN1 for_IF further_JJR solvent_NN1 flow_NN1 ._. 
This_DD1 pressure_NN1 is_VBZ the_AT osmotic_JJ pressure_NN1 ._. 
The_AT expression_NN1 for_IF the_AT reduced_JJ osmotic_JJ pressure_NN1 has_VHZ already_RR been_VBN derived_VVN in_II section_NN1 8.7_MC and_CC has_VHZ the_AT form_NN1 where_CS the_AT limiting_JJ form_NN1 ,_, valid_JJ only_RR at_II infinite_JJ dilution_NN1 ,_, is_VBZ Only_RR under_RG special_JJ conditions_NN2 ,_, when_CS the_AT polymer_NN1 is_VBZ dissolved_VVN in_II a_AT1 theta-solvent_NN1 ,_, will_VM be_VBI independent_JJ of_IO concentration_NN1 ._. 
Experimentally_RR ,_, a_AT1 series_NN of_IO concentrations_NN2 is_VBZ studied_VVN and_CC the_AT results_NN2 treated_VVN according_II21 to_II22 one_MC1 or_CC other_JJ of_IO the_AT following_JJ virial_NN1 expansions_NN2 ._. 
McMillan_NP1 and_CC Meyer_NP1 suggested_VVD ,_, while_CS alternative_JJ forms_NN2 are_VBR also_RR used_VVN :_: and_CC The_AT coefficients_NN2 B_ZZ1 ,_, A_ZZ1 2_MC ,_, 2_MC and_CC B_ZZ1 3_MC ,_, A_ZZ1 3_MC ,_, 3_MC ,_, are_VBR the_AT second_MD and_CC third_MD virial_NN1 coefficients_NN2 ._. 
When_RRQ solutions_NN2 are_VBR sufficiently_RR dilute_VV0 a_AT1 plot_NN1 of_IO against_II c_ZZ1 is_VBZ linear_JJ and_CC the_AT third_MD virial_NN1 coefficients_NN2 (_( B_ZZ1 3_MC ,_, A_ZZ1 3_MC ,_, 3_MC )_) can_VM be_VBI neglected_VVN ._. 
The_AT various_JJ forms_NN2 of_IO the_AT second_MD virial_NN1 coefficient_NN1 are_VBR interrelated_VVN by_RP Although_CS not_XX normally_RR detected_VVN ,_, the_AT third_MD virial_NN1 coefficient_NN1 occasionally_RR contributes_VVZ to_II the_AT non-ideal_JJ behaviour_NN1 in_II dilute_JJ solutions_NN2 and_CC a_AT1 curved_JJ plot_NN1 is_VBZ obtained_VVN (_( figure_NN1 9.2a_FO ._. )_) 
This_DD1 increases_VVZ the_AT uncertainty_NN1 of_IO the_AT extrapolation_NN1 ,_, but_CCB can_VM be_VBI overcome_VVN by_II recasting_VVG equation_NN1 (_( 9.19_MC )_) and_CC introducing_VVG a_AT1 polymer-solvent_NN1 interaction_NN1 parameter_NN1 g_ZZ1 It_PPH1 has_VHZ been_VBN found_VVN that_CST g_ZZ1 =_FO 0.25_MC in_II good_JJ solvents_NN2 so_CS21 that_CS22 equation_NN1 (_( 9.21_MC )_) becomes_VVZ A_AT1 plot_NN1 of_IO against_II c_ZZ1 is_VBZ now_RT linear_JJ and_CC this_DD1 extrapolation_NN1 is_VBZ illustrated_VVN in_II figure_NN1 9.2b_FO ._. 
This_DD1 example_NN1 (_( figure_NN1 9.2_MC )_) also_RR shows_VVZ the_AT differing_JJ solubility_NN1 of_IO poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) in_II the_AT three_MC solvents_NN2 ._. 
In_II a_AT1 good_JJ solvent_NN1 ,_, toluene_NN1 ,_, the_AT slope_NN1 or_CC A_ZZ1 2_MC is_VBZ large_JJ ,_, but_CCB as_CSA the_AT solvent_NN1 becomes_VVZ poorer_JJR (_( acetone_NN1 )_) A_ZZ1 2_MC decreases_VVZ ,_, until_CS it_PPH1 is_VBZ zero_MC in_II the_AT theta-solvent_NN1 acetonitrile_NN1 ._. 
Thus_RR A_ZZ1 2_MC provides_VVZ a_AT1 useful_JJ measure_NN1 of_IO the_AT thermodynamic_JJ quality_NN1 of_IO the_AT solvent_NN1 and_CC measures_VVZ the_AT deviation_NN1 from_II ideality_NN1 of_IO the_AT polymer_NN1 solution_NN1 ._. 
The_AT value_NN1 of_IO M_ZZ1 n_ZZ1 is_VBZ calculated_VVN from_II the_AT intercept_VV0 using_VVG equation_NN1 (_( 9.16_MC )_) ._. 
The_AT corresponding_JJ values_NN2 of_IO the_AT second_MD virial_NN1 coefficient_NN1 are_VBR obtained_VVN from_II the_AT slopes_NN2 of_IO the_AT plots_NN2 (_( table_NN1 9.1_MC )_) ._. 
PRACTICAL_JJ OSMOMETRY_NN1 The_AT static_JJ method_NN1 of_IO determining_VVG the_AT osmotic_JJ pressure_NN1 of_IO a_AT1 polymer_NN1 solution_NN1 ,_, using_VVG volumes_NN2 of_IO 3_MC to_II 20_MC cm_NNU 3_MC of_IO solution_NN1 ,_, is_VBZ a_AT1 relatively_RR slow_JJ process_NN1 which_DDQ requires_VVZ about_RG 24_MC h_ZZ1 to_TO equilibrate_VVI at_II each_DD1 concentration_NN1 ._. 
Several_DA2 designs_NN2 ,_, suitable_JJ for_IF this_DD1 type_NN1 of_IO measurement_NN1 ,_, are_VBR typified_VVN by_II the_AT Pinner_NP1 Stabin_NP1 instrument_NN1 shown_VVN schematically_RR in_II figure_NN1 9.1_MC ._. 
The_AT osmometer_NN1 is_VBZ assembled_VVN ,_, under_II a_AT1 layer_NN1 of_IO solvent_NN1 ,_, by_II clamping_VVG two_MC membranes_NN2 (_( kept_VVN continually_RR moist_JJ with_IW solvent_NN1 )_) on_II either_DD1 side_NN1 of_IO the_AT glass_NN1 cell_NN1 c_ZZ1 ._. 
These_DD2 are_VBR retained_VVN in_II position_NN1 by_II two_MC metal_NN1 plates_NN2 perforated_JJ and_CC grooved_VVD to_TO allow_VVI contact_NN1 between_II the_AT membrane_NN1 and_CC solvent_NN1 which_DDQ is_VBZ in_II the_AT outer_JJ container_NN1 ._. 
The_AT preparation_NN1 of_IO the_AT membranes_NN2 is_VBZ very_RG important_JJ and_CC must_VM be_VBI carefully_RR carried_VVN out_RP ._. 
They_PPHS2 are_VBR normally_RR made_VVN of_IO cellulose_NN1 or_CC a_AT1 cellulose_NN1 derivative_NN1 and_CC should_VM be_VBI slowly_RR conditioned_VVN from_II the_AT storage_NN1 liquid_NN1 to_II the_AT solvent_NN1 in_II use_NN1 ._. 
This_DD1 is_VBZ done_VDN by_II transferring_VVG the_AT membrane_NN1 to_II mixtures_NN2 progressively_RR richer_JJR in_II the_AT solvent_NN1 ,_, allowing_VVG them_PPHO2 time_NNT1 to_TO equilibrate_VVI with_IW the_AT mixture_NN1 ,_, then_RT transferring_VVG again_RT until_CS pure_JJ solvent_NN1 is_VBZ reached_VVN ._. 
Equilibration_NN1 in_II each_DD1 mixture_NN1 usually_RR takes_VVZ a_AT1 few_DA2 hours_NNT2 ._. 
When_CS assembled_VVN ,_, the_AT osmometer_NN1 is_VBZ placed_VVN in_II a_AT1 jacket_NN1 containing_VVG enough_DD solvent_NN1 to_TO cover_VVI the_AT bottom_JJ part_NN1 of_IO the_AT reference_NN1 capillary_NN1 s_ZZ1 ._. 
Solvent_NN1 is_VBZ then_RT withdrawn_VVN from_II the_AT cell_NN1 c_ZZ1 and_CC a_AT1 solution_NN1 of_IO polymer_NN1 added_VVN by_II means_NN of_IO a_AT1 syringe_NN1 ._. 
Care_NN1 is_VBZ taken_VVN during_II the_AT filling_NN1 stage_NN1 to_TO avoid_VVI trapping_VVG bubbles_NN2 in_II the_AT cell_NN1 ._. 
The_AT level_NN1 of_IO the_AT solution_NN1 is_VBZ then_RT adjusted_VVN to_II a_AT1 few_DA2 centimetres_NNU2 above_II the_AT level_NN1 of_IO solvent_NN1 in_II s_ZZ1 by_II31 means_II32 of_II33 a_AT1 levelling_JJ rod_NN1 1_MC1 ._. 
Mercury_NP1 is_VBZ added_VVN to_II the_AT cup_NN1 t_ZZ1 ,_, to_TO ensure_VVI a_AT1 leak_NN1 free_JJ system_NN1 ,_, and_CC the_AT osmometer_NN1 is_VBZ left_JJ undisturbed_JJ in_II a_AT1 thermostat_NN1 bath_NN1 controlled_VVN to_II 0.01_MC K_ZZ1 to_TO reach_VVI equilibrium_NN1 ._. 
The_AT osmotic_JJ pressure_NN1 can_VM be_VBI calculated_VVN from_II the_AT difference_NN1 in_II heights_NN2 h_ZZ1 between_II the_AT solvent_NN1 and_CC solution_NN1 in_II s_ZZ1 and_CC m_ZZ1 respectively_RR and_CC is_VBZ measured_VVN from_II =_FO hpg_NNU ,_, for_IF each_DD1 concentration_NN1 where_CS p_ZZ1 is_VBZ the_AT density_NN1 of_IO the_AT solution_NN1 and_CC g_ZZ1 the_AT acceleration_NN1 of_IO free_JJ fall_NN1 ._. 
Results_NN2 are_VBR plotted_VVN as_II21 against_II22 c_ZZ1 as_CSA described_VVN and_CC M_ZZ1 n_ZZ1 is_VBZ calculated_VVN from_II the_AT intercept_VV0 ._. 
The_AT method_NN1 suffers_VVZ from_II the_AT disadvantage_NN1 that_CST it_PPH1 is_VBZ slow_JJ and_CC consequently_RR diffusion_NN1 of_IO low_JJ molar_JJ mass_JJ material_NN1 could_VM be_VBI large_JJ enough_RR to_TO introduce_VVI serious_JJ error_NN1 in_II the_AT measurement_NN1 ._. 
Two_MC or_CC three_MC high-speed_JJ automatic_JJ membrane-osmometers_NN2 have_VH0 now_RT been_VBN designed_VVN to_TO reduce_VVI these_DD2 drawbacks_NN2 and_CC are_VBR commercially_RR available_JJ ._. 
The_AT Mechrolab_NP1 osmometer_NN1 ,_, shown_VVN schematically_RR in_II figure_NN1 9.3_MC ,_, consists_VVZ of_IO a_AT1 solution_NN1 +_FO solvent_NN1 cell_NN1 of_IO volume_NN1 approximately_RR 1_MC1 cm_NNU 3_MC ,_, with_IW the_AT solvent_NN1 side_NN1 connected_VVN to_II a_AT1 reservoir_NN1 attached_VVN to_II a_AT1 servo-driven_JJ elevator_NN1 ._. 
When_CS solution_NN1 is_VBZ added_VVN to_II the_AT top-half_MF of_IO the_AT cell_NN1 ,_, solvent_NN1 in_II the_AT lower-half_MF tends_VVZ to_TO flow_VVI into_II the_AT upper_JJ section_NN1 to_TO equalize_VVI the_AT chemical_JJ potentials_NN2 ._. 
The_AT flow_NN1 is_VBZ detected_VVN optically_RR by_II the_AT movement_NN1 of_IO a_AT1 bubble_NN1 in_II a_AT1 capillary_NN1 below_II the_AT cell_NN1 ._. 
The_AT movement_NN1 activates_VVZ the_AT servo-motor_NN1 ,_, which_DDQ alters_VVZ the_AT hydrostatic_JJ head_NN1 thereby_RR counteracting_VVG the_AT flow_NN1 ._. 
The_AT movement_NN1 of_IO the_AT solvent_NN1 reservoir_NN1 is_VBZ then_RT a_AT1 measure_NN1 of_IO the_AT osmotic_JJ pressure_NN1 of_IO the_AT solution_NN1 ._. 
Equalization_NN1 is_VBZ rapid_JJ (_( 5_MC to_II 30_MC min_NNU )_) and_CC permeation_NN1 is_VBZ readily_RR detected_VVN ,_, if_CS present_JJ ,_, by_II following_VVG the_AT change_NN1 of_IO head_NN1 as_II a_AT1 function_NN1 of_IO time_NNT1 on_II a_AT1 recorder_NN1 ._. 
There_EX is_VBZ no_AT actual_JJ flow_NN1 of_IO solvent_NN1 in_II the_AT Mechrolab_NP1 instrument_NN1 ._. 
A_AT1 slightly_RR different_JJ principle_NN1 ,_, which_DDQ allows_VVZ solvent_JJ flow_NN1 to_TO take_VVI place_NN1 ,_, forms_VVZ the_AT basis_NN1 of_IO the_AT Melab_NP1 and_CC Knauer_NP1 models_NN2 ._. 
The_AT Melab_NP1 osmometer_NN1 has_VHZ a_AT1 stainless_JJ steel_NN1 cell_NN1 (_( volume_NN1 0.5_MC cm_NNU 3_MC )_) ,_, with_IW solution_NN1 and_CC solvent_NN1 compartments_NN2 separated_VVN by_II the_AT membrane_NN1 ._. 
One_MC1 wall_NN1 of_IO the_AT cell_NN1 is_VBZ a_AT1 flexible_JJ stainless_JJ steel_NN1 diaphragm_NN1 connected_VVN through_II a_AT1 strain_NN1 gauge_NN1 to_II a_AT1 recorder_NN1 ._. 
As_CSA solvent_NN1 diffuses_VVZ through_II the_AT membrane_NN1 ,_, the_AT increase_NN1 in_II volume_NN1 causes_VVZ the_AT diaphragm_NN1 to_TO move_VVI ._. 
The_AT motion_NN1 is_VBZ detected_VVN by_II the_AT gauge_NN1 and_CC translated_VVN into_II a_AT1 pressure_NN1 ._. 
The_AT design_NN1 has_VHZ the_AT advantage_NN1 that_CST both_RR solvent_JJ and_CC solution_NN1 compartments_NN2 are_VBR easily_RR rinsed_VVN out_RP and_CC the_AT cell_NN1 does_VDZ not_XX have_VHI to_TO be_VBI dismantled_VVN if_CS contamination_NN1 by_II permeation_NN1 of_IO low_JJ molar_JJ mass_JJ solute_NN1 occurs_VVZ ._. 
All_DB osmotic_JJ pressure_NN1 measurements_NN2 are_VBR extremely_RR sensitive_JJ to_II temperature_NN1 and_CC must_VM be_VBI carried_VVN out_RP under_RG rigorously_RR controlled_VVN temperature_NN1 conditions_NN2 ._. 
This_DD1 is_VBZ allowed_VVN for_IF in_II each_DD1 instrument_NN1 and_CC in_RR21 addition_RR22 ,_, measurements_NN2 can_VM be_VBI made_VVN over_II a_AT1 range_NN1 of_IO temperatures_NN2 (_( 278_MC to_II 373_MC K_ZZ1 )_) ._. 
Solvents_NN2 should_VM be_VBI chosen_VVN which_DDQ are_VBR chemically_RR stable_JJ and_CC have_VH0 a_AT1 low_JJ to_II medium_JJ vapour_NN1 pressure_NN1 at_II the_AT temperature_NN1 of_IO operation_NN1 ,_, as_CSA this_DD1 avoids_VVZ problems_NN2 of_IO bubble_NN1 formation_NN1 in_II the_AT measuring_JJ chamber._NNU 9.8_MC Transport_NN1 methods_NN2 vapour_NN1 pressure_NN1 osmometer_NN1 In_II conventional_JJ osmometry_NN1 ,_, the_AT membrane_NN1 permeability_NN1 imposes_VVZ a_AT1 lower_JJR limit_NN1 of_IO about_II M_ZZ1 n_ZZ1 =_FO 15_MC 000_MC g_NNU mol_NN1 -1_MC ._. 
A_AT1 technique_NN1 ,_, based_VVN on_II the_AT lowering_NN1 of_IO the_AT vapour_NN1 pressure_NN1 ,_, called_VVN vapour_NN1 pressure_NN1 osmometry_NN1 is_VBZ a_AT1 useful_JJ method_NN1 of_IO measuring_VVG values_NN2 of_IO M_ZZ1 n_ZZ1 from_II 50_MC to_II 20_MC 000_MC g_NNU mol_NN1 -1_MC ._. 
It_PPH1 is_VBZ a_AT1 relative_JJ method_NN1 and_CC is_VBZ calibrated_VVN using_VVG such_DA low_JJ molar_JJ mass_JJ standards_NN2 as_CSA benzil_NN1 ,_, methyl_NN1 stearate_NN1 ,_, or_CC glucose_NN1 penta-acetate_NN1 ._. 
The_AT apparatus_NN1 consists_VVZ of_IO a_AT1 thermostatted_JJ chamber_NN1 ,_, saturated_VVN with_IW solvent_JJ vapour_NN1 at_II the_AT temperature_NN1 of_IO measurement_NN1 ,_, and_CC containing_VVG two_MC differential_JJ matched_JJ thermistors_NN2 which_DDQ are_VBR capable_JJ of_IO detecting_VVG temperature_NN1 differences_NN2 as_RG low_JJ as_CSA 10_MC -4_MC K._NP1 Two_MC syringes_NN2 ,_, one_MC1 for_IF solvent_NN1 and_CC one_PN1 for_IF solution_NN1 ,_, are_VBR used_VVN to_TO apply_VVI a_AT1 drop_NN1 of_IO solution_NN1 to_II one_MC1 thermistor_NN1 ,_, and_CC a_AT1 drop_NN1 of_IO solvent_NN1 to_II the_AT other_JJ ._. 
As_CSA there_EX is_VBZ a_AT1 difference_NN1 in_II vapour_NN1 pressure_NN1 between_II the_AT solution_NN1 and_CC the_AT solvent_NN1 drops_NN2 ,_, solvent_NN1 from_II the_AT vapour_NN1 phase_NN1 will_VM condense_VVI on_II the_AT solution_NN1 drop_NN1 causing_VVG its_APPGE temperature_NN1 to_TO rise_VVI ._. 
Because_II21 of_II22 the_AT large_JJ excess_NN1 of_IO solvent_NN1 present_NN1 ,_, evaporation_NN1 ,_, and_CC hence_RR cooling_VVG of_IO the_AT solvent_NN1 drop_NN1 ,_, is_VBZ negligible_JJ ._. 
When_CS equilibrium_NN1 is_VBZ attained_VVN ,_, the_AT temperature_NN1 difference_NN1 between_II the_AT two_MC drops_NN2 T_ZZ1 is_VBZ a_AT1 measure_NN1 of_IO the_AT extent_NN1 of_IO the_AT vapour_NN1 pressure_NN1 lowering_VVG by_II the_AT solute_NN1 ._. 
The_AT thermistors_NN2 form_VV0 part_NN1 of_IO a_AT1 Wheatstone_NP1 bridge_NN1 ,_, and_CC T_ZZ1 is_VBZ recorded_VVN as_II a_AT1 difference_NN1 in_II resistance_NN1 R._NP1 The_AT molar_JJ mass_NN1 can_VM then_RT be_VBI calculated_VVN from_II where_RRQ K_ZZ1 *_FU is_VBZ the_AT calibration_NN1 constant_NN1 ._. 
As_CSA with_IW other_JJ methods_NN2 M_ZZ1 n_ZZ1 is_VBZ obtained_VVN by_II extrapolating_VVG the_AT data_NN to_II ._. 
The_AT calibration_NN1 constant_NN1 is_VBZ estimated_VVN by_II measuring_VVG R_ZZ1 for_IF solutions_NN2 of_IO known_JJ concentration_NN1 prepared_VVN from_II standard_JJ compounds_NN2 of_IO known_JJ molar_JJ mass_NN1 M_ZZ1 k_ZZ1 then_RT In_II some_DD instances_NN2 an_AT1 additional_JJ correction_NN1 for_IF the_AT dilution_NN1 of_IO the_AT drop_NN1 of_IO solution_NN1 may_VM be_VBI necessary._NNU 9.9_MC Light_NN1 scattering_VVG Light_JJ scattering_NN1 is_VBZ one_MC1 of_IO the_AT most_RGT popular_JJ methods_NN2 for_IF determining_VVG the_AT weight_NN1 average_JJ molar_JJ mass_NN1 M_ZZ1 w_ZZ1 ._. 
The_AT phenomenon_NN1 of_IO light_NN1 scattering_VVG by_II small_JJ particles_NN2 is_VBZ familiar_JJ to_II us_PPIO2 all_DB ;_; the_AT blue_JJ colour_NN1 of_IO the_AT sky_NN1 or_CC the_AT varied_JJ colours_NN2 of_IO a_AT1 sunset_NN1 ,_, the_AT poor_JJ penetration_NN1 of_IO car_NN1 headlights_NN2 in_II a_AT1 fog_NN1 is_VBZ caused_VVN by_II water_NN1 droplets_NN2 scattering_VVG the_AT light_NN1 ,_, and_CC the_AT obvious_JJ presence_NN1 of_IO dust_NN1 in_II a_AT1 sunbeam_NN1 or_CC the_AT Tyndall_NP1 effect_NN1 in_II an_AT1 irradiated_JJ colloidal_JJ solution_NN1 are_VBR further_JJR examples_NN2 of_IO this_DD1 effect_NN1 ._. 
The_AT fundamentals_NN2 of_IO light_JJ scattering_NN1 were_VBDR expounded_VVN by_II Lord_NNB Rayleigh_NP1 in_II 1871_MC during_II his_APPGE studies_NN2 on_II gases_NN2 ,_, where_CS the_AT particle_NN1 is_VBZ small_RR compared_VVN with_IW the_AT wavelength_NN1 of_IO the_AT incident_NN1 radiation_NN1 ._. 
Light_NN1 is_VBZ an_AT1 electromagnetic_JJ wave_NN1 ,_, produced_VVN by_II the_AT interaction_NN1 of_IO a_AT1 magnetic_JJ and_CC electric_JJ field_NN1 ,_, both_DB2 oscillating_VVG at_II right_JJ angles_NN2 to_II one_PPX121 another_PPX122 in_II the_AT direction_NN1 of_IO propagation_NN1 ._. 
When_CS a_AT1 beam_NN1 of_IO light_NN1 strikes_VVZ the_AT atoms_NN2 or_CC molecules_NN2 of_IO the_AT medium_NN1 ,_, the_AT electrons_NN2 are_VBR perturbed_VVN or_CC displaced_JJ and_CC oscillate_VV0 about_II their_APPGE equilibrium_NN1 positions_NN2 with_IW the_AT same_DA frequency_NN1 as_CSA the_AT exciting_JJ beam_NN1 ._. 
This_DD1 induces_VVZ transient_JJ dipoles_NN2 in_II the_AT atoms_NN2 or_CC molecules_NN2 ,_, which_DDQ act_VV0 as_RG secondary_JJ scattering_NN1 centres_NN2 by_II re-emitting_VVG the_AT absorbed_JJ energy_NN1 in_II all_DB directions_NN2 ,_, i.e._REX scattering_NN1 takes_VVZ place_NN1 ._. 
For_IF gases_NN2 ,_, Rayleigh_NP1 showed_VVD that_CST the_AT reduced_JJ intensity_NN1 of_IO the_AT scattered_JJ light_NN1 R_ZZ1 at_II any_DD angle_NN1 to_II the_AT incident_NN1 beam_NN1 ,_, of_IO wavelength_NN1 could_VM be_VBI related_VVN to_II the_AT molar_JJ mass_NN1 of_IO the_AT gas_NN1 M_ZZ1 ,_, its_APPGE concentration_NN1 c_ZZ1 ,_, and_CC the_AT refractive_JJ index_NN1 increment_NN1 by_II The_AT quantity_NN1 R_ZZ1 is_VBZ often_RR referred_VVN to_II as_II the_AT Rayleigh_NP1 ratio_NN1 and_CC is_VBZ equal_JJ to_II where_RRQ I_ZZ1 is_VBZ the_AT intensity_NN1 of_IO the_AT incident_NN1 beam_NN1 ,_, i_ZZ1 ,_, is_VBZ the_AT quantity_NN1 of_IO light_NN1 scattered_VVN per_II unit_NN1 volume_NN1 by_II one_MC1 centre_NN1 at_II an_AT1 angle_NN1 to_II the_AT incident_NN1 beam_NN1 ,_, and_CC r_ZZ1 is_VBZ the_AT distance_NN1 of_IO the_AT centre_NN1 from_II the_AT observer_NN1 ._. 
This_DD1 is_VBZ valid_JJ for_IF a_AT1 gas_NN1 ,_, where_CS all_DB the_AT particles_NN2 are_VBR considered_VVN to_TO be_VBI independent_JJ scattering_NN1 centres_NN2 and_CC the_AT addition_NN1 of_IO more_DAR centres_NN2 ,_, which_DDQ increases_VVZ n_ZZ1 increases_VVZ the_AT scattering_NN1 ._. 
The_AT situation_NN1 changes_NN2 when_CS dealing_VVG with_IW a_AT1 liquid_JJ as_CSA remains_VVZ unaffected_JJ by_II the_AT addition_NN1 of_IO molecules_NN2 and_CC can_VM be_VBI expected_VVN to_TO be_VBI zero_MC ._. 
This_DD1 conceptual_JJ difficulty_NN1 was_VBDZ overcome_VVN in_II the_AT fluctuation_NN1 theories_NN2 of_IO Smoluchowski_NP1 and_CC Einstein_NP1 ;_; they_PPHS2 postulated_VVD that_CST optical_JJ discontinuities_NN2 exist_VV0 in_II the_AT liquid_NN1 arising_VVG from_II the_AT creation_NN1 and_CC destruction_NN1 of_IO holes_NN2 during_II Brownian_JJ motion_NN1 ._. 
Scattering_NN1 emanates_VVZ from_II these_DD2 centres_NN2 ,_, created_VVN by_II local_JJ density_NN1 fluctuations_NN2 ,_, which_DDQ produce_VV0 changes_NN2 in_RP in_II any_DD volume_NN1 element_NN1 ._. 
When_CS a_AT1 solute_NN1 is_VBZ dissolved_VVN in_II a_AT1 liquid_NN1 ,_, scattering_VVG from_II a_AT1 volume_NN1 element_NN1 again_RT arises_VVZ from_II liquid_JJ inhomogeneities_NN2 ,_, but_CCB now_RT an_AT1 additional_JJ contribution_NN1 from_II fluctuations_NN2 in_II the_AT solute_NN1 concentration_NN1 is_VBZ present_JJ and_CC for_IF polymer_NN1 solutions_NN2 the_AT problem_NN1 is_VBZ to_TO isolate_VVI and_CC measure_VVI these_DD2 additional_JJ effects_NN2 ._. 
This_DD1 was_VBDZ achieved_VVN by_II Debye_NP1 in_II 1944_MC ,_, who_PNQS showed_VVD that_CST for_IF a_AT1 solute_NN1 whose_DDQGE molecules_NN2 are_VBR small_RR compared_VVN with_IW the_AT wavelength_NN1 of_IO the_AT light_NN1 used_VVD ,_, the_AT reduced_JJ angular_JJ scattering_NN1 intensity_NN1 of_IO the_AT solute_NN1 is_VBZ and_CC that_CST this_DD1 is_VBZ related_VVN to_II the_AT change_NN1 in_II Gibbs_NP1 free_JJ energy_NN1 with_IW concentration_NN1 of_IO the_AT solute_NN1 ._. 
As_CSA G_ZZ1 is_VBZ related_VVN to_II the_AT osmotic_JJ pressure_NN1 ,_, we_PPIS2 have_VH0 Here_RL n_ZZ1 and_CC n_ZZ1 are_VBR the_AT refractive_JJ indices_NN2 of_IO solvent_NN1 and_CC solution_NN1 respectively_RR ,_, and_CC N_ZZ1 is_VBZ the_AT number_NN1 of_IO polymer_NN1 molecules_NN2 ._. 
Differentiation_NN1 of_IO the_AT virial_NN1 expansion_NN1 for_IF with_II31 respect_II32 to_II33 c_ZZ1 ,_, followed_VVN by_II substitution_NN1 in_II equation_NN1 (_( 9.27_MC )_) and_CC rearrangement_NN1 leads_VVZ to_II where_RRQ Alternatively_RR ,_, the_AT scattering_NN1 can_VM be_VBI expressed_VVN as_II a_AT1 turbidity_NN1 where_RRQ and_CC the_AT equation_NN1 becomes_VVZ The_AT new_JJ constant_NN1 is_VBZ ._. 
Both_DB2 equations_NN2 are_VBR valid_JJ for_IF molecules_NN2 smaller_JJR than_CSN when_CS the_AT angular_JJ scattering_NN1 is_VBZ symmetrical_JJ ._. 
Here_RL is_VBZ the_AT wavelength_NN1 of_IO light_NN1 in_II solution_NN1 ._. 
For_IF small_JJ particles_NN2 ,_, M_ZZ1 w_ZZ1 can_VM be_VBI calculated_VVN from_II either_RR equation_NN1 (_( 9.28_MC )_) or_CC (_( 9.31_MC )_) ._. 
The_AT important_JJ experimental_JJ point_NN1 to_TO remember_VVI is_VBZ that_DD1 dust_NN1 will_VM also_RR scatter_VVI light_JJ and_CC contribute_VVI to_II the_AT scattering_NN1 intensity_NN1 ._. 
Great_JJ care_NN1 must_VM be_VBI taken_VVN to_TO ensure_VVI that_CST solutions_NN2 are_VBR clean_JJ and_CC free_JJ of_IO extraneous_JJ matter_NN1 ._. 
Solutions_NN2 of_IO the_AT polymer_NN1 are_VBR prepared_VVN in_II a_AT1 concentration_NN1 series_NN and_CC clarified_VVN either_RR by_II centrifugation_NN1 for_IF a_AT1 few_DA2 hours_NNT2 at_II about_RG 25_MC 000_MC g_NNU ,_, or_CC filtered_VVN through_II a_AT1 grade_NN1 5_MC sinter_NN1 glass_NN1 filter_NN1 ._. 
Alternatively_RR ,_, a_AT1 millipore_NN1 filter_NN1 ,_, porosity_NN1 0.45_MC *_FU can_VM be_VBI used_VVN ._. 
A_AT1 number_NN1 of_IO instruments_NN2 are_VBR available_JJ commercially_RR ;_; only_RR one_PN1 is_VBZ described_VVN here_RL and_CC the_AT schematic_JJ diagram_NN1 9.5_MC provides_VVZ the_AT main_JJ features_NN2 of_IO the_AT model_NN1 ._. 
Light_NN1 is_VBZ obtained_VVN from_II a_AT1 water-cooled_JJ mercury_NN1 vapour_NN1 lamp_NN1 and_CC one_MC1 of_IO three_MC wavelengths_NN2 365_MC ,_, 436_MC ,_, or_CC 546_MC nm_FU can_VM be_VBI selected_VVN by_II31 means_II32 of_II33 an_AT1 appropriate_JJ filter_NN1 ._. 
As_II the_AT scattering_NN1 intensity_NN1 is_VBZ a_AT1 function_NN1 of_IO -4_MC ,_, use_NN1 of_IO a_AT1 lower_JJR wavelength_NN1 enhances_VVZ the_AT scattering_NN1 ,_, but_CCB the_AT choice_NN1 is_VBZ left_VVN to_II the_AT operator_NN1 ._. 
The_AT light_JJ beam_NN1 ,_, which_DDQ can_VM be_VBI polarized_VVN ,_, or_CC left_JJ unpolarized_JJ ,_, is_VBZ collimated_VVN before_II passing_VVG through_II the_AT cell_NN1 ._. 
The_AT measuring_JJ cell_NN1 is_VBZ immersed_VVN in_II a_AT1 vat_NN1 of_IO liquid_NN1 ,_, usually_RR benzene_NN1 or_CC xylene_NN1 which_DDQ can_VM be_VBI thermostatted_VVN at_II temperatures_NN2 between_II 273_MC and_CC 400_MC K._NP1 Scattering_NN1 is_VBZ detected_VVN by_II a_AT1 photomultiplier_JJR ,_, capable_JJ of_IO revolving_VVG round_II the_AT cell_NN1 and_CC the_AT intensity_NN1 is_VBZ recorded_VVN on_II a_AT1 galvanometer_NN1 ._. 
The_AT 90_MC scattering_NN1 is_VBZ plotted_VVN as_II21 against_II22 c_ZZ1 and_CC linear_JJ extrapolation_NN1 of_IO the_AT results_NN2 leads_VVZ to_II M_ZZ1 w_ZZ1 as_II the_AT intercept_VV0 at_II ._. 
Typical_JJ results_NN2 are_VBR shown_VVN in_II table_NN1 9.2_MC for_IF a_AT1 polystyrene_NN1 sample_NN1 dissolved_VVN in_II benzene_NN1 ._. 
The_AT relevant_JJ constants_NN2 are_VBR ,_, ,_, the_AT intercept_VV0 and_CC ._. 
SCATTERING_VVG FROM_II LARGE_JJ PARTICLES_NN2 When_RRQ polymer_NN1 dimensions_NN2 are_VBR greater_JJR than_CSN /20_MF ,_, ;_; intraparticle_NN1 interference_NN1 causes_VVZ the_AT scattered_JJ light_NN1 from_II two_MC or_CC more_DAR centres_NN2 to_TO arrive_VVI considerably_RR out_II21 of_II22 phase_NN1 at_II the_AT observation_NN1 point_NN1 ,_, and_CC the_AT scattering_NN1 envelope_NN1 becomes_VVZ dependent_JJ on_II the_AT molecular_JJ shape_NN1 ._. 
This_DD1 attenuation_NN1 ,_, produced_VVN by_II destructive_JJ interference_NN1 ,_, is_VBZ zero_MC in_II the_AT direction_NN1 of_IO the_AT incident_NN1 beam_NN1 ,_, but_CCB increases_VVZ as_RG 0_MC increases_NN2 because_CS the_AT path_NN1 length_NN1 difference_NN1 f_ZZ1 in_II the_AT forward_JJ direction_NN1 is_VBZ less_DAR than_CSN b_ZZ1 in_II the_AT backward_JJ (_( see_VV0 figure_NN1 9.6_MC )_) ._. 
This_DD1 difference_NN1 can_VM be_VBI measured_VVN from_II the_AT dissymmetry_NN1 coefficient_NN1 Z_ZZ1 which_DDQ is_VBZ unity_NN1 for_IF small_JJ particles_NN2 ,_, but_CCB greater_JJR than_CSN unity_NN1 for_IF large_JJ particles_NN2 ._. 
The_AT scattering_NN1 envelope_NN1 reflects_VVZ the_AT scattering_NN1 attenuation_NN1 and_CC is_VBZ compared_VVN with_IW that_DD1 for_IF small_JJ particles_NN2 in_II figure_NN1 9.7_MC ._. 
The_AT angular_JJ attenuation_NN1 of_IO scattering_VVG is_VBZ measured_VVN by_II the_AT particle_NN1 scattering_VVG factor_NN1 which_DDQ is_VBZ simply_RR the_AT ratio_NN1 of_IO the_AT scattering_NN1 intensity_NN1 to_II the_AT intensity_NN1 in_II the_AT absence_NN1 of_IO interference_NN1 ,_, measured_VVN at_II the_AT same_DA angle_NN1 ._. 
Gunier_NP1 showed_VVD that_CST a_AT1 characteristic_JJ shape-independent_JJ geometric_JJ function_NN1 ,_, called_VVN the_AT radius_NN1 of_IO gyration_NN1 can_VM be_VBI measured_VVN from_II large_JJ particle_NN1 scattering_VVG ._. 
It_PPH1 is_VBZ defined_VVN as_II an_AT1 average_JJ distance_NN1 from_II the_AT centre_NN1 of_IO gravity_NN1 of_IO a_AT1 polymer_NN1 coil_NN1 to_II the_AT chain_NN1 end_NN1 ._. 
The_AT function_NN1 is_VBZ size_NN1 dependent_NN1 and_CC can_VM be_VBI related_VVN to_II the_AT polymer_NN1 coil_NN1 size_NN1 by_II where_RRQ ,_, for_IF monodisperse_NN1 randomly_RR coiling_VVG polymers_NN2 ._. 
In_II the_AT limit_NN1 of_IO small_JJ the_AT expansion_NN1 can_VM be_VBI used_VVN ,_, and_CC the_AT coil_NN1 size_NN1 can_VM be_VBI estimated_VVN from_II without_IW assuming_VVG a_AT1 particular_JJ model_NN1 ._. 
Specific_JJ shapes_NN2 can_VM be_VBI related_VVN to_II if_CSW desired_VVN ,_, as_CSA shown_VVN in_II figure_NN1 9.8a_FO and_CC b_ZZ1 ._. 
Two_MC methods_NN2 can_VM be_VBI used_VVN to_TO calculate_VVI M_ZZ1 w_ZZ1 and_CC the_AT particle_NN1 size_NN1 for_IF large_JJ molecules._NNU (_( i_ZZ1 )_) Dissymmetry_NN1 method_NN1 ._. 
If_CS is_VBZ not_XX too_RG large_JJ ,_, one_PN1 need_VM only_RR measure_VVI the_AT scattering_NN1 intensity_NN1 at_II 90_MC and_CC two_MC angles_NN2 symmetrical_JJ about_RG 90_MC ,_, usually_RR 45_MC and_CC 135_MC ._. 
As_CSA Z_ZZ1 is_VBZ normally_RR concentration_NN1 dependent_NN1 ,_, the_AT value_NN1 at_II is_VBZ obtained_VVN by_II plotting_VVG against_II c_ZZ1 ._. 
From_II published_JJ tables_NN2 can_VM be_VBI related_VVN to_II ,_, and_CC M_ZZ1 w_ZZ1 is_VBZ calculated_VVN from_II the_AT 90_MC scattering_VVG then_RT corrected_VVN by_II multiplication_NN1 with_IW ._. 
Also_RR available_JJ in_II table_NN1 form_NN1 is_VBZ the_AT ratio_NN1 presented_VVN as_II a_AT1 function_NN1 of_IO Z_ZZ1 ,_, where_RRQ is_VBZ the_AT root_NN1 mean_VV0 square_JJ distance_NN1 between_II the_AT ends_NN2 of_IO the_AT polymer_NN1 coil_NN1 ._. 
The_AT corresponding_JJ functions_NN2 for_IF a_AT1 rod_NN1 and_CC a_AT1 sphere_NN1 have_VH0 different_JJ forms_NN2 (_( figure_NN1 9.8b_FO )_) ._. 
Polymer_NN1 dimensions_NN2 can_VM be_VBI calculated_VVN in_II this_DD1 way_NN1 if_CS an_AT1 assumption_NN1 is_VBZ made_VVN about_II the_AT best_JJT model_NN1 ._. 
A_AT1 much_RR more_RGR satisfying_JJ treatment_NN1 of_IO the_AT data_NN uses_VVZ the_AT double_JJ extrapolation_NN1 method_NN1 proposed_VVN by_II Zimm_NP1 ,_, which_DDQ leads_VVZ to_II the_AT shape_NN1 independent_JJ parameter._NNU (_( ii_MC )_) Zimm_NP1 plots_NN2 ._. 
This_DD1 is_VBZ based_VVN on_II the_AT knowledge_NN1 that_CST ,_, as_CSA the_AT scattering_NN1 at_II zero_MC angle_NN1 is_VBZ independent_JJ of_IO size_NN1 ,_, is_VBZ unity_NN1 when_RRQ ._. 
Experimentally_RR this_DD1 is_VBZ difficult_JJ to_TO measure_VVI ,_, and_CC an_AT1 extrapolation_NN1 procedure_NN1 has_VHZ been_VBN devised_VVN which_DDQ makes_VVZ use_NN1 of_IO a_AT1 modified_JJ form_NN1 of_IO equation_NN1 (_( 9.28_MC )_) for_IF large_JJ particles_NN2 ,_, Substituting_VVG for_IF leads_NN2 to_II If_CSW the_AT scattering_NN1 intensity_NN1 for_IF each_DD1 concentration_NN1 in_II a_AT1 dilution_NN1 series_NN is_VBZ measured_VVN over_II an_AT1 angular_JJ range_NN1 35_MC to_II 145_MC ,_, the_AT data_NN can_VM be_VBI plotted_VVN as_II21 against_II22 ,_, where_CS k_ZZ1 is_VBZ an_AT1 arbitrary_JJ constant_NN1 chosen_VVN to_TO provide_VVI a_AT1 convenient_JJ spread_NN1 of_IO the_AT data_NN in_II the_AT grid-like_JJ graph_NN1 which_DDQ is_VBZ obtained_VVN ._. 
A_AT1 double_JJ extrapolation_NN1 is_VBZ then_RT carried_VVN out_RP ,_, as_CSA shown_VVN in_II figure_NN1 9.9_MC ,_, by_II joining_VVG all_DB points_NN2 of_IO equal_JJ concentration_NN1 and_CC extrapolating_VVG to_TO zero_VVI angle_NN1 ,_, and_CC then_RT all_DB points_NN2 measured_VVN at_II equal_JJ angles_NN2 and_CC extrapolating_VVG these_DD2 to_TO zero_VVI concentration_NN1 ._. 
For_REX21 example_REX22 ,_, on_II the_AT diagram_NN1 the_AT points_NN2 corresponding_VVG to_II concentration_NN1 c_ZZ1 3_MC are_VBR joined_VVN and_CC extrapolated_VVN to_TO intersect_VVI with_IW an_AT1 imaginary_JJ line_NN1 corresponding_VVG to_II the_AT value_NN1 of_IO kc_NNU 3_MC on_II the_AT abscissa_NN1 ;_; similarly_RR all_DB points_NN2 measured_VVN at_II 90_MC are_VBR joined_VVN and_CC extrapolated_VVN until_CS the_AT point_NN1 corresponding_VVG to_TO is_VBZ reached_VVN ._. 
This_DD1 is_VBZ done_VDN for_IF each_DD1 concentration_NN1 and_CC each_DD1 angle_NN1 and_CC the_AT extrapolated_JJ points_NN2 are_VBR then_RT lines_NN2 of_IO and_CC ._. 
Both_DB2 lines_NN2 ,_, on_II extrapolation_NN1 to_II the_AT axis_NN1 ,_, should_VM intersect_VVI at_II the_AT same_DA point_NN1 ._. 
The_AT intercept_VV0 is_VBZ then_RT ,_, the_AT slope_NN1 of_IO the_AT line_NN1 yields_NN2 A_ZZ1 2_MC ,_, whereas_CS is_VBZ obtained_VVN from_II the_AT initial_JJ slope_NN1 s_ZZ1 i_MC1 of_IO the_AT line_NN1 i.e._REX The_AT radius_NN1 of_IO gyration_NN1 calculated_VVN in_II this_DD1 way_NN1 for_IF a_AT1 polydisperse_NN1 sample_NN1 is_VBZ a_AT1 z-average._JJ 9.10_MC Refractive_JJ index_NN1 increment_NN1 Before_CS M_ZZ1 w_ZZ1 can_VM be_VBI calculated_VVN from_II light_NN1 scattering_VVG measurements_NN2 ,_, the_AT specific_JJ refractive_JJ index_NN1 increment_NN1 must_VM be_VBI known_VVN for_IF the_AT particular_JJ polymer_NN1 +_FO solvent_NN1 system_NN1 under_II examination_NN1 ._. 
It_PPH1 is_VBZ defined_VVN as_CSA where_CS n_ZZ1 and_CC n_ZZ1 o_ZZ1 are_VBR the_AT refractive_JJ indices_NN2 of_IO the_AT solution_NN1 and_CC the_AT solvent_NN1 and_CC c_ZZ1 is_VBZ the_AT concentration_NN1 ._. 
Measurements_NN2 of_IO are_VBR made_VVN using_VVG a_AT1 differential_JJ refractometer_NN1 employing_VVG the_AT same_DA wavelength_NN1 of_IO light_JJ as_CSA used_VVN in_II the_AT light_JJ scattering_NN1 ._. 
The_AT monochromatic_JJ beam_NN1 (_( selected_VVN by_II filter_NN1 )_) from_II a_AT1 mercury_NN1 vapour_NN1 lamp_NN1 is_VBZ directed_VVN through_II a_AT1 differential_JJ cell_NN1 ,_, consisting_VVG of_IO a_AT1 solution_NN1 and_CC solvent_NN1 compartment_NN1 separated_VVN by_II a_AT1 diagonal_JJ glass_NN1 wall_NN1 ._. 
The_AT deflection_NN1 of_IO the_AT light_JJ beam_NN1 is_VBZ measured_VVN ,_, first_MD with_IW solvent_NN1 in_II the_AT forward_JJ compartment_NN1 and_CC solution_NN1 in_II the_AT rear_NN1 ,_, giving_VVG deflection_NN1 d_ZZ1 1_MC1 the_AT position_NN1 is_VBZ reversed_VVN and_CC deflection_NN1 d_ZZ1 2_MC measured_VVD ._. 
If_CS similar_JJ readings_NN2 for_IF solvent_NN1 alone_RR ,_, and_CC ,_, are_VBR obtained_VVN ,_, then_RT the_AT total_JJ displacement_NN1 d_ZZ1 is_VBZ If_CS the_AT instrument_NN1 is_VBZ calibrated_VVN with_IW aqueous_JJ KCI_JJ solutions_NN2 of_IO known_JJ n_ZZ1 a_AT1 relation_NN1 ,_, can_VM be_VBI obtained_VVN where_CS c_ZZ1 is_VBZ the_AT calibration_NN1 constant_NN1 ._. 
By_II measuring_VVG d_ZZ1 for_IF a_AT1 number_NN1 of_IO concentrations_NN2 of_IO polymer_NN1 ,_, n_ZZ1 is_VBZ obtained_VVN from_II a_AT1 knowledge_NN1 of_IO c_ZZ1 ,_, and_CC from_II the_AT slope_NN1 of_IO the_AT plot_NN1 of_IO n_ZZ1 against_II c._RG 9.11_MC Small_JJ angle_NN1 X-ray_NN1 scattering_VVG The_AT theoretical_JJ outline_NN1 presented_VVN for_IF light_NN1 scattering_VVG studies_NN2 is_VBZ valid_JJ for_IF electromagnetic_JJ radiation_NN1 of_IO all_DB wavelengths_NN2 ._. 
For_IF X-rays_NN2 ,_, is_VBZ as_RG low_JJ as_CSA 0.154_MC nm_FU ,_, and_CC as_CSA this_DD1 is_VBZ much_RR smaller_JJR than_CSN typical_JJ polymer_NN1 dimensions_NN2 structural_JJ information_NN1 over_II small_JJ distances_NN2 should_VM be_VBI available_JJ from_II X-ray_NN1 scattering_VVG ._. 
The_AT intensity_NN1 of_IO scattering_VVG is_VBZ a_AT1 function_NN1 of_IO the_AT electron_NN1 density_NN1 and_CC therefore_RR of_IO the_AT refractive_JJ index_NN1 ._. 
The_AT molar_JJ mass_NN1 is_VBZ then_RT related_VVN to_II the_AT excess_JJ electron_NN1 density_NN1 c_ZZ1 of_IO solute_NN1 over_II solvent_NN1 for_IF =0.154_FO ;_; nm_FU by_II where_RRQ R_ZZ1 o_ZZ1 is_VBZ the_AT Rayleigh_NP1 ratio_NN1 at_II ._. 
Experimental_JJ techniques_NN2 are_VBR difficult_JJ because_II21 of_II22 the_AT weak_JJ scattering_NN1 ,_, but_CCB the_AT method_NN1 has_VHZ provided_VVN useful_JJ information_NN1 on_II macromolecules_NN2 with_IW dimensions_NN2 in_II the_AT range_NN1 1_MC1 to_II 100_MC nm_FU and_CC ,_, as_II such_DA ,_, is_VBZ complementary_JJ to_TO light_VVI scattering._NNU 9.12_MC Ultracentrifuge_NP1 When_CS macroscopic_JJ particles_NN2 are_VBR allowed_VVN to_TO settle_VVI in_II a_AT1 liquid_NN1 under_II gravity_NN1 it_PPH1 is_VBZ possible_JJ to_TO determine_VVI their_APPGE size_NN1 and_CC weight_NN1 ._. 
Macromolecules_NP1 in_II solution_NN1 are_VBR usually_RR much_RR smaller_JJR and_CC it_PPH1 would_VM take_VVI years_NNT2 for_IF them_PPHO2 to_TO overcome_VVI the_AT Brownian_JJ motion_NN1 and_CC form_VVI a_AT1 sediment_NN1 ._. 
This_DD1 problem_NN1 can_VM be_VBI overcome_VVN by_II subjecting_VVG them_PPHO2 to_II an_AT1 external_JJ force_NN1 ,_, strong_JJ enough_RR to_TO alter_VVI their_APPGE spatial_JJ distribution_NN1 by_II a_AT1 significant_JJ amount_NN1 in_II a_AT1 short_JJ time_NNT1 ._. 
In_II 1925_MC ,_, Svedberg_NP1 first_MD achieved_VVN this_DD1 by_II subjecting_VVG polymer_NN1 solutions_NN2 to_II large_JJ force_NN1 fields_NN2 ,_, generated_VVN at_II high_JJ speeds_NN2 of_IO rotation_NN1 ._. 
The_AT technique_NN1 is_VBZ now_RT a_AT1 well_NN1 established_VVD method_NN1 for_IF measuring_VVG M_ZZ1 w_ZZ1 and_CC M_ZZ1 z_ZZ1 for_IF both_DB2 synthetic_JJ and_CC biological_JJ macromolecules_NN2 and_CC has_VHZ the_AT added_JJ advantage_NN1 that_CST measurements_NN2 require_VV0 only_RR small_JJ quantities_NN2 of_IO material_NN1 ._. 
The_AT dilute_JJ solution_NN1 of_IO polymer_NN1 is_VBZ placed_VVN in_II a_AT1 cell_NN1 with_IW a_AT1 sector_NN1 shaped_JJ centre_NN1 piece_NN1 in_II the_AT form_NN1 of_IO a_AT1 truncated_JJ cone_NN1 ,_, whose_DDQGE peak_NN1 is_VBZ located_VVN at_II the_AT centre_NN1 of_IO the_AT rotation_NN1 ._. 
The_AT shape_NN1 ensures_VVZ that_CST convective_JJ disturbances_NN2 are_VBR minimized_VVN during_II the_AT transportation_NN1 of_IO molecules_NN2 to_II the_AT cell_NN1 bottom_NN1 ._. 
The_AT cells_NN2 are_VBR supported_VVN in_II a_AT1 rotor_NN1 of_IO either_RR titanium_NN1 or_CC aluminium_NN1 alloy_NN1 ,_, which_DDQ is_VBZ attached_VVN to_II the_AT drive_NN1 motor_NN1 by_II a_AT1 fine_JJ steel_NN1 wire_NN1 ,_, thereby_RR allowing_VVG limited_JJ self-balancing_NN1 to_TO take_VVI place_NN1 ._. 
The_AT rotor_NN1 is_VBZ spun_VVN in_II a_AT1 vacuum_NN1 chamber_NN1 to_TO minimize_VVI frictional_JJ heating_NN1 during_II high_JJ speed_NN1 rotations_NN2 ,_, as_CSA speeds_NN2 of_IO up_RG21 to_RG22 68_MC 000_MC r.p.m._NNU ,_, capable_JJ of_IO producing_VVG 372_MC 000_MC g_ZZ1 can_VM be_VBI generated_VVN ._. 
During_II rotation_NN1 the_AT cell_NN1 passes_VVZ through_II a_AT1 collimated_JJ beam_NN1 of_IO light_NN1 from_II a_AT1 high_JJ pressure_NN1 mercury_NN1 lamp_NN1 and_CC the_AT emergent_JJ beam_NN1 then_RT travels_VVZ through_II the_AT optical_JJ system_NN1 to_TO be_VBI recorded_VVN photographically_RR ._. 
Three_MC types_NN2 of_IO optical_JJ system_NN1 are_VBR available_JJ ,_, schlieren_NN1 ,_, interference_NN1 ,_, and_CC UV_JJ absorption_NN1 ._. 
Solvents_NN2 ,_, having_VHG densities_NN2 and_CC refractive_JJ indices_NN2 sufficiently_RR different_JJ from_II the_AT polymer_NN1 ,_, are_VBR chosen_VVN to_TO ensure_VVI movement_NN1 of_IO the_AT polymer_NN1 chains_NN2 in_II the_AT medium_NN1 and_CC the_AT optical_JJ detection_NN1 of_IO this_DD1 motion_NN1 ._. 
Most_DAT commercial_JJ instruments_NN2 are_VBR extremely_RR versatile_JJ ,_, with_IW an_AT1 extensive_JJ choice_NN1 of_IO rotor_NN1 speeds_NN2 and_CC a_AT1 temperature_NN1 control_NN1 system_NN1 ._. 
Molar_JJ masses_NN2 from_II 10_MC 2_MC to_II 10_MC 6_MC g_NNU mol_NN1 -1_MC can_VM be_VBI measured_VVN and_CC this_DD1 range_NN1 is_VBZ much_RR wider_JJR than_CSN any_DD other_JJ existing_JJ technique_NN1 ._. 
Two_MC general_JJ methods_NN2 are_VBR used_JJ to_II measure_NN1 M_ZZ1 ,_, (_( 1_MC1 )_) sedimentation_NN1 velocity_NN1 and_CC (_( 2_MC )_) sedimentation_NN1 equilibrium_NN1 ._. 
SEDIMENTATION_NP1 VELOCITY_NN1 The_AT centrifuge_NN1 is_VBZ operated_VVN at_II high_JJ speeds_NN2 to_TO transport_VVI the_AT polymer_NN1 molecules_NN2 through_II the_AT solvent_NN1 to_II the_AT cell_NN1 bottom_NN1 if_CS the_AT solvent_NN1 density_NN1 is_VBZ less_DAR than_CSN the_AT polymer_NN1 ,_, or_CC to_II the_AT top_NN1 (_( flotation_NN1 )_) if_CS the_AT reverse_NN1 is_VBZ true_JJ ._. 
The_AT rate_NN1 of_IO movement_NN1 can_VM be_VBI measured_VVN by_II following_VVG the_AT change_NN1 in_II refractive_JJ index_NN1 n_ZZ1 through_II the_AT boundary_NN1 region_NN1 ._. 
As_II the_AT molecules_NN2 sediment_NN1 ,_, a_AT1 layer_NN1 of_IO pure_JJ solvent_NN1 is_VBZ left_VVN whose_DDQGE refractive_JJ index_NN1 differs_VVZ from_II the_AT solution_NN1 ._. 
The_AT boundary_NN1 is_VBZ located_VVN by_II the_AT sharp_JJ change_NN1 in_II n_ZZ1 and_CC its_APPGE movement_NN1 is_VBZ followed_VVN as_II a_AT1 function_NN1 of_IO time_NNT1 using_VVG one_MC1 or_CC other_JJ of_IO the_AT optical_JJ methods_NN2 available_JJ ._. 
When_CS moving_VVG through_II the_AT solution_NN1 ,_, the_AT polymer_NN1 will_VM experience_VVI a_AT1 centrifugal_JJ force_NN1 which_DDQ is_VBZ ,_, but_CCB as_CSA the_AT molecule_NN1 displaces_VVZ a_AT1 mass_NN1 of_IO solution_NN1 it_PPH1 will_VM be_VBI subject_II21 to_II22 a_AT1 buoyancy_NN1 effect_NN1 and_CC an_AT1 opposing_JJ force_NN1 ._. 
The_AT net_JJ force_NN1 is_VBZ then_RT where_CS r_ZZ1 is_VBZ the_AT distance_NN1 between_II the_AT boundary_NN1 and_CC the_AT centre_NN1 of_IO rotation_NN1 ,_, is_VBZ the_AT partial_JJ specific_JJ volume_NN1 of_IO the_AT polymer_NN1 ,_, is_VBZ the_AT angular_JJ velocity_NN1 ,_, the_AT mass_NN1 of_IO the_AT molecule_NN1 and_CC is_VBZ the_AT density_NN1 of_IO the_AT solution_NN1 ._. 
This_DD1 force_NN1 will_VM be_VBI balanced_VVN by_II the_AT frictional_JJ resistance_NN1 of_IO the_AT medium_NN1 F_ZZ1 for_IF a_AT1 particular_JJ velocity_NN1 and_CC where_RRQ R_ZZ1 s_ZZ1 is_VBZ the_AT spherical_JJ radius_NN1 of_IO the_AT polymer_NN1 particle_NN1 and_CC is_VBZ the_AT viscosity_NN1 of_IO the_AT medium_NN1 ._. 
These_DD2 two_MC forces_NN2 are_VBR in_II equilibrium_NN1 when_CS a_AT1 uniform_JJ particle_NN1 velocity_NN1 is_VBZ attained_VVN and_CC The_AT steady-state_JJ velocity_NN1 in_II a_AT1 unit_NN1 gravitational_JJ field_NN1 can_VM then_RT be_VBI defined_VVN as_II the_AT sedimentation_NN1 constant_JJ S_ZZ1 ,_, and_CC where_RRQ f_ZZ1 is_VBZ the_AT frictional_JJ coefficient_NN1 of_IO the_AT molecule_NN1 and_CC is_VBZ related_VVN to_II the_AT diffusion_NN1 constant_JJ D_ZZ1 by_II Substitution_NN1 gives_VVZ the_AT Svedberg_NP1 equation_NN1 ,_, From_II this_DD1 a_AT1 molar_JJ mass_NN1 M_ZZ1 SD_NP1 ,_, is_VBZ calculated_VVN if_CS both_RR S_ZZ1 and_CC D_ZZ1 are_VBR known_VVN ._. 
This_DD1 average_NN1 is_VBZ close_JJ to_II M_ZZ1 w_ZZ1 but_CCB is_VBZ usually_RR smaller_JJR and_CC depends_VVZ on_II the_AT method_NN1 used_VVN to_II measure_NN1 D._NP1 The_AT term_NN1 is_VBZ called_VVN the_AT buoyancy_NN1 factor_NN1 and_CC determines_VVZ the_AT direction_NN1 of_IO macromolecular_JJ transport_NN1 in_II the_AT cell_NN1 ._. 
If_CS the_AT factor_NN1 is_VBZ positive_JJ ,_, the_AT polymer_NN1 chains_NN2 sediment_NN1 away_II21 from_II22 the_AT centre_NN1 of_IO rotation_NN1 to_II the_AT cell_NN1 bottom_NN1 ,_, if_CS negative_JJ ,_, they_PPHS2 move_VV0 in_II the_AT opposite_JJ direction_NN1 and_CC float_VV0 to_II the_AT top_NN1 ._. 
The_AT determination_NN1 of_IO M_ZZ1 is_VBZ absolute_JJ when_CS S_ZZ1 and_CC D_ZZ1 are_VBR known_VVN ,_, but_CCB more_RGR commonly_RR a_AT1 relation_NN1 of_IO the_AT form_NN1 is_VBZ established_VVN ,_, using_VVG polymer_NN1 fractions_NN2 of_IO known_JJ M_NN1 ,_, for_IF a_AT1 given_JJ solvent_NN1 +_FO polymer_NN1 system_NN1 ._. 
This_DD1 approach_NN1 is_VBZ similar_JJ to_II that_DD1 used_JJ for_IF the_AT limiting_JJ viscosity_NN1 number_NN1 ,_, which_DDQ is_VBZ a_AT1 non-absolute_JJ method_NN1 ._. 
SEDIMENTATION_NP1 EQUILIBRIUM_NN1 In_II the_AT sedimentation_NN1 equilibrium_NN1 experiments_VVZ the_AT condition_NN1 for_IF equilibrium_NN1 requires_VVZ that_CST the_AT total_JJ potential_NN1 ,_, ,_, must_VM be_VBI constant_JJ in_II all_DB parts_NN2 of_IO the_AT system_NN1 ._. 
For_IF a_AT1 polymer_NN1 of_IO molar_JJ mass_NN1 M_ZZ1 2_MC dissolved_VVD in_II a_AT1 solvent_NN1 (_( 1_MC1 )_) and_CC placed_VVN in_II a_AT1 centrifugal_JJ field_NN1 of_IO angular_JJ velocity_NN1 ,_, its_APPGE potential_JJ energy_NN1 at_II a_AT1 distance_NN1 r_ZZ1 from_II the_AT centre_NN1 of_IO rotation_NN1 is_VBZ and_CC its_APPGE chemical_JJ potential_NN1 is_VBZ ._. 
The_AT total_JJ potential_NN1 then_RT becomes_VVZ and_CC the_AT conditions_NN2 for_IF equilibrium_NN1 are_VBR Consider_VV0 now_RT the_AT transportation_NN1 of_IO J_ZZ1 moles_NN2 of_IO polymer_NN1 across_II unit_NN1 cross_VV0 sectional_JJ area_NN1 in_II unit_NN1 time_NNT1 ;_; the_AT transport_NN1 equation_NN1 for_IF this_DD1 is_VBZ where_RRQ ,_, is_VBZ the_AT mass_JJ conductivity_NN1 which_DDQ is_VBZ proportional_JJ to_II the_AT concentration_NN1 of_IO the_AT substance_NN1 and_CC inversely_RR proportional_JJ to_II the_AT resistance_NN1 offered_VVN by_II the_AT medium_NN1 to_TO transport_VVI i.e._REX the_AT frictional_JJ coefficient_NN1 per_II mole_NN1 f_ZZ1 ._. 
Now_RT is_VBZ a_AT1 function_NN1 of_IO T_ZZ1 ,_, P_ZZ1 and_CC c_ZZ1 2_MC but_CCB if_CS T_ZZ1 is_VBZ constant_JJ then_RT and_CC as_CSA where_CS 2_MC is_VBZ the_AT activity_NN1 coefficient_NN1 ._. 
We_PPIS2 have_VH0 for_IF an_AT1 ideal_JJ solution_NN1 Substitution_NN1 in_II equation_NN1 (_( 9.49_MC )_) yields_NN2 or_CC Using_VVG the_AT equations_NN2 (_( 9.45_MC )_) ,_, (_( 9.46_MC )_) and_CC the_AT definition_NN1 of_IO L_ZZ1 ,_, equation_NN1 (_( 9.50_MC )_) can_VM be_VBI written_VVN as_CSA This_DD1 shows_VVZ that_CST the_AT flux_NN1 J_ZZ1 is_VBZ then_RT a_AT1 net_JJ result_NN1 of_IO the_AT sedimentation_NN1 rate_NN1 and_CC the_AT back_NN1 diffusion_NN1 of_IO the_AT molecules_NN2 ._. 
At_II equilibrium_NN1 these_DD2 balance_NN1 ,_, and_CC the_AT flow_NN1 vanishes_VVZ so_CS that_DD1 ._. 
It_PPH1 follows_VVZ that_CST This_DD1 describes_VVZ the_AT concentration_NN1 gradient_NN1 at_II equilibrium_NN1 for_IF a_AT1 single_JJ solute_NN1 under_II ideal_JJ solution_NN1 conditions_NN2 ,_, and_CC integration_NN1 between_II the_AT meniscus_NN1 r_ZZ1 m_ZZ1 and_CC any_DD point_NN1 '_GE r_ZZ1 '_GE in_II the_AT cell_NN1 gives_VVZ A_AT1 graph_NN1 of_IO against_II r_ZZ1 2_MC can_VM be_VBI constructed_VVN by_II measuring_VVG the_AT concentrations_NN2 at_II different_JJ points_NN2 in_II the_AT cell_NN1 and_CC M_ZZ1 2_MC can_VM be_VBI calculated_VVN from_II the_AT slope_NN1 ._. 
The_AT main_JJ experimental_JJ problem_NN1 ,_, however_RR ,_, is_VBZ being_VBG able_JK to_TO calculate_VVI the_AT concentration_NN1 at_II each_DD1 point_NN1 in_II the_AT cell_NN1 which_DDQ is_VBZ not_XX always_RR easy_JJ ._. 
A_AT1 more_RGR widely_RR used_JJ method_NN1 is_VBZ to_TO calculate_VVI the_AT difference_NN1 in_II concentration_NN1 between_II that_DD1 at_II the_AT meniscus_NN1 (_( cm_NNU )_) and_CC the_AT cell_NN1 bottom_NN1 (_( cb_NNU )_) ._. 
Rearranging_VVG and_CC integrating_VVG equation_NN1 (_( 9.51_MC )_) gives_VVZ and_CC the_AT integral_JJ on_II the_AT r_ZZ1 h_ZZ1 s._NNU can_VM be_VBI evaluated_VVN by_II considering_VVG mass_JJ conservation_NN1 in_II a_AT1 sector_NN1 shaped_JJ cell_NN1 ,_, so_CS21 that_CS22 where_CS c_ZZ1 o_ZZ1 is_VBZ the_AT initial_JJ concentration_NN1 ._. 
For_IF a_AT1 polydisperse_NN1 polymer_NN1 this_DD1 gives_VVZ a_AT1 weight_NN1 average_JJ molar_JJ mass_NN1 M_ZZ1 w_ZZ1 and_CC the_AT z_ZZ1 average_JJ M_ZZ1 z_ZZ1 can_VM be_VBI calculated_VVN from_II the_AT concentration_NN1 gradients_NN2 at_II the_AT top_NN1 and_CC bottom_NN1 of_IO the_AT cell_NN1 ._. 
The_AT main_JJ disadvantage_NN1 of_IO the_AT method_NN1 lies_VVZ in_II the_AT long_JJ periods_NN2 of_IO time_NNT1 required_VVN to_TO reach_VVI equilibrium_NN1 ._. 
Several_DA2 variations_NN2 exist_VV0 which_DDQ reduce_VV0 this_DD1 time_NNT1 scale_NN1 such_II21 as_II22 studying_VVG the_AT approach_NN1 to_II equilibrium_NN1 ,_, using_VVG short_JJ columns_NN2 ,_, or_CC meniscus_NN1 depletion_NN1 techniques_NN2 can_VM be_VBI employed_VVN but_CCB all_DB are_VBR outside_II the_AT scope_NN1 of_IO this_DD1 text_NN1 ._. 
The_AT value_NN1 of_IO M_ZZ1 w_ZZ1 calculated_VVN from_II equation_NN1 (_( 9.53_MC )_) is_VBZ ,_, of_RR21 course_RR22 ,_, an_AT1 apparent_JJ value_NN1 relating_VVG to_II the_AT initial_JJ concentration_NN1 of_IO the_AT solution_NN1 ,_, and_CC extrapolation_NN1 to_TO zero_VVI concentration_NN1 is_VBZ necessary._NNU 9.13_MC Viscosity_NN1 When_CS a_AT1 polymer_NN1 dissolves_VVZ in_II a_AT1 liquid_NN1 ,_, the_AT interaction_NN1 of_IO the_AT two_MC components_NN2 stimulates_VVZ an_AT1 increase_NN1 in_II polymer_NN1 dimensions_NN2 over_II that_DD1 in_II the_AT unsolvated_JJ state_NN1 ._. 
Because_II21 of_II22 the_AT vast_JJ difference_NN1 in_II size_NN1 between_II solvent_NN1 and_CC solute_NN1 ,_, the_AT frictional_JJ properties_NN2 of_IO the_AT solvent_NN1 in_II the_AT mixture_NN1 are_VBR drastically_RR altered_VVN ,_, and_CC an_AT1 increase_NN1 in_II viscosity_NN1 occurs_VVZ which_DDQ should_VM reflect_VVI the_AT size_NN1 and_CC shape_NN1 of_IO the_AT dissolved_JJ solute_NN1 ,_, even_RR in_II dilute_JJ solutions_NN2 ._. 
This_DD1 was_VBDZ first_MD recognized_VVN in_II 1930_MC by_II Staudinger_NP1 ,_, who_PNQS found_VVD that_CST an_AT1 empirical_JJ relation_NN1 existed_VVD between_II the_AT relative_JJ magnitude_NN1 of_IO the_AT increase_NN1 in_II viscosity_NN1 and_CC the_AT molar_JJ mass_NN1 of_IO the_AT polymer_NN1 ._. 
One_MC1 of_IO the_AT simplest_JJT methods_NN2 of_IO examining_VVG this_DD1 effect_NN1 is_VBZ by_II capillary_JJ viscometry_NN1 ._. 
It_PPH1 has_VHZ been_VBN shown_VVN that_CST the_AT ratio_NN1 of_IO the_AT flow_NN1 time_NNT1 of_IO a_AT1 polymer_NN1 solution_NN1 t_ZZ1 to_II that_DD1 of_IO the_AT pure_JJ solvent_NN1 t_ZZ1 o_ZZ1 is_VBZ effectively_RR equal_JJ to_II the_AT ratio_NN1 of_IO their_APPGE viscosity_NN1 if_CS the_AT densities_NN2 are_VBR equal_JJ ._. 
This_DD1 latter_DA approximation_NN1 is_VBZ reasonable_JJ for_IF dilute_JJ solutions_NN2 and_CC provides_VVZ a_AT1 measure_NN1 of_IO the_AT relative_JJ viscosity_NN1 r_ZZ1 As_CSA this_DD1 has_VHZ a_AT1 limiting_JJ value_NN1 of_IO unity_NN1 ,_, a_AT1 more_RGR useful_JJ quantity_NN1 is_VBZ the_AT specific_JJ viscosity_NN1 Even_RR in_II dilute_JJ solutions_NN2 molecular_JJ interference_NN1 is_VBZ likely_JJ to_TO occur_VVI and_CC sp_NNU is_VBZ extrapolated_VVN to_TO zero_VVI concentration_NN1 to_TO obtain_VVI a_AT1 measure_NN1 of_IO the_AT influence_NN1 of_IO an_AT1 isolated_JJ polymer_NN1 coil_NN1 ._. 
This_DD1 is_VBZ accomplished_VVN in_II either_DD1 of_IO two_MC ways_NN2 ;_; sp_NNU can_VM be_VBI expressed_VVN as_II a_AT1 reduced_JJ quantity_NN1 and_CC extrapolated_VVN to_II according_II21 to_II22 the_AT relation_NN1 and_CC the_AT intercept_VV0 is_VBZ the_AT limiting_JJ viscosity_NN1 number_NN1 &lsqb;_( &rsqb;_) which_DDQ is_VBZ a_AT1 characteristic_JJ parameter_NN1 for_IF the_AT polymer_NN1 in_II a_AT1 particular_JJ solvent_NN1 ,_, k_ZZ1 is_VBZ a_AT1 shape_NN1 dependent_JJ factor_NN1 called_VVN the_AT Huggins_NP1 constant_JJ and_CC has_VHZ values_NN2 between_II 0.3_MC and_CC 0.9_MC for_IF randomly_RR coiling_VVG vinyl_NN1 polymers_NN2 ._. 
The_AT alternative_JJ extrapolation_NN1 method_NN1 uses_VVZ the_AT inherent_JJ viscosity_NN1 as_CSA where_CS k_ZZ1 is_VBZ another_DD1 shape_NN1 dependent_JJ factor_NN1 ._. 
The_AT dimensions_NN2 of_IO &lsqb;_( &rsqb;_) are_VBR the_AT same_DA as_CSA the_AT reciprocal_JJ of_IO the_AT concentration_NN1 ._. 
When_CS measuring_VVG &lsqb;_( &rsqb;_) solutions_NN2 are_VBR filtered_VVN to_TO remove_VVI spurious_JJ particles_NN2 ,_, then_RT flow_NN1 times_NNT2 for_IF solvent_NN1 and_CC solutions_NN2 are_VBR recorded_VVN in_II U-tube_JJ viscometers_NN2 such_II21 as_II22 the_AT '_GE Cannon-Fenske_NP1 '_GE or_CC the_AT '_GE Ubbelohde_NN1 suspended_VVD level_JJ dilution_NN1 '_GE models_NN2 ._. 
Dilution_NN1 viscometers_NN2 are_VBR most_RGT convenient_JJ when_CS a_AT1 concentration_NN1 series_NN is_VBZ to_TO be_VBI measured_VVN ._. 
In_II these_DD2 the_AT concentration_NN1 can_VM be_VBI changed_VVN in_RR21 situ_RR22 ,_, whereas_CS fresh_JJ solution_NN1 concentrations_NN2 of_IO exactly_RR the_AT same_DA volume_NN1 must_VM be_VBI introduced_VVN for_IF each_DD1 measurement_NN1 in_II the_AT non_FU dilution_NN1 Cannon-Fenske_NP1 ._. 
In_II the_AT Ubbelohde_NN1 viscometer_NN1 an_AT1 aliquot_NN1 of_IO solution_NN1 of_IO known_JJ volume_NN1 is_VBZ pipetted_VVN into_II bulb_NN1 D_ZZ1 through_II A._NP1 The_AT solution_NN1 is_VBZ then_RT pumped_VVN into_II E_ZZ1 ,_, by_II applying_VVG a_AT1 pressure_NN1 down_II A_ZZ1 with_IW C_NP1 closed_VVD off_RP ;_; the_AT pressure_NN1 is_VBZ released_VVN and_CC C_ZZ1 is_VBZ opened_VVN to_TO allow_VVI the_AT excess_JJ solution_NN1 to_TO drain_VVI back_RP into_II D._NP1 This_DD1 leaves_VVZ the_AT end_NN1 of_IO the_AT capillary_JJ open_JJ or_CC suspended_VVD ._. 
Solution_NN1 then_RT flows_VVZ down_RP the_AT capillary_NN1 and_CC drains_NN2 round_II the_AT sides_NN2 of_IO the_AT bulb_NN1 back_RP into_II D_ZZ1 ,_, but_CCB as_CSA no_AT back_NN1 pressure_NN1 from_II the_AT excess_JJ solution_NN1 exists_VVZ ,_, the_AT volume_NN1 in_II D_ZZ1 plays_VVZ no_AT part_NN1 in_II determining_VVG the_AT flow_NN1 time_NNT1 t_ZZ1 ._. 
This_DD1 suspended_JJ level_NN1 is_VBZ the_AT feature_NN1 which_DDQ allows_VVZ dilution_NN1 to_TO be_VBI carried_VVN out_RP in_II D_ZZ1 without_IW affecting_VVG t_ZZ1 ._. 
Thus_RR addition_NN1 of_IO a_AT1 known_JJ amount_NN1 of_IO solvent_NN1 to_II the_AT solution_NN1 in_II D_ZZ1 ,_, followed_VVN by_II mixing_NN1 ,_, gives_VVZ the_AT next_MD concentration_NN1 in_II the_AT series_NN ._. 
The_AT flow_NN1 time_NNT1 t_ZZ1 ,_, is_VBZ the_AT time_NNT1 taken_VVN for_IF the_AT solution_NN1 meniscus_NN1 to_TO pass_VVI from_II x_ZZ1 to_II y_ZZ1 in_II bulb_NN1 E._NP1 For_IF a_AT1 given_JJ polymer_NN1 +_FO solvent_NN1 system_NN1 at_II a_AT1 specified_JJ temperature_NN1 ,_, &lsqb;_( &rsqb;_) can_VM be_VBI related_VVN to_II M_NN1 through_II the_AT Mark-Houwink_NP1 equation_NN1 K_ZZ1 v_ZZ1 and_CC v_ZZ1 can_VM be_VBI established_VVN by_II calibrating_VVG with_IW polymer_NN1 fractions_NN2 of_IO known_JJ molar_JJ mass_NN1 ,_, and_CC once_RR this_DD1 has_VHZ been_VBN established_VVN for_IF a_AT1 system_NN1 ,_, &lsqb;_( &rsqb;_) alone_RR will_VM give_VVI M_NN1 for_IF an_AT1 unknown_JJ fraction_NN1 ._. 
This_DD1 is_VBZ normally_RR achieved_VVN by_II plotting_VVG log_NN1 &lsqb;_( &rsqb;_) against_II log_NN1 M_ZZ1 and_CC interpolation_NN1 is_VBZ then_RT quite_RG straightforward_JJ ._. 
Values_NN2 of_IO v_ZZ1 lie_VV0 between_II 0.5_MC for_IF a_AT1 polymer_NN1 dissolved_VVN in_II a_AT1 theta-solvent_NN1 to_II about_RG 0.8_MC in_II very_RG good_JJ solvents_NN2 for_IF linear_JJ randomly_RR coiling_VVG vinyl_NN1 polymers_NN2 ,_, and_CC typical_JJ values_NN2 for_IF systems_NN2 studied_VVN by_II viscosity_NN1 and_CC sedimentation_NN1 are_VBR given_VVN in_II table_NN1 9.3_MC ._. 
The_AT exponents_NN2 v_ZZ1 and_CC b_ZZ1 are_VBR indicative_JJ of_IO solvent_NN1 quality_NN1 ._. 
When_CS the_AT solvent_NN1 is_VBZ ideal_JJ ,_, i.e._REX a_AT1 theta-solvent_NN1 ,_, both_RR v_ZZ1 and_CC b_ZZ1 are_VBR 0.5_MC ,_, but_CCB as_CSA the_AT solvent_NN1 becomes_VVZ thermodynamically_RR better_RRR ,_, and_CC deviations_NN2 from_II ideality_NN1 larger_JJR ,_, then_RT v_ZZ1 increases_NN2 and_CC b_ZZ1 decreases_VVZ ._. 
VISCOSITY_NP1 AVERAGE_JJ MOLECULAR_JJ WEIGHT_NN1 Polymer_NN1 samples_NN2 are_VBR normally_RR polydisperse_VV0 and_CC it_PPH1 is_VBZ of_IO interest_NN1 to_TO examine_VVI the_AT type_NN1 of_IO average_JJ molecular_JJ weight_NN1 that_CST might_VM be_VBI expected_VVN from_II a_AT1 measurement_NN1 of_IO *lsqb_FO ;_; &rsqb;_) ._. 
As_II the_AT specific_JJ viscosity_NN1 will_VM depend_VVI on_II the_AT contributions_NN2 from_II each_DD1 of_IO the_AT polymer_NN1 molecules_NN2 in_II the_AT sample_NN1 we_PPIS2 can_VM write_VVI If_CS we_PPIS2 now_RT divide_VV0 by_II the_AT total_JJ concentration_NN1 and_CC substitute_VV0 for_IF then_RT or_CC Comparison_NN1 with_IW equation_NN1 (_( 9.59_MC )_) shows_VVZ that_CST the_AT viscosity_NN1 average_NN1 M_ZZ1 v_ZZ1 is_VBZ then_RT and_CC that_CST this_DD1 lies_VVZ between_II M_ZZ1 n_ZZ1 and_CC M_ZZ1 w_ZZ1 in_II magnitude_NN1 ,_, but_CCB will_VM be_VBI usually_RR closer_JJR to_II M_MC w._NNU 9.14_MC Gel_NN1 permeation_NN1 chromatography_NN1 The_AT molar_JJ mass_JJ distribution_NN1 (_( MMD_MC )_) of_IO a_AT1 polymer_NN1 sample_NN1 has_VHZ a_AT1 significant_JJ influence_NN1 on_II its_APPGE properties_NN2 and_CC a_AT1 knowledge_NN1 of_IO the_AT shape_NN1 of_IO this_DD1 distribution_NN1 is_VBZ fundamental_JJ to_II the_AT full_JJ characterization_NN1 of_IO a_AT1 polymer_NN1 ._. 
The_AT determination_NN1 of_IO the_AT MMD_MC by_II conventional_JJ fractionation_NN1 techniques_NN2 is_VBZ time_NNT1 consuming_VVG ,_, and_CC a_AT1 rapid_JJ ,_, efficient_JJ and_CC reliable_JJ method_NN1 which_DDQ can_VM provide_VVI a_AT1 measure_NN1 of_IO the_AT MMD_MC in_II a_AT1 matter_NN1 of_IO hours_NNT2 has_VHZ been_VBN developed_VVN ._. 
This_DD1 is_VBZ gel_NN1 permeation_NN1 chromatography_NN1 (_( GPC_NP1 )_) ._. 
Known_VVN alternatively_RR by_II its_APPGE more_RGR descriptive_JJ name_NN1 '_GE size_NN1 exclusion_NN1 chromatography_NN1 '_GE (_( SEC_NNU )_) ,_, the_AT method_NN1 depends_VVZ on_II the_AT use_NN1 of_IO mechanically_RR stable_JJ ,_, highly_RR crosslinked_VVD gels_NN2 which_DDQ have_VH0 a_AT1 distribution_NN1 of_IO different_JJ pore_NN1 sizes_NN2 and_CC can_VM ,_, by_II31 means_II32 of_II33 a_AT1 sieving_NN1 action_NN1 ,_, effect_NN1 separation_NN1 of_IO a_AT1 polymer_NN1 sample_NN1 into_II fractions_NN2 ,_, dictated_VVN by_II their_APPGE molecular_JJ volume_NN1 ._. 
The_AT non-ionic_JJ gel_NN1 stationary_JJ phase_NN1 is_VBZ commonly_RR composed_VVN of_IO crosslinked_JJ polystyrene_NN1 or_CC macroporous_JJ silica_NN1 particles_NN2 ,_, which_DDQ do_VD0 not_XX swell_VVI significantly_RR in_II the_AT carrier_NN1 solvents_NN2 ._. 
A_AT1 range_NN1 of_IO pore_NN1 sizes_NN2 is_VBZ fundamental_JJ to_II the_AT success_NN1 of_IO this_DD1 size_NN1 fractionation_NN1 procedure_NN1 which_DDQ depends_VVZ on_II two_MC processes_NN2 ._. 
These_DD2 are_VBR (_( a_ZZ1 )_) separation_NN1 by_II size_NN1 exclusion_NN1 alone_RR ,_, which_DDQ is_VBZ the_AT more_RGR important_JJ feature_NN1 ,_, and_CC (_( b_ZZ1 )_) a_AT1 dispersion_NN1 process_NN1 ,_, controlled_VVN by_II molecular_JJ diffusion_NN1 which_DDQ may_VM lead_VVI to_II an_AT1 artificial_JJ broadening_NN1 of_IO the_AT MMD_MC ._. 
Looking_VVG first_MD at_II the_AT mechanism_NN1 of_IO the_AT separation_NN1 process_NN1 (_( a_ZZ1 )_) ;_; in_II simple_JJ terms_NN2 ,_, the_AT large_JJ molecules_NN2 ,_, which_DDQ occupy_VV0 the_AT greatest_JJT effective_JJ volume_NN1 in_II solution_NN1 ,_, are_VBR excluded_VVN from_II the_AT smaller_JJR pore_NN1 sizes_NN2 in_II the_AT gel_NN1 and_CC pass_VV0 quickly_RR through_II the_AT larger_JJR channels_NN2 between_II the_AT gel_NN1 particles_NN2 ._. 
This_DD1 results_VVZ in_II their_APPGE being_VBG eluted_VVN first_MD from_II the_AT column_NN1 ._. 
As_II the_AT molecular_JJ size_NN1 of_IO the_AT polymer_NN1 decreases_VVZ there_EX is_VBZ an_AT1 increasing_JJ probability_NN1 that_CST the_AT molecules_NN2 can_VM diffuse_VVI into_II the_AT smaller_JJR pores_NN2 and_CC channels_NN2 in_II the_AT gel_NN1 which_DDQ slows_VVZ their_APPGE time_NNT1 of_IO passage_NN1 through_II the_AT column_NN1 by_II providing_VVG a_AT1 potentially_RR longer_JJR path_NN1 length_NN1 before_II being_VBG eluted_VVN ._. 
By_II choosing_VVG a_AT1 series_NN of_IO gel_NN1 columns_NN2 with_IW an_AT1 appropriate_JJ range_NN1 of_IO pore_NN1 sizes_NN2 ,_, an_AT1 effective_JJ size_NN1 separation_NN1 can_VM be_VBI obtained_VVN ._. 
The_AT efficiency_NN1 of_IO the_AT separation_NN1 process_NN1 is_VBZ then_RT a_AT1 function_NN1 of_IO the_AT dependence_NN1 of_IO the_AT retention_NN1 (_( or_CC elution_NN1 )_) volume_NN1 V_ZZ1 R_ZZ1 on_II the_AT molar_JJ mass_JJ M_NN1 ,_, and_CC a_AT1 reliable_JJ relationship_NN1 between_II the_AT two_MC parameters_NN2 must_VM be_VBI established_VVN ._. 
The_AT value_NN1 of_IO V_ZZ1 R_ZZ1 depends_VVZ on_II the_AT interstitial_JJ void_JJ volume_NN1 V_ZZ1 o_ZZ1 and_CC the_AT accessible_JJ part_NN1 of_IO the_AT pore_NN1 volume_NN1 in_II the_AT gel_NN1 ,_, where_CS V_ZZ1 i_ZZ1 is_VBZ the_AT total_JJ internal_JJ pore_NN1 volume_NN1 and_CC KD_NP1 is_VBZ the_AT partition_NN1 coefficient_NN1 between_II V_ZZ1 and_CC the_AT portion_NN1 accessible_JJ to_II a_AT1 given_JJ solute_NN1 ._. 
Thus_RR for_IF very_RG large_JJ molecules_NN2 ,_, and_CC rapid_JJ elution_NN1 takes_VVZ place_NN1 ,_, whereas_CS for_IF very_RG small_JJ molecules_NN2 which_DDQ can_VM penetrate_VVI all_DB the_AT available_JJ pore_NN1 volume_NN1 ._. 
This_DD1 is_VBZ shown_VVN schematically_RR in_II figure_NN1 9.12_MC and_CC clearly_RR the_AT technique_NN1 can_VM not_XX discriminate_VVI amongst_II molecular_JJ sizes_NN2 with_IW or_CC ._. 
For_IF samples_NN2 which_DDQ fall_VV0 within_II the_AT appropriate_JJ range_NN1 it_PPH1 has_VHZ been_VBN suggested_VVN that_CST a_AT1 universal_JJ calibration_NN1 curve_NN1 can_VM be_VBI constructed_VVN to_TO relate_VVI V_ZZ1 R_ZZ1 and_CC M_ZZ1 ,_, by_II assuming_VVG that_CST the_AT hydrodynamic_JJ volume_NN1 of_IO a_AT1 macromolecule_NN1 is_VBZ related_VVN to_II the_AT product_NN1 ,_, where_RRQ &lsqb;_( &rsqb;_) is_VBZ the_AT intrinsic_JJ viscosity_NN1 of_IO the_AT polymer_NN1 in_II the_AT carrier_NN1 solvent_NN1 used_VVD ,_, at_II the_AT temperature_NN1 of_IO measurement_NN1 ._. 
A_AT1 universal_JJ calibration_NN1 curve_NN1 is_VBZ then_RT obtained_VVN by_II plotting_VVG against_II V_ZZ1 R_ZZ1 for_IF a_AT1 given_JJ carrier_NN1 solvent_NN1 and_CC a_AT1 fixed_JJ temperature_NN1 ._. 
Experimental_JJ verification_NN1 of_IO this_DD1 is_VBZ shown_VVN in_II figure_NN1 9.13_MC for_IF a_AT1 variety_NN1 of_IO different_JJ polymers_NN2 and_CC can_VM be_VBI utilized_VVN in_II the_AT following_JJ way_NN1 ._. 
To_TO obtain_VVI the_AT MMD_MC ,_, the_AT mass_NN1 of_IO the_AT polymer_NN1 being_VBG eluted_VVN must_VM be_VBI measured_VVN ._. 
This_DD1 can_VM be_VBI achieved_VVN continuously_RR using_VVG refractive_JJ index_NN1 ,_, UV_JJ or_CC IR_NN1 detectors_NN2 ,_, which_DDQ will_VM give_VVI a_AT1 mass_JJ distribution_NN1 as_II a_AT1 function_NN1 of_IO VR_NP1 ._. 
It_PPH1 is_VBZ still_RR necessary_JJ to_TO estimate_VVI the_AT molar_JJ mass_NN1 of_IO each_DD1 fraction_NN1 before_II the_AT MMD_MC curve_NN1 can_VM be_VBI constructed_VVN ._. 
If_CS the_AT universal_JJ calibration_NN1 curve_NN1 is_VBZ valid_JJ for_IF the_AT system_NN1 then_RT where_CS the_AT subscripts_NN2 s_ZZ1 and_CC u_ZZ1 denote_VV0 the_AT standard_JJ calibration_NN1 and_CC the_AT polymer_NN1 under_II study_NN1 ,_, respectively_RR ._. 
As_II the_AT chains_NN2 of_IO the_AT polymer_NN1 under_II examination_NN1 may_VM swell_VVI in_II the_AT carrier_NN1 solvent_NN1 to_II a_AT1 different_JJ extent_NN1 compared_VVN to_II an_AT1 equal_JJ molar_JJ mass_JJ sample_NN1 of_IO the_AT standard_NN1 ,_, the_AT hydrodynamic_JJ volumes_NN2 will_VM not_XX necessarily_RR be_VBI equivalent_JJ ._. 
This_DD1 can_VM be_VBI compensated_VVN for_IF by_II applying_VVG a_AT1 correction_NN1 based_VVN on_II the_AT knowledge_NN1 of_IO the_AT appropriate_JJ Mark-Houwink_NP1 relations_NN2 for_IF each_DD1 ,_, the_AT standard_NN1 and_CC the_AT unknown_JJ ,_, measured_VVN in_II the_AT solvent_NN1 used_VVN for_IF elution_NN1 ._. 
The_AT molar_JJ mass_NN1 M_ZZ1 u_ZZ1 can_VM then_RT be_VBI obtained_VVN from_II Thus_RR a_AT1 calibration_NN1 curve_NN1 constructed_VVN for_IF standard_JJ samples_NN2 of_IO polystyrene_NN1 can_VM be_VBI used_VVN to_TO determine_VVI M_NN1 for_IF other_JJ polymers_NN2 if_CS the_AT Mark-Houwink_NP1 relations_NN2 are_VBR also_RR known_VVN ._. 
This_DD1 can_VM be_VBI avoided_VVN if_CS a_AT1 '_GE Viscotek_NP1 '_GE ,_, which_DDQ is_VBZ a_AT1 combined_JJ differential_JJ refractometer_NN1 and_CC viscometer_NN1 ,_, is_VBZ attached_VVN to_II the_AT end_NN1 of_IO the_AT column_NN1 ,_, as_CSA this_DD1 measures_VVZ both_RR concentration_NN1 and_CC the_AT sp_NNU for_IF the_AT fraction_NN1 ._. 
By_II assuming_VVG in_II dilute_JJ solutions_NN2 the_AT molar_JJ mass_NN1 can_VM be_VBI obtained_VVN from_II the_AT Mark-Houwink_NP1 relation_NN1 ._. 
Alternatively_RR ,_, a_AT1 low_JJ angle_NN1 laser_NN1 light_NN1 scattering_VVG (_( LALLS_NP2 )_) instrument_NN1 can_VM be_VBI attached_VVN in_II series_NN with_IW a_AT1 concentration_NN1 detector_NN1 ._. 
This_DD1 gives_VVZ a_AT1 direct_JJ measurement_NN1 of_IO M_ZZ1 w_ZZ1 ,_, using_VVG equation_NN1 (_( 9.28_MC )_) ,_, if_CS all_DB the_AT parameters_NN2 in_II the_AT equation_NN1 are_VBR known_VVN ._. 
When_CS using_VVG SEC_NNU ,_, care_NN1 must_VM be_VBI taken_VVN not_XX to_TO overload_VVI the_AT columns_NN2 with_IW too_RG large_JJ a_AT1 polymer_NN1 sample_NN1 as_CSA this_DD1 results_VVZ in_II a_AT1 non-linear_JJ response_NN1 ,_, characterized_VVN by_II losses_NN2 in_II resolution_NN1 and_CC column_NN1 efficiency_NN1 ._. 
Also_RR ,_, although_CS the_AT band_NN1 broadening_NN1 referred_VVN to_II earlier_JJR can_VM be_VBI minimized_VVN by_II using_VVG long_RR ,_, efficient_JJ columns_NN2 ,_, it_PPH1 may_VM never_RR be_VBI entirely_RR eliminated_VVN ._. 
The_AT MMD_MC curves_NN2 may_VM then_RT be_VBI broadened_VVN by_II this_DD1 phenomenon_NN1 and_CC appropriate_JJ corrections_NN2 must_VM be_VBI applied_VVN ._. 
Unfortunately_RR these_DD2 are_VBR often_RR difficult_JJ to_TO calculate_VVI accurately_RR ,_, although_CS it_PPH1 has_VHZ been_VBN shown_VVN by_II Tung_NP1 that_CST the_AT error_NN1 introduced_VVN by_II broadening_NN1 is_VBZ negligible_JJ if_CS for_IF the_AT sample_NN1 ._. 
CHAPTER_NN1 10_MC Polymer_NN1 Characterization_NN1 Chain_NN1 Dimensions_NN2 and_CC Structures_NN2 As_II the_AT size_NN1 and_CC shape_NN1 of_IO a_AT1 polymer_NN1 chain_NN1 are_VBR of_IO considerable_JJ interest_NN1 to_II the_AT polymer_NN1 scientist_NN1 it_PPH1 is_VBZ useful_JJ to_TO know_VVI how_RRQ these_DD2 factors_NN2 can_VM be_VBI assessed_VVN ._. 
Much_DA1 of_IO the_AT information_NN1 can_VM be_VBI derived_VVN from_II studies_NN2 of_IO dilute_JJ solutions_NN2 ;_; an_AT1 absolute_JJ measurement_NN1 of_IO polymer_NN1 chain_NN1 size_NN1 can_VM be_VBI obtained_VVN from_II light_JJ scattering_NN1 ,_, when_CS the_AT polymer_NN1 is_VBZ large_JJ compared_VVN with_IW the_AT wavelength_NN1 of_IO the_AT incident_NN1 light_NN1 ._. 
Sometimes_RT the_AT absolute_JJ measurement_NN1 can_VM not_XX be_VBI used_VVN but_CCB the_AT size_NN1 can_VM be_VBI deduced_VVN indirectly_RR from_II viscosity_NN1 measurements_NN2 ,_, which_DDQ are_VBR related_VVN to_II the_AT volume_NN1 occupied_VVN by_II the_AT chain_NN1 in_II solution_NN1 ._. 
Armed_VVN with_IW this_DD1 information_NN1 we_PPIS2 must_VM now_RT determine_VVI how_RGQ meaningful_JJ it_PPH1 is_VBZ and_CC to_TO do_VDI this_DD1 a_AT1 clearer_JJR understanding_NN1 of_IO the_AT factors_NN2 governing_VVG the_AT shape_NN1 of_IO the_AT polymer_NN1 is_VBZ required_VVN ._. 
We_PPIS2 can_VM confine_VVI ourselves_PPX2 to_II models_NN2 of_IO the_AT random_JJ coil_NN1 ,_, as_CSA this_DD1 is_VBZ usually_RR believed_VVN to_TO be_VBI most_RGT appropriate_JJ for_IF synthetic_JJ polymers_NN2 ;_; other_JJ models_NN2 rods_NN2 ,_, discs_NN2 ,_, spheres_NN2 ,_, spheroids_NN2 are_VBR also_RR postulated_VVN ,_, but_CCB need_VM not_XX concern_VVI us_PPIO2 at_II this_DD1 level._NNU 10.1_MC Average_JJ chain_NN1 dimensions_NN2 A_ZZ1 polymer_NN1 chain_NN1 in_II dilute_JJ solution_NN1 can_VM be_VBI pictured_VVN as_II a_AT1 coil_NN1 ,_, continuously_RR changing_VVG its_APPGE shape_NN1 under_II the_AT action_NN1 of_IO random_JJ thermal_JJ motions_NN2 ._. 
This_DD1 means_VVZ ,_, that_CST at_II any_DD time_NNT1 ,_, the_AT volume_NN1 occupied_VVN by_II a_AT1 chain_NN1 in_II solution_NN1 ,_, could_VM differ_VVI from_II that_DD1 occupied_VVN by_II its_APPGE neighbours_NN2 ,_, and_CC these_DD2 size_NN1 differences_NN2 are_VBR further_RRR accentuated_VVN by_II the_AT fact_NN1 that_CST each_DD1 sample_NN1 will_VM contain_VVI a_AT1 variety_NN1 of_IO chain_NN1 lengths_NN2 ._. 
Taking_VVG these_DD2 two_MC points_NN2 into_II consideration_NN1 leads_VVZ us_PPIO2 to_II the_AT conclusion_NN1 that_CST meaningful_JJ chain_NN1 dimensions_NN2 can_VM only_RR be_VBI values_NN2 averaged_VVN over_II the_AT many_DA2 conformations_NN2 assumed_VVD ._. 
Two_MC such_DA averages_NN2 have_VH0 been_VBN defined_VVN :_: (_( a_ZZ1 )_) the_AT average_JJ root_NN1 mean_VV0 square_JJ distance_NN1 between_II the_AT chain_NN1 ends_NN2 ;_; and_CC (_( b_ZZ1 )_) the_AT average_JJ root_NN1 mean_VV0 square_JJ radius_NN1 of_IO gyration_NN1 which_DDQ is_VBZ a_AT1 measure_NN1 of_IO the_AT average_JJ distance_NN1 of_IO a_AT1 chain_NN1 element_NN1 from_II the_AT centre_NN1 of_IO gravity_NN1 of_IO the_AT coil_NN1 ._. 
The_AT angular_JJ brackets_NN2 denote_VV0 averaging_VVG due_JJ to_TO chain_VVI polydispersity_NN1 in_II the_AT sample_NN1 and_CC the_AT bar_NN1 indicates_VVZ averaging_VVG for_IF the_AT many_DA2 conformational_JJ sizes_NN2 available_JJ to_II chains_NN2 of_IO the_AT same_DA molar_JJ mass_NN1 ._. 
The_AT two_MC quantities_NN2 are_VBR related_VVN ,_, in_II the_AT absence_NN1 of_IO excluded_JJ volume_NN1 effects_NN2 ,_, for_IF simple_JJ chains_NN2 by_II but_CCB as_CSA the_AT actual_JJ dimensions_NN2 obtained_VVN can_VM depend_VVI on_II the_AT conditions_NN2 of_IO the_AT measurement_NN1 ,_, other_JJ factors_NN2 must_VM also_RR be_VBI considered._NNU 10.2_MC Freely_RR jointed_VVN chain_NN1 model_NN1 The_AT initial_JJ attempts_NN2 to_TO arrive_VVI at_II a_AT1 theoretical_JJ representation_NN1 of_IO the_AT dimensions_NN2 of_IO a_AT1 linear_JJ chain_NN1 ,_, treated_VVN the_AT molecule_NN1 as_II a_AT1 number_NN1 n_ZZ1 of_IO chain_NN1 elements_NN2 ,_, joined_VVN by_II bonds_NN2 of_IO length_NN1 l_ZZ1 ._. 
By_II assuming_VVG the_AT bonds_NN2 act_VV0 like_II universal_JJ joints_NN2 ,_, complete_JJ freedom_NN1 of_IO rotation_NN1 about_II the_AT chain_NN1 bonds_NN2 can_VM be_VBI postulated_VVN ._. 
This_DD1 model_NN1 allows_VVZ the_AT chain_NN1 to_TO be_VBI pictured_VVN as_CSA in_II figure_NN1 10.1(a)_FO which_DDQ resembles_VVZ the_AT path_NN1 of_IO a_AT1 diffusing_JJ gas_NN1 molecule_NN1 and_CC as_RG random_JJ flight_NN1 statistics_NN have_VH0 proved_VVN useful_JJ in_II describing_VVG gases_NN2 ,_, a_AT1 similar_JJ approach_NN1 is_VBZ used_VVN here_RL ._. 
In_II two_MC dimensions_NN2 the_AT diagram_NN1 is_VBZ more_RGR picturesquely_RR called_VVN the_AT '_GE drunkard_NN1 's_GE walk_NN1 '_GE and_CC r_ZZ1 f_ZZ1 is_VBZ estimated_VVN by_II considering_VVG first_MD the_AT simplest_JJT case_NN1 of_IO two_MC links_NN2 ._. 
The_AT end-to-end_JJ distance_NN1 r_ZZ1 f_ZZ1 follows_VVZ from_II the_AT cosine_NN1 law_NN1 that_CST see_VV0 figure_NN1 10.1(b)_FO ,_, or_CC When_CS the_AT number_NN1 of_IO bonds_NN2 ,_, n_ZZ1 is_VBZ large_JJ ,_, the_AT angle_NN1 will_VM vary_VVI over_RP all_DB possible_JJ values_NN2 so_CS21 that_CS22 the_AT sum_NN1 of_IO all_DB these_DD2 terms_NN2 will_VM be_VBI zero_MC ,_, and_CC as_CSA equation(10.3)_FO will_VM reduce_VVI to_II This_DD1 shows_VVZ that_CST the_AT distance_NN1 between_II the_AT chain_NN1 ends_NN2 ,_, for_IF this_DD1 model_NN1 is_VBZ proportional_JJ to_II the_AT square_JJ root_NN1 of_IO the_AT number_NN1 of_IO bonds_NN2 and_CC so_RR ,_, is_VBZ considerably_RR shorter_JJR than_CSN a_AT1 fully_RR extended_VVN chain_NN1 ._. 
The_AT result_NN1 is_VBZ the_AT same_DA if_CS the_AT molecule_NN1 is_VBZ thought_VVN to_TO occupy_VVI three-dimensional_JJ space_NN1 ,_, but_CCB if_CS it_PPH1 is_VBZ centred_VVN on_II a_AT1 co-ordinate_NN1 system_NN1 both_RR positive_JJ and_CC negative_JJ contributions_NN2 occur_VV0 with_IW equal_JJ probability_NN1 ._. 
To_TO overcome_VVI this_DD1 the_AT dimension_NN1 is_VBZ expressed_VVN always_RR as_CSA the_AT square_NN1 which_DDQ eliminates_VVZ negative_JJ signs_NN2 ._. 
This_DD1 model_NN1 is_VBZ ,_, however_RR ,_, unrealistic_JJ ._. 
Polymer_NN1 chains_NN2 occupy_VV0 a_AT1 volume_NN1 in_II space_NN1 ,_, and_CC the_AT dimensions_NN2 of_IO any_DD macromolecule_NN1 are_VBR influenced_VVN by_II the_AT bond_NN1 angles_NN2 and_CC by_II interactions_NN2 between_II the_AT chain_NN1 elements_NN2 ._. 
These_DD2 interactions_NN2 can_VM be_VBI classified_VVN into_II two_MC groups_NN2 :_: (_( i_ZZ1 )_) Short_NP1 range_NN1 interactions_NN2 which_DDQ occur_VV0 between_II neighbouring_JJ atoms_NN2 or_CC groups_NN2 ,_, and_CC are_VBR usually_RR forces_NN2 of_IO steric_JJ repulsion_NN1 caused_VVN by_II the_AT overlapping_JJ of_IO electron_NN1 clouds_NN2 ;_; (_( ii_MC )_) Long_JJ range_NN1 interactions_NN2 which_DDQ are_VBR comprised_VVN of_IO attractive_JJ and_CC repulsive_JJ forces_NN2 between_II segments_NN2 ,_, widely_RR separated_VVN in_II a_AT1 chain_NN1 ,_, that_DD1 occasionally_RR approach_VV0 one_PPX121 another_PPX122 during_II molecular_JJ flexing_NN1 ,_, and_CC between_II segments_NN2 and_CC solvent_NN1 molecules_NN2 ._. 
These_DD2 are_VBR often_RR termed_VVN excluded_JJ volume_NN1 effects._NNU 10.3_MC Short_JJ range_NN1 effects_NN2 The_AT expansion_NN1 of_IO a_AT1 covalently_RR bonded_JJ polymer_NN1 chain_NN1 will_VM be_VBI restricted_VVN by_II the_AT valence_NN1 angles_NN2 between_II each_DD1 chain_NN1 atom_NN1 ._. 
In_RR21 general_RR22 this_DD1 angle_NN1 is_VBZ for_IF a_AT1 homoatomic_JJ chain_NN1 and_CC equation_NN1 (_( 10.4_MC )_) can_VM be_VBI modified_VVN to_TO allow_VVI for_IF these_DD2 short_JJ range_NN1 interactions_NN2 ._. 
For_IF the_AT simplest_JJT case_NN1 of_IO an_AT1 all_DB carbon_NN1 backbone_NN1 chain_NN1 such_II21 as_II22 polyethylene_NN1 ,_, and_CC so_CS21 that_CS22 equation_NN1 (_( 10.5_MC )_) reduces_VVZ to_II This_DD1 indicates_VVZ that_CST the_AT polyethylene_NN1 chain_NN1 is_VBZ twice_RR as_RG extended_JJ as_II the_AT freely_RR jointed_VVN chain_NN1 model_NN1 when_CS short_JJ range_NN1 interactions_NN2 are_VBR considered._NNU 10.4_MC Chain_NN1 stiffness_NN1 As_CSA we_PPIS2 have_VH0 already_RR seen_VVN in_II chapter_NN1 1_MC1 for_IF butane_NN1 and_CC polyethylene_NN1 ,_, steric_JJ repulsions_NN2 impose_VV0 restrictions_NN2 to_TO bond_VVI rotation_NN1 ._. 
This_DD1 means_VVZ that_DD1 equation_NN1 (_( 10.5_MC )_) has_VHZ to_TO be_VBI modified_VVN further_RRR and_CC now_RT becomes_VVZ where_RRQ is_VBZ the_AT average_JJ cosine_NN1 of_IO the_AT angle_NN1 of_IO rotation_NN1 of_IO the_AT bonds_NN2 in_II the_AT backbone_NN1 chain_NN1 ._. 
The_AT parameter_NN1 is_VBZ the_AT average_JJ mean_JJ square_NN1 of_IO the_AT unperturbed_JJ dimension_NN1 ,_, which_DDQ is_VBZ a_AT1 characteristic_JJ parameter_NN1 for_IF a_AT1 given_JJ polymer_NN1 chain_NN1 ._. 
The_AT freely_RR jointed_VVN dimensions_NN2 are_VBR now_RT more_RGR realistic_JJ when_CS restricted_VVN by_II the_AT factor_NN1 the_AT skeletal_JJ factor_NN1 composed_VVN of_IO the_AT two_MC terms_NN2 where_RRQ is_VBZ known_VVN as_II the_AT steric_JJ parameter_NN1 and_CC is_VBZ for_IF simple_JJ chains_NN2 ._. 
For_IF more_RGR complex_JJ chains_NN2 ,_, containing_VVG rings_NN2 or_CC heteroatomic_JJ chains_NN2 ,_, e.g._REX polydienes_NN2 ,_, polyethers_VVZ ,_, polysaccharides_NN2 ,_, and_CC proteins_NN2 ,_, an_AT1 estimate_NN1 of_IO is_VBZ obtained_VVN from_II Values_NN2 of_IO the_AT unperturbed_JJ dimension_NN1 can_VM be_VBI obtained_VVN experimentally_RR from_II dilute_JJ solution_NN1 measurements_NN2 made_VVD either_RR directly_RR in_II a_AT1 theta-solvent_NN1 (_( see_VV0 section_NN1 9.9_MC )_) or_CC by_II using_VVG indirect_JJ measurements_NN2 in_II non-ideal_JJ solvents_NN2 and_CC employing_VVG an_AT1 extrapolation_NN1 procedure_NN1 ._. 
The_AT geometry_NN1 of_IO each_DD1 chain_NN1 allows_VVZ the_AT calculation_NN1 of_IO ,_, and_CC results_NN2 are_VBR expressed_VVN either_RR as_CSA or_CC as_II the_AT characteristic_JJ ratio_NN1 ._. 
Both_DB2 provide_VV0 a_AT1 measure_NN1 of_IO chain_NN1 stiffness_NN1 in_II dilute_JJ solution_NN1 ._. 
The_AT range_NN1 of_IO values_NN2 normally_RR found_VVN for_IF is_VBZ from_II about_RG 1.5_MC to_II 2.5_MC as_CSA shown_VVN in_II table_NN1 10.1._MC 10.5_MC Treatment_NN1 of_IO dilute_JJ solution_NN1 data_NN We_PPIS2 can_VM now_RT examine_VVI some_DD of_IO the_AT ways_NN2 of_IO calculating_VVG the_AT polymer_NN1 dimensions_NN2 from_II experimental_JJ data_NN ._. 
THE_AT SECOND_MD VIRIAL_NN1 COEFFICIENT_NN1 An_AT1 investigation_NN1 of_IO the_AT dilute_JJ solution_NN1 behaviour_NN1 of_IO a_AT1 polymer_NN1 can_VM provide_VVI useful_JJ information_NN1 about_II the_AT size_NN1 and_CC shape_NN1 of_IO the_AT coil_NN1 ,_, the_AT extent_NN1 of_IO polymer-solvent_NN1 interaction_NN1 and_CC the_AT molar_JJ mass_NN1 ._. 
Deviations_NN2 from_II ideality_NN1 ,_, as_CSA we_PPIS2 have_VH0 seen_VVN in_II section_NN1 9.7_MC ,_, are_VBR conveniently_RR expressed_VVN in_II31 terms_II32 of_II33 virial_NN1 expansions_NN2 ,_, and_CC when_RRQ solutions_NN2 are_VBR sufficiently_RR dilute_VV0 ,_, the_AT results_NN2 can_VM be_VBI adequately_RR described_VVN by_II the_AT terms_NN2 up_RG21 to_RG22 the_AT second_MD virial_NN1 coefficient_NN1 A_ZZ1 2_MC while_CS neglecting_VVG higher_JJR terms_NN2 ._. 
The_AT value_NN1 of_IO A_ZZ1 2_MC is_VBZ a_AT1 measure_NN1 of_IO solvent-polymer_JJ compatibility_NN1 ,_, as_CSA the_AT parameter_NN1 reflects_VVZ the_AT tendency_NN1 of_IO a_AT1 polymer_NN1 segment_NN1 to_TO exclude_VVI its_APPGE neighbours_NN2 from_II the_AT volume_NN1 it_PPH1 occupies_VVZ ._. 
Thus_RR a_AT1 large_JJ positive_JJ A_ZZ1 2_MC indicates_VVZ a_AT1 good_JJ solvent_NN1 for_IF the_AT polymer_NN1 while_CS a_AT1 low_JJ value_NN1 (_( sometimes_RT even_RR negative_JJ )_) shows_VVZ that_CST the_AT solvent_NN1 is_VBZ relatively_RR poor_JJ ._. 
The_AT virial_NN1 coefficient_NN1 can_VM be_VBI related_VVN to_II the_AT Flory_JJ dilute_JJ solution_NN1 parameters_NN2 by_II where_RRQ is_VBZ a_AT1 molar_JJ mass_JJ dependent_JJ function_NN1 of_IO the_AT excluded_JJ volume_NN1 ._. 
The_AT exact_JJ form_NN1 of_IO can_VM be_VBI defined_VVN explicitly_RR by_II one_MC1 of_IO several_DA2 theories_NN2 ,_, and_CC while_CS each_DD1 leads_VVZ to_II a_AT1 slightly_RR different_JJ form_NN1 ,_, all_DB predict_VV0 that_DD1 is_VBZ unity_NN1 when_CS theta_NN1 conditions_NN2 are_VBR attained_VVN and_CC the_AT excluded_JJ volume_NN1 effect_NN1 vanishes_VVZ ._. 
Equation_NN1 (_( 10.10_MC )_) can_VM be_VBI used_VVN to_TO analyse_VVI data_NN such_II21 as_II22 that_DD1 in_II figure_NN1 8.7_MC ._. 
Once_RR has_VHZ been_VBN located_VVN ,_, the_AT entropy_NN1 parameter_NN1 1_MC1 can_VM be_VBI calculated_VVN by_II replotting_VVG the_AT data_NN as_II21 against_II22 T._NP1 Extrapolation_NN1 to_II ,_, where_RRQ ,_, allows_VVZ 1_MC1 to_TO be_VBI estimated_VVN for_IF the_AT system_NN1 under_II theta_NN1 conditions_NN2 ._. 
This_DD1 method_NN1 of_IO measuring_VVG and_CC 1_MC1 ,_, is_VBZ only_RR accurate_JJ when_CS the_AT solvent_NN1 is_VBZ poor_JJ ,_, and_CC extrapolations_NN2 are_VBR short_JJ ._. 
The_AT dependence_NN1 of_IO A_ZZ1 2_MC on_II M_ZZ1 can_VM often_RR be_VBI predicted_VVN ,_, for_IF good_JJ solvents_NN2 ,_, by_II a_AT1 simple_JJ equation_NN1 where_RRQ varies_VVZ from_II 0.15_MC to_II 0.4_MC ,_, depending_II21 on_II22 the_AT system_NN1 and_CC k_ZZ1 is_VBZ a_AT1 constant_JJ ._. 
EXPANSION_NN1 FACTOR_NN1 The_AT value_NN1 of_IO A_ZZ1 2_MC will_VM tell_VVI us_PPIO2 whether_CSW31 or_CSW32 not_CSW33 the_AT size_NN1 of_IO the_AT polymer_NN1 coil_NN1 ,_, which_DDQ is_VBZ dissolved_VVN in_II a_AT1 particular_JJ solvent_NN1 ,_, will_VM be_VBI perturbed_VVN or_CC expanded_VVD over_RP that_DD1 of_IO the_AT unperturbed_JJ state_NN1 ,_, but_CCB the_AT extent_NN1 of_IO this_DD1 expansion_NN1 is_VBZ best_RRT estimated_VVN by_II calculating_VVG the_AT expansion_NN1 factor_NN1 ._. 
If_CS the_AT temperature_NN1 of_IO a_AT1 system_NN1 ,_, containing_VVG a_AT1 polymer_NN1 of_IO finite_JJ M_NN1 ,_, drops_VVZ much_RR below_II the_AT number_NN1 of_IO polymer-polymer_JJ contacts_NN2 increases_VVZ until_CS precipitation_NN1 of_IO the_AT polymer_NN1 occurs_VVZ ._. 
Above_II this_DD1 temperature_NN1 ,_, the_AT chains_NN2 are_VBR expanded_VVN ,_, or_CC perturbed_JJ ,_, from_II the_AT equilibrium_NN1 size_NN1 attained_VVN under_II pseudo-ideal_JJ conditions_NN2 ,_, by_II long_JJ range_NN1 interactions_NN2 ._. 
The_AT extent_NN1 of_IO this_DD1 coil_NN1 expansion_NN1 is_VBZ determined_VVN by_II two_MC long_JJ range_NN1 effects_NN2 ._. 
The_AT first_MD results_NN2 from_II the_AT physical_JJ exclusion_NN1 of_IO one_MC1 polymer_NN1 segment_NN1 by_II another_DD1 from_II a_AT1 hypothetical_JJ lattice_NN1 site_NN1 which_DDQ reduces_VVZ the_AT number_NN1 of_IO possible_JJ conformations_NN2 available_JJ to_II the_AT chain_NN1 ._. 
This_DD1 serves_VVZ to_TO lower_VVI the_AT probability_NN1 that_CST tightly_RR coiled_VVD conformations_NN2 will_VM be_VBI favoured_VVN ._. 
The_AT second_NNT1 is_VBZ observed_VVN in_II very_RG good_JJ solvents_NN2 ,_, where_CS the_AT tendency_NN1 is_VBZ for_IF polymer-solvent_NN1 interactions_NN2 to_TO predominate_VVI ,_, and_CC leads_VVZ to_II a_AT1 preference_NN1 for_IF even_RR more_RGR extended_JJ conformations_NN2 ._. 
In_II a_AT1 given_JJ solvent_NN1 an_AT1 equilibrium_NN1 conformation_NN1 is_VBZ eventually_RR achieved_VVN when_CS the_AT forces_NN2 of_IO expansion_NN1 are_VBR balanced_VVN by_II forces_NN2 of_IO contraction_NN1 in_II the_AT molecule_NN1 ._. 
The_AT tendency_NN1 to_TO contract_VVI arises_VVZ from_II the_AT both_RR the_AT polymer-polymer_JJ interactions_NN2 and_CC the_AT resistance_NN1 to_II expansion_NN1 of_IO the_AT chain_NN1 into_II over_II extended_JJ and_CC energetically_RR less_RGR favoured_JJ conformations_NN2 ._. 
The_AT extent_NN1 of_IO this_DD1 coil_NN1 perturbation_NN1 by_II long_JJ range_NN1 effects_NN2 is_VBZ measured_VVN by_II an_AT1 expansion_NN1 factor_NN1 ,_, introduced_VVN by_II Flory_NP1 ._. 
This_DD1 relates_VVZ the_AT perturbed_JJ and_CC unperturbed_JJ dimensions_NN2 by_II In_II good_JJ solvents_NN2 (_( large_JJ ,_, positive_JJ A_ZZ1 2_MC )_) the_AT coil_NN1 is_VBZ more_RGR extended_JJ than_CSN in_II poor_JJ solvents_NN2 (_( low_JJ A_ZZ1 2_MC )_) and_CC is_VBZ correspondingly_RR larger_JJR ._. 
Since_CS is_VBZ solvent_JJ and_CC temperature_NN1 dependent_NN1 a_AT1 more_RGR characteristic_JJ dimension_NN1 to_TO measure_VVI for_IF the_AT polymer_NN1 is_VBZ which_DDQ can_VM be_VBI calculated_VVN from_II light_NN1 scattering_VVG in_II a_AT1 theta-solvent_NN1 ,_, or_CC indirectly_RR as_CSA next_MD described_VVN ._. 
FLORY-FOX_JJ THEORY_NN1 The_AT molecular_JJ dimensions_NN2 of_IO a_AT1 polymer_NN1 chain_NN1 in_II any_DD solvent_NN1 can_VM be_VBI calculated_VVN directly_RR from_II light_NN1 scattering_VVG measurements_NN2 ,_, using_VVG equation_NN1 (_( 9.36_MC )_) ,_, if_CS the_AT coil_NN1 is_VBZ large_JJ enough_RR to_TO scatter_VVI light_NN1 in_II an_AT1 asymmetric_JJ manner_NN1 ,_, but_CCB when_RRQ the_AT chain_NN1 is_VBZ too_RG short_JJ to_TO be_VBI measured_VVN accurately_RR in_II this_DD1 way_NN1 an_AT1 alternative_JJ technique_NN1 has_VHZ to_TO be_VBI used_VVN ._. 
Flory_NN1 and_CC Fox_NP1 suggested_VVD that_CST as_II the_AT viscosity_NN1 of_IO a_AT1 polymer_NN1 solution_NN1 will_VM depend_VVI on_II the_AT volume_NN1 occupied_VVN by_II the_AT polymer_NN1 chain_NN1 ,_, it_PPH1 should_VM be_VBI feasible_JJ to_TO relate_VVI coil_NN1 size_NN1 and_CC &lsqb;_( &rsqb;_) ._. 
They_PPHS2 assumed_VVD that_CST if_CS the_AT unperturbed_JJ polymer_NN1 is_VBZ approximated_VVN by_II a_AT1 hydrodynamic_JJ sphere_NN1 ,_, then_RT &lsqb;_( &rsqb;_) ,_, the_AT limiting_JJ viscosity_NN1 number_NN1 in_II a_AT1 theta_NN1 solvent_NN1 ,_, could_VM be_VBI related_VVN to_II the_AT square_JJ root_NN1 of_IO the_AT molar_JJ mass_NN1 by_II where_RRQ Equations_NN2 (_( 10.13_MC )_) and_CC (_( 10.14_MC )_) are_VBR actually_RR derived_VVN for_IF monodisperse_NN1 samples_NN2 ,_, and_CC when_RRQ measurements_NN2 are_VBR performed_VVN with_IW heterodisperse_NN1 polymers_NN2 ,_, the_AT appropriate_JJ averages_NN2 to_TO use_VVI are_VBR M_ZZ1 n_ZZ1 and_CC ._. 
The_AT parameter_NN1 was_VBDZ originally_RR considered_VVN to_TO be_VBI a_AT1 universal_JJ constant_JJ ,_, but_CCB experimental_JJ work_NN1 suggests_VVZ that_CST it_PPH1 is_VBZ a_AT1 function_NN1 of_IO the_AT solvent_NN1 ,_, molar_JJ mass_NN1 ,_, and_CC heterogeneity_NN1 ._. 
Values_NN2 can_VM vary_VVI from_II an_AT1 experimental_JJ one_PN1 of_IO 2.1_MC 10_MC 23_MC to_II a_AT1 theoretical_JJ limit_NN1 of_IO about_RG 2.84_MC 10_MC 23_MC when_RRQ &lsqb;_( &rsqb;_) is_VBZ expressed_VVN in_II cm_NNU 3_MC g_ZZ1 -1_MC ._. 
A_AT1 most_RGT probable_JJ value_NN1 of_IO 2.5_MC 10_MC 23_MC has_VHZ been_VBN found_VVN to_TO be_VBI acceptable_JJ for_IF most_DAT flexible_JJ heterodisperse_NN1 polymers_NN2 in_II good_JJ solvents_NN2 ._. 
For_IF non-ideal_JJ solvents_NN2 equation_NN1 (_( 10.13_MC )_) can_VM be_VBI expanded_VVN to_TO give_VVI where_RRQ a_AT1 is_VBZ the_AT linear_JJ expansion_NN1 factor_NN1 ,_, pertaining_II21 to_II22 viscosity_NN1 measurements_NN2 ,_, and_CC is_VBZ a_AT1 measure_NN1 of_IO long_JJ range_NN1 interactions_NN2 ._. 
As_II the_AT derivation_NN1 is_VBZ based_VVN on_II an_AT1 unrealistic_JJ Gaussian_JJ distribution_NN1 of_IO segments_NN2 in_II good_JJ solvents_NN2 ,_, it_PPH1 has_VHZ been_VBN suggested_VVN that_DD1 is_VBZ related_VVN to_II the_AT more_RGR direct_JJ measurement_NN1 of_IO in_II equation_NN1 (_( 10.12_MC )_) by_II Considerable_JJ experimental_JJ evidence_NN1 exists_VVZ to_TO support_VVI this_DD1 conclusion_NN1 ._. 
INDIRECT_JJ ESTIMATES_NN2 OF_IO It_PPH1 is_VBZ not_XX always_RR possible_JJ to_TO find_VVI a_AT1 suitable_JJ theta-solvent_NN1 for_IF a_AT1 polymer_NN1 and_CC methods_NN2 have_VH0 been_VBN developed_VVN which_DDQ allow_VV0 unperturbed_JJ dimensions_NN2 to_TO be_VBI estimated_VVN in_II non-ideal_JJ (_( good_JJ )_) solvents_NN2 ._. 
Several_DA2 methods_NN2 of_IO extrapolating_VVG data_NN for_IF &lsqb;_( &rsqb;_) have_VH0 been_VBN suggested_VVN ._. 
The_AT most_RGT useful_JJ of_IO these_DD2 was_VBDZ proposed_VVN by_II Stockmayer_NP1 and_CC Fixman_NP1 ,_, using_VVG the_AT equation_NN1 :_: when_RRQ is_VBZ assumed_VVN to_TO adopt_VVI its_APPGE limiting_JJ theoretical_JJ value_NN1 ,_, B_ZZ1 is_VBZ related_VVN to_II the_AT thermodynamic_JJ interaction_NN1 parameter_NN1 1_MC1 by_II and_CC examination_NN1 of_IO equation_NN1 (_( 10.10_MC )_) shows_VVZ that_CST B_ZZ1 is_VBZ also_RR proportional_JJ to_II A_ZZ1 2_MC ._. 
The_AT unperturbed_JJ dimension_NN1 can_VM be_VBI estimated_VVN by_II plotting_VVG against_II M_MC ;_; K_ZZ1 ,_, is_VBZ obtained_VVN from_II the_AT intercept_VV0 and_CC is_VBZ calculated_VVN from_II equation_NN1 (_( 10.14_MC )_) ._. 
A_AT1 similar_JJ procedure_NN1 has_VHZ been_VBN proposed_VVN by_II Cowie_NP1 and_CC Bywater_NP1 ,_, in_II which_DDQ the_AT intrinsic_JJ frictional_JJ coefficient_NN1 &lsqb;_( f_ZZ1 &rsqb;_) measured_VVD from_II sedimentation_NN1 or_CC diffusion_NN1 experiments_NN2 ,_, will_VM provide_VVI the_AT same_DA information_NN1 using_VVG where_RRQ and_CC P_ZZ1 o_ZZ1 is_VBZ a_AT1 '_GE constant_NN1 '_GE with_IW a_AT1 limiting_JJ value_NN1 of_IO 5.2_MC ._. 
These_DD2 extrapolation_NN1 procedures_NN2 all_DB depend_VV0 on_II the_AT validity_NN1 of_IO the_AT theoretical_JJ treatment_NN1 and_CC reliability_NN1 must_VM be_VBI judged_VVN in_II this_DD1 light_NN1 ._. 
Fortunately_RR ,_, it_PPH1 has_VHZ been_VBN demonstrated_VVN that_CST most_DAT non-polar_JJ polymers_NN2 can_VM be_VBI treated_VVN in_II this_DD1 way_NN1 and_CC results_NN2 agree_VV0 well_RR with_IW direct_JJ measurements_NN2 of_IO ._. 
For_IF more_RGR polar_JJ polymers_NN2 ,_, specific_JJ solvent_NN1 effects_NN2 become_VV0 more_RGR pronounced_JJ and_CC extrapolations_NN2 have_VH0 to_TO be_VBI regarded_VVN with_IW corresponding_JJ caution_NN1 ._. 
INFLUENCE_NN1 OF_IO TACTICITY_NN1 ON_II CHAIN_NN1 DIMENSIONS_NN2 Studies_NN2 of_IO the_AT dilute_JJ solution_NN1 behaviour_NN1 of_IO polymers_NN2 with_IW a_AT1 specific_JJ stereostructure_NN1 have_VH0 revealed_VVN that_CST the_AT unperturbed_JJ dimensions_NN2 may_VM depend_VVI on_II the_AT chain_NN1 configuration_NN1 ._. 
This_DD1 can_VM be_VBI seen_VVN from_II the_AT data_NN in_II table_NN1 10.1_MC where_RRQ isotactic_JJ ,_, syndiotactic_JJ ,_, and_CC atactic_JJ poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) have_VH0 different_JJ values_NN2 ._. 
If_CS the_AT size_NN1 of_IO a_AT1 polymer_NN1 chain_NN1 can_VM be_VBI affected_VVN by_II its_APPGE configuration_NN1 ,_, the_AT microstructure_NN1 must_VM be_VBI well_RR characterized_VVN before_II an_AT1 accurate_JJ assessment_NN1 of_IO experimental_JJ data_NN can_VM be_VBI made_VVN ._. 
This_DD1 can_VM be_VBI achieved_VVN using_VVG n.m.r._NNU and_CC infrared_JJ techniques._NNU 10.6_MC Nuclear_JJ magnetic_JJ resonance_NN1 (_( n.m.r_NNU ._. )_) 
High_JJ resolution_NN1 n.m.r._NNU has_VHZ proved_VVN to_TO be_VBI a_AT1 particularly_RR useful_JJ tool_NN1 in_II the_AT study_NN1 of_IO the_AT microstructure_NN1 of_IO polymers_NN2 in_II solution_NN1 ,_, where_CS the_AT extensive_JJ molecular_JJ motion_NN1 reduces_VVZ the_AT effect_NN1 of_IO long_JJ range_NN1 interactions_NN2 and_CC allows_VVZ the_AT short_JJ range_NN1 effects_NN2 to_TO dominate_VVI ._. 
Interpretation_NN1 of_IO chain_NN1 tacticity_NN1 ,_, based_VVN on_II the_AT work_NN1 of_IO Bovey_NP1 and_CC Tiers_NN2 ,_, can_VM be_VBI illustrated_VVN using_VVG poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) ._. 
The_AT three_MC possible_JJ steric_JJ configurations_NN2 are_VBR shown_VVN in_II figure_NN1 10.2_MC where_RRQ R_ZZ1 is_VBZ the_AT group_NN1 ._. 
For_IF the_AT purposes_NN2 of_IO n.m.r._NNU measurements_NN2 three_MC consecutive_JJ monomer_NN1 units_NN2 in_II a_AT1 chain_NN1 are_VBR considered_VVN to_TO define_VVI a_AT1 configuration_NN1 and_CC called_VVN a_AT1 triad_NN1 ._. 
The_AT term_NN1 heterotactic_NN1 is_VBZ used_VVN now_RT to_TO define_VVI a_AT1 triad_NN1 which_DDQ is_VBZ neither_RR isotatic_JJ nor_CC syndiotactic_JJ ._. 
In_II the_AT structures_NN2 shown_VVN ,_, the_AT three_MC equivalent_JJ protons_NN2 of_IO the_AT -methyl_JJ group_NN1 absorb_VV0 radiation_NN1 at_II a_AT1 single_JJ frequency_NN1 ,_, but_CCB this_DD1 frequency_NN1 will_VM be_VBI different_JJ for_IF each_DD1 of_IO the_AT three_MC kinds_NN2 of_IO triad_NN1 ,_, because_CS the_AT environment_NN1 of_IO the_AT -methyl_JJ groups_NN2 in_II each_DD1 is_VBZ different_JJ ._. 
For_IF poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) samples_NN2 ,_, which_DDQ were_VBDR prepared_VVN under_II different_JJ conditions_NN2 to_TO give_VVI the_AT three_MC forms_NN2 ,_, resonances_NN2 at_II =_FO 8.78_MC ,_, 8.95_MC and_CC 9.09_MC were_VBDR observed_VVN ,_, which_DDQ were_VBDR assigned_VVN to_II the_AT isotatic_JJ ,_, heterotactic_JJ ,_, and_CC syndiotactic_JJ triads_NN2 respectively_RR ._. 
Thus_RR in_II a_AT1 sample_NN1 with_IW a_AT1 mixture_NN1 of_IO configurations_NN2 a_AT1 triple_JJ peak_NN1 will_VM be_VBI observed_VVN and_CC the_AT area_NN1 under_II each_DD1 of_IO these_DD2 peaks_NN2 will_VM correspond_VVI to_II the_AT amount_NN1 of_IO each_DD1 triad_NN1 present_NN1 in_II the_AT polymer_NN1 chain_NN1 ._. 
This_DD1 is_VBZ illustrated_VVN in_II figure_NN1 10.3_MC ,_, where_CS one_MC1 sample_NN1 is_VBZ predominantly_RR isotactic_JJ ,_, but_CCB also_RR contains_VVZ smaller_JJR percentages_NN2 of_IO the_AT heterotactic_JJ and_CC syndiotactic_JJ configurations_NN2 ._. 
The_AT analysis_NN1 can_VM be_VBI carried_VVN further_RRR ._. 
The_AT fraction_NN1 of_IO each_DD1 configuration_NN1 ,_, P_ZZ1 i_ZZ1 ,_, P_ZZ1 h_ZZ1 ,_, and_CC P_ZZ1 s_ZZ1 ,_, measured_VVN from_II the_AT respective_JJ peak_NN1 areas_NN2 ,_, can_VM be_VBI related_VVN to_II m_ZZ1 the_AT probability_NN1 that_CST a_AT1 monomer_NN1 adding_VVG on_RP to_II the_AT end_NN1 of_IO a_AT1 growing_JJ chain_NN1 will_VM have_VHI the_AT same_DA configuration_NN1 as_CSA the_AT unit_NN1 it_PPH1 is_VBZ joining_VVG ._. 
This_DD1 leads_VVZ to_II the_AT relations_NN2 Curves_NN2 plotted_VVN according_II21 to_II22 this_DD1 simple_JJ analysis_NN1 are_VBR shown_VVN in_II figure_NN1 10.4_MC where_RRQ they_PPHS2 are_VBR compared_VVN with_IW experimental_JJ data_NN obtained_VVN for_IF various_JJ tactic_NN1 forms_NN2 of_IO poly_NN1 (_( -methyl_JJ styrene_NN1 )_) ._. 
Differences_NN2 in_II the_AT microstructure_NN1 of_IO polydienes_NN2 and_CC copolymers_NN2 can_VM also_RR be_VBI made_VVN using_VVG n.m.r_NNU ._. 
In_II the_AT polydienes_NN2 the_AT difference_NN1 between_II 1,2-_FO and_CC 1,4-addition_FO can_VM be_VBI distinguished_VVN on_II examination_NN1 of_IO the_AT resonance_NN1 peaks_NN2 corresponding_VVG to_II terminal_JJ olefinic_JJ protons_NN2 ,_, found_VVN at_II ,_, and_CC non-terminal_JJ olefinic_JJ protons_NN2 observed_VVN at_II ._. 
Not_XX only_RR is_VBZ the_AT local_JJ field_NN1 acting_VVG on_II the_AT nucleus_NN1 altered_VVN by_II environment_NN1 ,_, it_PPH1 is_VBZ also_RR sensitive_JJ to_II molecular_JJ motion_NN1 ,_, and_CC it_PPH1 has_VHZ been_VBN observed_VVN that_CST as_II the_AT molecular_JJ motion_NN1 within_II a_AT1 sample_NN1 increases_NN2 ,_, the_AT resonance_NN1 lines_NN2 become_VV0 narrower_JJR ._. 
Determination_NN1 of_IO the_AT width_NN1 ,_, or_CC second_MD moment_NN1 ,_, of_IO an_AT1 n.m.r._NNU resonance_NN1 line_NN1 ,_, then_RT provides_VVZ a_AT1 sensitive_JJ measure_NN1 of_IO low_JJ frequency_NN1 internal_JJ motions_NN2 in_II solid_JJ polymers_NN2 and_CC can_VM be_VBI used_VVN to_TO study_VVI transitions_NN2 and_CC segmental_JJ rotations_NN2 in_II the_AT polymer_NN1 sample_NN1 ._. 
Line_NN1 widths_NN2 are_VBR also_RR altered_VVN by_II the_AT polymer_NN1 crystallinity_NN1 ._. 
Partially_RR crystalline_JJ polymers_NN2 present_VV0 complex_JJ spectra_NN2 as_CSA they_PPHS2 are_VBR multi-phase_JJ materials_NN2 ,_, in_II which_DDQ the_AT molecular_JJ motions_NN2 are_VBR more_RRR restricted_VVN in_II the_AT crystalline_JJ phase_NN1 than_CSN in_II the_AT amorphous_JJ phase_NN1 ._. 
However_RR ,_, attempts_NN2 to_TO estimate_VVI percentage_NN1 crystallinity_NN1 in_II a_AT1 sample_NN1 using_VVG n.m.r._NNU have_VH0 not_XX been_VBN particularly_RR successful_JJ ._. 
The_AT method_NN1 is_VBZ illustrated_VVN in_II figure_NN1 10.5_MC for_IF where_CS glass_NN1 and_CC other_JJ transitions_NN2 are_VBR readily_RR detected_VVN ._. 
Below_RG 200_MC K_ZZ1 the_AT chains_NN2 are_VBR virtually_RR immobile_JJ ,_, but_CCB above_II 200_MC K_ZZ1 the_AT lines_NN2 sharpen_VV0 as_CSA rotation_NN1 begins_VVZ ._. 
This_DD1 is_VBZ associated_VVN with_IW the_AT glass_NN1 transition_NN1 ,_, but_CCB the_AT way_NN1 the_AT line_NN1 width_NN1 increases_VVZ in_II this_DD1 region_NN1 is_VBZ governed_VVN by_II sample_NN1 crystallinity._NNU 10.7_MC Infrared_JJ spectroscopy_NN1 Infrared_JJ spectroscopy_NN1 can_VM be_VBI used_VVN to_TO characterize_VVI long_JJ chain_NN1 polymers_NN2 because_CS the_AT infrared_JJ active_JJ groups_NN2 ,_, present_NN1 along_II the_AT chain_NN1 ,_, absorb_VV0 as_CS21 if_CS22 each_DD1 was_VBDZ a_AT1 localized_JJ group_NN1 in_II a_AT1 simple_JJ molecule_NN1 ._. 
Identification_NN1 of_IO polymer_NN1 samples_NN2 can_VM be_VBI made_VVN by_II making_VVG use_NN1 of_IO the_AT '_GE finger-print_NN1 '_GE region_NN1 ,_, where_CS it_PPH1 is_VBZ least_RGT likely_JJ for_IF one_MC1 polymer_NN1 to_TO exhibit_VVI exactly_RR the_AT same_DA spectrum_NN1 as_CSA another_DD1 ._. 
This_DD1 region_NN1 lies_VVZ within_II the_AT range_NN1 6.67_MC to_II 12.50_MC m_NNO ._. 
In_II31 addition_II32 to_II33 identification_NN1 ,_, the_AT technique_NN1 has_VHZ been_VBN used_VVN to_TO elucidate_VVI certain_JJ aspects_NN2 of_IO polymer_NN1 microstructure_NN1 ,_, such_II21 as_II22 branching_JJ ,_, crystallinity_NN1 ,_, tacticity_NN1 ,_, and_CC cis-trans_NN2 isomerism_NN1 ._. 
The_AT relative_JJ proportions_NN2 of_IO ,_, and_CC ;_; addition_NN1 in_II polybutadienes_NN2 can_VM be_VBI ascertained_VVN by_II making_VVG use_NN1 of_IO the_AT differences_NN2 in_II absorption_NN1 between_II (_( CH_NN1 )_) out_II21 of_II22 plane_NN1 bending_VVG vibrations_NN2 ,_, which_DDQ depend_VV0 on_II the_AT type_NN1 of_IO substitution_NN1 at_II the_AT olefinic_JJ bond_NN1 ._. 
Terminal_NN1 and_CC internal_JJ groups_NN2 can_VM also_RR be_VBI distinguished_VVN ,_, as_CSA an_AT1 absorption_NN1 band_NN1 at_II about_RG 11.0_MC m_ZZ1 is_VBZ characteristic_JJ of_IO a_AT1 vinyl_NN1 group_NN1 and_CC indicates_VVZ ._. 
The_AT is_VBZ characterized_VVN by_II an_AT1 absorption_NN1 band_NN1 at_II about_RG 13.6_MC m_NNO ,_, whereas_CS the_AT configuration_NN1 exhibits_VVZ a_AT1 band_NN1 at_II about_RG 10.4_MC m_NNO ._. 
An_AT1 estimate_NN1 of_IO cis-trans_NN2 isomerism_NN1 can_VM be_VBI made_VVN by_II measuring_VVG the_AT absorbance_NN1 A_ZZ1 of_IO each_DD1 band_NN1 ,_, where_RRQ and_CC I_ZZ1 o_ZZ1 and_CC I_PPIS1 are_VBR the_AT intensities_NN2 of_IO the_AT incident_NN1 and_CC transmitted_JJ radiation_NN1 respectively_RR ._. 
This_DD1 is_VBZ calculated_VVN by_II locating_VVG a_AT1 base_NN1 line_NN1 across_II the_AT minima_NN2 on_II either_DD1 side_NN1 of_IO the_AT absorption_NN1 band_NN1 and_CC the_AT vertical_JJ height_NN1 to_II the_AT top_NN1 of_IO the_AT band_NN1 from_II the_AT base_NN1 line_NN1 is_VBZ converted_VVN into_II a_AT1 composition_NN1 using_VVG the_AT equation_NN1 where_CS P_ZZ1 cis_NN1 is_VBZ the_AT fraction_NN1 of_IO cis_NN1 configuration_NN1 ,_, A_ZZ1 cis_NN1 is_VBZ the_AT absorbance_NN1 at_II 13.6_MC m_NNO ,_, A_ZZ1 trans_NN2 the_AT absorbance_NN1 at_II 10.4_MC m_NNO ,_, and_CC if_CS we_PPIS2 assume_VV0 that_CST the_AT 1,2_MC content_NN1 is_VBZ negligible_JJ ._. 
Polyisoprenes_NN2 can_VM also_RR be_VBI analyzed_VVN in_II this_DD1 way_NN1 ,_, only_RR now_RT the_AT bands_NN2 at_II 11.0_MC and_CC 11.25_MC ,_, m_ZZ1 are_VBR used_VVN to_TO estimate_VVI the_AT and_CC ,_, while_CS a_AT1 band_NN1 at_II 8.7_MC m_ZZ1 corresponds_VVZ to_II the_AT linkage_NN1 ._. 
The_AT infrared_JJ spectra_NN2 of_IO highly_RR stereoregular_JJ polymers_NN2 are_VBR distinguishable_JJ from_II those_DD2 of_IO their_APPGE less_RGR regular_JJ counterparts_NN2 ,_, but_CCB many_DA2 of_IO the_AT differences_NN2 can_VM be_VBI attributed_VVN to_II crystallinity_NN1 rather_II21 than_II22 tacticity_NN1 as_II such_DA ._. 
The_AT application_NN1 of_IO infrared_JJ to_TO stereostructure_VVI determination_NN1 in_II polymers_NN2 is_VBZ less_RGR reliable_JJ than_CSN n.m.r._NNU ,_, but_CCB has_VHZ achieved_VVN moderate_JJ success_NN1 for_IF poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) and_CC polypropylene_NN1 ._. 
In_II poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) ,_, a_AT1 methyl_NN1 deformation_NN1 at_II 7.25_MC m_ZZ1 is_VBZ unaffected_JJ by_II microstructure_NN1 ,_, and_CC comparison_NN1 of_IO this_DD1 with_IW a_AT1 band_NN1 at_II 9.40_MC m_NNO ,_, which_DDQ is_VBZ present_JJ only_RR in_II atactic_JJ or_CC syndiotactic_JJ polymers_NN2 allows_VVZ an_AT1 estimate_NN1 of_IO the_AT syndiotacticity_NN1 to_TO be_VBI made_VVN from_II the_AT ratio_NN1 ._. 
Similarly_RR provides_VVZ a_AT1 measure_NN1 of_IO the_AT isotactic_JJ content_NN1 ._. 
An_AT1 alternative_JJ method_NN1 is_VBZ to_TO calculate_VVI the_AT quantity_NN1 J_ZZ1 as_II an_AT1 average_NN1 of_IO the_AT two_MC equations_NN2 where_RRQ the_AT absorption_NN1 band_NN1 at_II 10.10_MC m_ZZ1 is_VBZ now_RT used_VVN ._. 
If_CS J_ZZ1 lies_VVZ between_II 100_MC and_CC 115_MC a_AT1 highly_RR syndiotactic_JJ polymer_NN1 is_VBZ indicated_VVN ,_, if_CS between_II 25_MC and_CC 30_MC the_AT polymer_NN1 is_VBZ highly_RR isotactic_JJ ._. 
For_IF polypropylene_NN1 ,_, the_AT characteristic_JJ band_NN1 for_IF the_AT syndiotactic_JJ polymer_NN1 appears_VVZ at_II 11.53_MC m_NNO ,_, and_CC the_AT syndiotactic_JJ index_NN1 I_ZZ1 s_ZZ1 is_VBZ ._. 
Values_NN2 of_IO I_ZZ1 s_ZZ1 about_RG 0.8_MC indicate_VV0 highly_RR syndiotactic_JJ samples_NN2 ._. 
Spectra_NN2 can_VM be_VBI measured_VVN in_II a_AT1 number_NN1 of_IO ways_NN2 ;_; for_IF soluble_JJ polymers_NN2 a_AT1 film_NN1 can_VM be_VBI cast_VVN ,_, perhaps_RR even_RR on_II the_AT N_NNU a_AT1 Cl_FO plate_NN1 to_TO be_VBI used_VVN and_CC examined_VVN directly_RR ._. 
Measurements_NN2 can_VM also_RR be_VBI made_VVN in_II solution_NN1 ,_, if_CS the_AT solvent_NN1 absorption_NN1 in_II any_DD important_JJ region_NN1 is_VBZ low_JJ ,_, or_CC by_II a_AT1 differential_JJ method._NNU 10.8_MC X-ray_NN1 diffraction_NN1 The_AT extent_NN1 of_IO sample_NN1 crystallinity_NN1 can_VM influence_VVI the_AT behaviour_NN1 of_IO a_AT1 polymer_NN1 sample_NN1 greatly_RR ._. 
A_AT1 particularly_RR effective_JJ way_NN1 of_IO examining_VVG partially_RR crystalline_JJ polymers_NN2 is_VBZ by_II X-ray_NN1 diffraction_NN1 ._. 
The_AT crystallites_NN2 present_VV0 in_II a_AT1 powdered_JJ or_CC unoriented_JJ polymer_NN1 sample_NN1 diffract_NN1 X-ray_NN1 beams_NN2 from_II parallel_JJ planes_NN2 for_IF incident_NN1 angles_NN2 which_DDQ are_VBR determined_VVN by_II the_AT Bragg_NP1 equation_NN1 where_RRQ is_VBZ the_AT wavelength_NN1 of_IO the_AT radiation_NN1 ,_, d_ZZ1 is_VBZ the_AT distance_NN1 between_II the_AT parallel_JJ planes_NN2 in_II the_AT crystallites_NN2 ,_, and_CC n_ZZ1 is_VBZ an_AT1 integer_NN1 ._. 
The_AT reinforced_JJ waves_NN2 reflected_VVN by_II all_DB the_AT small_JJ crystallites_NN2 produce_VV0 diffraction_NN1 rings_NN2 ,_, or_CC haloes_NN2 ,_, which_DDQ are_VBR sharply_RR defined_VVN for_IF highly_RR crystalline_JJ materials_NN2 and_CC become_VV0 increasingly_RR diffuse_VV0 when_RRQ the_AT amorphous_JJ content_NN1 is_VBZ high_JJ ._. 
If_CS the_AT polymer_NN1 sample_NN1 is_VBZ oriented_VVN ,_, by_II drawing_VVG a_AT1 fibre_NN1 ,_, or_CC by_II applying_VVG tension_NN1 to_II a_AT1 film_NN1 ,_, the_AT crystallites_NN2 tend_VV0 to_TO become_VVI aligned_VVN in_II the_AT direction_NN1 of_IO the_AT stress_NN1 and_CC the_AT X-ray_NN1 pattern_NN1 is_VBZ improved_VVN ._. 
In_II some_DD samples_NN2 of_IO stereoregular_JJ or_CC symmetrical_JJ polymers_NN2 ,_, the_AT degree_NN1 of_IO three-dimensional_JJ ordering_NN1 of_IO the_AT chains_NN2 may_VM be_VBI sufficiently_RR high_JJ to_TO allow_VVI a_AT1 structural_JJ analysis_NN1 of_IO the_AT polymer_NN1 to_TO be_VBI accomplished_VVN ._. 
Sample_NN1 crystallinity_NN1 can_VM be_VBI estimated_VVN from_II the_AT X-ray_NN1 patterns_NN2 by_II plotting_VVG the_AT density_NN1 of_IO the_AT scattered_JJ beam_NN1 against_II the_AT angle_NN1 of_IO incidence_NN1 ._. 
If_CS this_DD1 can_VM be_VBI done_VDN for_IF an_AT1 amorphous_JJ sample_NN1 and_CC a_AT1 corresponding_JJ sample_NN1 which_DDQ is_VBZ highly_RR crystalline_JJ ,_, a_AT1 relative_JJ measure_NN1 of_IO crystallinity_NN1 for_IF other_JJ samples_NN2 of_IO the_AT same_DA polymer_NN1 can_VM be_VBI obtained_VVN ._. 
In_II figure_NN1 10.6_MC the_AT shaded_JJ portion_NN1 is_VBZ the_AT amorphous_JJ polypropylene_NN1 ,_, while_CS the_AT maxima_NN2 arise_VV0 from_II the_AT crystallites._NNU 10.9_MC Thermal_JJ analysis_NN1 When_CS a_AT1 substance_NN1 undergoes_VVZ a_AT1 physical_JJ or_CC chemical_NN1 change_VV0 a_AT1 corresponding_JJ change_NN1 in_II enthalpy_NN1 is_VBZ observed_VVN ._. 
This_DD1 forms_VVZ the_AT basis_NN1 of_IO the_AT technique_NN1 known_VVN as_II differential_JJ thermal_JJ analysis_NN1 (_( DTA_NP1 )_) in_II which_DDQ the_AT change_NN1 is_VBZ detected_VVN by_II measuring_VVG the_AT enthalpy_NN1 difference_NN1 between_II the_AT material_NN1 under_II study_NN1 and_CC an_AT1 inert_JJ standard_NN1 ._. 
The_AT sample_NN1 is_VBZ placed_VVN in_II a_AT1 heating_NN1 block_NN1 and_CC warmed_VVN at_II a_AT1 uniform_JJ rate_NN1 ._. 
The_AT sample_NN1 temperature_NN1 is_VBZ then_RT monitored_VVN by_II31 means_II32 of_II33 a_AT1 thermocouple_NN1 and_CC compared_VVN with_IW the_AT temperature_NN1 of_IO an_AT1 inert_JJ reference_NN1 such_II21 as_II22 powdered_JJ alumina_NN1 ,_, or_CC simply_RR an_AT1 empty_JJ sample_NN1 pan_NN1 ,_, which_DDQ is_VBZ subjected_VVN to_II the_AT same_DA linear_JJ heating_NN1 programme_NN1 ._. 
As_CSA the_AT temperature_NN1 of_IO the_AT block_NN1 is_VBZ raised_VVN at_II a_AT1 constant_JJ rate_NN1 the_AT sample_NN1 temperature_NN1 T_ZZ1 s_ZZ1 and_CC that_DD1 of_IO the_AT reference_NN1 ,_, T_ZZ1 r_ZZ1 will_VM keep_VVI pace_NN1 until_CS a_AT1 change_NN1 in_II the_AT sample_NN1 takes_VVZ place_NN1 ._. 
If_CS the_AT change_NN1 is_VBZ exothermic_JJ T_ZZ1 s_ZZ1 will_VM exceed_VVI T_ZZ1 r_ZZ1 for_IF a_AT1 short_JJ period_NN1 ,_, but_CCB if_CS it_PPH1 is_VBZ endothermic_JJ T_ZZ1 s_ZZ1 will_VM temporarily_RR lag_VVI behind_II T_ZZ1 r_ZZ1 ._. 
This_DD1 temperature_NN1 difference_NN1 T_ZZ1 is_VBZ recorded_VVN and_CC transmitted_VVN to_II a_AT1 chart_NN1 recorder_NN1 where_RRQ changes_NN2 such_II21 as_II22 melting_NN1 or_CC crystallization_NN1 are_VBR recorded_VVN as_CSA peaks_NN2 ._. 
A_AT1 third_MD type_NN1 of_IO change_NN1 can_VM be_VBI detected_VVN ._. 
Since_CS the_AT heat_NN1 capacities_NN2 of_IO sample_NN1 and_CC reference_NN1 are_VBR different_JJ T_ZZ1 is_VBZ never_RR actually_RR zero_MC ,_, and_CC a_AT1 change_NN1 in_II heat_NN1 capacity_NN1 ,_, such_II21 as_II22 that_DD1 associated_VVN with_IW a_AT1 glass_NN1 transition_NN1 ,_, will_VM cause_VVI a_AT1 shift_NN1 in_II the_AT base_NN1 line_NN1 ._. 
All_DB three_MC possibilities_NN2 are_VBR shown_VVN in_II figure_NN1 10.7_MC for_IF quenched_JJ terylene_NN1 ._. 
Other_JJ changes_NN2 such_II21 as_II22 sample_NN1 decomposition_NN1 ,_, crosslinking_VVG ,_, and_CC the_AT existence_NN1 of_IO polymorphic_JJ forms_NN2 can_VM also_RR be_VBI detected_VVN ._. 
As_CSA T_ZZ1 measured_VVN in_II DTA_NP1 is_VBZ a_AT1 function_NN1 of_IO the_AT thermal_JJ conductivity_NN1 and_CC bulk_VV0 density_NN1 of_IO the_AT sample_NN1 ,_, it_PPH1 is_VBZ non-quantitative_JJ and_CC relatively_RR uninformative_JJ ._. 
To_TO overcome_VVI these_DD2 drawbacks_NN2 an_AT1 alternative_JJ procedure_NN1 known_VVN as_II differential_JJ scanning_NN1 calorimetry_NN1 (_( DSC_NN1 )_) is_VBZ used_VVN ._. 
This_DD1 technique_NN1 retains_VVZ the_AT constant_JJ mean_JJ heat_NN1 input_NN1 but_CCB instead_II21 of_II22 measuring_VVG the_AT temperature_NN1 difference_NN1 during_II a_AT1 change_NN1 a_AT1 servo-system_NN1 immediately_RR increases_VVZ the_AT energy_NN1 input_NN1 to_TO either_RR sample_VVI or_CC reference_NN1 to_TO maintain_VVI both_RR at_II the_AT same_DA temperature_NN1 ._. 
The_AT thermograms_NN2 obtained_VVN are_VBR similar_JJ to_II DTA_NP1 ,_, but_CCB actually_RR represent_VV0 the_AT amount_NN1 of_IO electrical_JJ energy_NN1 supplied_VVN to_II the_AT system_NN1 ,_, not_XX T_NP1 ,_, and_CC so_RR the_AT areas_NN2 under_II the_AT peaks_NN2 will_VM be_VBI proportional_JJ to_II the_AT change_NN1 in_II enthalpy_NN1 which_DDQ occurred_VVD ._. 
An_AT1 actual_JJ reference_NN1 sample_NN1 can_VM be_VBI dispensed_VVN with_IW in_II practice_NN1 and_CC an_AT1 empty_JJ sample_NN1 pan_NN1 used_VVD instead_RR ._. 
Calibration_NN1 of_IO the_AT instrument_NN1 will_VM allow_VVI the_AT heat_NN1 capacity_NN1 of_IO a_AT1 sample_NN1 to_TO be_VBI calculated_VVN in_II a_AT1 quantitative_JJ manner_NN1 ._. 
This_DD1 information_NN1 is_VBZ additional_JJ to_II that_DD1 gained_VVD on_II crystallization_NN1 ,_, melting_VVG ,_, glass_NN1 transitions_NN2 ,_, and_CC decompositions_NN2 ._. 
CHAPTER_NN1 11_MC The_AT Crystalline_JJ State_NN1 11.1_MC Introduction_NN1 When_RRQ polymers_NN2 are_VBR irradiated_VVN by_II a_AT1 beam_NN1 of_IO X-rays_NN2 ,_, scattering_VVG produces_VVZ diffuse_JJ haloes_NN2 on_II the_AT photographic_JJ plate_NN1 for_IF some_DD polymers_NN2 ,_, while_CS for_IF others_NN2 a_AT1 series_NN of_IO sharply_RR defined_VVN rings_NN2 superimposed_VVN on_II a_AT1 diffuse_JJ background_NN1 is_VBZ recorded_VVN ._. 
The_AT former_DA are_VBR characteristic_JJ of_IO amorphous_JJ polymers_NN2 ,_, and_CC illustrate_VV0 that_CST a_AT1 limited_JJ amount_NN1 of_IO short_JJ range_NN1 order_NN1 exists_VVZ in_II most_DAT polymeric_JJ solids_NN2 ._. 
The_AT latter_DA patterns_NN2 are_VBR indicative_JJ of_IO considerable_JJ three-dimensional_JJ order_NN1 and_CC are_VBR typical_JJ of_IO polycrystalline_JJ samples_NN2 containing_VVG a_AT1 large_JJ number_NN1 of_IO unoriented_JJ crystallites_NN2 associated_VVN with_IW amorphous_JJ regions_NN2 ._. 
The_AT rings_NN2 are_VBR observed_VVN to_TO sharpen_VVI into_II arcs_NN2 ,_, or_CC discrete_JJ spots_NN2 ,_, if_CS the_AT polymer_NN1 is_VBZ drawn_VVN or_CC stretched_VVD ,_, a_AT1 process_NN1 which_DDQ orients_VVZ the_AT axes_NN2 of_IO the_AT crystallites_NN2 in_II one_MC1 direction_NN1 ._. 
The_AT occurrence_NN1 of_IO significant_JJ crystallinity_NN1 in_II a_AT1 polymer_NN1 sample_NN1 is_VBZ of_IO considerable_JJ consequence_NN1 to_II a_AT1 materials_NN2 scientist_NN1 ._. 
The_AT properties_NN2 of_IO the_AT sample_NN1 the_AT density_NN1 ,_, optical_JJ clarity_NN1 ,_, modulus_NN1 ,_, and_CC general_JJ mechanical_JJ response_NN1 all_DB change_VV0 dramatically_RR when_CS crystallites_NN2 are_VBR present_JJ and_CC the_AT polymer_NN1 is_VBZ no_RR21 longer_RR22 subject_II21 to_II22 the_AT rules_NN2 of_IO linear_JJ visco-elasticity_NN1 ,_, which_DDQ apply_VV0 to_II amorphous_JJ polymers_NN2 as_CSA outlined_VVN in_II Chapter_NN1 13_MC ._. 
However_RR ,_, a_AT1 polymer_NN1 sample_NN1 is_VBZ rarely_RR completely_RR crystalline_JJ and_CC the_AT properties_NN2 also_RR depend_VV0 on_II the_AT amount_NN1 of_IO crystalline_JJ order_NN1 ._. 
It_PPH1 is_VBZ important_JJ then_RT to_TO examine_VVI crystallinity_NN1 in_II polymers_NN2 and_CC determine_VVI the_AT factors_NN2 which_DDQ control_VV0 the_AT extent_NN1 of_IO crystallinity._NNU 11.2_MC Mechanism_NN1 of_IO crystallization_NN1 A_ZZ1 polymer_NN1 in_II very_RG dilute_JJ solution_NN1 can_VM be_VBI effectively_RR regarded_VVN as_II an_AT1 isolated_JJ chain_NN1 whose_DDQGE shape_NN1 is_VBZ governed_VVN by_II short_JJ and_CC long_JJ range_NN1 inter-_NN1 and_CC intra-molecular_JJ interactions_NN2 ._. 
In_II the_AT aggregated_JJ state_NN1 this_DD1 is_VBZ no_RR21 longer_RR22 true_JJ ,_, the_AT behaviour_NN1 of_IO the_AT chain_NN1 is_VBZ now_RT influenced_VVN largely_RR by_II the_AT proximity_NN1 of_IO the_AT neighbouring_JJ chains_NN2 and_CC the_AT secondary_JJ valence_NN1 forces_NN2 which_DDQ act_VV0 between_II them_PPHO2 ._. 
These_DD2 factors_NN2 determine_VV0 the_AT orientation_NN1 of_IO chains_NN2 relative_II21 to_II22 each_PPX221 other_PPX222 in_II the_AT undiluted_JJ state_NN1 ,_, and_CC this_DD1 is_VBZ essentially_RR an_AT1 interplay_NN1 between_II the_AT entropy_NN1 and_CC internal_JJ energy_NN1 of_IO the_AT system_NN1 which_DDQ is_VBZ expressed_VVN in_II the_AT usual_JJ thermodynamic_JJ form_NN1 In_II the_AT melt_NN1 ,_, polymers_NN2 normally_RR attain_VV0 a_AT1 state_NN1 of_IO maximum_JJ entropy_NN1 consistent_JJ with_IW a_AT1 stable_JJ state_NN1 of_IO minimum_JJ free_JJ energy_NN1 ._. 
Crystallization_NN1 is_VBZ a_AT1 process_NN1 involving_VVG the_AT orderly_JJ arrangement_NN1 of_IO chains_NN2 and_CC is_VBZ consequently_RR associated_VVN with_IW a_AT1 large_JJ negative_JJ entropy_NN1 of_IO activation_NN1 ._. 
If_CS a_AT1 favourable_JJ free_JJ energy_NN1 change_NN1 is_VBZ to_TO be_VBI obtained_VVN for_IF crystallite_NN1 formation_NN1 ,_, the_AT entropy_NN1 term_NN1 has_VHZ to_TO be_VBI offset_VVN by_II a_AT1 large_JJ negative_JJ energy_NN1 contribution_NN1 ._. 
The_AT alignment_NN1 of_IO polymer_NN1 chains_NN2 at_II specific_JJ distances_NN2 from_II one_PPX121 another_PPX122 to_TO form_VVI crystalline_JJ nuclei_NN2 will_VM be_VBI assisted_VVN when_CS intermolecular_JJ forces_NN2 are_VBR strong_JJ ._. 
The_AT greater_JJR this_DD1 interaction_NN1 between_II chains_NN2 the_AT more_RGR favourable_JJ will_VM be_VBI the_AT energy_NN1 parameter_NN1 and_CC this_DD1 provides_VVZ some_DD indication_NN1 of_IO the_AT type_NN1 of_IO chain_NN1 which_DDQ might_VM be_VBI expected_VVN to_TO crystallize_VVI from_II the_AT melt_NN1 ,_, viz._REX (_( 1_MC1 )_) Symmetrical_JJ chains_NN2 which_DDQ allow_VV0 the_AT regular_JJ close_JJ packing_NN1 required_VVN for_IF crystallite_NN1 formation._NNU (_( 2_MC )_) Chains_NN2 possessing_VVG groups_NN2 which_DDQ encourage_VV0 strong_JJ intermolecular_JJ attraction_NN1 thereby_RR stabilizing_VVG the_AT alignment_NN1 ._. 
In_II31 addition_II32 to_II33 the_AT thermodynamic_JJ requirements_NN2 ,_, kinetic_JJ factors_NN2 relating_VVG to_II the_AT flexibility_NN1 and_CC mobility_NN1 of_IO a_AT1 chain_NN1 in_II the_AT melt_NN1 must_VM also_RR be_VBI considered_VVN ._. 
Thus_RR polyisobutylene_NN1 might_VM be_VBI expected_VVN to_TO crystallize_VVI because_CS the_AT chain_NN1 is_VBZ symmetrical_JJ ,_, but_CCB it_PPH1 will_VM only_RR do_VDI so_RR if_CS maintained_VVN at_II an_AT1 optimum_JJ temperature_NN1 for_IF several_DA2 months_NNT2 ._. 
This_DD1 is_VBZ presumably_RR a_AT1 result_NN1 of_IO the_AT flexibility_NN1 of_IO the_AT chain_NN1 which_DDQ allows_VVZ extensive_JJ convolution_NN1 thereby_RR impeding_VVG stabilization_NN1 of_IO the_AT required_JJ long_JJ range_NN1 alignment_NN1 ._. 
The_AT creation_NN1 of_IO a_AT1 three-dimensional_JJ ordered_JJ phase_NN1 from_II a_AT1 disordered_JJ state_NN1 is_VBZ a_AT1 two_MC stage_NN1 process_NN1 ._. 
Just_RR above_II its_APPGE melting_NN1 temperature_NN1 a_AT1 polymer_NN1 behaves_VVZ like_II a_AT1 highly_RR viscous_JJ liquid_NN1 in_II which_DDQ the_AT chains_NN2 are_VBR all_DB tangled_VVD up_RP with_IW their_APPGE neighbours_NN2 ._. 
Each_DD1 chain_NN1 pervades_VVZ a_AT1 given_JJ volume_NN1 in_II the_AT sample_NN1 ,_, but_CCB as_CSA the_AT temperature_NN1 decreases_VVZ the_AT volume_NN1 available_JJ to_II the_AT molecule_NN1 also_RR decreases_VVZ ._. 
This_DD1 in_II turn_NN1 restricts_VVZ the_AT number_NN1 of_IO disordered_JJ conformational_JJ states_NN2 available_JJ to_II the_AT chain_NN1 due_II21 to_II22 the_AT constraining_JJ influence_NN1 of_IO intramolecular_JJ interactions_NN2 among_II chains_NN2 in_II juxtaposition_NN1 ._. 
As_II a_AT1 result_NN1 there_EX is_VBZ an_AT1 increasing_JJ tendency_NN1 for_IF the_AT polymer_NN1 to_TO assume_VVI an_AT1 ordered_JJ conformation_NN1 in_II which_DDQ the_AT chain_NN1 bonds_NN2 are_VBR in_II the_AT rotational_JJ states_NN2 of_IO lowest_JJT energy_NN1 ._. 
However_RR ,_, various_JJ other_JJ factors_NN2 will_VM tend_VVI to_TO oppose_VVI crystallization_NN1 ;_; chain_NN1 entanglements_NN2 will_VM hinder_VVI the_AT diffusion_NN1 of_IO chains_NN2 into_II suitable_JJ orientations_NN2 and_CC if_CS the_AT temperature_NN1 is_VBZ above_II the_AT melting_NN1 temperature_NN1 ,_, thermal_JJ motions_NN2 will_VM be_VBI sufficient_JJ to_TO disrupt_VVI the_AT potential_JJ nuclei_NN2 before_II significant_JJ growth_NN1 can_VM take_VVI place_NN1 ._. 
This_DD1 restricts_VVZ crystallization_NN1 to_II a_AT1 range_NN1 of_IO temperatures_NN2 between_II T_ZZ1 g_ZZ1 and_CC T_ZZ1 m_ZZ1 ._. 
The_AT first_MD step_NN1 in_II crystallite_NN1 formation_NN1 is_VBZ the_AT creation_NN1 of_IO a_AT1 stable_JJ nucleus_NN1 brought_VVN about_RP by_II the_AT ordering_NN1 of_IO chains_NN2 in_II a_AT1 parallel_JJ array_NN1 ,_, stimulated_VVN by_II intramolecular_JJ forces_NN2 ,_, followed_VVN by_II the_AT stabilization_NN1 of_IO long_JJ range_NN1 order_NN1 by_II the_AT secondary_JJ valence_NN1 forces_NN2 which_DDQ aid_VV0 the_AT packing_NN1 of_IO molecules_NN2 into_II a_AT1 three-dimensional_JJ ordered_JJ structure_NN1 ._. 
The_AT second_MD stage_NN1 is_VBZ the_AT growth_NN1 of_IO the_AT crystalline_JJ region_NN1 ,_, the_AT size_NN1 of_IO which_DDQ is_VBZ governed_VVN by_II the_AT rate_NN1 of_IO addition_NN1 of_IO other_JJ chains_NN2 to_II the_AT nucleus_NN1 ._. 
As_CSA this_DD1 growth_NN1 is_VBZ counteracted_VVN by_II thermal_JJ redispersion_NN1 of_IO the_AT chains_NN2 at_II the_AT crystal-melt_JJ interface_NN1 ,_, the_AT temperature_NN1 must_VM be_VBI low_JJ enough_RR to_TO ensure_VVI that_CST this_DD1 disordering_JJ process_NN1 is_VBZ minimal._NNU 11.3_MC Temperature_NN1 and_CC growth_NN1 rate_NN1 Measurable_JJ rates_NN2 of_IO crystallization_NN1 occur_VV0 between_II and_CC ,_, a_AT1 range_NN1 in_II which_DDQ the_AT thermal_JJ motion_NN1 of_IO the_AT polymer_NN1 chains_NN2 is_VBZ conducive_JJ to_II the_AT formation_NN1 of_IO stable_JJ ordered_JJ regions_NN2 ._. 
The_AT growth_NN1 rate_NN1 of_IO crystalline_JJ areas_NN2 passes_VVZ through_II a_AT1 maximum_NN1 in_II this_DD1 range_NN1 as_CSA illustrated_VVN in_II figure_NN1 11.1_MC for_IF isotactic_JJ polystyrene_NN1 ._. 
Close_RR to_II T_ZZ1 m_ZZ1 the_AT segmental_JJ motion_NN1 is_VBZ too_RG great_JJ to_TO allow_VVI many_DA2 stable_JJ nuclei_NN2 to_TO form_VVI ,_, while_CS near_II T_ZZ1 g_ZZ1 the_AT melt_NN1 is_VBZ so_RG viscous_JJ that_CST molecular_JJ motion_NN1 is_VBZ extremely_RR slow_JJ ._. 
As_II the_AT temperature_NN1 drops_VVZ from_II T_ZZ1 m_ZZ1 ,_, the_AT melt_NN1 viscosity_NN1 ,_, which_DDQ is_VBZ a_AT1 function_NN1 of_IO the_AT molar_JJ mass_NN1 ,_, increases_NN2 and_CC the_AT diffusion_NN1 rate_NN1 decreases_VVZ ,_, thereby_RR giving_VVG the_AT chains_NN2 greater_JJR opportunity_NN1 to_TO rearrange_VVI themselves_PPX2 to_TO form_VVI a_AT1 nucleus_NN1 ._. 
This_DD1 means_VVZ that_CST there_EX will_VM exist_VVI an_AT1 optimum_JJ temperature_NN1 of_IO crystallization_NN1 ,_, which_DDQ depends_VVZ largely_RR on_II the_AT interval_NN1 T_ZZ1 m_ZZ1 to_II T_ZZ1 g_ZZ1 ,_, but_CCB also_RR on_II the_AT molar_JJ mass_NN1 of_IO the_AT sample_NN1 ._. 
The_AT melt_VV0 usually_RR has_VHZ to_TO be_VBI supercooled_VVN by_RP about_II 5_MC to_II 20_MC K_ZZ1 before_II a_AT1 significant_JJ number_NN1 of_IO nuclei_NN2 appear_VV0 which_DDQ possess_VV0 the_AT critical_JJ dimensions_NN2 required_VVN for_IF stability_NN1 and_CC further_JJR growth_NN1 ._. 
If_CS a_AT1 nucleating_JJ agent_NN1 is_VBZ added_VVN to_II the_AT system_NN1 ,_, crystallization_NN1 can_VM be_VBI induced_VVN at_II higher_JJR temperatures_NN2 ._. 
This_DD1 is_VBZ known_VVN as_II heterogeneous_JJ nucleation_NN1 and_CC only_RR affects_VVZ the_AT crystallization_NN1 rate_NN1 ,_, not_XX the_AT spherulitic_JJ growth_NN1 rate_NN1 ,_, at_II a_AT1 given_JJ temperature._NNU 11.4_MC Melting_VVG The_AT melting_NN1 of_IO a_AT1 perfectly_RR crystalline_JJ substance_NN1 is_VBZ an_AT1 equilibrium_NN1 process_NN1 characterized_VVN by_II a_AT1 marked_JJ volume_NN1 change_NN1 and_CC a_AT1 well-defined_JJ melting_NN1 temperature_NN1 ._. 
Polymers_NN2 are_VBR never_RR perfectly_RR crystalline_JJ ,_, but_CCB contain_VV0 disordered_JJ regions_NN2 and_CC crystallites_NN2 of_IO varying_JJ size_NN1 ._. 
The_AT process_NN1 is_VBZ normally_RR incomplete_JJ because_CS crystallization_NN1 takes_VVZ place_NN1 when_CS the_AT polymer_NN1 is_VBZ a_AT1 viscous_JJ liquid_NN1 ._. 
In_II this_DD1 state_NN1 ,_, the_AT chains_NN2 are_VBR highly_RR entangled_VVN ,_, and_CC as_RG sufficient_JJ time_NNT1 must_VM be_VBI allowed_VVN for_IF the_AT chains_NN2 to_TO diffuse_VVI into_II the_AT three-dimensional_JJ order_NN1 required_VVN for_IF crystallite_NN1 formation_NN1 ,_, the_AT crystalline_JJ perfection_NN1 of_IO the_AT sample_NN1 is_VBZ affected_VVN by_II the_AT thermal_JJ history_NN1 ._. 
Thus_RR ,_, rapid_JJ cooling_NN1 from_II the_AT melt_VV0 usually_RR prevents_VVZ the_AT development_NN1 of_IO significant_JJ crystallinity_NN1 ._. 
The_AT result_NN1 is_VBZ that_DD1 melting_NN1 takes_VVZ place_NN1 over_II a_AT1 range_NN1 of_IO temperatures_NN2 ,_, and_CC this_DD1 range_NN1 is_VBZ a_AT1 useful_JJ indication_NN1 of_IO sample_NN1 crystallinity_NN1 ._. 
Effect_NN1 of_IO crystallite_NN1 size_NN1 on_II melting_NN1 ._. 
The_AT range_NN1 of_IO temperature_NN1 ,_, which_DDQ covers_VVZ the_AT melting_NN1 of_IO a_AT1 polymer_NN1 ,_, is_VBZ indicative_JJ of_IO the_AT size_NN1 and_CC perfection_NN1 of_IO the_AT crystallites_NN2 in_II the_AT sample_NN1 ._. 
This_DD1 is_VBZ illustrated_VVN in_II a_AT1 study_NN1 of_IO the_AT melting_NN1 of_IO natural_JJ rubber_JJ samples_NN2 ,_, which_DDQ has_VHZ shown_VVN that_CST the_AT melting_NN1 range_NN1 is_VBZ a_AT1 function_NN1 of_IO the_AT temperature_NN1 of_IO crystallization_NN1 ._. 
At_II low_JJ crystallization_NN1 temperatures_NN2 the_AT nucleation_NN1 density_NN1 in_II the_AT rubber_JJ melt_NN1 is_VBZ high_JJ ,_, segmental_JJ diffusion_NN1 rates_NN2 are_VBR low_JJ ,_, and_CC small_JJ imperfect_JJ crystalline_JJ regions_NN2 are_VBR formed_VVN ._. 
Thus_RR broad_JJ melting_NN1 ranges_NN2 are_VBR measured_VVN for_IF samples_NN2 crystallized_VVN at_II these_DD2 lower_JJR temperatures_NN2 ,_, and_CC these_DD2 become_VV0 narrower_JJR as_II the_AT crystallization_NN1 temperature_NN1 increases_NN2 ._. 
This_DD1 suggests_VVZ that_CST careful_JJ annealing_NN1 at_II the_AT appropriate_JJ temperature_NN1 could_VM produce_VVI samples_NN2 with_IW a_AT1 high_JJ degree_NN1 of_IO crystallinity_NN1 ._. 
These_DD2 samples_NN2 might_VM then_RT exhibit_VVI almost_RR perfect_JJ first_MD order_NN1 phase_NN1 changes_NN2 at_II the_AT melting_NN1 temperature_NN1 ._. 
A_AT1 close_JJ approximation_NN1 to_II these_DD2 conditions_NN2 has_VHZ been_VBN attained_VVN by_II Mandelkern_NP1 ,_, who_PNQS annealed_VVD a_AT1 linear_JJ polyethylene_NN1 for_IF 40_MC days_NNT2 ._. 
The_AT improvement_NN1 in_II the_AT crystalline_JJ organization_NN1 is_VBZ obvious_JJ from_II examination_NN1 of_IO the_AT resulting_JJ fusion_NN1 curves_NN2 in_II figure_NN1 11.2_MC ,_, where_CS the_AT variation_NN1 of_IO specific_JJ volume_NN1 with_IW temperature_NN1 for_IF this_DD1 sample_NN1 is_VBZ compared_VVN with_IW that_DD1 for_IF a_AT1 branched_JJ polyethylene_NN1 of_IO low_JJ crystallinity_NN1 ._. 
The_AT effect_NN1 of_IO branching_JJ is_VBZ to_TO decrease_VVI the_AT percentage_NN1 crystallinity_NN1 ,_, broaden_VV0 the_AT melting_NN1 range_NN1 ,_, and_CC reduce_VV0 the_AT average_JJ melting_NN1 temperature_NN1 ._. 
The_AT points_NN2 A_ZZ1 and_CC B_ZZ1 in_II the_AT diagram_NN1 represent_VV0 the_AT temperatures_NN2 at_II which_DDQ the_AT largest_JJT crystallites_NN2 disappear_VV0 and_CC are_VBR regarded_VVN as_II the_AT respective_JJ melting_NN1 temperatures_NN2 T_ZZ1 m_ZZ1 for_IF the_AT samples_NN2 ._. 
The_AT effect_NN1 of_IO crystal_NN1 size_NN1 on_II T_ZZ1 m_ZZ1 is_VBZ shown_VVN more_RGR clearly_RR in_II figure_NN1 11.3_MC ._. 
The_AT small_JJ crystals_NN2 melt_VV0 about_RG 30_MC K_ZZ1 lower_RRR than_CSN the_AT large_JJ ones_NN2 due_II21 to_II22 the_AT greater_JJR contribution_NN1 from_II the_AT interfacial_JJ free_JJ energy_NN1 in_II the_AT smaller_JJR crystallites_NN2 ,_, i.e._REX there_EX is_VBZ an_AT1 excess_NN1 of_IO free_JJ energy_NN1 associated_VVN with_IW the_AT disordered_JJ chains_NN2 emerging_VVG from_II the_AT ends_NN2 of_IO ordered_JJ crystallites_NN2 and_CC this_DD1 is_VBZ relatively_RR greater_JJR for_IF the_AT small_JJ crystallites_NN2 ,_, resulting_VVG in_II lower_JJR melting_NN1 temperatures._NNU 11.5_MC Thermodynamic_JJ parameters_NN2 Even_RR with_IW carefully_RR annealed_VVD specimens_NN2 ,_, it_PPH1 is_VBZ thought_VVN that_CST the_AT equilibrium_NN1 melting_NN1 temperature_NN1 of_IO the_AT completely_RR crystalline_JJ polymer_NN1 T_ZZ1 m_ZZ1 is_VBZ never_RR actually_RR attained_VVN ._. 
The_AT temperature_NN1 T_ZZ1 m_ZZ1 is_VBZ related_VVN to_II the_AT change_NN1 in_II enthalpy_NN1 H_ZZ1 u_ZZ1 and_CC the_AT entropy_NN1 change_NN1 S_ZZ1 u_ZZ1 ,_, for_IF the_AT first_MD order_NN1 melting_NN1 transition_NN1 of_IO pure_JJ crystalline_JJ polymer_NN1 to_II pure_JJ amorphous_JJ melt_NN1 ,_, by_II The_AT enthalpy_NN1 change_NN1 can_VM be_VBI estimated_VVN by_II adding_VVG varying_JJ quantities_NN2 of_IO a_AT1 diluent_JJ to_II the_AT polymer_NN1 ,_, which_DDQ serves_VVZ to_TO depress_VVI the_AT observed_JJ melting_NN1 temperature_NN1 ,_, and_CC measuring_VVG T_ZZ1 m_ZZ1 for_IF each_DD1 polymer_NN1 +_FO diluent_JJ mixture_NN1 ._. 
The_AT results_NN2 are_VBR then_RT plotted_VVN according_II21 to_II22 the_AT Flory_JJ equation_NN1 where_RRQ is_VBZ the_AT ratio_NN1 of_IO the_AT molar_JJ volume_NN1 of_IO the_AT repeating_JJ unit_NN1 in_II the_AT chain_NN1 to_II that_DD1 of_IO the_AT diluent_JJ ,_, and_CC 1_MC1 ,_, is_VBZ the_AT volume_NN1 fraction_NN1 of_IO the_AT diluent_JJ ._. 
The_AT factor_NN1 is_VBZ equivalent_JJ to_II the_AT Flory_JJ interaction_NN1 parameter_NN1 1_MC1 ,_, indicating_VVG that_DD1 equation_NN1 (_( 11.2_MC )_) is_VBZ dependent_JJ on_II the_AT polymer-diluent_JJ interaction_NN1 ._. 
For_IF practical_JJ purposes_NN2 T_ZZ1 m_ZZ1 is_VBZ taken_VVN to_TO be_VBI the_AT melting_NN1 temperature_NN1 of_IO the_AT undiluted_JJ polymer_NN1 irrespective_II21 of_II22 the_AT crystalline_JJ content_NN1 ._. 
Typical_JJ values_NN2 obtained_VVN in_II this_DD1 way_NN1 are_VBR shown_VVN in_II table_NN1 11.1_MC In_II many_DA2 cases_NN2 the_AT entropy_NN1 change_NN1 is_VBZ the_AT most_RGT important_JJ influence_NN1 on_II the_AT magnitude_NN1 of_IO the_AT melting_NN1 temperature_NN1 of_IO a_AT1 polymer_NN1 ._. 
A_AT1 large_JJ part_NN1 of_IO this_DD1 entropy_NN1 is_VBZ due_II21 to_II22 the_AT additional_JJ freedom_NN1 which_DDQ allows_VVZ the_AT chain_NN1 conformational_JJ changes_NN2 to_TO occur_VVI in_II the_AT melt_NN1 ,_, after_CS the_AT restrictions_NN2 of_IO the_AT crystalline_JJ lattice_NN1 ._. 
In_II the_AT crystalline_JJ phase_NN1 the_AT chain_NN1 bonds_NN2 are_VBR in_II their_APPGE lowest_JJT energy_NN1 state_NN1 ._. 
If_CS the_AT energy_NN1 difference_NN1 between_II the_AT rotational_JJ states_NN2 is_VBZ low_JJ ,_, the_AT population_NN1 of_IO the_AT higher_JJR energy_NN1 states_NN2 will_VM increase_VVI in_II the_AT melt_NN1 and_CC considerable_JJ flexing_NN1 of_IO the_AT chain_NN1 is_VBZ achieved_VVN ._. 
The_AT contribution_NN1 of_IO S_ZZ1 u_ZZ1 is_VBZ then_RT high_JJ ._. 
When_RRQ is_VBZ large_JJ ,_, the_AT tendency_NN1 to_TO populate_VVI the_AT high_JJ energy_NN1 states_NN2 is_VBZ not_XX too_RG great_JJ ,_, consequently_RR the_AT chain_NN1 is_VBZ less_RGR flexible_JJ and_CC S_ZZ1 u_ZZ1 is_VBZ lower_JJR ._. 
Two_MC polymers_NN2 which_DDQ exist_VV0 in_II the_AT all_DB trans_NN2 state_VV0 in_II the_AT crystal_NN1 are_VBR polyethylene_NN1 and_CC ._. 
For_IF polyethylene_NN1 is_VBZ about_RG 3.0_MC kJ_NNU mol_NN1 -_- ,_, but_CCB it_PPH1 is_VBZ as_RG high_JJ as_CSA 18.0_MC kJ_NNU mol_NN1 -_- for_IF ._. 
Hence_RR the_AT polyethylene_NN1 chain_NN1 is_VBZ much_RR more_RGR flexible_JJ in_II the_AT melt_NN1 and_CC gains_NN2 considerably_RR more_DAR entropy_NN1 on_II melting_NN1 ,_, so_CS21 that_CS22 T_ZZ1 m_ZZ1 is_VBZ correspondingly_RR lower._NNU 11.6_MC Crystalline_JJ arrangement_NN1 of_IO polymers_NN2 The_AT formation_NN1 of_IO stable_JJ crystalline_JJ regions_NN2 in_II a_AT1 polymer_NN1 requires_VVZ that_CST ,_, (_( i_ZZ1 )_) an_AT1 economical_JJ close_JJ packed_JJ arrangement_NN1 of_IO the_AT chains_NN2 can_VM be_VBI achieved_VVN in_II three_MC dimensions_NN2 ,_, and_CC that_DD1 (_( ii_MC )_) a_AT1 favourable_JJ change_NN1 in_II internal_JJ energy_NN1 is_VBZ obtained_VVN during_II this_DD1 process_NN1 ._. 
This_DD1 imposes_VVZ restrictions_NN2 on_II the_AT type_NN1 of_IO chain_NN1 which_DDQ can_VM be_VBI crystallized_VVN with_IW ease_NN1 and_CC ,_, as_CSA mentioned_VVN earlier_RRR ,_, one_PN1 would_VM expect_VVI symmetrical_JJ linear_JJ chains_NN2 such_II21 as_II22 polyesters_NN2 ,_, polyamides_NN2 ,_, and_CC polyethylene_NN1 to_TO crystallize_VVI most_RGT readily_RR ._. 
FACTORS_NN2 AFFECTING_VVG CRYSTALLINITY_NN1 AND_CC T_ZZ1 m_ZZ1 These_DD2 can_VM be_VBI dealt_VVN with_IW under_II the_AT general_JJ headings_NN2 ,_, symmetry_NN1 ,_, intermolecular_JJ bonding_NN1 ,_, tacticity_NN1 ,_, branching_JJ and_CC molar_JJ mass_NN1 ._. 
Symmetry_NN1 ._. 
The_AT symmetry_NN1 of_IO the_AT chain_NN1 shape_NN1 influences_NN2 both_RR T_ZZ1 m_ZZ1 and_CC the_AT ability_NN1 to_TO form_VVI crystallites_NN2 ._. 
Polyethylene_NN1 and_CC are_VBR both_RR sufficiently_RR symmetrical_JJ to_TO be_VBI considered_VVN as_CSA smooth_JJ stiff_JJ cylindrical_JJ rods_NN2 ._. 
In_II the_AT crystal_NN1 these_DD2 rods_NN2 tend_VV0 to_TO roll_VVI over_II each_PPX221 other_PPX222 and_CC change_VV0 position_NN1 when_CS thermally_RR agitated_VVN ._. 
This_DD1 motion_NN1 within_II the_AT crystal_NN1 lattice_NN1 ,_, called_VVN premelting_VVG ,_, increases_VVZ the_AT entropy_NN1 of_IO the_AT crystal_NN1 and_CC effectively_RR stabilizes_VVZ it_PPH1 ._. 
Consequently_RR ,_, more_RGR thermal_JJ energy_NN1 is_VBZ required_VVN before_II the_AT crystal_NN1 becomes_VVZ unstable_JJ ,_, and_CC T_ZZ1 m_ZZ1 is_VBZ raised_VVN ._. 
Flat_NN1 or_CC irregularly_RR shaped_JJ polymers_NN2 ,_, with_IW bends_NN2 and_CC bumps_NN2 in_II the_AT chain_NN1 ,_, can_VM not_XX move_VVI in_II this_DD1 way_NN1 without_IW disrupting_VVG the_AT crystal_NN1 lattice_NN1 ,_, and_CC so_RR have_VH0 lower_JJR T_ZZ1 m_ZZ1 values_NN2 ._. 
This_DD1 is_VBZ only_RR one_MC1 aspect_NN1 ._. 
For_IF crystallite_NN1 formation_NN1 in_II a_AT1 polymer_NN1 ,_, easy_JJ close-packing_NN1 of_IO the_AT chains_NN2 in_II a_AT1 regular_JJ three-dimensional_JJ fashion_NN1 is_VBZ required_VVN ._. 
Again_RT linear_JJ symmetrical_JJ molecules_NN2 are_VBR best_JJT ._. 
Polyethylene_NN1 ,_, and_CC other_JJ chains_NN2 with_IW more_RGR complex_JJ backbones_NN2 containing_VVG ,_, ,_, and_CC groups_NN2 all_DB possess_VV0 a_AT1 suitable_JJ symmetry_NN1 for_IF crystallite_NN1 formation_NN1 and_CC usually_RR assume_VV0 extended_JJ zig-zag_JJ conformations_NN2 when_CS aligned_VVN in_II the_AT lattice_NN1 ._. 
Chains_NN2 containing_VVG irregular_JJ units_NN2 ,_, which_DDQ detract_VV0 from_II the_AT linear_JJ geometry_NN1 ,_, reduce_VV0 the_AT ability_NN1 of_IO a_AT1 polymer_NN1 to_TO crystallize_VVI ._. 
Thus_RR cis-double_JJ bonds_NN2 (_( I_ZZ1 )_) ,_, o-_JJ and_CC m-phenylene_JJ groups_NN2 (_( 11_MC )_) ,_, or_CC cis-oriented_JJ puckered_JJ rings_NN2 (_( III_MC )_) ,_, all_DB encourage_VV0 bending_VVG and_CC twisting_VVG in_II the_AT chains_NN2 and_CC make_VV0 regular_JJ close-packing_NN1 very_RG difficult_JJ ._. 
If_CS ,_, however_RR ,_, the_AT phenylene_NN1 rings_NN2 are_VBR para-oriented_JJ ,_, the_AT chains_NN2 retain_VV0 their_APPGE axial_JJ symmetry_NN1 and_CC can_VM crystallize_VVI more_RGR readily_RR ._. 
Similarly_RR ,_, incorporation_NN1 of_IO a_AT1 trans-double_JJ bond_NN1 maintains_VVZ the_AT chain_NN1 symmetry_NN1 ._. 
This_DD1 is_VBZ highlighted_VVN when_CS comparing_VVG the_AT amorphous_JJ elastomeric_JJ cis-polyisoprene_NN1 with_IW the_AT highly_RR crystalline_JJ trans-polyisoprene_NN1 which_DDQ has_VHZ no_AT virtue_NN1 as_II an_AT1 elastomer_NN1 ,_, or_CC cis-poly(1,3-butadiene)_FO ,_, with_IW ,_, ._. 
Intermolecular_JJ bonding_NN1 ._. 
In_II polyethylene_NN1 crystallites_NN2 ,_, the_AT close_JJ packing_NN1 achieved_VVN by_II the_AT chains_NN2 allows_VVZ the_AT van_NP1 der_NP1 Waals_NP1 forces_VVZ to_TO act_VVI co-operatively_RR and_CC provide_VVI additional_JJ stability_NN1 to_II the_AT crystallite_NN1 ._. 
Any_DD interaction_NN1 between_II chains_NN2 in_II the_AT crystal_NN1 lattice_NN1 will_VM help_VVI to_TO hold_VVI the_AT structure_NN1 together_RL more_DAR firmly_RR and_CC raise_VV0 the_AT melting_NN1 temperature_NN1 ._. 
Polymers_NN2 containing_VVG polar_JJ groups_NN2 ,_, e.g._REX Cl_FO ,_, CN_NP1 ,_, or_CC OH_UH ,_, can_VM be_VBI held_VVN rigid_JJ ,_, and_CC aligned_VVN ,_, in_II a_AT1 polymer_NN1 matrix_NN1 by_II the_AT strong_JJ dipole-dipole_JJ interactions_NN2 between_II the_AT substituents_NN2 ,_, but_CCB the_AT effect_NN1 is_VBZ most_RGT obvious_JJ in_II the_AT symmetrical_JJ polyamides_NN2 ._. 
These_DD2 polymers_NN2 can_VM form_VVI intermolecular_JJ hydrogen_NN1 bonds_NN2 which_DDQ greatly_RR enhance_VV0 crystallite_NN1 stability_NN1 ._. 
This_DD1 is_VBZ illustrated_VVN in_II figure_NN1 11.4_MC for_IF nylon-6,6_FO ,_, where_CS the_AT extended_JJ zig-zag_JJ conformation_NN1 is_VBZ ideally_RR suited_VVN to_TO allow_VVI regular_JJ intermolecular_JJ hydrogen_NN1 bonding_NN1 ._. 
The_AT increased_JJ stability_NN1 is_VBZ reflected_VVN in_II T_ZZ1 m_ZZ1 which_DDQ for_IF nylon-6,6_FO is_VBZ 540_MC K_ZZ1 compared_VVN with_IW 410_MC K_ZZ1 for_IF polyethylene_NN1 ._. 
The_AT structures_NN2 of_IO related_JJ polyamides_NN2 do_VD0 not_XX always_RR lead_VVI to_II this_DD1 neat_JJ arrangement_NN1 of_IO intermolecular_JJ bond_NN1 formation_NN1 ;_; for_REX21 example_REX22 the_AT geometry_NN1 of_IO an_AT1 extended_JJ nylon-7,7_FO chain_NN1 allows_VVZ the_AT formation_NN1 of_IO only_RR every_AT1 second_MD possible_JJ hydrogen_NN1 bond_NN1 when_CS the_AT chains_NN2 are_VBR aligned_VVN and_CC fully_RR extended_VVN ._. 
However_RR ,_, the_AT process_NN1 is_VBZ so_RG favourable_JJ energetically_RR ,_, that_DD1 sufficient_JJ deformation_NN1 of_IO the_AT chain_NN1 takes_VVZ place_NN1 to_TO enable_VVI formation_NN1 of_IO all_DB possible_JJ hydrogen_NN1 bonds_NN2 ._. 
The_AT added_JJ stability_NN1 that_CST this_DD1 imparts_VVZ to_II the_AT crystallite_NN1 far_RR outweighs_VVZ the_AT limited_JJ loss_NN1 of_IO energy_NN1 caused_VVN by_II chain_NN1 flexing_VVG ._. 
Secondary_JJ bonds_NN2 can_VM therefore_RR lead_VVI to_II a_AT1 stimulation_NN1 of_IO the_AT crystallization_NN1 process_NN1 in_II the_AT appropriate_JJ polymers_NN2 ._. 
Tacticity_NN1 ._. 
Chain_NN1 symmetry_NN1 and_CC flexibility_NN1 both_RR affect_VV0 the_AT crystallinity_NN1 of_IO a_AT1 polymer_NN1 sample_NN1 ._. 
If_CS a_AT1 chain_NN1 possesses_VVZ large_JJ pendant_NN1 groups_NN2 ,_, these_DD2 will_VM increase_VVI the_AT rigidity_NN1 but_CCB also_RR increase_VV0 the_AT difficulty_NN1 of_IO close_JJ packing_NN1 to_TO form_VVI a_AT1 crystalline_JJ array_NN1 ._. 
This_DD1 latter_DA problem_NN1 can_VM be_VBI overcome_VVN if_CS the_AT groups_NN2 are_VBR arranged_VVN in_II a_AT1 regular_JJ fashion_NN1 along_II the_AT chain_NN1 ._. 
Isotactic_JJ polymers_NN2 tend_VV0 to_TO form_VVI helices_NN2 to_TO accommodate_VVI the_AT substituents_NN2 in_II the_AT most_RGT stable_JJ steric_JJ positions_NN2 ;_; these_DD2 helices_NN2 are_VBR regular_JJ forms_NN2 capable_JJ of_IO regular_JJ alignment_NN1 ._. 
Thus_RR atactic_JJ polystyrene_NN1 is_VBZ amorphous_JJ but_CCB isotactic_JJ polystyrene_NN1 is_VBZ semi-crystalline_JJ ._. 
Syndiotactic_JJ polymers_NN2 are_VBR also_RR sufficiently_RR regular_JJ to_TO crystallize_VVI ,_, but_CCB not_XX necessarily_RR as_II a_AT1 helix_NN1 ,_, rather_RR in_II glide_NN1 planes_NN2 ._. 
Branching_JJ in_II the_AT side_NN1 group_NN1 tends_VVZ to_TO stiffen_VVI the_AT chain_NN1 and_CC raise_VVI T_ZZ1 m_ZZ1 as_CSA shown_VVN in_II the_AT series_NN poly(but-1-ene)_NN1 ,_, ;_; poly_NN1 (_( 3-methyl_JJ but-1-ene_NN1 )_) ,_, ;_; poly_NN1 (_( 3,3-dimethyl_FO but-1-ene_NN1 )_) ,_, ._. 
If_CS the_AT side_NN1 group_NN1 is_VBZ flexible_JJ and_CC non-polar_JJ ,_, T_ZZ1 m_ZZ1 is_VBZ lowered_VVN ._. 
Branching_JJ and_CC molar_JJ mass_NN1 ._. 
If_CS the_AT chain_NN1 is_VBZ substantially_RR branched_VVN ,_, the_AT packing_NN1 efficiency_NN1 deteriorates_VVZ and_CC the_AT crystalline_JJ content_NN1 is_VBZ lowered_VVN ._. 
Polyethylene_NN1 provides_VVZ a_AT1 good_JJ example_NN1 of_IO this_DD1 (_( figure_NN1 11.2_MC )_) where_RRQ extensive_JJ branching_JJ lowers_VVZ the_AT density_NN1 and_CC T_ZZ1 m_ZZ1 of_IO the_AT polymer_NN1 ._. 
Molar_JJ mass_NN1 can_VM also_RR alter_VVI T_ZZ1 m_ZZ1 ._. 
Chain_NN1 ends_NN2 are_VBR relatively_RR free_JJ to_TO move_VVI and_CC if_CS the_AT number_NN1 of_IO chain_NN1 ends_NN2 is_VBZ increased_VVN by_II reducing_VVG the_AT molar_JJ mass_NN1 ,_, then_RT T_ZZ1 m_ZZ1 is_VBZ lowered_VVN because_II21 of_II22 the_AT decrease_NN1 in_II energy_NN1 required_VVN to_TO stimulate_VVI chain_NN1 motion_NN1 and_CC melting_NN1 ._. 
For_REX21 example_REX22 ,_, polypropylene_NN1 ,_, with_IW M_ZZ1 =_FO 2000_MC g_NNU mol_NN1 -1_MC ,_, has_VHZ ,_, whereas_CS a_AT1 sample_NN1 with_IW M_ZZ1 =_FO 30_MC 000_MC g_NNU mol_NN1 -1_MC ,_, has._NNU 11.7_MC Morphology_NP1 and_CC kinetics_NN1 Having_VHG once_RR established_VVN that_CST certain_JJ polymeric_JJ materials_NN2 are_VBR capable_JJ of_IO crystallizing_VVG ,_, fundamental_JJ studies_NN2 are_VBR directed_VVN along_II two_MC main_JJ channels_NN2 of_IO interest_NN1 centred_VVN on_II (_( a_ZZ1 )_) the_AT mode_NN1 and_CC kinetics_NN1 of_IO crystallization_NN1 ,_, and_CC (_( b_ZZ1 )_) the_AT morphology_NN1 of_IO the_AT sample_NN1 on_II completion_NN1 of_IO the_AT process_NN1 ._. 
Although_CS the_AT morphology_NN1 depends_VVZ largely_RR on_II the_AT crystallizing_JJ conditions_NN2 ,_, we_PPIS2 shall_VM consider_VVI the_AT macro-_NN1 and_CC microscopic_JJ structure_NN1 first_MD before_II dealing_VVG with_IW the_AT kinetics_NN1 of_IO formation._NNU 11.8_MC Morphology_NP1 A_ZZ1 number_NN1 of_IO distinct_JJ morphological_JJ units_NN2 have_VH0 been_VBN identified_VVN during_II the_AT crystallization_NN1 of_IO polymers_NN2 from_II the_AT melt_NN1 ,_, which_DDQ have_VH0 helped_VVN to_TO clarify_VVI the_AT mechanism_NN1 ._. 
We_PPIS2 shall_VM now_RT discuss_VVI the_AT ordered_JJ forms_NN2 which_DDQ have_VH0 been_VBN identified_VVN ._. 
Crystallites_NN2 ._. 
In_II an_AT1 X-ray_NN1 pattern_NN1 produced_VVN by_II a_AT1 semicrystalline_NN1 polymer_NN1 ,_, the_AT discrete_JJ maxima_NN2 observed_VVN arise_VV0 from_II the_AT scattering_NN1 by_II small_JJ regions_NN2 of_IO three-dimensional_JJ order_NN1 ,_, which_DDQ are_VBR called_VVN crystallites_NN2 ._. 
They_PPHS2 are_VBR formed_VVN in_II the_AT melt_NN1 by_II diffusion_NN1 of_IO molecules_NN2 ,_, or_CC sections_NN2 of_IO molecules_NN2 ,_, into_II close_JJ packed_JJ ordered_JJ arrays_NN2 ;_; these_DD2 then_RT crystallize_VV0 ._. 
The_AT sizes_NN2 of_IO these_DD2 crystallites_NN2 are_VBR small_JJ relative_II21 to_II22 the_AT length_NN1 of_IO a_AT1 fully_RR extended_VVN polymer_NN1 chain_NN1 ,_, but_CCB they_PPHS2 are_VBR also_RR found_VVN to_TO be_VBI independent_JJ of_IO the_AT molar_JJ mass_NN1 and_CC rarely_RR exceed_VV0 1_MC1 to_II 100_MC nm_FU ._. 
As_II a_AT1 result_NN1 ,_, various_JJ portions_NN2 of_IO one_MC1 chain_NN1 may_VM become_VVI incorporated_VVN in_II more_DAR than_CSN one_MC1 crystallite_NN1 during_II growth_NN1 ,_, thereby_RR imposing_VVG a_AT1 strain_NN1 on_II the_AT polymer_NN1 which_DDQ retards_VVZ the_AT process_NN1 of_IO crystallite_NN1 formation_NN1 ._. 
This_DD1 will_VM also_RR introduce_VVI imperfections_NN2 in_II the_AT crystallites_NN2 which_DDQ continue_VV0 growing_JJ until_CS the_AT strains_NN2 imposed_VVN by_II the_AT surrounding_JJ crystallites_NN2 eventually_RR stop_VV0 further_JJR enlargement_NN1 ._. 
Thus_RR a_AT1 matrix_NN1 of_IO ordered_JJ regions_NN2 with_IW disordered_JJ interfacial_JJ areas_NN2 is_VBZ formed_VVN ,_, but_CCB ,_, unlike_JJ materials_NN2 with_IW small_JJ molar_JJ masses_NN2 ,_, the_AT ordered_JJ and_CC disordered_JJ regions_NN2 are_VBR not_XX discrete_JJ entities_NN2 and_CC can_VM not_XX be_VBI separated_VVN by_II differential_JJ solution_NN1 techniques_NN2 unless_CS the_AT solvent_NN1 causes_VVZ selective_JJ degradation_NN1 of_IO the_AT primary_JJ bonds_NN2 in_II the_AT amorphous_JJ regions_NN2 ._. 
Crystallites_NN2 of_IO cellulose_NN1 have_VH0 been_VBN isolated_VVN from_II wood_NN1 pulp_NN1 in_II this_DD1 way_NN1 by_II treatment_NN1 with_IW acid_NN1 to_TO hydrolyse_VVI and_CC remove_VVI the_AT amorphous_JJ regions_NN2 ._. 
Typical_JJ dimensions_NN2 of_IO the_AT remaining_JJ crystallites_NN2 were_VBDR 46_MC nm_FU long_RR by_II 7.3_MC nm_FU wide_JJ corresponding_JJ to_II bundles_NN2 of_IO about_RG 100_MC to_II 150_MC chains_NN2 in_II each_DD1 crystallite_NN1 ._. 
The_AT first_MD attempts_NN2 to_TO explain_VVI the_AT crystalline_JJ structure_NN1 of_IO a_AT1 polymer_NN1 sample_NN1 produced_VVD a_AT1 model_NN1 called_VVN the_AT fringe-micelle_JJ structure_NN1 ._. 
The_AT chain_NN1 was_VBDZ envisaged_VVN as_CSA meandering_VVG throughout_II the_AT system_NN1 ,_, entering_VVG and_CC leaving_VVG several_DA2 ordered_JJ regions_NN2 along_II its_APPGE length_NN1 ._. 
The_AT whole_JJ structure_NN1 was_VBDZ thus_RR made_VVN up_RP of_IO crystalline_JJ regions_NN2 imbedded_VVD randomly_RR in_II a_AT1 continuous_JJ amorphous_JJ matrix_NN1 ._. 
This_DD1 model_NN1 has_VHZ now_RT been_VBN virtually_RR discarded_VVN in_II41 the_II42 light_II43 of_II44 more_RGR recent_JJ research_NN1 which_DDQ has_VHZ revealed_VVN features_NN2 incompatible_JJ with_IW this_DD1 picture_NN1 ._. 
Single_JJ crystals_NN2 ._. 
When_CS a_AT1 polymer_NN1 is_VBZ crystallized_VVN from_II the_AT melt_NN1 ,_, imperfect_JJ polycrystalline_JJ aggregations_NN2 are_VBR formed_VVN in_II31 association_II32 with_II33 a_AT1 substantial_JJ amorphous_JJ content_NN1 ._. 
This_DD1 is_VBZ a_AT1 consequence_NN1 of_IO chain_NN1 entanglement_NN1 and_CC the_AT high_JJ viscosity_NN1 of_IO the_AT melt_NN1 combining_VVG to_TO hinder_VVI the_AT diffusion_NN1 of_IO chains_NN2 into_II the_AT ordered_JJ arrays_NN2 necessary_JJ for_IF crystallite_NN1 formation_NN1 ._. 
If_CS these_DD2 restrictions_NN2 to_TO free_VVI movement_NN1 are_VBR reduced_VVN and_CC a_AT1 polymer_NN1 is_VBZ allowed_VVN to_TO crystallize_VVI from_II a_AT1 dilute_JJ solution_NN1 ,_, it_PPH1 is_VBZ possible_JJ to_TO obtain_VVI well-defined_JJ single_JJ crystals_NN2 ._. 
By_II working_VVG with_IW solutions_NN2 in_II which_DDQ the_AT amount_NN1 of_IO polymer_NN1 is_VBZ considerably_RR less_DAR than_CSN 0.1_MC per_NNU21 cent_NNU22 the_AT chance_NN1 of_IO a_AT1 chain_NN1 being_VBG incorporated_VVN in_II more_DAR than_CSN one_MC1 crystal_NN1 is_VBZ greatly_RR reduced_VVN ,_, thereby_RR increasing_VVG the_AT possibility_NN1 of_IO isolated_JJ single_JJ crystals_NN2 being_VBG formed_VVN ._. 
These_DD2 crystals_NN2 are_VBR usually_RR very_RG small_JJ ,_, but_CCB they_PPHS2 have_VH0 been_VBN detected_VVN for_IF a_AT1 range_NN1 of_IO polymers_NN2 including_II polyesters_NN2 ,_, polyamides_NN2 ,_, polyethylene_NN1 ,_, cellulose_VV0 acetate_NN1 ,_, and_CC poly_NN1 (_( 4-methyl_NN1 pentene-1_MC1 )_) ._. 
Although_CS small_JJ ,_, these_DD2 single_JJ crystals_NN2 can_VM be_VBI studied_VVN using_VVG an_AT1 electron_NN1 microscope_NN1 ._. 
This_DD1 reveals_VVZ that_CST they_PPHS2 are_VBR made_VVN up_RP of_IO thin_JJ lamellae_NN2 ,_, often_RR lozenge_NN1 shaped_VVD ,_, sometimes_RT oval_JJ ,_, about_RG 10_MC to_II 20_MC nm_FU thick_JJ ,_, depending_II21 on_II22 the_AT temperature_NN1 of_IO crystallization_NN1 ._. 
The_AT most_RGT surprising_JJ feature_NN1 of_IO these_DD2 lamellae_NN2 is_VBZ that_CST while_CS the_AT molecular_JJ chains_NN2 may_VM be_VBI as_CS31 long_CS32 as_CS33 1000_MC nm_FU ,_, the_AT direction_NN1 of_IO the_AT chain_NN1 axis_NN1 is_VBZ across_II the_AT thickness_NN1 of_IO the_AT platelet_NN1 ._. 
This_DD1 means_VVZ that_CST the_AT chain_NN1 must_VM be_VBI folded_VVN many_DA2 times_NNT2 like_II a_AT1 concertina_NN1 to_TO be_VBI accommodated_VVN in_II the_AT crystal_NN1 ._. 
For_IF a_AT1 polymer_NN1 such_II21 as_II22 polyethylene_NN1 ,_, the_AT fold_NN1 in_II the_AT chain_NN1 is_VBZ completed_VVN using_VVG only_RR 3_MC or_CC 4_MC monomer_NN1 units_NN2 with_IW bonds_NN2 in_II the_AT gauche_JJ conformation_NN1 ._. 
The_AT extended_JJ portions_NN2 in_RL21 between_RL22 have_VH0 about_RG 40_MC monomers_NN2 units_NN2 all_DB in_II the_AT trans_NN2 conformation_NN1 ._. 
The_AT crystals_NN2 ,_, thus_RR formed_VVN ,_, have_VH0 a_AT1 hollow_JJ pyramid_NN1 shape_NN1 ,_, because_II21 of_II22 the_AT requirement_NN1 that_CST the_AT chain_NN1 folding_NN1 must_VM involve_VVI a_AT1 staggering_JJ of_IO the_AT chains_NN2 if_CS the_AT most_RGT efficient_JJ packing_NN1 is_VBZ to_TO be_VBI achieved_VVN ._. 
There_EX is_VBZ also_RR a_AT1 remarkable_JJ constancy_NN1 of_IO lamellar_JJ thickness_NN1 ,_, but_CCB this_DD1 increases_VVZ as_II the_AT temperature_NN1 increases_NN2 ._. 
While_CS opinions_NN2 vary_VV0 between_II kinetic_JJ and_CC thermodynamic_JJ reasons_NN2 for_IF this_DD1 constancy_NN1 of_IO fold_NN1 distance_NN1 ,_, it_PPH1 is_VBZ suggested_VVN that_CST the_AT fold_NN1 structure_NN1 allows_VVZ the_AT maximum_JJ amount_NN1 of_IO crystallization_NN1 of_IO the_AT molecule_NN1 at_II a_AT1 length_NN1 which_DDQ produces_VVZ a_AT1 free_JJ energy_NN1 minimum_NN1 in_II the_AT crystal_NN1 ._. 
One_MC1 suggestion_NN1 is_VBZ that_CST the_AT folding_NN1 maintains_VVZ the_AT appropriate_JJ kinetic_JJ unit_NN1 of_IO the_AT chain_NN1 at_II any_DD given_JJ temperature_NN1 ;_; as_CSA this_DD1 would_VM be_VBI expected_VVN to_TO lengthen_VVI with_IW increasing_JJ temperature_NN1 ,_, it_PPH1 would_VM account_VVI for_IF the_AT observed_JJ thickening_NN1 of_IO the_AT lamellae_NN2 ._. 
Hedrites_NN2 ._. 
If_CS the_AT concentration_NN1 of_IO the_AT polymer_NN1 solution_NN1 is_VBZ increased_VVN a_AT1 crystalline_JJ polyhedral_JJ structure_NN1 emerges_VVZ composed_VVN of_IO lamellae_NN2 joined_VVN together_RL along_II a_AT1 common_JJ plane_NN1 ._. 
These_DD2 have_VH0 also_RR been_VBN detected_VVN growing_VVG from_II a_AT1 melt_NN1 which_DDQ suggests_VVZ that_CST lamellar_JJ growth_NN1 can_VM take_VVI place_NN1 in_II the_AT melt_NN1 and_CC may_VM be_VBI a_AT1 sub-unit_NN1 of_IO the_AT spherulite_NN1 ._. 
Crystallization_NN1 from_II the_AT melt_NN1 ._. 
Whereas_CS crystallization_NN1 from_II dilute_JJ solutions_NN2 may_VM result_VVI in_II the_AT formation_NN1 of_IO single_JJ polymer_NN1 crystals_NN2 ,_, this_DD1 perfection_NN1 is_VBZ not_XX achieved_VVN when_CS dealing_VVG with_IW polymers_NN2 cooled_VVN from_II the_AT melt_NN1 ._. 
The_AT basic_JJ characteristic_JJ feature_NN1 is_VBZ still_RR the_AT lamellar-like_JJ crystallite_NN1 with_IW amorphous_JJ surfaces_NN2 or_CC interfaces_NN2 ,_, but_CCB the_AT way_NN1 these_DD2 are_VBR formed_VVN may_VM be_VBI different_JJ based_VVN on_II the_AT careful_JJ investigation_NN1 of_IO melt_VV0 crystallized_JJ polymers_NN2 using_VVG neutron_NN1 scattering_VVG techniques_NN2 ._. 
The_AT two_MC models_NN2 that_CST have_VH0 been_VBN proposed_VVN to_TO describe_VVI the_AT fine_JJ structure_NN1 of_IO these_DD2 lamellae_NN2 and_CC their_APPGE surface_NN1 characteristics_NN2 in_II semicrystalline_NN1 polymers_NN2 ,_, differ_VV0 mainly_RR in_II the_AT way_NN1 the_AT chains_NN2 are_VBR thought_VVN to_TO enter_VVI and_CC leave_VVI the_AT ordered_JJ lamellae_NN2 regions_NN2 ._. 
These_DD2 are_VBR :_: (_( a_ZZ1 )_) the_AT regular_JJ folded_JJ array_NN1 with_IW adjacent_JJ re-entry_NN1 of_IO the_AT chains_NN2 ,_, but_CCB with_IW some_DD loose_JJ folding_NN1 and_CC emergent_JJ chain_NN1 ends_NN2 or_CC cilia_NN1 that_CST contribute_VV0 to_II the_AT disordered_JJ surface_NN1 ,_, or_CC (_( b_ZZ1 )_) the_AT switchboard_NN1 model_NN1 ,_, where_CS there_EX is_VBZ some_DD folding_NN1 of_IO the_AT chains_NN2 but_CCB re-entry_NN1 is_VBZ now_RT quite_RG random_JJ ._. 
Both_DB2 are_VBR represented_VVN schematically_RR in_II figure_NN1 11.5_MC but_CCB the_AT exact_JJ nature_NN1 of_IO the_AT structure_NN1 has_VHZ been_VBN the_AT subject_NN1 of_IO considerable_JJ controversy_NN1 ._. 
While_CS the_AT morphology_NN1 of_IO the_AT single_JJ crystals_NN2 grown_VVN from_II dilute_JJ solutions_NN2 may_VM be_VBI more_RGR regular_JJ and_CC resemble_VV0 the_AT first_MD model_NN1 ,_, for_IF polymers_NN2 that_CST are_VBR crystallized_VVN from_II the_AT melt_NN1 (_( and_CC this_DD1 is_VBZ by_RR21 far_RR22 the_AT more_RGR important_JJ procedure_NN1 technically_RR )_) the_AT mass_NN1 of_IO evidence_NN1 tends_VVZ to_TO favour_VVI a_AT1 form_NN1 of_IO the_AT switchboard_NN1 model_NN1 ._. 
Measurements_NN2 of_IO the_AT densities_NN2 of_IO several_DA2 semicrystalline_VV0 polymers_NN2 points_VVZ to_II the_AT fact_NN1 that_CST a_AT1 significant_JJ fraction_NN1 of_IO the_AT chain_NN1 units_NN2 are_VBR in_II a_AT1 non_FU crystalline_JJ environment_NN1 ._. 
This_DD1 is_VBZ not_XX consistent_JJ with_IW the_AT regular_JJ folded_JJ form_NN1 of_IO the_AT crystallites_NN2 where_RRQ the_AT amorphous_JJ part_NN1 is_VBZ associated_VVN only_RR with_IW loose_JJ folding_NN1 of_IO the_AT chains_NN2 and_CC cilia_NN1 ._. 
Even_RR more_RGR persuasive_JJ are_VBR small_JJ angle_NN1 neutron_NN1 scattering_VVG studies_NN2 ._. 
These_DD2 have_VH0 demonstrated_VVN that_CST the_AT radii_NN2 of_IO gyration_NN1 of_IO several_DA2 semicrystalline_VV0 polymers_NN2 remain_VV0 essentially_RR unchanged_JJ on_II moving_VVG from_II the_AT melt_NN1 phase_NN1 to_II the_AT semicrystalline_NN1 phase_NN1 (_( table_NN1 11.2_MC )_) ._. 
This_DD1 means_VVZ that_CST there_EX is_VBZ no_AT significant_JJ reordering_NN1 of_IO the_AT chain_NN1 conformation_NN1 when_CS crystallization_NN1 takes_VVZ place_NN1 after_II cooling_VVG from_II the_AT melt_NN1 ,_, which_DDQ would_VM be_VBI required_VVN if_CS a_AT1 regularly_RR folded_JJ chain_NN1 structure_NN1 was_VBDZ to_TO be_VBI constructed_VVN in_II the_AT lamellae_NN2 ._. 
To_TO explain_VVI these_DD2 observations_NN2 Fischer_NP1 has_VHZ proposed_VVN the_AT solidification_NN1 model_NN1 in_II which_DDQ crystallization_NN1 is_VBZ believed_VVN to_TO take_VVI place_NN1 by_II the_AT straightening_NN1 of_IO sections_NN2 of_IO the_AT polymer_NN1 coil_NN1 followed_VVN by_II alignment_NN1 of_IO these_DD2 sequences_NN2 in_II regular_JJ arrays_NN2 forming_VVG the_AT lamellar_JJ structure_NN1 ._. 
This_DD1 precludes_VVZ the_AT need_NN1 for_IF the_AT extensive_JJ ,_, long_JJ range_NN1 ,_, diffusion_NN1 of_IO the_AT chain_NN1 through_II a_AT1 highly_RR viscous_JJ medium_NN1 that_CST would_VM be_VBI necessary_JJ if_CS a_AT1 regular_JJ chain_NN1 folded_JJ structure_NN1 was_VBDZ to_TO be_VBI constructed_VVN ._. 
The_AT process_NN1 is_VBZ shown_VVN schematically_RR in_II figure_NN1 11.6_MC and_CC the_AT resulting_JJ structure_NN1 is_VBZ a_AT1 variation_NN1 of_IO the_AT switchboard_NN1 model_NN1 ._. 
This_DD1 hypothesis_NN1 can_VM account_VVI for_IF the_AT fact_NN1 that_CST on_II cooling_VVG ,_, rapid_JJ crystal_NN1 growth_NN1 is_VBZ seen_VVN to_TO occur_VVI which_DDQ is_VBZ inconsistent_JJ with_IW the_AT need_NN1 for_RR21 long_RR22 range_VVI diffusion_NN1 if_CS the_AT regularly_RR folding_JJ lamellae_NN2 were_VBDR forming_VVG ._. 
The_AT solidification_NN1 model_NN1 shows_VVZ that_CST the_AT chains_NN2 can_VM be_VBI incorporated_VVN into_II the_AT basic_JJ lamellar_JJ form_NN1 with_IW the_AT minimum_JJ amount_NN1 of_IO movement_NN1 and_CC that_CST there_EX will_VM be_VBI extensive_JJ meandering_NN1 of_IO chains_NN2 between_II the_AT lamellae_NN2 forming_VVG the_AT interfacial_JJ amorphous_JJ regions_NN2 ._. 
Spherulites_NN2 ._. 
Examination_NN1 of_IO thin_JJ sections_NN2 of_IO semicrystalline_NN1 polymers_NN2 reveals_VVZ that_CST the_AT crystallites_NN2 themselves_PPX2 are_VBR not_XX arranged_VVN randomly_RR ,_, but_CCB form_VV0 regular_JJ birefringent_JJ structures_NN2 with_IW circular_JJ symmetry_NN1 ._. 
These_DD2 structures_NN2 ,_, which_DDQ exhibit_VV0 a_AT1 characteristic_JJ Maltese_JJ cross_NN1 optical_JJ extinction_NN1 pattern_NN1 ,_, are_VBR called_VVN spherulites_NN2 ._. 
While_CS spherulites_NN2 are_VBR characteristic_JJ of_IO crystalline_JJ polymers_NN2 ,_, they_PPHS2 have_VH0 also_RR been_VBN observed_VVN to_TO form_VVI in_II low_JJ molar_JJ mass_JJ compounds_NN2 which_DDQ are_VBR crystallized_VVN from_II highly_RR viscous_JJ media_NN ._. 
Each_DD1 spherulite_NN1 grows_VVZ radially_RR from_II a_AT1 nucleus_NN1 formed_VVD either_RR by_II the_AT density_NN1 fluctuations_NN2 which_DDQ result_VV0 in_II the_AT initial_JJ chain_NN1 ordering_NN1 process_NN1 or_CC from_II an_AT1 impurity_NN1 in_II the_AT system_NN1 ._. 
As_II the_AT structure_NN1 is_VBZ not_XX a_AT1 single_JJ crystal_NN1 ,_, the_AT sizes_NN2 found_VVN vary_VV0 from_II somewhat_RR greater_JJR than_CSN a_AT1 crystallite_NN1 to_II diameters_NN2 of_IO a_AT1 few_DA2 millimetres_NNU2 ._. 
The_AT number_NN1 ,_, size_NN1 ,_, and_CC fine_JJ structure_NN1 depend_VV0 on_II the_AT temperature_NN1 of_IO crystallization_NN1 ,_, which_DDQ determines_VVZ the_AT critical_JJ size_NN1 of_IO the_AT nucleating_JJ centre_NN1 ._. 
This_DD1 means_VVZ that_CST large_JJ fibrous_JJ structures_NN2 form_VV0 near_RL T_ZZ1 m_ZZ1 ,_, whereas_CS greater_JJR numbers_NN2 of_IO small_JJ spherulites_NN2 grow_VV0 at_II lower_JJR temperatures_NN2 ._. 
When_CS the_AT nucleation_NN1 density_NN1 is_VBZ high_JJ ,_, the_AT spherical_JJ symmetry_NN1 tends_VVZ to_TO be_VBI lost_VVN as_II the_AT spherulite_NN1 edges_NN2 impinge_VV0 on_II their_APPGE neighbours_NN2 to_TO form_VVI a_AT1 mass_NN1 such_II21 as_II22 shown_VVN in_II figure_NN1 11.7_MC ._. 
A_AT1 study_NN1 of_IO the_AT fine_JJ structure_NN1 of_IO a_AT1 spherulite_NN1 shows_VVZ that_CST it_PPH1 is_VBZ built_VVN up_RP of_IO fibrous_JJ sub-units_NN2 ,_, growth_NN1 takes_VVZ place_NN1 by_II the_AT formation_NN1 of_IO fibrils_NN2 which_DDQ spread_VV0 outwards_RL from_II the_AT nucleus_NN1 in_II bundles_NN2 ,_, into_II the_AT surrounding_JJ amorphous_JJ phase_NN1 ._. 
As_CSA this_DD1 fibrillar_JJ growth_NN1 advances_NN2 ,_, branching_JJ takes_VVZ place_NN1 ,_, and_CC at_II some_DD intermediate_JJ stage_NN1 in_II the_AT development_NN1 ,_, the_AT spherulite_NN1 often_RR resembles_VVZ a_AT1 sheaf_NN1 of_IO grain_NN1 ._. 
This_DD1 forms_VVZ as_II the_AT fibrils_NN2 fan_VV0 out_RP and_CC begin_VV0 to_TO create_VVI the_AT spherical_JJ outline_NN1 ._. 
Although_CS the_AT fibrils_NN2 are_VBR arranged_VVN radially_RR ,_, the_AT molecular_JJ chains_NN2 lie_VV0 at_II right_JJ angles_NN2 to_II the_AT fibril_NN1 axis_NN1 ._. 
This_DD1 has_VHZ led_VVN to_II the_AT suggestion_NN1 that_CST the_AT fine_JJ structure_NN1 is_VBZ created_VVN from_II a_AT1 series_NN of_IO lamellar_JJ crystals_NN2 winding_VVG helically_RR along_II the_AT spherulite_NN1 radius_NN1 ._. 
Growth_NN1 proceeds_VVZ from_II a_AT1 small_JJ crystal_NN1 nucleus_NN1 which_DDQ develops_VVZ into_II a_AT1 fibril_NN1 ._. 
Low_JJ branching_JJ and_CC twisting_VVG then_RT produces_VVZ bundles_NN2 of_IO diverging_JJ and_CC spreading_JJ fibrils_NN2 which_DDQ eventually_RR fill_VV0 out_RP into_II the_AT characteristic_JJ spherical_JJ structure_NN1 ._. 
In_RL21 between_RL22 the_AT branches_NN2 of_IO the_AT fibrils_NN2 are_VBR amorphous_JJ areas_NN2 and_CC these_DD2 ,_, along_II21 with_II22 the_AT amorphous_JJ interfacial_JJ regions_NN2 between_II the_AT lamellae_NN2 ,_, make_VV0 up_RP the_AT disordered_JJ content_NN1 of_IO the_AT semi-crystalline_JJ polymer_NN1 (_( figure_NN1 11.8_MC )_) ._. 
Spherulites_NN2 are_VBR classified_VVN as_CSA positive_JJ when_CS the_AT refractive_JJ index_NN1 of_IO the_AT polymer_NN1 chain_NN1 is_VBZ greater_JJR across_II the_AT chain_NN1 than_CSN along_II the_AT axis_NN1 ,_, and_CC negative_JJ when_CS the_AT greater_JJR refractive_JJ index_NN1 is_VBZ in_II the_AT axial_JJ direction_NN1 ._. 
They_PPHS2 also_RR show_VV0 various_JJ other_JJ features_NN2 such_II21 as_II22 zig-zag_JJ patterns_NN2 ,_, concentric_JJ rings_NN2 ,_, and_CC dendritic_JJ structures._NNU 11.9_MC Kinetics_NN1 of_IO crystallization_NN1 The_AT crystalline_JJ content_NN1 of_IO a_AT1 polymer_NN1 has_VHZ a_AT1 profound_JJ effect_NN1 on_II its_APPGE properties_NN2 and_CC it_PPH1 is_VBZ important_JJ to_TO know_VVI how_RRQ the_AT rate_NN1 of_IO crystallization_NN1 will_VM vary_VVI with_IW the_AT temperature_NN1 ,_, especially_RR during_II the_AT processing_NN1 and_CC manufacturing_NN1 of_IO polymeric_JJ articles_NN2 ._. 
The_AT chemical_JJ structure_NN1 of_IO the_AT polymer_NN1 is_VBZ also_RR an_AT1 important_JJ feature_NN1 in_II the_AT crystallization_NN1 ;_; for_REX21 example_REX22 ,_, polyethylene_NN1 crystallizes_VVZ readily_RR and_CC can_VM not_XX be_VBI quenched_VVN rapidly_RR enough_RR to_TO give_VVI a_AT1 largely_RR amorphous_JJ sample_NN1 whereas_CS this_DD1 is_VBZ readily_RR accomplished_VVN for_IF isotactic_JJ polystyrene_NN1 ._. 
However_RR ,_, this_DD1 aspect_NN1 will_VM be_VBI discussed_VVN more_RGR fully_RR later_RRR ._. 
Isothermal_JJ crystallization_NN1 ._. 
Two_MC main_JJ factors_NN2 influence_VV0 the_AT rate_NN1 of_IO crystallization_NN1 at_II any_DD given_JJ temperature_NN1 :_: (_( i_ZZ1 )_) the_AT rate_NN1 of_IO nucleation_NN1 ;_; and_CC (_( ii_MC )_) the_AT subsequent_JJ rate_NN1 of_IO growth_NN1 of_IO these_DD2 nuclei_NN2 to_II macroscopic_JJ dimensions_NN2 ._. 
The_AT kinetic_JJ treatment_NN1 of_IO crystallization_NN1 from_II the_AT melt_NN1 is_VBZ based_VVN on_II the_AT radial_JJ growth_NN1 of_IO a_AT1 front_NN1 through_II space_NN1 and_CC can_VM be_VBI likened_VVN to_II someone_PN1 scattering_VVG a_AT1 handful_NN1 of_IO gravel_NN1 onto_II the_AT surface_NN1 of_IO a_AT1 pond_NN1 ._. 
Each_DD1 stone_NN1 is_VBZ a_AT1 nucleus_NN1 which_DDQ ,_, when_CS it_PPH1 strikes_VVZ the_AT surface_NN1 ,_, generates_VVZ expanding_JJ circles_NN2 (_( similar_JJ to_II spherulites_NN2 in_II two_MC dimensions_NN2 )_) ._. 
These_DD2 grow_VV0 unimpeded_JJ for_IF a_AT1 while_NNT1 but_CCB the_AT leading_JJ edges_NN2 eventually_RR collide_VV0 with_IW others_NN2 and_CC growth_NN1 rates_NN2 are_VBR altered_VVN ._. 
When_CS a_AT1 similar_JJ picture_NN1 is_VBZ adopted_VVN for_IF the_AT crystallization_NN1 of_IO a_AT1 polymer_NN1 certain_JJ basic_JJ assumptions_NN2 are_VBR made_VVN first_MD ._. 
The_AT formation_NN1 of_IO ordered_JJ growth_NN1 centres_NN2 by_II the_AT alignment_NN1 of_IO chains_NN2 from_II the_AT melt_NN1 is_VBZ called_VVN spontaneous_JJ nucleation_NN1 ._. 
When_CS the_AT temperature_NN1 of_IO crystallization_NN1 is_VBZ close_JJ to_II the_AT melting_NN1 temperature_NN1 ,_, nucleation_NN1 is_VBZ sporadic_JJ and_CC only_RR a_AT1 few_DA2 large_JJ spherulites_NN2 will_VM grow_VVI ._. 
At_II lower_JJR temperatures_NN2 ,_, nucleation_NN1 is_VBZ rapid_JJ and_CC a_AT1 large_JJ number_NN1 of_IO small_JJ spherulites_NN2 are_VBR formed_VVN ._. 
The_AT growth_NN1 of_IO the_AT spherulites_NN2 may_VM occur_VVI in_II one_MC1 ,_, two_MC ,_, or_CC three_MC dimensions_NN2 and_CC the_AT rate_NN1 of_IO radial_JJ growth_NN1 is_VBZ taken_VVN to_TO be_VBI linear_JJ at_II any_DD temperature_NN1 ._. 
Finally_RR the_AT density_NN1 c_ZZ1 of_IO the_AT crystalline_JJ phase_NN1 is_VBZ considered_VVN to_TO be_VBI uniform_JJ throughout_RL but_CCB different_JJ from_II that_DD1 of_IO the_AT melt_VV0 L._NP1 A_ZZ1 kinetic_JJ treatment_NN1 has_VHZ been_VBN developed_VVN taking_VVG account_NN1 of_IO these_DD2 points_NN2 ._. 
The_AT Avrami_JJ equation_NN1 ._. 
The_AT kinetic_JJ approach_NN1 relies_VVZ on_II the_AT establishment_NN1 of_IO a_AT1 relation_NN1 between_II the_AT density_NN1 of_IO the_AT crystalline_JJ and_CC melt_VV0 phases_NN2 and_CC the_AT time_NNT1 ._. 
This_DD1 provides_VVZ a_AT1 measure_NN1 of_IO the_AT overall_JJ crystallization_NN1 rate_NN1 ._. 
It_PPH1 is_VBZ assumed_VVN that_CST the_AT spherulites_NN2 grow_VV0 from_II nuclei_NN2 whose_DDQGE relative_JJ positions_NN2 in_II the_AT melt_NN1 remain_VV0 unaltered_JJ ,_, and_CC the_AT analysis_NN1 allows_VVZ for_IF the_AT eventual_JJ impingement_NN1 of_IO the_AT growing_JJ discs_NN2 on_II one_PPX121 another_PPX122 ._. 
The_AT final_JJ relation_NN1 describing_VVG the_AT process_NN1 is_VBZ known_VVN as_II the_AT Avrami_JJ equation_NN1 expressed_VVN as_CSA where_CS k_ZZ1 is_VBZ the_AT rate_NN1 constant_NN1 ,_, w_ZZ1 o_ZZ1 and_CC w_ZZ1 L_ZZ1 are_VBR the_AT masses_NN2 of_IO the_AT melt_NN1 at_II zero_MC time_NNT1 and_CC that_CST left_VVD after_II time_NNT1 t_ZZ1 ._. 
The_AT exponent_NN1 n_ZZ1 is_VBZ the_AT Avrami_JJ exponent_NN1 and_CC is_VBZ an_AT1 integer_NN1 which_DDQ can_VM provide_VVI information_NN1 on_II the_AT geometric_JJ form_NN1 of_IO the_AT growth_NN1 ._. 
Sporadic_JJ nucleation_NN1 is_VBZ assumed_VVN to_TO be_VBI a_AT1 first-order_JJ mechanism_NN1 and_CC if_CS we_PPIS2 consider_VV0 that_CST a_AT1 two-dimensional_JJ disc_NN1 is_VBZ formed_VVN ,_, then_RT ._. 
Rapid_JJ nucleation_NN1 is_VBZ a_AT1 zeroth-order_JJ process_NN1 in_II which_DDQ the_AT growth_NN1 centres_NN2 are_VBR formed_VVN at_II the_AT same_DA time_NNT1 ,_, and_CC for_IF each_DD1 growth_NN1 unit_NN1 listed_VVN in_II table_NN1 11.3_MC ,_, the_AT corresponding_JJ values_NN2 of_IO the_AT exponent_NN1 would_VM be_VBI ._. 
Thus_RR the_AT Avrami_JJ exponent_NN1 is_VBZ the_AT sum_NN1 of_IO the_AT order_NN1 of_IO the_AT rate_NN1 process_NN1 and_CC the_AT number_NN1 of_IO dimensions_NN2 the_AT morphological_JJ unit_NN1 possesses_VVZ ._. 
Dilatometry_NN1 ._. 
As_CSA crystallization_NN1 involves_VVZ the_AT close_JJ packing_NN1 of_IO chains_NN2 in_II regular_JJ three-dimensional_JJ structures_NN2 ,_, the_AT economical_JJ use_NN1 of_IO space_NN1 is_VBZ accompanied_VVN by_II an_AT1 increase_NN1 in_II density_NN1 ._. 
Thus_RR the_AT rate_NN1 of_IO crystallization_NN1 can_VM be_VBI followed_VVN by_II recording_VVG the_AT density_NN1 changes_NN2 which_DDQ are_VBR readily_RR detected_VVN in_II a_AT1 dilatometer_NN1 ._. 
This_DD1 is_VBZ achieved_VVN by_II placing_VVG the_AT polymer_NN1 in_II a_AT1 dilatometer_NN1 with_IW a_AT1 confining_JJ liquid_NN1 ,_, such_II21 as_II22 mercury_NN1 ,_, so_CS21 that_CS22 any_DD volume_NN1 change_NN1 can_VM be_VBI recorded_VVN as_II a_AT1 movement_NN1 of_IO the_AT liquid_JJ meniscus_NN1 in_II a_AT1 capillary_NN1 ._. 
A_AT1 typical_JJ design_NN1 is_VBZ shown_VVN in_II figure_NN1 11.9_MC ._. 
The_AT polymer_NN1 is_VBZ introduced_VVN into_II the_AT dilatometer_NN1 between_II the_AT point_NN1 A_ZZ1 and_CC the_AT capillary_NN1 ._. 
The_AT apparatus_NN1 is_VBZ then_RT pumped_VVN out_RP and_CC sealed_VVN under_II vacuum_NN1 at_II the_AT point_NN1 A._NNU Sufficient_JJ mercury_NN1 is_VBZ then_RT added_VVN to_TO enclose_VVI the_AT polymer_NN1 and_CC extend_VVI into_II the_AT capillary_NN1 ,_, after_II which_DDQ the_AT tube_NN1 is_VBZ sealed_VVN at_II B_ZZ1 ,_, and_CC placed_VVN in_II a_AT1 thermostat_NN1 at_II a_AT1 temperature_NN1 somewhat_RR higher_JJR than_CSN the_AT melting_NN1 temperature_NN1 of_IO the_AT polymer_NN1 ._. 
When_CS the_AT sample_NN1 is_VBZ completely_RR molten_JJ the_AT dilatometer_NN1 is_VBZ transferred_VVN to_II a_AT1 second_MD thermostat_NN1 set_VVN at_II the_AT temperature_NN1 selected_VVN for_IF crystallization_NN1 to_TO take_VVI place_NN1 and_CC allowed_VVD to_TO equilibrate_VVI ._. 
The_AT initial_JJ period_NN1 of_IO temperature_NN1 adjustment_NN1 to_II the_AT second_MD temperature_NN1 may_VM make_VVI the_AT initial_JJ height_NN1 h_ZZ1 o_ZZ1 rather_RR difficult_JJ to_TO locate_VVI ,_, but_CCB usually_RR a_AT1 plot_NN1 such_II21 as_II22 shown_VVN in_II figure_NN1 10.9(b)_FO is_VBZ recorded_VVN ._. 
If_CS secondary_JJ crystallization_NN1 takes_VVZ place_NN1 the_AT final_JJ portion_NN1 of_IO the_AT curve_NN1 may_VM tail_NN1 away_RL making_VVG h_ZZ1 more_RRR difficulty_NN1 to_TO measure_VVI ._. 
The_AT mass_JJ fraction_NN1 of_IO the_AT uncrystallized_JJ polymer_NN1 can_VM be_VBI related_VVN to_II the_AT volume_NN1 changes_NN2 and_CC to_II the_AT heights_NN2 measured_VVN in_II the_AT dilatometer_NN1 by_II where_RRQ h_ZZ1 t_ZZ1 ,_, h_ZZ1 o_ZZ1 ,_, and_CC h_ZZ1 are_VBR the_AT heights_NN2 at_II time_NNT1 t_ZZ1 ,_, the_AT beginning_NN1 ,_, and_CC the_AT end_NN1 of_IO the_AT process_NN1 respectively_RR ,_, with_IW V_ZZ1 t_ZZ1 ,_, V_ZZ1 o_ZZ1 ,_, and_CC V_II the_AT corresponding_JJ volumes_NN2 ._. 
The_AT slope_NN1 of_IO a_AT1 plot_NN1 of_IO against_II t_ZZ1 allows_VVZ evaluation_NN1 of_IO the_AT Avrami_JJ exponent_NN1 n_ZZ1 while_CS k_ZZ1 can_VM be_VBI calculated_VVN from_II the_AT intercept_VV0 ._. 
Deviations_NN2 from_II Avrami_JJ equation_NN1 ._. 
The_AT Avrami_JJ equation_NN1 can_VM describe_VVI some_DD but_CCB not_XX all_DB systems_NN2 investigated_VVD ._. 
The_AT crystallization_NN1 isotherms_NN2 of_IO poly_NN1 (_( ethylene_NN1 terephthalate_NN1 )_) can_VM be_VBI fitted_VVN by_II equation_NN1 (_( 11.3_MC )_) using_VVG n_ZZ1 =_FO 4_MC above_II 473_MC K_ZZ1 and_CC n_ZZ1 =_FO 2_MC at_II 383_MC K._NP1 The_AT equation_NN1 should_VM be_VBI used_VVN with_IW caution_NN1 ,_, however_RR ,_, as_CSA non-integer_JJ values_NN2 have_VH0 been_VBN reported_VVN and_CC the_AT geometric_JJ shape_NN1 of_IO the_AT morphological_JJ unit_NN1 is_VBZ not_XX always_RR that_CST predicted_VVD by_II the_AT value_NN1 of_IO n_ZZ1 calculated_VVN from_II the_AT experimental_JJ data_NN ._. 
Secondary_JJ crystallization_NN1 ._. 
Deviations_NN2 from_II the_AT Avrami_JJ treatment_NN1 may_VM also_RR be_VBI observed_VVN towards_II the_AT end_NN1 of_IO the_AT crystallization_NN1 process_NN1 and_CC values_NN2 of_IO h_ZZ1 are_VBR often_RR difficult_JJ to_TO determine_VVI accurately_RR ,_, as_CSA shown_VVN in_II the_AT curve_NN1 derived_VVN from_II dilatometric_JJ data_NN ._. 
The_AT tailing_NN1 of_IO the_AT curve_NN1 is_VBZ a_AT1 result_NN1 of_IO a_AT1 secondary_JJ crystallization_NN1 process_NN1 which_DDQ is_VBZ a_AT1 slower_JJR reorganization_NN1 of_IO the_AT crystalline_JJ regions_NN2 to_TO produce_VVI more_RGR perfectly_RR formed_VVN crystallites_NN2 ._. 
CHAPTER_NN1 12_MC The_AT Amorphous_JJ State_NN1 12.1_MC Molecular_JJ motion_NN1 A_ZZ1 linear_JJ polymer_NN1 chain_NN1 can_VM be_VBI treated_VVN as_II a_AT1 '_GE one-dimensional_JJ co-operative_JJ system_NN1 '_GE in_II which_DDQ the_AT rotation_NN1 of_IO a_AT1 chain_NN1 segment_NN1 is_VBZ restricted_VVN or_CC aided_VVN by_II the_AT neighbouring_JJ segments_NN2 ._. 
For_IF long_JJ chains_NN2 ,_, co-operative_JJ motion_NN1 can_VM not_XX be_VBI expected_VVN to_TO extend_VVI along_II the_AT entire_JJ length_NN1 ,_, and_CC the_AT polymer_NN1 tends_VVZ to_TO act_VVI as_CS21 if_CS22 it_PPH1 were_VBDR composed_VVN of_IO a_AT1 series_NN of_IO interconnected_JJ ,_, but_CCB independent_JJ ,_, kinetic_JJ units_NN2 ._. 
Any_DD significant_JJ movement_NN1 of_IO such_DA a_AT1 chain_NN1 is_VBZ generated_VVN by_II rotation_NN1 about_II the_AT single_JJ bonds_NN2 connecting_VVG the_AT atoms_NN2 in_II the_AT chain_NN1 ,_, and_CC depends_VVZ on_II the_AT ease_NN1 of_IO interchange_NN1 of_IO any_DD element_NN1 from_II one_MC1 rotational_JJ state_NN1 to_II another_DD1 ._. 
The_AT height_NN1 of_IO the_AT potential_JJ energy_NN1 barrier_NN1 E_ZZ1 (_( c.f._VV0 figure_NN1 1.3_MC )_) will_VM determine_VVI the_AT rapidity_NN1 of_IO conformational_JJ change_NN1 at_II any_DD temperature_NN1 ,_, and_CC when_CS the_AT temperature_NN1 of_IO the_AT polymer_NN1 increases_NN2 ,_, the_AT additional_JJ thermal_JJ energy_NN1 allows_VVZ E_NP1 to_TO be_VBI overcome_VVN more_RGR often_RR ._. 
This_DD1 encourages_VVZ increasing_JJ molecular_JJ motion_NN1 until_CS eventually_RR the_AT polymer_NN1 behaves_VVZ like_II a_AT1 viscous_JJ liquid_NN1 (_( assuming_VVG that_CST no_AT thermal_JJ degradation_NN1 takes_VVZ place_NN1 )_) ._. 
In_II the_AT amorphous_JJ state_NN1 the_AT distribution_NN1 of_IO polymer_NN1 chains_NN2 in_II the_AT matrix_NN1 is_VBZ completely_RR random_JJ ,_, with_IW none_PN of_IO the_AT strictures_NN2 imposed_VVN by_II the_AT ordering_NN1 encountered_VVN in_II the_AT crystallites_NN2 of_IO partially_RR crystalline_JJ polymers_NN2 ._. 
This_DD1 allows_VVZ the_AT onset_NN1 of_IO molecular_JJ motion_NN1 in_II amorphous_JJ polymers_NN2 to_TO take_VVI place_NN1 at_II temperatures_NN2 below_II the_AT melting_NN1 temperature_NN1 of_IO such_DA crystallites_NN2 ._. 
Consequently_RR ,_, as_CSA the_AT molecular_JJ motion_NN1 in_II an_AT1 amorphous_JJ polymer_NN1 increases_NN2 ,_, the_AT sample_NN1 passes_VVZ from_II a_AT1 glass_NN1 ,_, through_II a_AT1 rubber-like_JJ state_NN1 ,_, until_CS finally_RR it_PPH1 becomes_VVZ molten_JJ ._. 
These_DD2 transitions_NN2 lead_VV0 to_II changes_NN2 in_II the_AT physical_JJ properties_NN2 and_CC material_NN1 application_NN1 of_IO a_AT1 polymer_NN1 ,_, and_CC it_PPH1 is_VBZ important_JJ to_TO examine_VVI physical_JJ changes_NN2 wrought_VVN in_II an_AT1 amorphous_JJ polymer_NN1 as_II a_AT1 result_NN1 of_IO variations_NN2 in_II the_AT molecular_JJ motion._NNU 12.2_MC The_AT five_MC regions_NN2 of_IO viscoelastic_JJ behaviour_NN1 The_AT physical_JJ nature_NN1 of_IO an_AT1 amorphous_JJ polymer_NN1 is_VBZ related_VVN to_II the_AT extent_NN1 of_IO the_AT molecular_JJ motion_NN1 in_II the_AT sample_NN1 ,_, which_DDQ in_II turn_NN1 is_VBZ governed_VVN by_II the_AT chain_NN1 flexibility_NN1 and_CC the_AT temperature_NN1 of_IO the_AT system_NN1 ._. 
Examination_NN1 of_IO the_AT mechanical_JJ behaviour_NN1 shows_VVZ that_CST there_EX are_VBR five_MC distinguishable_JJ states_NN2 in_II which_DDQ a_AT1 linear_JJ amorphous_JJ polymer_NN1 can_VM exist_VVI and_CC these_DD2 are_VBR readily_RR displayed_VVN if_CS a_AT1 parameter_NN1 such_II21 as_II22 the_AT elastic_JJ modulus_NN1 is_VBZ measured_VVN over_II a_AT1 range_NN1 of_IO temperatures_NN2 ._. 
The_AT general_JJ behaviour_NN1 of_IO a_AT1 polymer_NN1 can_VM be_VBI typified_VVN by_II results_NN2 obtained_VVN for_IF an_AT1 amorphous_JJ atactic_JJ polystyrene_NN1 sample_NN1 ._. 
The_AT relaxation_NN1 modulus_NN1 E_ZZ1 r_ZZ1 was_VBDZ measured_VVN at_II a_AT1 standard_JJ time_NNT1 interval_NN1 of_IO 10_MC s_ZZ1 and_CC ,_, is_VBZ shown_VVN as_II a_AT1 function_NN1 of_IO temperature_NN1 in_II figure_NN1 12.1_MC ._. 
Five_MC distinct_JJ regions_NN2 can_VM be_VBI identified_VVN on_II this_DD1 curve._NNU (_( i_ZZ1 )_) The_AT glassy_JJ state_NN1 ._. 
This_DD1 is_VBZ section_NN1 A_ZZ1 to_II B_ZZ1 lying_VVG below_RG 363_MC K_ZZ1 and_CC it_PPH1 is_VBZ characterized_VVN by_II a_AT1 modulus_NN1 between_II 10_MC 9.5_MC ;_; and_CC ._. 
Here_RL co-operative_JJ molecular_JJ motion_NN1 along_II the_AT chain_NN1 is_VBZ frozen_VVN ,_, causing_VVG the_AT material_NN1 to_TO respond_VVI like_II an_AT1 elastic_JJ solid_NN1 to_II a_AT1 stress_NN1 ,_, and_CC the_AT strain_NN1 time_NNT1 curve_NN1 is_VBZ of_IO the_AT form_NN1 shown_VVN in_II figure_NN1 12.1(a)._FO (_( ii_MC )_) Leathery_NN1 or_CC retarded_JJ highly_RR elastic_JJ state_NN1 ._. 
This_DD1 is_VBZ the_AT transition_NN1 region_NN1 B_ZZ1 to_II C_ZZ1 where_RRQ the_AT modulus_NN1 drops_VVZ sharply_RR from_II about_RG 10_MC 9_MC to_II about_RP over_II the_AT temperature_NN1 range_NN1 363_MC to_II 393_MC K._NP1 The_AT glass_NN1 transition_NN1 temperature_NN1 T_ZZ1 g_ZZ1 is_VBZ located_VVN in_II this_DD1 area_NN1 and_CC the_AT rapid_JJ change_NN1 in_II modulus_NN1 reflects_VVZ the_AT constant_JJ increase_NN1 in_II molecular_JJ motion_NN1 as_II the_AT temperature_NN1 rises_VVZ from_II T_ZZ1 g_ZZ1 to_II about_RP ._. 
Just_RR above_II T_ZZ1 g_ZZ1 the_AT movement_NN1 of_IO the_AT chain_NN1 segments_NN2 is_VBZ still_RR rather_RG slow_JJ ,_, imparting_VVG what_DDQ can_VM best_RRT be_VBI described_VVN as_CSA leathery_JJ properties_NN2 to_II the_AT material_NN1 ._. 
The_AT strain-time_JJ curve_NN1 is_VBZ that_DD1 shown_VVN in_II figure_NN1 12.1(b)._FO (_( iii_MC )_) The_AT rubbery_JJ state_NN1 ._. 
At_II approximately_RR 30_MC K_ZZ1 above_II the_AT glass_NN1 transition_NN1 the_AT modulus_NN1 curve_NN1 begins_VVZ to_TO flatten_VVI out_RP into_II the_AT plateau_NN1 region_NN1 C_ZZ1 to_II D_ZZ1 in_II the_AT modulus_NN1 interval_NN1 10_MC 5.7_MC ;_; to_II and_CC extends_VVZ up_II21 to_II22 about_II 420_MC K._NP1 (_( iv_MC )_) Rubbery_JJ flow_NN1 ._. 
After_CS the_AT rubbery_JJ plateau_NN1 the_AT modulus_NN1 again_RT decreases_VVZ from_II 10_MC 5.4_MC to_II in_II the_AT section_NN1 D_ZZ1 to_II E._NP1 The_AT effect_NN1 of_IO applied_JJ stress_NN1 to_II a_AT1 polymer_NN1 in_II states_NN2 (_( iii_MC )_) and_CC (_( iv_MC )_) is_VBZ shown_VVN in_II figure_NN1 12.1(c)_FO where_RRQ there_EX is_VBZ instantaneous_JJ elastic_JJ response_NN1 followed_VVN by_II a_AT1 region_NN1 of_IO flow._NNU (_( v_ZZ1 )_) Viscous_JJ state_NN1 ._. 
Above_II a_AT1 temperature_NN1 of_IO 450_MC K_ZZ1 ,_, in_II the_AT section_NN1 E_ZZ1 to_II F_ZZ1 ,_, there_EX is_VBZ little_DA1 evidence_NN1 of_IO any_DD elastic_JJ recovery_NN1 in_II the_AT polymer_NN1 and_CC all_DB the_AT characteristics_NN2 of_IO a_AT1 viscous_JJ liquid_NN1 become_VV0 evident_JJ (_( figure_NN1 12.1(d)_FO )_) ._. 
Here_RL there_EX is_VBZ a_AT1 steady_JJ decrease_NN1 of_IO the_AT modulus_NN1 from_II as_II the_AT temperature_NN1 increases_NN2 ._. 
The_AT overall_JJ shape_NN1 of_IO the_AT curve_NN1 shown_VVN in_II figure_NN1 12.1_MC is_VBZ typical_JJ for_IF linear_JJ amorphous_JJ polymers_NN2 in_RR21 general_RR22 ,_, although_CS the_AT temperatures_NN2 quoted_VVN are_VBR specific_JJ to_II polystyrene_NN1 and_CC will_VM differ_VVI for_IF other_JJ polymers_NN2 ._. 
Variations_NN2 in_II shape_NN1 are_VBR found_VVN for_IF different_JJ molar_JJ masses_NN2 and_CC when_CS the_AT sample_NN1 is_VBZ crosslinked_VVN or_CC partly_RR crystalline_JJ ._. 
The_AT value_NN1 of_IO the_AT modulus_NN1 provides_VVZ a_AT1 good_JJ indication_NN1 of_IO the_AT state_NN1 of_IO the_AT polymer_NN1 and_CC can_VM be_VBI obtained_VVN from_II the_AT curve._NNU 12.3_MC The_AT viscous_JJ region_NN1 Before_II considering_VVG the_AT flow_NN1 in_II polymer_NN1 melts_VVZ ,_, the_AT viscous_JJ behaviour_NN1 of_IO simple_JJ liquids_NN2 will_VM be_VBI examined_VVN ._. 
The_AT application_NN1 of_IO a_AT1 force_NN1 to_II a_AT1 simple_JJ liquid_NN1 of_IO low_JJ molar_JJ mass_NN1 is_VBZ relieved_VVN by_II the_AT flow_NN1 of_IO molecules_NN2 past_II one_PPX121 another_PPX122 into_II new_JJ positions_NN2 in_II the_AT system_NN1 ._. 
A_AT1 liquid_NN1 ,_, forced_VVN to_TO flow_VVI in_II this_DD1 way_NN1 by_II a_AT1 shearing_JJ force_NN1 ,_, experiences_NN2 a_AT1 viscous_JJ resistance_NN1 expressed_VVN by_II where_RRQ v_ZZ1 is_VBZ the_AT velocity_NN1 of_IO flow_NN1 along_II a_AT1 tube_NN1 of_IO radius_NN1 x_ZZ1 ,_, so_CS that_DD1 is_VBZ the_AT velocity_NN1 gradient_NN1 or_CC shear_VV0 rate_NN1 ,_, and_CC is_VBZ the_AT viscosity_NN1 coefficient_NN1 of_IO the_AT liquid_NN1 ._. 
A_AT1 liquid_NN1 is_VBZ said_VVN to_TO exhibit_VVI Newtonian_JJ flow_NN1 if_CS is_VBZ independent_JJ of_IO but_CCB substances_NN2 which_DDQ show_VV0 deviations_NN2 from_II this_DD1 flow_NN1 pattern_NN1 ,_, with_IW either_RR decreasing_JJ or_CC increasing_JJ ratios_NN2 ,_, are_VBR termed_VVN non-Newtonian_JJ ._. 
(_( See_VV0 figure_NN1 12.2_MC ._. )_) 
Most_DAT polymers_NN2 fall_VV0 into_II this_DD1 latter_DA category_NN1 ,_, with_IW decreasing_VVG as_II the_AT shear_VV0 rate_NN1 increases_NN2 ._. 
The_AT temperature_NN1 dependence_NN1 of_IO can_NN1 normally_RR be_VBI expressed_VVN in_II the_AT form_NN1 where_CS A_ZZ1 is_VBZ a_AT1 constant_JJ and_CC E_ZZ1 D_ZZ1 represents_VVZ the_AT activation_NN1 energy_NN1 required_VVN to_TO create_VVI a_AT1 hole_NN1 big_JJ enough_RR for_IF a_AT1 molecule_NN1 to_TO translate_VVI or_CC '_GE jump_NN1 '_GE into_II during_II flow_NN1 ._. 
In_II liquids_NN2 with_IW larger_JJR or_CC irregularly_RR shaped_JJ molecules_NN2 ,_, the_AT deformation_NN1 is_VBZ slower_JJR as_II the_AT molecules_NN2 restrict_VV0 the_AT easy_JJ translation_NN1 of_IO one_PN1 past_II the_AT other_JJ ._. 
This_DD1 results_VVZ in_II a_AT1 high_JJ value_NN1 of_IO 12.4_MC Kinetic_JJ units_NN2 in_II polymer_NN1 chains_NN2 Resistance_NN1 to_TO flow_VVI in_II polymer_NN1 systems_NN2 is_VBZ even_RR greater_JJR ,_, because_CS now_RT the_AT molecules_NN2 are_VBR covalently_RR bonded_VVN into_II long_JJ chains_NN2 which_DDQ are_VBR coiled_VVN and_CC entangled_JJ and_CC translational_JJ motion_NN1 must_VM ,_, of_IO necessity_NN1 ,_, be_VBI a_AT1 co-operative_JJ process_NN1 ._. 
It_PPH1 would_VM be_VBI unreasonable_JJ to_TO expect_VVI easy_JJ co-operative_JJ motion_NN1 along_II the_AT entire_JJ polymer_NN1 chain_NN1 ,_, but_CCB as_CSA there_EX is_VBZ normally_RR some_DD degree_NN1 of_IO flexibility_NN1 in_II the_AT chain_NN1 ,_, local_JJ segmental_JJ motion_NN1 can_VM take_VVI place_NN1 more_RGR readily_RR ._. 
The_AT polymer_NN1 can_VM then_RT be_VBI considered_VVN as_II a_AT1 series_NN of_IO kinetic_JJ units_NN2 ;_; each_DD1 of_IO these_DD2 moves_NN2 in_II an_AT1 independent_JJ manner_NN1 and_CC involves_VVZ the_AT co-operative_JJ movement_NN1 of_IO a_AT1 number_NN1 of_IO consecutive_JJ chain_NN1 atoms_NN2 ._. 
Crankshaft_VV0 motion_NN1 ._. 
If_CS we_PPIS2 now_RT consider_VV0 an_AT1 arbitrary_JJ kinetic_JJ unit_NN1 which_DDQ involves_VVZ the_AT movement_NN1 of_IO six_MC atoms_NN2 by_II rotation_NN1 about_RG two_MC chain_NN1 bonds_NN2 ,_, the_AT movement_NN1 can_VM be_VBI visualized_VVN as_CSA shown_VVN diagrammatically_RR in_II figure_NN1 12.3_MC ._. 
The_AT amorphous_JJ or_CC molten_JJ polymer_NN1 is_VBZ a_AT1 conglomeration_NN1 of_IO badly_RR packed_VVN interlacing_VVG chains_NN2 and_CC the_AT extra_JJ empty_JJ space_NN1 caused_VVN by_II this_DD1 random_JJ molecular_JJ arrangement_NN1 is_VBZ called_VVN the_AT free_JJ volume_NN1 which_DDQ essentially_RR consists_VVZ of_IO all_DB the_AT holes_NN2 in_II the_AT matrix_NN1 ._. 
When_CS sufficient_JJ thermal_JJ energy_NN1 is_VBZ present_JJ in_II the_AT system_NN1 the_AT vibrations_NN2 can_VM cause_VVI a_AT1 segment_NN1 to_TO jump_VVI into_II a_AT1 hole_NN1 by_II co-operative_JJ bond_NN1 rotation_NN1 and_CC a_AT1 series_NN of_IO such_DA jumps_NN2 will_VM enable_VVI the_AT complete_JJ polymer_NN1 chain_NN1 eventually_RR to_TO change_VVI its_APPGE position_NN1 ._. 
Heating_NN1 will_VM cause_VVI a_AT1 polymer_NN1 sample_NN1 to_TO expand_VVI thereby_RR creating_VVG more_DAR room_NN1 for_IF movement_NN1 of_IO each_DD1 kinetic_JJ unit_NN1 and_CC the_AT application_NN1 of_IO a_AT1 stress_NN1 in_II a_AT1 particular_JJ direction_NN1 will_VM encourage_VVI flow_NN1 by_II segmental_JJ motion_NN1 in_II the_AT direction_NN1 of_IO the_AT stress_NN1 ._. 
The_AT segmental_JJ transposition_NN1 involving_VVG six_MC carbon_NN1 atoms_NN2 is_VBZ called_VVN crankshaft_NN1 motion_NN1 and_CC is_VBZ believed_VVN to_TO require_VVI an_AT1 activation_NN1 energy_NN1 of_IO about_RG 25_MC kJ_NNU mol_NN1 -1._MC 12.5_MC Effect_NN1 of_IO chain_NN1 length_NN1 Although_CS it_PPH1 is_VBZ thought_VVN that_DD1 translation_NN1 of_IO a_AT1 polymer_NN1 chain_NN1 proceeds_VVZ by_II31 means_II32 of_II33 a_AT1 series_NN of_IO segmental_JJ jumps_NN2 involving_VVG short_JJ kinetic_JJ units_NN2 ,_, which_DDQ may_VM each_DD1 consist_VVI of_IO between_II 15_MC and_CC 30_MC chain_NN1 atoms_NN2 ,_, the_AT complete_JJ movement_NN1 of_IO a_AT1 chain_NN1 can_VM not_XX remain_VVI unaffected_JJ by_II the_AT surrounding_JJ chains_NN2 ._. 
As_CSA stated_VVN previously_RR ,_, considerable_JJ entanglement_NN1 exists_VVZ in_II the_AT melt_NN1 and_CC any_DD motion_NN1 will_VM be_VBI retarded_VVN by_II other_JJ chains_NN2 ._. 
According_II21 to_II22 Bueche_NP1 the_AT polymer_NN1 molecule_NN1 may_VM drag_VVI along_RP several_DA2 others_NN2 during_II flow_NN1 and_CC the_AT energy_NN1 dissipation_NN1 is_VBZ then_RT a_AT1 combination_NN1 of_IO the_AT friction_NN1 between_II the_AT chain_NN1 plus_II those_DD2 which_DDQ are_VBR entangled_VVN and_CC the_AT neighbouring_JJ chains_NN2 as_CSA they_PPHS2 slip_VV0 past_II each_PPX221 other_PPX222 ._. 
It_PPH1 would_VM seem_VVI reasonable_JJ to_TO assume_VVI from_II this_DD1 ,_, that_CST the_AT length_NN1 of_IO the_AT chains_NN2 in_II the_AT sample_NN1 must_VM play_VVI a_AT1 significant_JJ role_NN1 in_II determining_VVG the_AT resistance_NN1 to_TO flow_VVI and_CC the_AT effect_NN1 of_IO chain_NN1 length_NN1 on_II log_NN1 ,_, measured_VVN at_II low_RR shear_VV0 rates_NN2 to_TO ensure_VVI Newtonian_JJ flow_NN1 ,_, is_VBZ illustrated_VVN in_II figure_NN1 12.4_MC ._. 
The_AT plot_NN1 comprises_VVZ two_MC linear_JJ portions_NN2 meeting_VVG at_II a_AT1 critical_JJ chain_NN1 length_NN1 &lsqb;_( formula_NN1 &rsqb;_) c_ZZ1 ._. 
Above_RL &lsqb;_( formula_NN1 &rsqb;_) c_ZZ1 the_AT relation_NN1 describing_VVG the_AT flow_NN1 behaviour_NN1 is_VBZ and_CC K_ZZ1 1_MC1 is_VBZ proportional_JJ to_II the_AT 3.4_MC power_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
Below_RL &lsqb;_( formula_NN1 &rsqb;_) c_ZZ1 ,_, is_VBZ directly_RR proportional_JJ to_II and_CC the_AT expression_NN1 becomes_VVZ where_RRQ K_ZZ1 1_MC1 and_CC K_ZZ1 2_MC are_VBR temperature_NN1 dependent_JJ constants_NN2 ._. 
The_AT critical_JJ chain_NN1 length_NN1 &lsqb;_( formula_NN1 &rsqb;_) c_ZZ1 is_VBZ interpreted_VVN as_CSA representing_VVG the_AT dividing_JJ point_NN1 between_II chains_NN2 which_DDQ are_VBR too_RG short_JJ to_TO provide_VVI a_AT1 significant_JJ contribution_NN1 to_II from_II entanglement_NN1 effects_NN2 and_CC those_DD2 large_JJ enough_RR to_TO cause_VVI retardation_NN1 of_IO flow_NN1 by_II intertwining_JJ with_IW their_APPGE neighbours_NN2 ._. 
If_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ defined_VVN as_II the_AT number_NN1 of_IO atoms_NN2 in_II the_AT backbone_NN1 chain_NN1 of_IO a_AT1 polymer_NN1 then_RT typical_JJ values_NN2 for_IF &lsqb;_( formula_NN1 &rsqb;_) c_ZZ1 are_VBR 610_MC for_IF polyisobutylene_NN1 ,_, 730_MC for_IF polystyrene_NN1 ,_, and_CC 208_MC for_IF poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) ._. 
In_RR21 general_RR22 &lsqb;_( formula_NN1 &rsqb;_) c_ZZ1 is_VBZ lower_JJR for_IF polar_JJ polymers_NN2 than_CSN for_IF non-polar_JJ polymers._NNU 12.6_MC The_AT reptation_NN1 model_NN1 The_AT theory_NN1 proposed_VVN by_II Bueche_NP1 tends_VVZ to_TO suggest_VVI a_AT1 very_RG clear-cut_JJ distinction_NN1 between_II the_AT movement_NN1 of_IO chains_NN2 of_IO length_NN1 less_RRR than_CSN &lsqb;_( formula_NN1 &rsqb;_) c_ZZ1 and_CC the_AT relative_JJ immobility_NN1 of_IO the_AT entangled_JJ chains_NN2 with_IW lengths_NN2 greater_JJR than_CSN &lsqb;_( formula_NN1 &rsqb;_) c_ZZ1 ._. 
As_CSA independent_JJ chain_NN1 mobility_NN1 can_VM not_XX be_VBI discounted_VVN for_IF these_DD2 longer_JJR chains_NN2 after_II the_AT onset_NN1 of_IO entanglement_NN1 ,_, a_AT1 modified_JJ model_NN1 is_VBZ required_VVN to_TO account_VVI for_IF the_AT ability_NN1 of_IO long_JJ chains_NN2 to_TO translate_VVI and_CC diffuse_VVI through_II the_AT polymer_NN1 matrix_NN1 ,_, i.e._REX the_AT entanglement_NN1 network_NN1 must_VM be_VBI considered_VVN as_CSA being_VBG transient_JJ ._. 
Such_DA a_AT1 concept_NN1 is_VBZ embodied_VVN in_II the_AT '_GE reptation_NN1 '_GE model_NN1 proposed_VVN by_II de_NP1 Gennes_NP2 ._. 
In_II this_DD1 approach_NN1 the_AT chain_NN1 is_VBZ assumed_VVN to_TO be_VBI contained_VVN in_II a_AT1 hypothetical_JJ tube_NN1 which_DDQ is_VBZ placed_VVN initially_RR in_II a_AT1 three_MC dimensional_JJ network_NN1 formed_VVN from_II the_AT other_JJ entangled_JJ chains_NN2 ._. 
Although_CS for_IF simplicity_NN1 ,_, these_DD2 network_NN1 '_GE knots_NN2 '_GE are_VBR regarded_VVN as_II a_AT1 set_NN1 of_IO fixed_JJ obstacles_NN2 round_II which_DDQ the_AT isolated_JJ chain_NN1 under_II consideration_NN1 must_VM wriggle_VVI during_II translation_NN1 ,_, in_II practice_NN1 the_AT network_NN1 '_GE knots_NN2 '_GE would_VM also_RR be_VBI in_II motion_NN1 ._. 
The_AT contours_NN2 of_IO the_AT tube_NN1 are_VBR then_RT defined_VVN by_II the_AT position_NN1 of_IO the_AT entanglement_NN1 points_VVZ in_II the_AT network_NN1 ._. 
Two_MC types_NN2 of_IO chain_NN1 motion_NN1 can_VM be_VBI envisaged_VVN ,_, a_AT1 conformational_JJ change_NN1 taking_VVG place_NN1 within_II the_AT confines_NN2 of_IO the_AT tube_NN1 ,_, and_CC more_RGR importantly_RR ,_, reptation_NN1 ._. 
The_AT latter_DA is_VBZ imagined_VVN to_TO be_VBI a_AT1 snake-like_JJ movement_NN1 that_CST translates_VVZ the_AT chain_NN1 through_II the_AT tube_NN1 and_CC allows_VVZ it_PPH1 to_TO escape_VVI at_II the_AT tube_NN1 ends_NN2 ._. 
Mechanistically_RR it_PPH1 can_VM be_VBI regarded_VVN as_II the_AT movement_NN1 of_IO a_AT1 kink_NN1 in_II the_AT chain_NN1 along_II its_APPGE length_NN1 (_( see_VV0 figure_NN1 12.5_MC )_) until_CS this_DD1 reaches_VVZ the_AT end_NN1 of_IO the_AT chain_NN1 and_CC leaves_VVZ it_PPH1 ._. 
Motion_NN1 of_IO this_DD1 kind_NN1 translates_VVZ the_AT chain_NN1 through_II the_AT tube_NN1 ,_, like_II a_AT1 snake_NN1 moving_VVG through_II grass_NN1 ,_, and_CC successive_JJ defects_NN2 moving_VVG the_AT chain_NN1 in_II this_DD1 way_NN1 will_VM eventually_RR carry_VVI it_PPH1 completely_RR out_II21 of_II22 the_AT hypothetical_JJ tube_NN1 ._. 
The_AT motion_NN1 can_VM be_VBI characterized_VVN by_II a_AT1 reptation_NN1 time_NNT1 ,_, or_CC more_RGR accurately_RR by_II a_AT1 relaxation_NN1 time_NNT1 ,_, ,_, that_DD1 is_VBZ a_AT1 measure_NN1 of_IO the_AT time_NNT1 required_VVN for_IF a_AT1 chain_NN1 to_TO escape_VVI completely_RR from_II its_APPGE tube_NN1 ._. 
If_CS the_AT tube_NN1 is_VBZ defined_VVN as_CSA having_VHG the_AT same_DA length_NN1 as_CSA the_AT unperturbed_JJ chain_NN1 ,_, nl_NNU o_ZZ1 ,_, where_CS l_ZZ1 o_ZZ1 is_VBZ the_AT bond_NN1 length_NN1 under_II conditions_NN2 (_( corrected_VVN for_IF short_JJ range_NN1 interactions_NN2 )_) ,_, then_RT the_AT time_NNT1 required_VVN for_IF the_AT chain_NN1 to_TO reptate_VVI out_II21 of_II22 the_AT tube_NN1 is_VBZ proportional_JJ to_II the_AT square_NN1 of_IO the_AT distance_NN1 travelled_VVD ,_, i.e._REX Here_RL D_ZZ1 1_MC1 is_VBZ the_AT diffusion_NN1 constant_NN1 within_II the_AT tube_NN1 ,_, and_CC is_VBZ distinguished_VVN from_II translation_NN1 outside_II the_AT tube_NN1 which_DDQ will_VM be_VBI slower_JJR and_CC more_RGR difficult_JJ ._. 
This_DD1 can_VM be_VBI expressed_VVN as_II the_AT frictional_JJ coefficient_NN1 for_IF the_AT chain_NN1 ,_, again_RT within_II the_AT tube_NN1 confines_NN2 ._. 
However_RR ,_, because_CS the_AT reptation_NN1 is_VBZ assumed_VVN to_TO occur_VVI by_II migration_NN1 of_IO a_AT1 segmental_JJ kink_NN1 along_II the_AT chain_NN1 ,_, the_AT force_NN1 needed_VVD to_TO do_VDI this_DD1 is_VBZ applied_VVN one_MC1 segment_NN1 at_II a_AT1 time_NNT1 and_CC so_RR it_PPH1 is_VBZ more_RGR appropriate_JJ to_TO use_VVI the_AT frictional_JJ factor_NN1 per_II segment_NN1 ._. 
Thus_RR or_CC Equation_NN1 (_( 12.6b_FO )_) shows_VVZ that_CST the_AT relaxation_NN1 time_NNT1 is_VBZ proportional_JJ to_II the_AT cube_NN1 of_IO the_AT chain_NN1 length_NN1 ._. 
This_DD1 is_VBZ the_AT fundamental_JJ result_NN1 of_IO the_AT reptation_NN1 model_NN1 ._. 
The_AT cube_NN1 dependence_NN1 is_VBZ not_XX a_AT1 precise_JJ match_NN1 with_IW the_AT 3.4_MC exponent_NN1 obtained_VVN from_II viscosity_NN1 measurements_NN2 of_IO long_JJ chains_NN2 ,_, but_CCB it_PPH1 is_VBZ acceptable_JJ ,_, particularly_RR as_CSA the_AT model_NN1 gives_VVZ a_AT1 satisfactory_JJ picture_NN1 of_IO how_RRQ a_AT1 polymer_NN1 chain_NN1 can_VM overcome_VVI the_AT restraining_JJ influence_NN1 of_IO entanglements_NN2 and_CC move_VV0 within_II the_AT matrix_NN1 ._. 
Typically_RR o_ZZ1 is_VBZ of_IO the_AT order_NN1 of_IO 10_MC -10_MC seconds_NNT2 for_IF and_CC so_RR the_AT relaxation_NN1 time_NNT1 for_IF a_AT1 polymer_NN1 chain_NN1 with_IW would_VM be_VBI about_RG 100_MC seconds_NNT2 ._. 
Reptation_NN1 theory_NN1 has_VHZ been_VBN developed_VVN further_RRR by_II Doi_NP1 and_CC Edwards_NP1 and_CC is_VBZ being_VBG applied_VVN to_II both_RR viscoelastic_JJ and_CC solution_NN1 behaviour_NN1 ._. 
It_PPH1 has_VHZ been_VBN shown_VVN that_CST for_IF a_AT1 chain_NN1 moving_VVG in_II the_AT melt_NN1 ,_, over_II time-scales_NN2 that_CST greatly_RR exceed_VV0 the_AT lifetime_NNT1 of_IO the_AT tube_NN1 ,_, a_AT1 reptation_NN1 self-diffusion_NN1 coefficient_NN1 D_ZZ1 rept_VV0 ,_, can_VM be_VBI measured_VVN which_DDQ is_VBZ inversely_RR proportional_JJ to_II n_ZZ1 2_MC ,_, i.e._REX the_AT diffusion_NN1 law_NN1 is_VBZ This_DD1 law_NN1 holds_VVZ for_IF the_AT '_GE welding_NN1 '_GE of_IO polymers_NN2 at_II an_AT1 interface_NN1 which_DDQ can_VM be_VBI explained_VVN by_II reptation_NN1 ._. 
When_CS two_MC blocks_NN2 of_IO the_AT same_DA polymer_NN1 are_VBR brought_VVN together_RL and_CC held_VVN at_II a_AT1 temperature_NN1 just_RR above_II the_AT T_ZZ1 g_ZZ1 for_IF a_AT1 time_NNT1 t_ZZ1 ,_, interdiffusion_NN1 of_IO the_AT chains_NN2 takes_VVZ place_NN1 from_II each_DD1 block_NN1 across_II the_AT interface_NN1 (_( see_VV0 figure_NN1 12.6_MC )_) thereby_RR joining_VVG the_AT blocks_NN2 together_RL ._. 
The_AT strength_NN1 of_IO the_AT junction_NN1 formed_VVN will_VM depend_VVI on_II t_ZZ1 which_DDQ should_VM be_VBI smaller_JJR than_CSN the_AT reptation_NN1 time_NNT1 ,_, i.e._REX the_AT mixed_JJ layer_NN1 ought_VMK to_TO be_VBI smaller_JJR than_CSN the_AT size_NN1 of_IO the_AT coil_NN1 if_CS an_AT1 interfacial_JJ link_NN1 is_VBZ to_TO be_VBI formed_VVN ._. 
The_AT situation_NN1 changes_NN2 if_CS the_AT blocks_NN2 are_VBR composed_VVN of_IO two_MC different_JJ polymers_NN2 which_DDQ ,_, as_CSA a_AT1 pair_NN ,_, can_VM form_VVI a_AT1 miscible_JJ blend_NN1 ._. 
Although_CS welding_NN1 can_VM again_RT take_VVI place_NN1 ,_, the_AT diffusion_NN1 law_NN1 (_( equation_NN1 12.7_MC )_) is_VBZ now_RT altered_VVN ._. 
It_PPH1 has_VHZ been_VBN found_VVN that_CST if_CS a_AT1 block_NN1 of_IO poly(vinylchloride)_NN1 is_VBZ brought_VVN in_II31 contact_II32 with_II33 a_AT1 block_NN1 of_IO polycaprolactone_NN1 ,_, at_II temperatures_NN2 above_II T_ZZ1 g_ZZ1 ,_, then_RT D_ZZ1 rept_NN1 is_VBZ higher_JJR than_CSN expected_JJ and_CC is_VBZ proportional_JJ to_TO ._. 
This_DD1 has_VHZ been_VBN interpreted_VVN as_CSA being_VBG a_AT1 consequence_NN1 of_IO the_AT negative_JJ enthalpy_NN1 of_IO mixing_VVG in_II the_AT system_NN1 which_DDQ acts_VVZ as_II an_AT1 additional_JJ driving_JJ force_NN1 for_IF the_AT chains_NN2 on_II either_DD1 side_NN1 of_IO the_AT boundary_NN1 to_TO cross_VVI into_II the_AT other_JJ matrix_NN1 ._. 
This_DD1 driving_JJ force_NN1 will_VM be_VBI proportional_JJ to_II the_AT number_NN1 of_IO monomers_NN2 in_II a_AT1 chain_NN1 ,_, hence_RR the_AT change_NN1 in_II the_AT diffusion_NN1 law_NN1 ._. 
Reptation_NN1 theory_NN1 can_VM also_RR be_VBI applied_VVN to_II polymer_NN1 dissolution_NN1 processes._NNU 12.7_MC Temperature_NN1 dependence_NN1 of_IO When_RRQ a_AT1 polymer_NN1 is_VBZ transformed_VVN into_II a_AT1 melt_NN1 without_IW degradation_NN1 and_CC is_VBZ stable_JJ at_II even_RR higher_JJR temperatures_NN2 ,_, is_VBZ observed_VVN to_TO decrease_VVI rapidly_RR as_CSA the_AT temperature_NN1 increases_NN2 ._. 
If_CS it_PPH1 is_VBZ still_RR stable_JJ at_II temperatures_NN2 in_II31 excess_II32 of_II33 100_MC K_ZZ1 above_II T_ZZ1 g_ZZ1 ,_, the_AT temperature_NN1 dependence_NN1 has_VHZ an_AT1 exponential_NN1 form_NN1 where_CS according_II21 to_II22 the_AT Eyring_JJ rate_NN1 theory_NN1 H_ZZ1 is_VBZ the_AT activation_NN1 enthalpy_NN1 of_IO viscous_JJ flow_NN1 and_CC is_VBZ a_AT1 more_RGR representative_JJ parameter_NN1 than_CSN the_AT energy_NN1 ._. 
Values_NN2 of_IO H_ZZ1 vary_VV0 slowly_RR over_II a_AT1 range_NN1 from_II 20_MC to_II 120_MC kJ_NNU mol_NN1 -1_MC ._. 
When_CS the_AT temperature_NN1 is_VBZ lowered_VVN towards_II T_ZZ1 g_ZZ1 ,_, H_ZZ1 changes_VVZ dramatically_RR and_CC a_AT1 simple_JJ equation_NN1 such_II21 as_II22 (_( 12.8_MC )_) is_VBZ no_RR21 longer_RR22 valid_JJ ._. 
The_AT increase_NN1 in_II H_ZZ1 ,_, observed_VVN with_IW temperature_NN1 lowering_NN1 ,_, can_VM be_VBI equated_VVN with_IW a_AT1 rapid_JJ loss_NN1 of_IO free_JJ volume_NN1 as_CSA T_ZZ1 g_ZZ1 is_VBZ approached_VVN ._. 
Hence_RR H_ZZ1 becomes_VVZ dependent_JJ on_II the_AT availability_NN1 of_IO a_AT1 suitable_JJ hole_NN1 for_IF a_AT1 segment_NN1 to_TO move_VVI into_II ,_, rather_II21 than_II22 being_VBG representative_JJ of_IO the_AT potential_JJ energy_NN1 barrier_NN1 to_II rotation_NN1 ._. 
This_DD1 approach_NN1 suggests_VVZ that_CST the_AT jump_NN1 frequency_NN1 decreases_VVZ when_RRQ there_EX is_VBZ an_AT1 increasing_JJ co-operative_JJ motion_NN1 among_II the_AT chains_NN2 needed_VVN to_TO produce_VVI holes._NNU 12.8_MC Rubbery_JJ state_NN1 With_IW a_AT1 decrease_NN1 in_II temperature_NN1 ,_, the_AT flow_NN1 of_IO a_AT1 polymer_NN1 melt_NN1 becomes_VVZ increasingly_RR sluggish_JJ as_II the_AT chain_NN1 motion_NN1 becomes_VVZ too_RG slow_JJ to_TO effect_VVI complete_JJ untangling_NN1 of_IO the_AT polymer_NN1 coils_NN2 ._. 
The_AT viscosity_NN1 increases_VVZ rapidly_RR to_II a_AT1 value_NN1 of_IO about_II as_CSA T_ZZ1 g_ZZ1 is_VBZ approached_VVN ,_, but_CCB on_II passing_VVG from_II the_AT melt_NN1 to_II the_AT glass_NN1 a_AT1 region_NN1 of_IO rubbery_JJ flow_NN1 and_CC elasticity_NN1 is_VBZ traversed_VVN ._. 
In_II this_DD1 state_NN1 the_AT polymer_NN1 exhibits_VVZ several_DA2 unique_JJ properties_NN2 which_DDQ are_VBR dealt_VVN with_IW in_II chapter_NN1 14_MC and_CC only_RR a_AT1 brief_JJ description_NN1 of_IO the_AT chain_NN1 behaviour_NN1 in_II this_DD1 region_NN1 is_VBZ given_VVN here_RL ._. 
Long_JJ range_NN1 elasticity_NN1 ._. 
The_AT rubber-like_JJ region_NN1 ,_, which_DDQ lies_VVZ above_II T_ZZ1 g_ZZ1 ,_, appears_VVZ when_RRQ the_AT rotation_NN1 about_II the_AT segment_NN1 links_NN2 is_VBZ free_JJ enough_RR to_TO enable_VVI the_AT chains_NN2 to_TO assume_VVI any_DD of_IO the_AT immense_JJ number_NN1 of_IO equi-energetic_JJ conformations_NN2 available_JJ ,_, without_IW significant_JJ chain_NN1 untangling_VVG taking_VVG place_NN1 ._. 
The_AT majority_NN1 of_IO these_DD2 shapes_NN2 will_VM be_VBI compact_JJ coils_NN2 because_CS the_AT possibility_NN1 of_IO their_APPGE occurrence_NN1 is_VBZ much_RR greater_JJR than_CSN for_IF the_AT more_RGR extended_JJ forms_NN2 ._. 
When_CS a_AT1 polymer_NN1 ,_, which_DDQ is_VBZ not_XX too_RG crystalline_JJ and_CC has_VHZ a_AT1 reasonably_RR high_JJ molar_JJ mass_NN1 (_( &lt;20_FO 000_MC g_NNU mol_NN1 -1_MC )_) ,_, is_VBZ in_II this_DD1 elastic_NN1 state_VV0 it_PPH1 will_VM elongate_VVI quite_RG readily_RR in_II the_AT direction_NN1 of_IO an_AT1 applied_JJ stress_NN1 ,_, e.g._REX natural_JJ rubber_NN1 will_VM stretch_VVI easily_RR when_CS pulled_VVN ._. 
If_CS the_AT stress_NN1 is_VBZ applied_VVN for_IF a_AT1 short_JJ time_NNT1 ,_, then_RT removed_VVD ,_, the_AT sample_NN1 snaps_VVZ back_RP to_II its_APPGE original_JJ length_NN1 suggesting_VVG that_CST some_DD '_GE memory_NN1 '_GE of_IO its_APPGE initial_JJ unstretched_JJ condition_NN1 is_VBZ retained_VVN ._. 
The_AT ability_NN1 of_IO an_AT1 elastomer_NN1 to_TO regain_VVI its_APPGE former_DA size_NN1 ,_, when_CS extensions_NN2 of_IO up_RG21 to_RG22 400_MC per_NNU21 cent_NNU22 have_VH0 been_VBN experienced_VVN ,_, is_VBZ associated_VVN with_IW the_AT long_JJ chain_NN1 character_NN1 of_IO the_AT material_NN1 ._. 
This_DD1 retractive_JJ action_NN1 of_IO linear_JJ uncrosslinked_JJ polymers_NN2 can_VM be_VBI observed_VVN if_CS the_AT time_NNT1 interval_NN1 between_II extension_NN1 and_CC release_NN1 is_VBZ short_JJ ,_, but_CCB if_CS the_AT stress_NN1 is_VBZ maintained_VVN for_IF some_DD time_NNT1 ,_, then_RT a_AT1 relaxation_NN1 process_NN1 takes_VVZ place_NN1 allowing_VVG the_AT tension_NN1 to_TO decay_VVI eventually_RR to_TO zero_VVI ._. 
This_DD1 can_VM be_VBI explained_VVN quite_RG simply_RR ._. 
The_AT molecules_NN2 are_VBR initially_RR in_II highly_RR coiled_VVD shapes_NN2 but_CCB application_NN1 of_IO a_AT1 force_NN1 causes_VVZ rotation_NN1 about_II the_AT chain_NN1 bonds_NN2 resulting_VVG in_II an_AT1 elongation_NN1 of_IO the_AT molecules_NN2 in_II the_AT direction_NN1 of_IO the_AT stress_NN1 ._. 
This_DD1 produces_VVZ a_AT1 distribution_NN1 of_IO chain_NN1 conformations_NN2 which_DDQ differs_VVZ significantly_RR from_II the_AT most_RGT probable_JJ distribution_NN1 ,_, and_CC as_CSA this_DD1 is_VBZ an_AT1 unstable_JJ state_NN1 the_AT chains_NN2 will_VM rapidly_RR recoil_VVI when_RRQ the_AT stress_NN1 is_VBZ released_VVN in_II an_AT1 attempt_NN1 to_TO regain_VVI their_APPGE original_JJ shape_NN1 distribution_NN1 ._. 
For_IF short_JJ periods_NN2 of_IO stress_NN1 in_II an_AT1 amorphous_JJ elastomer_NN1 ,_, the_AT entanglement_NN1 and_CC intertwining_JJ of_IO chains_NN2 with_IW their_APPGE neighbours_NN2 acts_VVZ as_II a_AT1 physical_JJ restraint_NN1 to_II excessive_JJ chain_NN1 movement_NN1 and_CC the_AT elastomer_NN1 regains_VVZ its_APPGE original_JJ length_NN1 when_CS the_AT stress_NN1 is_VBZ removed_VVN ._. 
If_CS however_RR ,_, the_AT stress_NN1 is_VBZ maintained_VVN for_IF a_AT1 sufficient_JJ time_NNT1 ,_, there_EX is_VBZ a_AT1 general_JJ tendency_NN1 for_IF chains_NN2 to_TO unravel_VVI and_CC slip_VVI past_II one_PPX121 another_PPX122 into_II new_JJ positions_NN2 where_RRQ the_AT segments_NN2 can_VM relax_VVI and_CC regain_VVI a_AT1 stable_JJ coiled_JJ form_NN1 ._. 
The_AT resultant_JJ flow_NN1 relieves_VVZ the_AT tension_NN1 and_CC produces_VVZ the_AT observed_JJ stress_NN1 decay_NN1 ._. 
The_AT process_NN1 is_VBZ shown_VVN schematically_RR in_II figure_NN1 12.7_MC ._. 
When_CS the_AT molar_JJ mass_NN1 is_VBZ too_RG low_JJ to_TO produce_VVI sufficient_JJ entanglement_NN1 ,_, the_AT material_NN1 will_VM flow_VVI more_RGR readily_RR and_CC behave_VVI like_II a_AT1 viscous_JJ liquid_NN1 ._. 
Similarly_RR ,_, as_CSA the_AT temperature_NN1 increases_VVZ further_RRR and_CC further_RRR above_II the_AT glass_NN1 transition_NN1 ,_, the_AT enhanced_JJ segmental_JJ movement_NN1 facilitates_VVZ stress_NN1 decay_NN1 because_II21 of_II22 the_AT greater_JJR ease_NN1 of_IO chain_NN1 disentanglement._NNU 12.9_MC Glass_NP1 transition_NN1 region_NN1 When_CS the_AT polymer_NN1 is_VBZ at_II a_AT1 temperature_NN1 below_II its_APPGE glass_NN1 temperature_NN1 ,_, chain_NN1 motion_NN1 is_VBZ frozen_VVN ._. 
The_AT polymer_NN1 then_RT behaves_VVZ like_II a_AT1 stiff_JJ spring_NN1 storing_VVG all_DB the_AT available_JJ energy_NN1 in_II stretching_VVG as_RG potential_JJ energy_NN1 ,_, when_CS work_NN1 is_VBZ performed_VVN on_II it_PPH1 ._. 
If_CS sufficient_JJ thermal_JJ energy_NN1 is_VBZ supplied_VVN to_II the_AT system_NN1 to_TO allow_VVI the_AT chain_NN1 segments_NN2 to_TO move_VVI co-operatively_RR ,_, a_AT1 transition_NN1 from_II the_AT glass_NN1 to_II the_AT rubber-like_JJ state_NN1 begins_VVZ to_TO take_VVI place_NN1 ._. 
Motion_NN1 is_VBZ still_RR restricted_VVN at_II this_DD1 stage_NN1 ,_, but_CCB as_CSA the_AT temperature_NN1 increases_VVZ further_RRR a_AT1 larger_JJR number_NN1 of_IO chains_NN2 begin_VV0 to_TO move_VVI with_IW greater_JJR freedom_NN1 ._. 
In_II mechanical_JJ terms_NN2 the_AT transition_NN1 can_VM be_VBI likened_VVN to_II the_AT transformation_NN1 of_IO a_AT1 stiff_JJ spring_NN1 to_II a_AT1 weak_JJ spring_NN1 ._. 
As_CSA weak_JJ springs_NN2 can_VM only_RR stored_VVN a_AT1 fraction_NN1 of_IO the_AT potential_JJ energy_NN1 that_CST a_AT1 strong_JJ spring_NN1 can_VM hold_VVI ,_, the_AT remainder_NN1 is_VBZ lost_VVN as_CSA heat_NN1 and_CC if_CS the_AT change_NN1 from_II a_AT1 strong_JJ to_II a_AT1 weak_JJ spring_NN1 takes_VVZ place_NN1 over_II a_AT1 period_NN1 of_IO time_NNT1 ,_, equivalent_JJ to_II the_AT observation_NN1 time_NNT1 ,_, then_RT the_AT energy_NN1 loss_NN1 is_VBZ detected_VVN as_CSA mechanical_JJ damping_NN1 ._. 
Finally_RR when_CS molecular_JJ motion_NN1 increases_VVZ to_II a_AT1 sufficiently_RR high_JJ level_NN1 ,_, all_DB the_AT chains_NN2 behave_VV0 like_II weak_JJ springs_NN2 the_AT whole_JJ time_NNT1 ._. 
This_DD1 means_VVZ that_CST the_AT modulus_NN1 is_VBZ much_RR lower_JJR ,_, but_CCB so_RG too_RR is_VBZ the_AT damping_NN1 ,_, which_DDQ passes_VVZ through_II a_AT1 maximum_NN1 in_II the_AT vicinity_NN1 of_IO T_ZZ1 g_ZZ1 ._. 
The_AT maximum_NN1 appears_VVZ because_CS the_AT polymer_NN1 is_VBZ passing_VVG from_II the_AT low-damping_JJ glassy_JJ state_NN1 ,_, through_II the_AT high-damping_JJ transition_NN1 region_NN1 ,_, to_II the_AT lower-damping_JJR rubber-like_JJ state_NN1 ._. 
Treloar_NN1 has_VHZ described_VVN a_AT1 very_RG apt_JJ demonstration_NN1 of_IO the_AT transition_NN1 ._. 
A_AT1 thin_JJ rubber_JJ rod_NN1 is_VBZ wound_VVN round_II a_AT1 cylinder_NN1 to_TO create_VVI the_AT shape_NN1 of_IO a_AT1 spring_NN1 and_CC then_RT frozen_VVN in_II this_DD1 shape_NN1 using_VVG liquid_JJ nitrogen_NN1 ._. 
The_AT cylinder_NN1 (_( possibly_RR of_IO paper_NN1 )_) is_VBZ then_RT removed_VVN leaving_VVG the_AT rubber_JJ spring_NN1 ._. 
The_AT rubber_NN1 is_VBZ now_RT in_II the_AT glassy_JJ state_NN1 and_CC it_PPH1 acts_VVZ like_II a_AT1 stiff_JJ metal_NN1 spring_NN1 by_II regaining_VVG its_APPGE shape_NN1 rapidly_RR after_II an_AT1 extension_NN1 ._. 
As_II the_AT temperature_NN1 is_VBZ raised_VVN a_AT1 gradual_JJ loss_NN1 in_II the_AT elastic_JJ recovery_NN1 is_VBZ observed_VVN after_II each_DD1 applied_JJ stress_NN1 ,_, until_CS a_AT1 stage_NN1 is_VBZ reached_VVN when_CS there_EX is_VBZ no_AT recovery_NN1 and_CC the_AT rubber_NN1 remains_VVZ in_II the_AT deformed_JJ shape_NN1 ._. 
With_IW a_AT1 further_JJR increase_NN1 in_II temperature_NN1 the_AT rod_NN1 straightens_VVZ under_II its_APPGE own_DA weight_NN1 and_CC eventually_RR regains_VVZ its_APPGE rubber-like_JJ elasticity_NN1 at_II slightly_RR higher_JJR temperatures_NN2 ._. 
THE_AT GLASS_NN1 TRANSITION_NN1 TEMPERATURE_NN1 ,_, T_ZZ1 g_ZZ1 The_AT transition_NN1 from_II the_AT glass_NN1 to_II the_AT rubber-like_JJ state_NN1 is_VBZ an_AT1 important_JJ feature_NN1 of_IO polymer_NN1 behaviour_NN1 ,_, marking_VVG as_CSA it_PPH1 does_VDZ a_AT1 region_NN1 where_CS dramatic_JJ changes_NN2 in_II the_AT physical_JJ properties_NN2 ,_, such_II21 as_II22 hardness_NN1 and_CC elasticity_NN1 ,_, are_VBR observed_VVN ._. 
The_AT changes_NN2 are_VBR completely_RR reversible_JJ ,_, however_RR ,_, and_CC the_AT transition_NN1 from_II a_AT1 glass_NN1 to_II a_AT1 rubber_NN1 is_VBZ a_AT1 function_NN1 of_IO molecular_JJ motion_NN1 ,_, not_XX polymer_NN1 structure_NN1 ._. 
In_II the_AT rubber-like_JJ state_NN1 or_CC in_II the_AT melt_VV0 the_AT chains_NN2 are_VBR in_II relatively_RR rapid_JJ motion_NN1 ,_, but_CCB as_CSA the_AT temperature_NN1 is_VBZ lowered_VVN the_AT movement_NN1 becomes_VVZ progressively_RR slower_RRR until_CS eventually_RR the_AT available_JJ thermal_JJ energy_NN1 is_VBZ insufficient_JJ to_TO overcome_VVI the_AT rotational_JJ energy_NN1 barriers_NN2 in_II the_AT chain_NN1 ._. 
At_II this_DD1 temperature_NN1 ,_, which_DDQ is_VBZ known_VVN as_II the_AT glass_NN1 transition_NN1 temperature_NN1 T_ZZ1 g_ZZ1 ,_, the_AT chains_NN2 become_VV0 locked_VVN in_II whichever_DDQV conformation_NN1 they_PPHS2 possessed_VVD when_RRQ T_ZZ1 g_ZZ1 was_VBDZ reached_VVN ._. 
Below_II T_ZZ1 g_ZZ1 the_AT polymer_NN1 is_VBZ in_II the_AT glassy_JJ state_NN1 and_CC is_VBZ ,_, in_II effect_NN1 ,_, a_AT1 frozen_JJ liquid_NN1 with_IW a_AT1 completely_RR random_JJ structure_NN1 ._. 
Although_CS the_AT glass-rubber_JJ transition_NN1 itself_PPX1 does_VDZ not_XX depend_VVI on_II polymer_NN1 structure_NN1 ,_, the_AT temperature_NN1 at_II which_DDQ T_ZZ1 g_ZZ1 is_VBZ observed_VVN depends_VVZ largely_RR on_II the_AT chemical_JJ nature_NN1 of_IO the_AT polymer_NN1 chain_NN1 and_CC for_IF most_DAT common_JJ synthetic_JJ polymers_NN2 lies_VVZ between_II 170_MC and_CC 500_MC K._NP1 It_PPH1 is_VBZ quite_RG obvious_JJ that_CST T_ZZ1 g_ZZ1 is_VBZ an_AT1 important_JJ characteristic_JJ property_NN1 of_IO any_DD polymer_NN1 as_CSA it_PPH1 has_VHZ an_AT1 important_JJ bearing_NN1 on_II the_AT potential_JJ application_NN1 of_IO a_AT1 polymer_NN1 ._. 
Thus_RR for_IF a_AT1 polymer_NN1 with_IW a_AT1 flexible_JJ chain_NN1 ,_, such_II21 as_II22 polyisoprene_NN1 ,_, the_AT thermal_JJ energy_NN1 available_JJ at_II about_RG 300_MC K_ZZ1 is_VBZ sufficient_JJ to_TO cause_VVI the_AT chain_NN1 to_TO change_VVI shape_NN1 many_DA2 thousands_NNO2 of_IO times_NNT2 in_II a_AT1 second_NNT1 ._. 
This_DD1 polymer_NN1 has_VHZ ._. 
On_II the_AT other_JJ hand_NN1 ,_, virtually_RR no_AT motion_NN1 can_VM be_VBI detected_VVN in_II atactic_JJ poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) at_II 300_MC K_ZZ1 ,_, but_CCB at_II 450_MC K_ZZ1 ,_, the_AT chains_NN2 are_VBR in_II rapid_JJ motion_NN1 ._. 
In_II this_DD1 case_NN1 ._. 
This_DD1 means_VVZ that_CST at_II 300_MC K_ZZ1 polyisoprene_NN1 is_VBZ likely_JJ to_TO exhibit_VVI rubber-like_JJ behaviour_NN1 and_CC be_VBI useful_JJ as_II an_AT1 elastomer_NN1 ,_, whereas_CS poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) will_VM be_VBI a_AT1 glassy_JJ material_NN1 ._. 
If_CS the_AT operating_NN1 temperature_NN1 was_VBDZ lowered_VVN to_II 100_MC K_ZZ1 ,_, both_DB2 polymers_NN2 would_VM be_VBI glasses_NN2 ._. 
EXPERIMENTAL_JJ DEMONSTRATION_NN1 OF_IO T_ZZ1 g_ZZ1 The_AT glass_NN1 transition_NN1 is_VBZ not_XX specific_JJ to_TO long_JJ chain_NN1 polymers_NN2 ._. 
Any_DD substance_NN1 ,_, which_DDQ can_VM be_VBI cooled_VVN to_II a_AT1 sufficient_JJ degree_NN1 below_II its_APPGE melting_NN1 temperature_NN1 without_IW crystallizing_VVG ,_, will_VM form_VVI a_AT1 glass_NN1 ._. 
The_AT phenomenon_NN1 can_VM be_VBI conveniently_RR demonstrated_VVN using_VVG glucose_NN1 penta-acetate_NN1 (_( GPA_NP1 )_) ._. 
A_AT1 crystalline_JJ sample_NN1 of_IO GPA_NN1 is_VBZ melted_VVN ,_, then_RT chilled_VVN rapidly_RR in_II ice-water_JJ to_TO form_VVI a_AT1 brittle_JJ amorphous_JJ mass_NN1 ._. 
By_II working_VVG the_AT hard_JJ material_NN1 between_II one_PN1 's_GE fingers_NN2 ,_, the_AT transition_NN1 from_II glass_NN1 to_II rubber_NN1 will_VM be_VBI felt_VVN as_II the_AT sample_NN1 warms_VVZ up_RP ._. 
A_AT1 little_JJ perseverance_NN1 ,_, with_IW further_JJR rubbing_NN1 and_CC pulling_VVG ,_, will_VM eventually_RR result_VVI in_II the_AT recrystallization_NN1 of_IO the_AT rubbery_JJ phase_NN1 ,_, which_DDQ then_RT crumbles_VVZ to_II a_AT1 powder_NN1 ._. 
DETECTION_NN1 OF_IO T_ZZ1 g_ZZ1 The_AT transition_NN1 from_II a_AT1 glass_NN1 to_II a_AT1 rubber-like_JJ state_NN1 is_VBZ accompanied_VVN by_II marked_JJ changes_NN2 in_II the_AT specific_JJ volume_NN1 ,_, the_AT modulus_NN1 ,_, the_AT heat_NN1 capacity_NN1 ,_, the_AT refractive_JJ index_NN1 ,_, and_CC other_JJ physical_JJ properties_NN2 of_IO the_AT polymer_NN1 ._. 
The_AT glass_NN1 transition_NN1 is_VBZ not_XX a_AT1 first-order_JJ transition_NN1 ,_, in_II the_AT thermodynamic_JJ sense_NN1 ,_, as_CSA no_AT discontinuities_NN2 are_VBR observed_VVN when_CS the_AT entropy_NN1 or_CC volume_NN1 of_IO the_AT polymer_NN1 are_VBR measured_VVN as_II a_AT1 function_NN1 of_IO temperature_NN1 ._. 
If_CS the_AT first_MD derivative_NN1 of_IO the_AT property-temperature_JJ curve_NN1 is_VBZ measured_VVN ,_, a_AT1 change_NN1 in_II the_AT vicinity_NN1 of_IO T_ZZ1 g_ZZ1 is_VBZ found_VVN ;_; for_IF this_DD1 reason_NN1 it_PPH1 is_VBZ sometimes_RT called_VVN a_AT1 second-order_JJ transition_NN1 ._. 
Thus_RR while_CS the_AT change_NN1 in_II a_AT1 physical_JJ property_NN1 can_VM be_VBI used_VVN to_TO locate_VVI T_ZZ1 g_ZZ1 ,_, the_AT transition_NN1 bears_VVZ many_DA2 of_IO the_AT characteristics_NN2 of_IO a_AT1 relaxation_NN1 process_NN1 and_CC the_AT precise_JJ value_NN1 of_IO T_ZZ1 g_ZZ1 can_VM depend_VVI on_II the_AT method_NN1 used_VVN and_CC the_AT rate_NN1 of_IO the_AT measurement_NN1 ._. 
Techniques_NN2 for_IF locating_VVG T_ZZ1 g_ZZ1 can_VM be_VBI divided_VVN into_II two_MC categories_NN2 ,_, dynamic_JJ and_CC static_JJ ._. 
In_II the_AT static_JJ methods_NN2 ,_, changes_NN2 in_II the_AT temperature_NN1 dependence_NN1 of_IO an_AT1 intensive_JJ property_NN1 ,_, such_II21 as_II22 density_NN1 or_CC heat_NN1 capacity_NN1 are_VBR followed_VVN and_CC measurements_NN2 are_VBR carried_VVN out_RP slowly_RR ,_, to_TO allow_VVI the_AT sample_NN1 to_TO equilibrate_VVI and_CC relax_VVI at_II each_DD1 observation_NN1 temperature_NN1 ._. 
In_II dynamic_JJ mechanical_JJ methods_NN2 a_AT1 rapid_JJ change_NN1 in_II modulus_NN1 is_VBZ indicative_JJ of_IO the_AT glass_NN1 transition_NN1 ,_, but_CCB now_RT the_AT transition_NN1 region_NN1 is_VBZ dependent_JJ on_II the_AT frequency_NN1 of_IO the_AT applied_JJ force_NN1 ._. 
If_CS we_PPIS2 assume_VV0 that_CST ,_, in_II the_AT transition_NN1 region_NN1 ,_, the_AT restrictions_NN2 to_II motion_NN1 still_RR present_JJ in_II the_AT sample_NN1 ,_, allow_VV0 only_RR a_AT1 few_DA2 segments_NN2 to_TO move_VVI in_II some_DD time_NNT1 interval_NN1 ,_, say_VV0 10_MC s_ZZ1 ,_, then_RT considerably_RR fewer_DAR will_VM have_VHI moved_VVN if_CS the_AT observation_NN1 time_NNT1 is_VBZ less_DAR than_CSN 10_MC s_ZZ1 ._. 
This_DD1 means_VVZ that_CST the_AT location_NN1 of_IO the_AT transition_NN1 region_NN1 and_CC T_ZZ1 g_ZZ1 will_VM depend_VVI on_II the_AT experimental_JJ approach_NN1 used_VVD ,_, and_CC T_ZZ1 g_ZZ1 is_VBZ found_VVN to_TO increase_VVI 5_MC to_II 7_MC K_ZZ1 for_IF every_AT1 tenfold_JJ increase_NN1 in_II the_AT frequency_NN1 of_IO the_AT measuring_JJ techniques_NN2 ._. 
This_DD1 time_NNT1 dependence_NN1 of_IO segmental_JJ motion_NN1 corresponds_VVZ to_II the_AT strong-weak_JJ transformation_NN1 of_IO a_AT1 hypothetical_JJ spring_NN1 and_CC results_NN2 in_II the_AT high_JJ damping_NN1 which_DDQ imparts_VVZ the_AT lifeless_JJ leathery_JJ consistency_NN1 to_II the_AT polymer_NN1 in_II this_DD1 region_NN1 ._. 
The_AT temperature_NN1 of_IO maximum_JJ damping_NN1 is_VBZ usually_RR associated_VVN with_IW T_ZZ1 g_ZZ1 ,_, and_CC at_II low_JJ frequencies_NN2 the_AT value_NN1 assigned_VVN to_II T_ZZ1 g_ZZ1 is_VBZ within_II a_AT1 few_DA2 kelvins_NNU2 of_IO that_DD1 obtained_VVN from_II the_AT static_JJ methods_NN2 ._. 
As_II the_AT static_JJ methods_NN2 lead_VV0 to_II more_RGR consistent_JJ values_NN2 some_DD of_IO these_DD2 can_VM be_VBI described_VVN ._. 
Measurement_NN1 of_IO T_ZZ1 g_ZZ1 from_II V-T_NP1 curves_NN2 ._. 
One_MC1 of_IO the_AT most_RGT frequently_RR used_JJ methods_NN2 of_IO locating_VVG T_ZZ1 g_ZZ1 is_VBZ to_TO follow_VVI the_AT change_NN1 in_II the_AT volume_NN1 of_IO the_AT polymer_NN1 as_II a_AT1 function_NN1 of_IO the_AT temperature_NN1 ._. 
The_AT polymer_NN1 sample_NN1 is_VBZ placed_VVN in_II the_AT bulb_NN1 of_IO a_AT1 dilatometer_NN1 ,_, degassed_VVD ,_, and_CC a_AT1 confining_JJ liquid_NN1 such_II21 as_II22 mercury_NN1 added_VVD ._. 
If_CS the_AT bulb_NN1 is_VBZ attached_VVN to_II a_AT1 capillary_NN1 the_AT change_NN1 in_II polymer_NN1 volume_NN1 can_VM be_VBI traced_VVN by_II noting_VVG the_AT overall_JJ change_NN1 in_II volume_NN1 registered_VVN by_II the_AT movement_NN1 of_IO the_AT mercury_NN1 level_NN1 in_II the_AT capillary_NN1 ._. 
A_AT1 variation_NN1 of_IO this_DD1 method_NN1 makes_VVZ use_NN1 of_IO a_AT1 density_NN1 gradient_NN1 column_NN1 ._. 
A_AT1 small_JJ sample_NN1 of_IO polymer_NN1 suspended_VVN in_II this_DD1 column_NN1 provides_VVZ a_AT1 direct_JJ measure_NN1 of_IO the_AT polymer_NN1 density_NN1 which_DDQ can_VM be_VBI measured_VVN easily_RR as_CSA the_AT temperature_NN1 is_VBZ varied_VVN ._. 
Typical_JJ curves_NN2 are_VBR shown_VVN in_II figure_NN1 12.8_MC for_IF poly_NN1 (_( vinyl_NN1 acetate_NN1 )_) ._. 
These_DD2 consist_VV0 of_IO two_MC linear_JJ portions_NN2 whose_DDQGE slopes_NN2 differ_VV0 and_CC closer_JJR inspection_NN1 reveals_VVZ that_CST over_II a_AT1 narrow_JJ range_NN1 of_IO temperature_NN1 of_IO between_II 2_MC and_CC 5_MC K_ZZ1 the_AT slope_NN1 changes_NN2 continuously_RR ._. 
To_TO locate_VVI T_ZZ1 g_ZZ1 ,_, the_AT linear_JJ portions_NN2 are_VBR extrapolated_VVN and_CC intersect_VV0 at_II the_AT point_NN1 which_DDQ is_VBZ taken_VVN to_TO be_VBI the_AT characteristic_JJ transition_NN1 temperature_NN1 of_IO the_AT material_NN1 ._. 
Each_DD1 point_NN1 on_II the_AT curve_NN1 is_VBZ normally_RR recorded_VVN after_II allowing_VVG the_AT polymer_NN1 time_NNT1 to_TO equilibrate_VVI at_II the_AT chosen_JJ temperature_NN1 and_CC as_II the_AT rate_NN1 of_IO measurement_NN1 affects_VVZ the_AT magnitude_NN1 of_IO T_ZZ1 g_ZZ1 quite_RR noticeably_RR the_AT equilibration_NN1 time_NNT1 should_VM be_VBI several_DA2 hours_NNT2 at_RR21 least_RR22 ._. 
The_AT effect_NN1 of_IO the_AT measuring_JJ rate_NN1 on_II T_ZZ1 g_ZZ1 was_VBDZ demonstrated_VVN by_II Kovacs_NN2 who_PNQS recorded_VVD the_AT volume_NN1 of_IO a_AT1 polymer_NN1 at_II each_DD1 temperature_NN1 ,_, over_II a_AT1 range_NN1 including_II the_AT transition_NN1 ,_, using_VVG two_MC rates_NN2 of_IO cooling_VVG ._. 
If_CS the_AT sample_NN1 was_VBDZ cooled_VVN rapidly_RR (_( 0.02_MC h_ZZ1 )_) to_II each_DD1 temperature_NN1 the_AT value_NN1 of_IO T_ZZ1 g_ZZ1 derived_VVN from_II the_AT resulting_JJ curve_NN1 was_VBDZ some_DD 8_MC K_ZZ1 higher_RRR than_CSN that_DD1 measured_VVD from_II results_NN2 obtained_VVD using_VVG a_AT1 slow_JJ cooling_JJ rate_NN1 (_( 100_MC h_ZZ1 )_) ._. 
Refractive_JJ index_NN1 measurements_NN2 ._. 
The_AT change_NN1 in_II refractive_JJ index_NN1 of_IO the_AT polymer_NN1 with_IW temperature_NN1 has_VHZ been_VBN used_VVN by_II several_DA2 workers_NN2 to_TO establish_VVI T_ZZ1 g_ZZ1 ._. 
A_AT1 linear_JJ decrease_NN1 in_II refractive_JJ index_NN1 is_VBZ observed_VVN as_II the_AT temperature_NN1 increases_NN2 ,_, and_CC as_II the_AT transition_NN1 is_VBZ passed_VVN ,_, the_AT rate_NN1 of_IO decrease_NN1 becomes_VVZ greater_JJR ;_; T_ZZ1 g_ZZ1 is_VBZ again_RT taken_VVN as_II the_AT intersection_NN1 of_IO the_AT linear_JJ extrapolation_NN1 ._. 
Heat_NN1 capacity_NN1 and_CC other_JJ methods_NN2 ._. 
The_AT glass_NN1 transition_NN1 temperature_NN1 can_VM be_VBI detected_VVN calorimetrically_RR by_II following_VVG the_AT change_NN1 in_II heat_NN1 capacity_NN1 with_IW change_NN1 in_II temperature_NN1 ._. 
The_AT curve_NN1 for_IF atactic_JJ polypropylene_NN1 is_VBZ shown_VVN in_II figure_NN1 12.9_MC where_RRQ the_AT abrupt_JJ increase_NN1 in_II c_ZZ1 p_ZZ1 at_II about_RG 260_MC K_ZZ1 ,_, corresponds_VVZ to_II the_AT glass_NN1 transition_NN1 ._. 
Among_II other_JJ reported_JJ techniques_NN2 the_AT most_RGT useful_JJ include_VV0 differential_JJ thermal_JJ analysis_NN1 ,_, dielectric_JJ loss_NN1 measurements_NN2 ,_, X-_JJ and_CC -ray_JJ absorption_NN1 ,_, and_CC gas_NN1 permeability_NN1 studies_NN2 ._. 
All_DB indicate_VV0 the_AT existence_NN1 of_IO the_AT phenomenon_NN1 which_DDQ we_PPIS2 call_VV0 the_AT glass_NN1 transition._NNU 12.10_MC Factors_NN2 affecting_VVG T_ZZ1 g_ZZ1 We_PPIS2 have_VH0 seen_VVN that_CST the_AT magnitude_NN1 of_IO T_ZZ1 g_ZZ1 varies_VVZ over_RP a_AT1 wide_JJ temperature_NN1 range_NN1 for_IF different_JJ polymers_NN2 ._. 
As_CSA T_ZZ1 g_ZZ1 depends_VVZ largely_RR on_II the_AT amount_NN1 of_IO thermal_JJ energy_NN1 required_VVN to_TO keep_VVI the_AT polymer_NN1 chains_NN2 moving_VVG ,_, a_AT1 number_NN1 of_IO factors_NN2 which_DDQ affect_VV0 rotation_NN1 about_II chain_NN1 links_NN2 ,_, will_VM also_RR influence_VVI T_ZZ1 g_ZZ1 ._. 
These_DD2 include_VV0 (_( 1_MC1 )_) chain_NN1 flexibility_NN1 ,_, (_( 2_MC )_) molecular_JJ structure_NN1 (_( steric_JJ effects_NN2 )_) ,_, (_( 3_MC )_) molar_JJ mass_NN1 (_( see_VV0 section_NN1 12.12_MC )_) ,_, (_( 4_MC )_) branching_JJ and_CC crosslinking_VVG Chain_NN1 flexibility_NN1 ._. 
The_AT flexibility_NN1 of_IO the_AT chain_NN1 is_VBZ undoubtedly_RR the_AT most_RGT important_JJ factor_NN1 influencing_VVG T_ZZ1 g_ZZ1 ._. 
It_PPH1 is_VBZ a_AT1 measure_NN1 of_IO the_AT ability_NN1 of_IO a_AT1 chain_NN1 to_TO rotate_VVI about_II the_AT constituent_NN1 chain_NN1 bonds_NN2 ,_, hence_RR a_AT1 flexible_JJ chain_NN1 has_VHZ a_AT1 low_JJ T_ZZ1 g_ZZ1 whereas_CS a_AT1 rigid_JJ chain_NN1 has_VHZ a_AT1 high_JJ T_ZZ1 g_ZZ1 ._. 
For_IF symmetrical_JJ polymers_NN2 ,_, the_AT chemical_JJ nature_NN1 of_IO the_AT chain_NN1 backbone_NN1 is_VBZ all_DB important_JJ ._. 
Flexibility_NN1 is_VBZ obtained_VVN when_CS the_AT chains_NN2 are_VBR made_VVN up_RP of_IO bond_NN1 sequences_NN2 which_DDQ are_VBR able_JK to_TO rotate_VVI easily_RR ,_, and_CC polymers_NN2 containing_VVG ,_, ,_, or_CC links_NN2 will_VM have_VHI correspondingly_RR low_JJ values_NN2 of_IO T_ZZ1 g_ZZ1 ._. 
The_AT value_NN1 of_IO T_ZZ1 g_ZZ1 is_VBZ raised_VVN markedly_RR by_II inserting_VVG groups_NN2 which_DDQ stiffen_VV0 the_AT chain_NN1 by_II impeding_VVG rotation_NN1 ,_, so_CS21 that_CS22 more_RGR thermal_JJ energy_NN1 is_VBZ required_VVN to_TO set_VVI the_AT chain_NN1 in_II motion_NN1 ._. 
The_AT p-phenylene_JJ ring_NN1 is_VBZ particularly_RR effective_JJ in_II this_DD1 respect_NN1 ,_, but_CCB when_CS carried_VVN to_II extremes_NN2 ,_, produces_VVZ a_AT1 highly_RR intractable_JJ ,_, rigid_JJ structure_NN1 ,_, poly(p-phenylene)_NN1 with_IW no_AT softening_JJ point_NN1 ._. 
The_AT basic_JJ structure_NN1 can_VM be_VBI modified_VVN by_II introducing_VVG flexible_JJ groups_NN2 in_II the_AT chain_NN1 and_CC some_DD examples_NN2 are_VBR given_VVN in_II table_NN1 12.1_MC ._. 
Steric_JJ effects_NN2 ._. 
When_CS the_AT polymer_NN1 chains_NN2 are_VBR unsymmetrical_JJ ,_, with_IW repeat_VV0 units_NN2 of_IO the_AT type_NN1 ,_, an_AT1 additional_JJ restriction_NN1 to_II rotation_NN1 is_VBZ imposed_VVN by_II steric_JJ effects_NN2 ._. 
These_DD2 arise_VV0 when_RRQ bulky_JJ pendant_NN1 groups_NN2 hinder_VV0 the_AT rotation_NN1 about_II the_AT backbone_NN1 and_CC cause_VV0 T_ZZ1 g_ZZ1 to_TO increase_VVI ._. 
The_AT effect_NN1 is_VBZ accentuated_VVN by_II increasing_VVG the_AT size_NN1 of_IO the_AT side_NN1 group_NN1 and_CC there_EX is_VBZ some_DD evidence_NN1 of_IO a_AT1 correlation_NN1 between_II T_ZZ1 g_ZZ1 and_CC the_AT molar_JJ volume_NN1 V_ZZ1 of_IO the_AT pendant_NN1 group_NN1 ._. 
It_PPH1 can_VM be_VBI seen_VVN in_II table_NN1 12.2_MC ,_, that_CST T_ZZ1 g_ZZ1 increases_VVZ with_IW increasing_JJ V_ZZ1 in_II the_AT progressive_JJ series_NN ,_, polyethylene_NN1 ,_, polypropylene_NN1 ,_, polystyrene_NN1 ,_, and_CC poly_NN1 (_( vinyl_NN1 naphthalene_NN1 )_) ._. 
Superimposed_VVN on_II this_DD1 group_NN1 size_NN1 factor_NN1 are_VBR the_AT effects_NN2 of_IO polarity_NN1 and_CC the_AT intrinsic_JJ flexibility_NN1 of_IO the_AT pendant_NN1 group_NN1 itself_PPX1 ._. 
An_AT1 increase_NN1 in_II the_AT lateral_JJ forces_NN2 in_II the_AT bulk_NN1 state_NN1 will_VM hinder_VVI molecular_JJ motion_NN1 and_CC increase_VVI T_ZZ1 g_ZZ1 ._. 
Thus_RR polar_JJ groups_NN2 tend_VV0 to_TO encourage_VVI a_AT1 higher_JJR T_ZZ1 g_NNU than_CSN non-polar_JJ groups_NN2 of_IO similar_JJ size_NN1 ,_, as_CSA seen_VVN when_CS comparing_VVG polypropylene_NN1 ,_, poly_NN1 (_( vinyl_NN1 chloride_NN1 )_) and_CC polyacrylonitrile_NN1 ._. 
The_AT influence_NN1 of_IO side_NN1 chain_NN1 flexibility_NN1 is_VBZ evident_JJ on_II examination_NN1 of_IO the_AT polyacrylate_NN1 series_NN from_II methyl_NN1 through_II butyl_NN1 ,_, and_CC also_RR in_II the_AT polypropylene_NN1 to_II poly(hex-1-ene)_JJ series_NN ._. 
A_AT1 further_JJR increase_NN1 in_II steric_JJ hindrance_NN1 is_VBZ imposed_VVN by_II substituting_VVG an_AT1 -methyl_JJ group_NN1 ,_, which_DDQ restricts_VVZ rotation_NN1 even_RR further_RRR and_CC leads_VVZ to_II higher_JJR T_ZZ1 g_ZZ1 ._. 
For_IF the_AT pair_NN polystyrene_NN1 poly_NN1 (_( -methyl_JJ styrene_NN1 )_) ,_, the_AT increase_NN1 in_II T_ZZ1 g_ZZ1 is_VBZ 70_MC K_ZZ1 ,_, while_CS the_AT difference_NN1 between_II poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) and_CC poly_NN1 (_( methyl_NN1 acrylate_NN1 )_) is_VBZ 100_MC K._NP1 These_DD2 steric_JJ factors_NN2 all_DB affect_VV0 the_AT chain_NN1 flexibility_NN1 and_CC are_VBR simply_RR additional_JJ contributions_NN2 to_II the_AT main_JJ chain_NN1 effects_NN2 ._. 
Configurational_JJ effects_NN2 ._. 
Cis-trans_NN2 isomerism_NN1 in_II polydienes_NN2 and_CC tacticity_NN1 variations_NN2 in_II certain_JJ -methyl_NN1 substituted_VVD polymers_NN2 alter_VV0 chain_NN1 flexibility_NN1 and_CC affect_VV0 T_ZZ1 g_ZZ1 ._. 
Some_DD examples_NN2 are_VBR shown_VVN in_II table_NN1 12.3_MC ._. 
It_PPH1 is_VBZ interesting_JJ to_TO note_VVI that_CST when_CS no_AT -methyl_JJ group_NN1 is_VBZ present_JJ in_II a_AT1 polymer_NN1 ,_, tacticity_NN1 has_VHZ little_RR influence_VV0 on_II T_ZZ1 g_ZZ1 ._. 
Effect_NN1 of_IO crosslinks_NN2 on_II T_ZZ1 g_ZZ1 When_RRQ crosslinks_NN2 are_VBR introduced_VVN into_II a_AT1 polymer_NN1 ,_, the_AT density_NN1 of_IO the_AT sample_NN1 is_VBZ increased_VVN proportionally_RR ._. 
As_II the_AT density_NN1 increases_NN2 ,_, the_AT molecular_JJ motion_NN1 in_II the_AT sample_NN1 is_VBZ restricted_VVN and_CC T_ZZ1 g_ZZ1 rises_VVZ ._. 
For_IF a_AT1 high_JJ crosslink_NN1 density_NN1 the_AT transition_NN1 is_VBZ broad_JJ and_CC ill-defined_NN1 ,_, but_CCB at_II lower_JJR values_NN2 ,_, T_ZZ1 g_ZZ1 is_VBZ found_VVN to_TO increase_VVI linearly_RR with_IW the_AT number_NN1 of_IO crosslinks._NNU 12.11_MC Theoretical_JJ treatments_NN2 Before_II embarking_VVG on_II a_AT1 rather_RG brief_JJ description_NN1 of_IO the_AT theoretical_JJ interpretations_NN2 of_IO the_AT glass_NN1 transition_NN1 a_AT1 word_NN1 of_IO caution_NN1 should_VM be_VBI given_VVN ._. 
In_II the_AT foregoing_JJ sections_NN2 several_DA2 features_NN2 of_IO the_AT results_NN2 point_VV0 to_II the_AT fact_NN1 that_CST ,_, in_II the_AT vicinity_NN1 of_IO T_ZZ1 g_ZZ1 ,_, rate_NN1 effects_NN2 are_VBR closely_RR associated_VVN with_IW changes_NN2 in_II certain_JJ thermodynamic_JJ properties_NN2 ._. 
This_DD1 has_VHZ engendered_VVN two_MC schools_NN2 of_IO thought_NN1 on_II the_AT origins_NN2 of_IO this_DD1 phenomenon_NN1 ,_, together_RL with_IW variations_NN2 on_II each_DD1 theme_NN1 ._. 
The_AT elementary_JJ level_NN1 of_IO this_DD1 text_NN1 precludes_VVZ detailed_JJ critical_JJ discussion_NN1 of_IO the_AT relative_JJ merits_NN2 of_IO any_DD particular_JJ treatment_NN1 ,_, and_CC to_TO avoid_VVI prejudicing_VVG the_AT issue_NN1 with_IW personal_JJ comment_NN1 the_AT main_JJ ideas_NN2 of_IO each_DD1 are_VBR outlined_VVN together_RL with_IW a_AT1 more_RGR recent_JJ and_CC possibly_RR unifying_JJ approach_NN1 to_TO complete_VVI the_AT picture_NN1 ._. 
THE_AT FREE_JJ VOLUME_NN1 THEORY_NN1 The_AT free_JJ volume_NN1 concept_NN1 has_VHZ been_VBN touched_VVN on_RP in_II previous_JJ sections_NN2 but_CCB it_PPH1 is_VBZ instructive_JJ now_RT to_TO consider_VVI this_DD1 idea_NN1 more_RGR closely_RR and_CC to_TO draw_VVI together_RL the_AT various_JJ points_NN2 alluded_VVD to_II earlier_JJR ._. 
The_AT free_JJ volume_NN1 ,_, V_ZZ1 f_ZZ1 ,_, is_VBZ defined_VVN as_II the_AT unoccupied_JJ space_NN1 in_II a_AT1 sample_NN1 ,_, arising_VVG from_II the_AT inefficient_JJ packing_NN1 of_IO disordered_JJ chains_NN2 in_II the_AT amorphous_JJ regions_NN2 of_IO a_AT1 polymer_NN1 sample_NN1 ._. 
The_AT presence_NN1 of_IO these_DD2 empty_JJ spaces_NN2 can_VM be_VBI inferred_VVN from_II the_AT fact_NN1 that_CST when_CS a_AT1 polystyrene_NN1 glass_NN1 is_VBZ dissolved_VVN in_II benzene_NN1 there_EX is_VBZ a_AT1 contraction_NN1 in_II the_AT total_JJ volume_NN1 ._. 
This_DD1 and_CC similar_JJ observations_NN2 indicate_VV0 that_CST the_AT polymer_NN1 can_VM occupy_VVI less_DAR volume_NN1 when_CS surrounded_VVN by_II benzene_NN1 molecules_NN2 and_CC that_CST there_EX must_VM have_VHI been_VBN unused_JJ space_NN1 in_II the_AT glassy_JJ matrix_NN1 to_TO allow_VVI this_DD1 increase_NN1 in_II packing_VVG efficiency_NN1 to_TO occur_VVI ._. 
On_II that_DD1 basis_NN1 ,_, the_AT observed_JJ specific_JJ volume_NN1 of_IO a_AT1 sample_NN1 ,_, V_ZZ1 ,_, will_VM be_VBI composed_VVN of_IO the_AT volume_NN1 actually_RR occupied_VVN by_II the_AT polymer_NN1 molecules_NN2 ,_, V_ZZ1 o_ZZ1 ,_, and_CC the_AT free_JJ volume_NN1 in_II the_AT system._NNU i.e._REX Each_DD1 term_NN1 will_VM ,_, of_RR21 course_RR22 ,_, be_VBI temperature_NN1 dependent_NN1 ._. 
The_AT free_JJ volume_NN1 is_VBZ a_AT1 measure_NN1 of_IO the_AT space_NN1 available_JJ for_IF the_AT polymer_NN1 to_TO undergo_VVI rotation_NN1 and_CC translation_NN1 ,_, and_CC when_CS the_AT polymer_NN1 is_VBZ in_II the_AT liquid_NN1 or_CC rubber-like_JJ states_NN2 the_AT amount_NN1 of_IO free_JJ volume_NN1 will_VM increase_VVI with_IW temperature_NN1 as_II the_AT molecular_JJ motion_NN1 increases_NN2 ._. 
If_CS the_AT temperature_NN1 is_VBZ decreased_VVN ,_, this_DD1 free_JJ volume_NN1 will_VM contract_VVI and_CC eventually_RR reach_VVI a_AT1 critical_JJ value_NN1 when_CS there_EX is_VBZ insufficient_JJ free_JJ space_NN1 to_TO allow_VVI large_JJ scale_NN1 segmental_JJ motion_NN1 to_TO take_VVI place_NN1 ._. 
The_AT temperature_NN1 at_II which_DDQ this_DD1 critical_JJ value_NN1 is_VBZ reached_VVN is_VBZ the_AT glass_NN1 transition_NN1 temperature_NN1 ._. 
Below_II T_ZZ1 g_ZZ1 the_AT free_JJ volume_NN1 will_VM remain_VVI essentially_RR constant_JJ as_II the_AT temperature_NN1 decreases_VVZ further_RRR since_CS the_AT chains_NN2 have_VH0 now_RT been_VBN immobilized_VVN and_CC frozen_VVN in_II position_NN1 ._. 
In_II contrast_NN1 ,_, the_AT occupied_JJ volume_NN1 will_VM alter_VVI because_II21 of_II22 the_AT changing_JJ amplitude_NN1 of_IO thermal_JJ vibrations_NN2 in_II the_AT chains_NN2 and_CC ,_, to_II the_AT first_MD approximation_NN1 ,_, will_VM be_VBI a_AT1 linear_JJ function_NN1 of_IO temperature_NN1 irrespective_II21 of_II22 whether_CSW the_AT polymer_NN1 is_VBZ in_II the_AT liquid_NN1 or_CC glassy_JJ state_NN1 ._. 
The_AT glass_NN1 transition_NN1 can_VM then_RT be_VBI visualized_VVN as_II the_AT onset_NN1 of_IO co-ordinated_JJ segmental_JJ motion_NN1 made_VVD possible_JJ by_II an_AT1 increase_NN1 of_IO the_AT holes_NN2 in_II the_AT polymer_NN1 matrix_NN1 to_II a_AT1 size_NN1 sufficient_JJ to_TO allow_VVI this_DD1 type_NN1 of_IO motion_NN1 to_TO occur_VVI ._. 
This_DD1 is_VBZ manifest_JJ as_II a_AT1 change_NN1 in_II the_AT specific_JJ volume_NN1 due_JJ solely_RR to_II an_AT1 increase_NN1 in_II the_AT free_JJ volume_NN1 and_CC is_VBZ shown_VVN schematically_RR as_CSA the_AT cross_NN1 hatched_VVD area_NN1 in_II figure_NN1 12.10_MC ,_, where_CS the_AT broken_JJ line_NN1 indicates_VVZ the_AT temperature_NN1 dependence_NN1 of_IO V_ZZ1 o_ZZ1 ._. 
The_AT precise_JJ definition_NN1 of_IO the_AT average_JJ amount_NN1 of_IO free_JJ volume_NN1 present_NN1 in_II a_AT1 totally_RR amorphous_JJ polymer_NN1 remains_VVZ unclear_JJ ,_, but_CCB it_PPH1 must_VM also_RR depend_VVI to_II some_DD extent_NN1 on_II the_AT thermal_JJ history_NN1 of_IO the_AT sample_NN1 ._. 
A_AT1 number_NN1 of_IO suggestions_NN2 have_VH0 been_VBN made_VVN ._. 
Simha_NN1 and_CC Boyer_NP1 observed_VVD that_CST a_AT1 general_JJ empirical_JJ relationship_NN1 exists_VVZ between_II the_AT T_ZZ1 g_ZZ1 and_CC the_AT difference_NN1 in_II expansion_NN1 coefficients_NN2 of_IO the_AT liquid_NN1 and_CC glass_NN1 states_NN2 ._. 
From_II the_AT examination_NN1 of_IO a_AT1 wide_JJ range_NN1 of_IO polymers_NN2 they_PPHS2 concluded_VVD that_CST where_CS K_ZZ1 1_MC1 is_VBZ a_AT1 constant_JJ with_IW a_AT1 value_NN1 of_IO 0.113_MC ._. 
This_DD1 implies_VVZ that_CST the_AT free_JJ volume_NN1 fraction_NN1 is_VBZ the_AT same_DA for_IF all_DB polymers_NN2 ,_, i.e._REX 11.3_MC per_NNU21 cent_NNU22 of_IO the_AT total_JJ volume_NN1 in_II the_AT glassy_JJ state_NN1 ._. 
The_AT definition_NN1 of_IO the_AT S-B_NP1 free_JJ volume_NN1 can_VM be_VBI seen_VVN from_II figure_NN1 12.10_MC to_TO be_VBI where_CS V_ZZ1 o_ZZ1 is_VBZ the_AT hypothetical_JJ liquid_JJ volume_NN1 at_II absolute_JJ zero_NN1 ._. 
This_DD1 definition_NN1 is_VBZ perhaps_RR too_RG rigid_JJ and_CC discounts_NN2 differing_VVG chain_NN1 flexibilities_NN2 ,_, so_RR a_AT1 more_RGR accurate_JJ representation_NN1 is_VBZ thought_VVN to_TO be_VBI given_VVN by_II The_AT values_NN2 obtained_VVN are_VBR still_RR much_RR higher_JJR than_CSN the_AT estimates_NN2 from_II the_AT Williams_NP1 ,_, Landel_NP1 ,_, Ferry_NN1 (_( WLF_NP1 )_) equation_NN1 ._. 
This_DD1 is_VBZ an_AT1 empirical_JJ equation_NN1 but_CCB it_PPH1 can_VM also_RR be_VBI derived_VVN from_II free_JJ volume_NN1 considerations_NN2 by_II starting_VVG with_IW a_AT1 description_NN1 of_IO the_AT viscosity_NN1 of_IO the_AT system_NN1 ._. 
In_II section_NN1 12.7_MC the_AT Arrhenius_NP1 equation_NN1 was_VBDZ used_VVN to_TO describe_VVI the_AT temperature_NN1 dependence_NN1 of_IO viscous_JJ flow_NN1 ,_, but_CCB an_AT1 empirical_JJ equation_NN1 proposed_VVN by_II Doolittle_NP1 gives_VVZ a_AT1 much_RR better_JJR description_NN1 of_IO viscous_JJ flow_NN1 ,_, and_CC has_VHZ a_AT1 similar_JJ form_NN1 where_CS A_ZZ1 and_CC B_ZZ1 are_VBR constants_NN2 and_CC ._. 
On_II a_AT1 molecular_JJ level_NN1 ,_, the_AT ratio_NN1 is_VBZ then_RT a_AT1 measure_NN1 of_IO the_AT average_JJ volume_NN1 of_IO the_AT polymer_NN1 relative_II21 to_II22 that_DD1 of_IO the_AT holes_NN2 ._. 
Thus_RR when_RRQ ,_, i.e._REX the_AT polymer_NN1 chain_NN1 is_VBZ larger_JJR than_CSN the_AT average_JJ hole_NN1 size_NN1 ,_, the_AT viscosity_NN1 will_VM be_VBI correspondingly_RR high_JJ ,_, whereas_CS when_CS ,_, the_AT viscosity_NN1 will_VM be_VBI low_JJ ._. 
We_PPIS2 can_VM now_RT introduce_VVI a_AT1 free_JJ volume_NN1 fraction_NN1 f_ZZ1 and_CC substitute_VV0 in_II equation_NN1 (_( 12.13_MC )_) Next_MD ,_, a_AT1 comparison_NN1 can_VM be_VBI made_VVN between_II the_AT viscosity_NN1 of_IO a_AT1 polymer_NN1 melt_VV0 at_II a_AT1 temperature_NN1 ,_, and_CC that_CST at_II a_AT1 reference_NN1 temperature_NN1 such_II21 as_II22 and_CC so_RR Here_RL f_ZZ1 and_CC f_ZZ1 g_ZZ1 are_VBR the_AT fractional_JJ free_JJ volumes_NN2 at_II T_ZZ1 and_CC T_ZZ1 g_ZZ1 respectively_RR ._. 
From_II figure_NN1 12.10_MC it_PPH1 can_VM be_VBI seen_VVN that_CST V_ZZ1 f_ZZ1 is_VBZ assumed_VVN to_TO remain_VVI constant_JJ during_II expansion_NN1 of_IO the_AT polymer_NN1 in_II the_AT glassy_JJ state_NN1 but_CCB that_DD1 above_II T_ZZ1 g_ZZ1 there_EX is_VBZ a_AT1 steady_JJ increase_NN1 with_IW rising_VVG temperature_NN1 ._. 
If_CS f_ZZ1 is_VBZ the_AT expansion_NN1 coefficient_NN1 of_IO the_AT free_JJ volume_NN1 above_II T_ZZ1 g_ZZ1 ,_, then_RT the_AT temperature_NN1 dependence_NN1 of_IO f_ZZ1 can_VM be_VBI written_VVN Substitution_NN1 of_IO equation_NN1 (_( 12.17_MC )_) in_II equation_NN1 (_( 12.16_MC )_) gives_VVZ Rearranging_NP1 and_CC dividing_VVG by_II f_ZZ1 Equation_NN1 (_( 12.19_MC )_) is_VBZ one_MC1 form_NN1 of_IO the_AT WLF_NP1 equation_NN1 ,_, but_CCB as_CSA viscosity_NN1 is_VBZ a_AT1 time_NNT1 dependent_JJ quantity_NN1 and_CC is_VBZ proportional_JJ to_II the_AT flow_NN1 time_NNT1 ,_, t_ZZ1 ,_, and_CC density_NN1 ,_, ,_, then_RT and_CC where_CS the_AT small_JJ differences_NN2 in_II density_NN1 have_VH0 been_VBN neglected_VVN ._. 
This_DD1 can_VM be_VBI compared_VVN with_IW the_AT form_NN1 of_IO the_AT WLF_NP1 equation_NN1 where_RRQ is_VBZ the_AT reduced_JJ variables_NN2 shift_VV0 factor_NN1 ,_, C_ZZ1 1_MC1 and_CC C_ZZ1 2_MC are_VBR constants_NN2 that_CST can_VM be_VBI evaluated_VVN from_II experimental_JJ data_NN ,_, and_CC are_VBR found_VVN to_TO be_VBI and_CC when_RRQ T_ZZ1 g_ZZ1 is_VBZ the_AT reference_NN1 temperature_NN1 ._. 
A_AT1 more_RGR general_JJ description_NN1 can_VM be_VBI used_VVN where_CS T_ZZ1 s_ZZ1 is_VBZ an_AT1 arbitrary_JJ reference_NN1 temperature_NN1 usually_RR located_VVD 50_MC K_ZZ1 above_II T_ZZ1 g_ZZ1 ._. 
C_ZZ1 1_MC1 and_CC C_ZZ1 2_MC now_RT have_VH0 different_JJ values_NN2 ,_, and_CC the_AT shift_NN1 factor_NN1 is_VBZ expressed_VVN as_II a_AT1 ratio_NN1 of_IO relaxation_NN1 times_NNT2 ,_, ,_, at_II T_ZZ1 and_CC T_ZZ1 s_ZZ1 ._. 
As_CSA we_PPIS2 shall_VM see_VVI in_II chapter_NN1 13_MC ,_, the_AT relaxation_NN1 time_NNT1 is_VBZ a_AT1 function_NN1 of_IO the_AT viscosity_NN1 and_CC modulus_NN1 (_( G_ZZ1 )_) of_IO the_AT polymer_NN1 and_CC ,_, according_II21 to_II22 the_AT Maxwell_NP1 model_NN1 ,_, ._. 
The_AT modulus_NN1 will_VM be_VBI much_DA1 less_DAR temperature_NN1 dependent_JJ than_CSN the_AT viscosity_NN1 so_CS we_PPIS2 can_VM write_VVI which_DDQ demonstrates_VVZ the_AT equivalence_NN1 of_IO the_AT empirical_JJ equation_NN1 (_( 12.22_MC )_) with_IW that_DD1 derived_VVN from_II the_AT free_JJ volume_NN1 theory_NN1 ,_, equations_NN2 (_( 12.19_MC )_) and_CC 12.21_MC )_) ._. 
The_AT WLF_NP1 equation_NN1 can_VM be_VBI used_VVN to_TO describe_VVI the_AT temperature_NN1 dependence_NN1 of_IO dynamic_JJ mechanical_JJ ,_, and_CC dielectric_JJ relaxation_NN1 behaviour_NN1 of_IO polymers_NN2 near_II the_AT glass_NN1 transition_NN1 where_CS the_AT response_NN1 is_VBZ no_RR21 longer_RR22 described_VVN by_II an_AT1 Arrhenius_NN1 relation_NN1 ._. 
This_DD1 will_VM be_VBI dealt_VVN with_IW in_II chapter_NN1 13_MC ._. 
The_AT equations_NN2 (_( 12.19_MC )_) and_CC (_( 12.22_MC )_) can_VM be_VBI used_VVN to_TO evaluate_VVI f_ZZ1 g_ZZ1 as_CSA we_PPIS2 see_VV0 that_DD1 and_CC ._. 
On_II the_AT basis_NN1 of_IO viscosity_NN1 data_NN ,_, B_ZZ1 can_VM be_VBI assigned_VVN a_AT1 value_NN1 of_IO unity_NN1 ,_, leading_VVG to_II and_CC ._. 
If_CS f_ZZ1 is_VBZ assumed_VVN to_TO be_VBI equivalent_JJ to_TO ,_, this_DD1 value_NN1 compares_VVZ well_RR with_IW the_AT average_JJ value_NN1 of_IO determined_JJ for_IF 18_MC polymers_NN2 covering_VVG a_AT1 wide_JJ range_NN1 of_IO T_ZZ1 g_ZZ1 ._. 
The_AT free_JJ volume_NN1 fraction_NN1 of_IO 2.5_MC per_NNU21 cent_NNU22 is_VBZ low_RR compared_VVN with_IW the_AT S-B_NP1 estimation_NN1 but_CCB is_VBZ comparable_JJ to_II that_DD1 derived_VVN from_II the_AT Gibbs-Di_NP1 Marzio_NP1 theory_NN1 ._. 
Other_JJ values_NN2 of_IO 8_MC per_NNU21 cent_NNU22 from_II the_AT '_GE hole_NN1 '_GE theory_NN1 of_IO Hirai_NP1 and_CC Eyring_NP1 ,_, and_CC 12_MC per_NNU21 cent_NNU22 calculated_VVN by_II Miller_NP1 from_II heats_NN2 of_IO vaporization_NN1 and_CC liquid_JJ compressibilities_NN2 ,_, illustrate_VV0 the_AT uncertainty_NN1 surrounding_VVG the_AT magnitude_NN1 of_IO this_DD1 free_JJ volume_NN1 parameter_NN1 ._. 
The_AT free_JJ volume_NN1 theory_NN1 deals_VVZ with_IW the_AT need_NN1 for_IF space_NN1 to_TO be_VBI available_JJ before_II co-operative_JJ motion_NN1 ,_, characteristic_NN1 of_IO the_AT glass_NN1 transition_NN1 ,_, can_VM be_VBI initiated_VVN ,_, but_CCB it_PPH1 tells_VVZ us_PPIO2 little_RR about_II the_AT molecular_JJ motion_NN1 itself_PPX1 ._. 
Other_JJ approaches_NN2 have_VH0 chosen_VVN to_TO base_VVI their_APPGE description_NN1 of_IO the_AT glass_NN1 transition_NN1 on_II a_AT1 thermodynamic_JJ analysis_NN1 ._. 
GIBBS-DI_NP1 MARZIO_NP1 THERMODYNAMIC_NP1 THEORY_NN1 Comments_NN2 on_II the_AT thermodynamic_JJ theories_NN2 will_VM be_VBI restricted_VVN to_II the_AT proposals_NN2 of_IO Gibbs_NP1 and_CC Di_NP1 Marzio_NP1 (_( G-D_JJ )_) who_PNQS ,_, while_CS acknowledging_VVG that_DD1 kinetic_JJ effects_NN2 are_VBR inevitably_RR encountered_VVN when_CS measuring_VVG T_ZZ1 g_ZZ1 ,_, consider_VV0 the_AT fundamental_JJ transition_NN1 to_TO be_VBI a_AT1 true_JJ equilibrium_NN1 ._. 
The_AT data_NN reported_VVN by_II Kovacs_NP2 in_II section_NN1 12.9_MC imply_VV0 that_CST the_AT observed_JJ T_ZZ1 g_ZZ1 would_VM decrease_VVI further_RRR if_CS a_AT1 sufficiently_RR long_JJ time_NNT1 for_IF measurement_NN1 was_VBDZ allowed_VVN ._. 
This_DD1 aspect_NN1 is_VBZ considered_VVN in_II the_AT G-D_JJ theory_NN1 by_II defining_VVG a_AT1 new_JJ transition_NN1 temperature_NN1 T_ZZ1 2_MC at_II which_DDQ the_AT configurational_JJ entropy_NN1 of_IO the_AT system_NN1 is_VBZ zero_MC ._. 
This_DD1 temperature_NN1 can_VM be_VBI considered_VVN in_II effect_NN1 to_TO be_VBI the_AT limiting_JJ value_NN1 T_ZZ1 g_ZZ1 would_VM reach_VVI in_II a_AT1 hypothetical_JJ experiment_NN1 taking_VVG an_AT1 infinitely_RR long_JJ time_NNT1 ._. 
On_II this_DD1 basis_NN1 the_AT experimentally_RR detectable_JJ T_ZZ1 g_ZZ1 is_VBZ a_AT1 time_NNT1 dependent_JJ relaxation_NN1 process_NN1 and_CC the_AT observed_JJ value_NN1 is_VBZ a_AT1 function_NN1 of_IO the_AT time_NNT1 scale_NN1 of_IO the_AT measuring_JJ technique_NN1 ._. 
The_AT theoretical_JJ derivation_NN1 is_VBZ based_VVN on_II a_AT1 lattice_NN1 treatment_NN1 ._. 
The_AT configurational_JJ entropy_NN1 is_VBZ found_VVN by_II calculating_VVG the_AT number_NN1 of_IO ways_NN2 that_CST n_ZZ1 x_ZZ1 linear_JJ chains_NN2 each_DD1 x_ZZ1 segments_VVZ long_JJ can_VM be_VBI placed_VVN on_II a_AT1 diamond_NN1 lattice_NN1 ,_, for_IF which_DDQ the_AT coordination_NN1 number_NN1 z_ZZ1 =_FO 4_MC ,_, together_RL with_IW n_ZZ1 o_ZZ1 holes_NN2 ._. 
The_AT restrictions_NN2 imposed_VVN on_II the_AT placing_NN1 of_IO a_AT1 chain_NN1 on_II the_AT lattice_NN1 are_VBR embodied_VVN in_II the_AT hindered_JJ rotation_NN1 which_DDQ is_VBZ expressed_VVN as_II the_AT '_GE flex_NN1 energy_NN1 '_GE and_CC h_ZZ1 which_DDQ is_VBZ the_AT energy_NN1 of_IO formation_NN1 of_IO a_AT1 hole_NN1 ._. 
The_AT flex_NN1 energy_NN1 is_VBZ the_AT energy_NN1 difference_NN1 between_II the_AT potential_JJ energy_NN1 minimum_NN1 of_IO the_AT located_JJ bond_NN1 and_CC the_AT potential_JJ minima_NN2 of_IO the_AT remaining_JJ possible_JJ orientations_NN2 which_DDQ may_VM be_VBI used_VVN on_II the_AT lattice_NN1 ._. 
Thus_RR for_IF polyethylene_NN1 the_AT trans_NN2 position_NN1 is_VBZ considered_VVN most_DAT stable_JJ and_CC the_AT gauche_JJ positions_NN2 are_VBR the_AT flexed_VVN ones_NN2 with_IW the_AT energy_NN1 difference_NN1 between_II the_AT ground_NN1 and_CC flexed_VVD states_NN2 ._. 
This_DD1 of_RR21 course_RR22 varies_VVZ with_IW the_AT nature_NN1 of_IO the_AT polymer_NN1 ._. 
The_AT quantity_NN1 h_ZZ1 is_VBZ a_AT1 measure_NN1 of_IO the_AT cohesive_JJ energy_NN1 ._. 
The_AT configurational_JJ entropy_NN1 S_ZZ1 conf_NN1 is_VBZ derived_VVN from_II the_AT partition_NN1 function_NN1 describing_VVG the_AT location_NN1 of_IO holes_NN2 and_CC polymer_NN1 molecules_NN2 ._. 
As_II the_AT temperature_NN1 drops_VVZ towards_II T_ZZ1 2_MC the_AT number_NN1 of_IO available_JJ configurational_JJ states_NN2 in_II the_AT system_NN1 decreases_VVZ until_CS at_II the_AT temperature_NN1 T_ZZ1 2_MC the_AT system_NN1 possesses_VVZ only_RR one_MC1 degree_NN1 of_IO freedom_NN1 ._. 
This_DD1 leads_VVZ to_II where_RRQ and_CC The_AT fractions_NN2 of_IO unoccupied_JJ and_CC occupied_JJ sites_NN2 are_VBR f_ZZ1 o_ZZ1 and_CC f_ZZ1 x_ZZ1 respectively_RR while_CS S_ZZ1 o_ZZ1 is_VBZ a_AT1 function_NN1 of_IO f_ZZ1 o_ZZ1 ,_, f_ZZ1 x_ZZ1 ,_, and_CC z_ZZ1 ._. 
The_AT main_JJ weaknesses_NN2 of_IO this_DD1 theory_NN1 are_VBR (_( a_ZZ1 )_) that_CST a_AT1 chain_NN1 of_IO zero_NN1 stiffness_NN1 would_VM have_VHI a_AT1 T_ZZ1 g_ZZ1 of_IO 0_MC K_ZZ1 and_CC (_( b_ZZ1 )_) that_CST the_AT T_ZZ1 g_ZZ1 would_VM be_VBI essentially_RR independent_JJ of_IO any_DD intermolecular_JJ interactions_NN2 ._. 
In_II31 spite_II32 of_II33 these_DD2 limitations_NN2 ,_, various_JJ aspects_NN2 of_IO the_AT behaviour_NN1 of_IO copolymers_NN2 ,_, plasticized_VVD polymers_NN2 ,_, and_CC the_AT chain_NN1 length_NN1 dependence_NN1 of_IO T_ZZ1 g_ZZ1 ,_, can_VM be_VBI predicted_VVN in_II a_AT1 reasonably_RR satisfactory_JJ manner_NN1 ._. 
The_AT temperature_NN1 T_ZZ1 2_MC is_VBZ not_XX of_RR21 course_RR22 an_AT1 experimentally_RR measurable_JJ quantity_NN1 but_CCB is_VBZ calculated_VVN to_TO lie_VVI approximately_RR 50_MC K_ZZ1 below_II the_AT experimental_JJ T_ZZ1 g_ZZ1 and_CC can_VM be_VBI related_VVN to_II T_ZZ1 g_ZZ1 on_II this_DD1 basis_NN1 ._. 
ADAM-GIBBS_NP1 THEORY_NN1 While_CS the_AT kinetic_JJ approach_NN1 embodied_VVN in_II the_AT WLF_NP1 equation_NN1 and_CC the_AT equilibrium_NN1 treatment_NN1 of_IO the_AT G-D_JJ theory_NN1 have_VH0 both_RR been_VBN successful_JJ in_II their_APPGE way_NN1 ,_, the_AT one-sided_JJ aspect_NN1 of_IO each_DD1 probably_RR masks_VVZ the_AT fact_NN1 that_CST they_PPHS2 are_VBR not_XX entirely_RR incompatible_JJ with_IW one_PPX121 another_PPX122 ._. 
An_AT1 attempt_NN1 to_TO reunite_VVI both_DB2 channels_NN2 of_IO thought_NN1 has_VHZ been_VBN made_VVN by_II Adam_NP1 and_CC Gibbs_NP1 who_PNQS have_VH0 outlined_VVN a_AT1 molecular_JJ kinetic_JJ theory_NN1 ._. 
In_II this_DD1 they_PPHS2 relate_VV0 the_AT temperature_NN1 dependence_NN1 of_IO the_AT relaxation_NN1 process_NN1 to_II the_AT temperature_NN1 dependence_NN1 of_IO the_AT size_NN1 of_IO a_AT1 region_NN1 ,_, which_DDQ is_VBZ defined_VVN as_II a_AT1 volume_NN1 large_JJ enough_RR to_TO allow_VVI co-operative_JJ rearrangement_NN1 to_TO take_VVI place_NN1 without_IW affecting_VVG a_AT1 neighbouring_JJ region_NN1 ._. 
This_58 '_GE co-operatively_RR rearranging_VVG region_NN1 '_GE is_VBZ large_JJ enough_RR to_TO allow_VVI a_AT1 transition_NN1 to_II a_AT1 new_JJ conformation_NN1 ,_, hence_RR is_VBZ determined_VVN by_II the_AT chain_NN1 conformation_NN1 and_CC by_II definition_NN1 will_VM equal_VVI the_AT sample_NN1 size_NN1 at_II T_ZZ1 2_MC where_RRQ only_RR one_MC1 conformation_NN1 is_VBZ available_JJ to_II each_DD1 molecule_NN1 ._. 
Evaluation_NN1 of_IO the_AT temperature_NN1 dependence_NN1 of_IO the_AT size_NN1 of_IO such_DA a_AT1 region_NN1 leads_VVZ to_II an_AT1 expression_NN1 for_IF the_AT co-operative_JJ transition_NN1 probability_NN1 ,_, ,_, which_DDQ is_VBZ simply_RR the_AT reciprocal_JJ of_IO the_AT relaxation_NN1 time_NNT1 ._. 
The_AT polymer_NN1 sample_NN1 is_VBZ described_VVN as_II an_AT1 ensemble_NN1 of_IO co-operative_JJ regions_NN2 ,_, or_CC subsystems_NN2 ,_, each_DD1 containing_VVG monomeric_JJ segments_NN2 ._. 
The_AT transition_NN1 probability_NN1 of_IO such_DA a_AT1 co-operative_JJ region_NN1 is_VBZ then_RT calculated_VVN as_CSA a_AT1 function_NN1 of_IO its_APPGE size_NN1 ,_, to_TO be_VBI where_RRQ is_VBZ the_AT activation_NN1 energy_NN1 for_IF a_AT1 co-operative_JJ rearrangement_NN1 per_II monomer_NN1 segment_NN1 ._. 
A_AT1 lower_JJR limit_NN1 to_TO is_VBZ then_RT defined_VVN ;_; this_DD1 is_VBZ the_AT smallest_JJT size_NN1 *_FU ;_; capable_JJ of_IO having_VHG two_MC configurations_NN2 available_JJ to_II it_PPH1 ,_, with_IW a_AT1 critical_JJ configurational_JJ entropy_NN1 which_DDQ ,_, from_II the_AT definition_NN1 ,_, can_VM be_VBI approximated_VVN by_II ._. 
Thus_RR where_CS S_ZZ1 c_ZZ1 is_VBZ the_AT macroscopic_JJ configurational_JJ entropy_NN1 for_IF the_AT ensemble_NN1 ._. 
Substitution_NN1 gives_VVZ and_CC expressing_VVG this_DD1 in_II the_AT WLF_NP1 form_VV0 The_AT following_JJ approximations_NN2 for_IF S_ZZ1 c_ZZ1 can_VM now_RT be_VBI used_VVN and_CC ,_, remembering_VVG that_CST ,_, then_RT Substitution_NN1 in_II equation_NN1 (_( 12.29b_FO )_) gives_VVZ a_AT1 WLF_NP1 equation_NN1 where_RRQ Results_NN2 plotted_VVN according_II21 to_II22 the_AT WLF_NP1 equation_NN1 could_VM be_VBI predicted_VVN also_RR from_II the_AT molecular_JJ kinetic_JJ equation_NN1 and_CC show_VV0 that_CST the_AT two_MC approaches_NN2 are_VBR compatible_JJ ._. 
The_AT Adam-Gibbs_NP1 equations_NN2 also_RR lead_VV0 to_II a_AT1 value_NN1 of_IO ,_, so_CS the_AT theory_NN1 appears_VVZ to_TO resolve_VVI most_DAT of_IO the_AT differences_NN2 between_II the_AT kinetic_JJ and_CC thermodynamic_JJ interpretations_NN2 of_IO the_AT glass_NN1 transition_NN1 ._. 
These_DD2 theories_NN2 point_VV0 to_II the_AT fundamental_JJ importance_NN1 of_IO T_ZZ1 2_MC as_II a_AT1 true_JJ second-order_JJ transition_NN1 temperature_NN1 and_CC to_II the_AT experimental_JJ T_ZZ1 g_ZZ1 as_II the_AT temperature_NN1 governed_VVN by_II the_AT time_NNT1 scale_NN1 of_IO the_AT measuring_JJ technique_NN1 ._. 
The_AT latter_DA value_NN1 has_VHZ great_JJ practical_JJ significance_NN1 ,_, however_RR ,_, and_CC is_VBZ a_AT1 parameter_NN1 which_DDQ is_VBZ essential_JJ to_II the_AT understanding_NN1 of_IO the_AT physical_JJ behaviour_NN1 of_IO a_AT1 polymer._NNU 12.12_MC Dependence_NN1 of_IO T_ZZ1 g_ZZ1 on_II molar_JJ mass_NN1 The_AT value_NN1 of_IO T_ZZ1 g_ZZ1 depends_VVZ on_II the_AT way_NN1 in_II which_DDQ it_PPH1 is_VBZ measured_VVN but_CCB it_PPH1 is_VBZ also_RR found_VVN to_TO be_VBI a_AT1 function_NN1 of_IO the_AT polymer_NN1 chain_NN1 length_NN1 ._. 
At_II high_JJ molar_JJ masses_NN2 the_AT glass_NN1 temperature_NN1 is_VBZ essentially_RR constant_JJ when_CS measured_VVN by_II any_DD given_JJ method_NN1 ,_, but_CCB decreases_VVZ as_II the_AT molar_JJ mass_NN1 of_IO the_AT sample_NN1 is_VBZ lowered_VVN ._. 
In_II31 terms_II32 of_II33 the_AT simple_JJ free_JJ volume_NN1 concept_NN1 each_DD1 chain_NN1 end_NN1 requires_VVZ more_RGR free_JJ volume_NN1 in_II which_DDQ to_TO move_VVI about_II than_CSN a_AT1 segment_NN1 in_II the_AT chain_NN1 interior_NN1 ._. 
With_IW increasing_JJ thermal_JJ energy_NN1 the_AT chain_NN1 ends_NN2 will_VM be_VBI able_JK to_TO rotate_VVI more_RGR readily_RR than_CSN the_AT rest_NN1 of_IO the_AT chain_NN1 and_CC the_AT more_DAR chain_NN1 ends_VVZ a_AT1 sample_NN1 has_VHZ the_AT greater_JJR the_AT contribution_NN1 to_II the_AT free_JJ volume_NN1 when_CS these_DD2 begin_VV0 moving_VVG ,_, consequently_RR the_AT glass_NN1 transition_NN1 temperature_NN1 is_VBZ lowered_VVN ._. 
Bueche_NP1 expressed_VVD this_DD1 as_II where_RRQ is_VBZ the_AT glass_NN1 temperature_NN1 of_IO a_AT1 polymer_NN1 with_IW a_AT1 very_RG large_JJ molar_JJ mass_NN1 ,_, is_VBZ the_AT free_JJ volume_NN1 contribution_NN1 of_IO one_MC1 chain_NN1 end_NN1 and_CC is_VBZ 2_MC for_IF a_AT1 linear_JJ polymer_NN1 ,_, is_VBZ the_AT polymer_NN1 density_NN1 ,_, N_ZZ1 A_ZZ1 is_VBZ Avogradro_NP1 's_GE constant_JJ ,_, and_CC f_ZZ1 is_VBZ the_AT free_JJ volume_NN1 expansivity_NN1 defined_VVN as_II The_AT linear_JJ expression_NN1 in_II equation_NN1 (_( 12.33_MC )_) has_VHZ been_VBN widely_RR used_VVN and_CC describes_VVZ the_AT behaviour_NN1 of_IO many_DA2 polymer_NN1 systems_NN2 over_II a_AT1 reasonable_JJ range_NN1 of_IO molar_JJ mass_NN1 (_( &gt;_FO 5000_MC )_) ._. 
For_IF short_JJ chains_NN2 the_AT relationship_NN1 is_VBZ no_RR21 longer_RR22 valid_JJ and_CC it_PPH1 has_VHZ been_VBN shown_VVN that_CST if_CS T_ZZ1 g_ZZ1 is_VBZ plotted_VVN against_II log_NN1 x_ZZ1 ,_, where_CS x_ZZ1 is_VBZ the_AT number_NN1 of_IO atoms_NN2 or_CC bonds_NN2 in_II the_AT polymer_NN1 backbone_NN1 ,_, then_RT three_MC distinct_JJ regions_NN2 can_VM be_VBI identified_VVN for_IF a_AT1 number_NN1 of_IO common_JJ amorphous_JJ polymers_NN2 (_( figure_NN1 12.11_MC )_) ._. 
Region_NN1 I_PPIS1 denotes_VVZ the_AT range_NN1 of_IO chain_NN1 lengths_NN2 at_II which_DDQ T_ZZ1 g_ZZ1 reaches_VVZ its_APPGE asymptotic_JJ value_NN1 and_CC the_AT critical_JJ value_NN1 x_ZZ1 c_ZZ1 at_II which_DDQ this_DD1 occurs_VVZ increases_NN2 as_II the_AT chain_NN1 becomes_VVZ more_RGR rigid_JJ ._. 
Thus_RR x_ZZ1 c_ZZ1 is_VBZ approximately_RR 90_MC for_IF a_AT1 flexible_JJ polymer_NN1 poly_NN1 (_( dimethyl_NN1 siloxane_NN1 )_) but_CCB nearer_II 600_MC for_IF the_AT more_RGR rigid_JJ poly_NN1 (_( -methyl_JJ styrene_NN1 )_) ._. 
The_AT relationship_NN1 between_II and_CC x_ZZ1 c_ZZ1 is_VBZ then_RT In_II region_NN1 II_MC ,_, T_ZZ1 g_ZZ1 is_VBZ dependent_JJ on_II the_AT molar_JJ mass_NN1 and_CC can_VM be_VBI described_VVN by_II equation_NN1 (_( 12.35_MC )_) ,_, but_CCB ,_, on_II entering_VVG region_NN1 III_MC where_RRQ the_AT decrease_NN1 in_II T_ZZ1 g_ZZ1 accelerates_VVZ ,_, this_DD1 is_VBZ no_RR21 longer_RR22 true_JJ ._. 
The_AT latter_DA region_NN1 incorporates_VVZ the_AT material_NN1 that_CST is_VBZ oligomeric_JJ ,_, and_CC the_AT line_NN1 separating_VVG II_MC and_CC III_MC represents_VVZ the_AT oligomer-polymer_JJ transition_NN1 where_CS the_AT chains_NN2 begin_VV0 to_TO become_VVI long_RR enough_RR to_TO be_VBI considered_VVN capable_JJ of_IO adopting_VVG a_AT1 gaussian_JJ coil_NN1 conformation._NNU 12.13_MC The_AT glassy_JJ state_NN1 When_CS a_AT1 linear_JJ amorphous_JJ polymer_NN1 is_VBZ in_II the_AT glassy_JJ state_NN1 ,_, the_AT material_NN1 is_VBZ rigid_JJ and_CC brittle_JJ because_CS the_AT flow_NN1 units_NN2 of_IO the_AT chain_NN1 are_VBR co-operatively_RR immobile_JJ and_CC effectively_RR frozen_VVN in_II position_NN1 ._. 
The_AT polymer_NN1 sample_NN1 is_VBZ also_RR optically_RR transparent_JJ ,_, as_CSA the_AT chains_NN2 are_VBR distributed_VVN in_II a_AT1 random_JJ fashion_NN1 and_CC present_VV0 no_AT definite_JJ boundaries_NN2 or_CC discontinuities_NN2 from_II which_DDQ light_NN1 can_VM be_VBI reflected_VVN ._. 
An_AT1 amorphous_JJ polymer_NN1 in_II this_DD1 state_NN1 has_VHZ been_VBN likened_VVN to_II a_AT1 plate_NN1 of_IO frozen_JJ spaghetti_NN1 ._. 
If_CS a_AT1 small_JJ stress_NN1 is_VBZ applied_VVN to_II a_AT1 polymer_NN1 glass_NN1 ,_, it_PPH1 exhibits_VVZ a_AT1 rapid_JJ elastic_JJ response_NN1 resulting_VVG from_II purely_RR local_JJ ,_, bond_VV0 angle_NN1 ,_, deformation_NN1 ._. 
Consequently_RR ,_, although_CS the_AT modulus_NN1 is_VBZ high_JJ ,_, specimen_NN1 deformation_NN1 is_VBZ limited_VVN to_II about_II 1_MC1 per_NNU21 cent_NNU22 ,_, due_II21 to_II22 the_AT lack_NN1 of_IO glide_NN1 planes_NN2 in_II the_AT disordered_JJ mass_NN1 ._. 
This_DD1 means_VVZ that_CST the_AT sample_NN1 has_VHZ no_AT way_NN1 of_IO dissipating_VVG a_AT1 large_JJ applied_JJ stress_NN1 ,_, other_II21 than_II22 by_II bond_NN1 rupture_NN1 ,_, and_CC so_RR a_AT1 polymer_NN1 glass_NN1 is_VBZ prone_JJ to_II brittle_JJ fracture._NNU 12.14_MC Relaxation_NN1 processes_NN2 in_II the_AT glassy_JJ state_NN1 Polymers_NN2 do_VD0 not_XX form_VVI perfectly_RR elastic_JJ solids_NN2 ,_, as_CSA a_AT1 limited_JJ amount_NN1 of_IO bond_NN1 rotation_NN1 can_VM occur_VVI in_II the_AT glass_NN1 which_DDQ allows_VVZ slight_JJ plastic_NN1 deformation_NN1 ;_; this_DD1 makes_VVZ them_PPHO2 somewhat_RR tougher_JJR than_CSN an_AT1 inorganic_JJ glass_NN1 ._. 
There_EX is_VBZ now_RT ample_JJ evidence_NN1 to_TO support_VVI the_AT suggestion_NN1 that_CST relaxation_NN1 processes_NN2 can_VM be_VBI active_JJ in_II polymer_NN1 glasses_NN2 at_II temperature_NN1 well_RR below_II T_ZZ1 g_ZZ1 ._. 
While_CS the_AT co-operative_JJ ,_, long-range_JJ ,_, chain_NN1 motion_NN1 which_DDQ is_VBZ released_VVN on_II passing_VVG from_II the_AT glass_NN1 to_II the_AT rubber-like_JJ state_NN1 is_VBZ not_XX possible_JJ at_II ,_, other_JJ relaxations_NN2 can_VM take_VVI place_NN1 ._. 
Many_DA2 of_IO these_DD2 processes_NN2 can_VM be_VBI identified_VVN as_CSA secondary_JJ loss_NN1 peaks_NN2 in_II dynamic_JJ mechanical_JJ ,_, or_CC dielectric_JJ measurements_NN2 ,_, as_CSA will_VM be_VBI seen_VVN in_II chapter_NN1 13_MC ._. 
They_PPHS2 often_RR find_VV0 their_APPGE origin_NN1 in_II the_AT movement_NN1 of_IO groups_NN2 that_CST are_VBR attached_VVN pendant_NN1 to_II the_AT main_JJ chain_NN1 ,_, but_CCB relaxation_NN1 of_IO limited_JJ sections_NN2 of_IO the_AT main_JJ chain_NN1 can_VM also_RR be_VBI identified_VVN ._. 
The_AT molecular_JJ mechanisms_NN2 for_IF a_AT1 number_NN1 of_IO these_DD2 sub-glass_JJ transition_NN1 relaxations_NN2 have_VH0 now_RT been_VBN established_VVN ,_, and_CC by_II31 way_II32 of_II33 illustration_NN1 some_DD examples_NN2 of_IO group_NN1 motions_NN2 that_CST have_VH0 been_VBN found_VVN to_TO be_VBI active_JJ in_II a_AT1 series_NN of_IO poly_NN1 (_( alkyl_NN1 methacrylate_NN1 )_) s_ZZ1 will_VM be_VBI described_VVN ._. 
For_IF the_AT methyl_NN1 ,_, ethyl_NN1 ,_, and_CC propyl_NN1 derivatives_NN2 a_AT1 broad_JJ ,_, mechanically_RR active_JJ ,_, damping_VVG peak_NN1 is_VBZ observed_VVN at_II 280_MC K_ZZ1 (_( 1_MC1 Hz_NNU )_) ,_, which_DDQ is_VBZ below_II the_AT T_ZZ1 g_ZZ1 of_IO each_DD1 polymer_NN1 ._. 
Rotation_NN1 of_IO the_AT oxycarbonyl_NN1 group_NN1 about_II has_VHZ been_VBN identified_VVN as_II the_AT cause_NN1 ._. 
However_RR ,_, if_CS the_AT group_NN1 R_ZZ1 is_VBZ an_AT1 alkyl_NN1 or_CC cycloalkyl_NN1 unit_NN1 then_RT these_DD2 can_VM relax_VVI at_II even_RR lower_JJR temperatures_NN2 ._. 
Thus_RR when_CS R_ZZ1 is_VBZ a_AT1 methyl_NN1 unit_NN1 ,_, rotation_NN1 is_VBZ possible_JJ in_II the_AT glass_NN1 at_II temperatures_NN2 below_RG 100_MC K_ZZ1 ,_, and_CC the_AT -methyl_JJ unit_NN1 will_VM also_RR be_VBI capable_JJ of_IO rotation_NN1 at_II low_JJ temperatures_NN2 ._. 
When_CS R_ZZ1 is_VBZ larger_JJR ,_, ,_, another_DD1 relaxation_NN1 process_NN1 is_VBZ seen_VVN at_II around_RG 120_MC K_ZZ1 ,_, which_DDQ is_VBZ common_JJ to_II all_DB of_IO these_DD2 polymers_NN2 with_IW ._. 
This_DD1 is_VBZ believed_VVN to_TO involve_VVI the_AT relaxation_NN1 of_IO a_AT1 four_MC atom_NN1 unit_NN1 (_( -O-C-C-C-_NN1 )_) or_CC (_( -C-C-C-C-_NN1 )_) which_DDQ has_VHZ been_VBN variously_RR described_VVN by_II mechanisms_NN2 including_VVG ,_, and_CC similar_JJ to_TO ,_, the_AT Schatzki_NN1 or_CC Boyer_NP1 crankshaft_NN1 motions_NN2 shown_VVN schematically_RR in_II figures_NN2 12.3_MC and_CC 12.12_MC ._. 
The_AT latter_DA mechanisms_NN2 have_VH0 also_RR been_VBN used_VVN to_TO account_VVI for_IF limited_JJ segmental_JJ relaxations_NN2 in_II the_AT backbone_NN1 of_IO all_DB carbon_NN1 chain_NN1 ,_, single_JJ strand_NN1 ,_, polymers_NN2 ._. 
Even_RR larger_JJR units_NN2 can_VM relax_VVI ,_, and_CC Heijboer_NP1 has_VHZ demonstrated_VVN that_CST if_CS R_ZZ1 is_VBZ a_AT1 cyclohexyl_NN1 ring_NN1 ,_, a_AT1 relaxation_NN1 at_II 180_MC K_ZZ1 (_( 1_MC1 Hz_NNU )_) can_VM be_VBI located_VVN in_II the_AT glass_NN1 ._. 
This_DD1 can_VM be_VBI attributed_VVN to_II an_AT1 intramolecular_JJ chair-chair_JJ transition_NN1 in_II the_AT ring_NN1 ._. 
As_CSA these_DD2 relaxations_NN2 require_VV0 energy_NN1 ,_, and_CC are_VBR associated_VVN with_IW a_AT1 characteristic_JJ activation_NN1 energy_NN1 ,_, it_PPH1 has_VHZ been_VBN suggested_VVN that_CST they_PPHS2 may_VM improve_VVI the_AT impact_NN1 resistance_NN1 of_IO some_DD materials_NN2 ._. 
This_DD1 point_NN1 still_RR requires_VVZ confirmation_NN1 as_II a_AT1 general_JJ phenomenon_NN1 ,_, but_CCB there_EX is_VBZ little_RR doubt_VV0 that_DD1 polymer_NN1 molecules_NN2 are_VBR not_XX totally_RR frozen_VVN or_CC immobile_JJ when_CS in_II the_AT glassy_JJ state_NN1 and_CC that_DD1 small_JJ sub_JJ units_NN2 in_II the_AT chain_NN1 can_VM remain_VVI mechanically_RR and_CC dielectrically_RR active_JJ below_II T_ZZ1 g_ZZ1 ._. 
CHAPTER_NN1 13_MC Mechanical_JJ Properties_NN2 13.1_MC Viscoelastic_JJ state_NN1 The_AT fabrication_NN1 of_IO an_AT1 article_NN1 from_II a_AT1 polymeric_JJ material_NN1 in_II the_AT bulk_NN1 state_NN1 ,_, whether_CSW it_PPH1 be_VBI the_AT moulding_NN1 of_IO a_AT1 thermosetting_JJ plastic_NN1 or_CC the_AT spinning_NN1 of_IO a_AT1 fibre_NN1 from_II the_AT melt_NN1 ,_, involves_VVZ deformation_NN1 of_IO the_AT material_NN1 by_II applied_JJ forces_NN2 ._. 
Afterwards_RT ,_, the_AT finished_JJ article_NN1 is_VBZ inevitably_RR subjected_VVN to_II stresses_NN2 ,_, hence_RR it_PPH1 is_VBZ important_JJ to_TO be_VBI aware_JJ of_IO the_AT mechanical_JJ and_CC rheological_JJ properties_NN2 of_IO each_DD1 material_NN1 and_CC understand_VV0 the_AT basic_JJ principles_NN2 underlying_VVG their_APPGE response_NN1 to_II such_DA forces_NN2 ._. 
In_II classical_JJ terms_NN2 the_AT mechanical_JJ properties_NN2 of_IO elastic_JJ solids_NN2 can_VM be_VBI described_VVN by_II Hooke_NP1 's_GE law_NN1 ,_, which_DDQ states_VVZ that_CST an_AT1 applied_JJ stress_NN1 is_VBZ proportional_JJ to_II the_AT resultant_JJ strain_NN1 ,_, but_CCB is_VBZ independent_JJ of_IO the_AT rate_NN1 of_IO strain_NN1 ._. 
For_IF liquids_NN2 the_AT corresponding_JJ statement_NN1 is_VBZ known_VVN as_II Newton_NP1 's_GE law_NN1 ,_, with_IW the_AT stress_NN1 now_RT independent_JJ of_IO the_AT strain_NN1 ,_, but_CCB proportional_JJ to_II the_AT rate_NN1 of_IO strain_NN1 ._. 
Both_DB2 are_VBR limiting_JJ laws_NN2 ,_, valid_JJ only_RR for_IF small_JJ strains_NN2 or_CC rates_NN2 of_IO strain_NN1 ,_, and_CC while_CS it_PPH1 is_VBZ essential_JJ that_CST conditions_NN2 involving_VVG large_JJ stresses_NN2 ,_, leading_VVG to_II eventual_JJ mechanical_JJ failure_NN1 ,_, be_VBI studied_VVN ,_, it_PPH1 is_VBZ also_RR important_JJ to_TO examine_VVI the_AT response_NN1 to_II small_JJ mechanical_JJ stresses_NN2 ._. 
Both_DB2 laws_NN2 can_VM prove_VVI useful_JJ under_II these_DD2 circumstances_NN2 ._. 
In_II many_DA2 cases_NN2 ,_, a_AT1 material_NN1 may_VM exhibit_VVI the_AT characteristics_NN2 of_IO both_RR a_AT1 liquid_JJ and_CC a_AT1 solid_JJ and_CC neither_DD1 of_IO the_AT limiting_JJ laws_NN2 will_VM adequately_RR describe_VVI its_APPGE behaviour_NN1 ._. 
The_AT system_NN1 is_VBZ then_RT said_VVN to_TO be_VBI in_II a_AT1 viscoelastic_JJ state_NN1 ._. 
A_AT1 particularly_RR good_JJ illustration_NN1 of_IO a_AT1 viscoelastic_JJ material_NN1 is_VBZ provided_VVN by_II a_AT1 silicone_NN1 polymer_NN1 known_VVN as_98 '_GE bouncing_JJ putty_NN1 '_GE ._. 
If_CS a_AT1 sample_NN1 is_VBZ rolled_VVN into_II the_AT shape_NN1 of_IO a_AT1 sphere_NN1 it_PPH1 can_VM be_VBI bounced_VVN like_II a_AT1 rubber_JJ ball_NN1 ,_, i.e._REX the_AT rapid_JJ application_NN1 and_CC removal_NN1 of_IO a_AT1 stress_NN1 causes_VVZ the_AT material_NN1 to_TO behave_VVI like_II an_AT1 elastic_JJ body_NN1 ._. 
If_CS on_II the_AT other_JJ hand_NN1 ,_, a_AT1 stress_NN1 is_VBZ applied_VVN slowly_RR over_II a_AT1 longer_JJR period_NN1 the_AT material_NN1 flows_VVZ like_II a_AT1 viscous_JJ liquid_NN1 so_CS21 that_CS22 the_AT spherical_JJ shape_NN1 is_VBZ soon_RR lost_VVN if_CS left_VVN to_TO stand_VVI for_IF some_DD time_NNT1 ._. 
Pitch_NN1 behaves_VVZ in_II a_AT1 similar_JJ ,_, if_CS less_RGR spectacular_JJ ,_, manner_NN1 ._. 
Before_II examining_VVG the_AT viscoelastic_JJ behaviour_NN1 of_IO amorphous_JJ polymeric_JJ substances_NN2 in_II more_DAR detail_NN1 ,_, some_DD of_IO the_AT fundamental_JJ terms_NN2 used_VVN will_VM be_VBI defined._NNU 13.2_MC Mechanical_JJ properties_NN2 Homogeneous_JJ ,_, isotropic_JJ ,_, elastic_JJ materials_NN2 possess_VV0 the_AT simplest_JJT mechanical_JJ properties_NN2 and_CC three_MC elementary_JJ types_NN2 of_IO elastic_JJ deformation_NN1 can_VM be_VBI observed_VVN when_CS such_DA a_AT1 body_NN1 is_VBZ subjected_VVN to_II (_( i_ZZ1 )_) simple_JJ tension_NN1 ,_, (_( ii_MC )_) simple_JJ shear_VV0 ,_, and_CC (_( iii_MC )_) uniform_JJ compression_NN1 ._. 
Simple_JJ tension_NN1 ._. 
Consider_VV0 a_AT1 parallelepiped_JJ of_IO length_NN1 x_ZZ1 o_ZZ1 and_CC cross-sectional_JJ area_NN1 ._. 
If_CS this_DD1 is_VBZ subjected_VVN to_II a_AT1 balanced_JJ pair_NN of_IO tensile_JJ forces_NN2 F_ZZ1 ,_, its_APPGE length_NN1 changes_NN2 by_II an_AT1 increment_NN1 dx_MC so_CS that_DD1 ._. 
When_CS dx_MC is_VBZ small_JJ ,_, Hooke_NP1 's_GE law_NN1 is_VBZ obeyed_VVN ,_, and_CC the_AT tensile_JJ stress_NN1 is_VBZ proportional_JJ to_II the_AT tensile_JJ strain_NN1 ._. 
The_AT constant_NN1 of_IO proportionality_NN1 is_VBZ known_VVN as_II the_AT modulus_NN1 ,_, and_CC for_IF elastic_JJ solids_NN2 where_RRQ E_ZZ1 is_VBZ Young_NP1 's_GE modulus_NN1 The_AT stress_NN1 is_VBZ a_AT1 measure_NN1 of_IO the_AT force_NN1 per_II unit_NN1 area_NN1 ,_, and_CC the_AT strain_NN1 or_CC elongation_NN1 is_VBZ defined_VVN as_II the_AT extension_NN1 per_II unit_NN1 length_NN1 ,_, i.e_REX .._... 
It_PPH1 should_VM be_VBI pointed_VVN out_RP ,_, however_RR ,_, that_DD1 other_JJ definitions_NN2 of_IO strain_NN1 will_VM be_VBI met_VVN with_IW in_II the_AT literature_NN1 ,_, most_RGT notably_RR ,_, is_VBZ often_RR called_VVN the_AT true_JJ strain_NN1 ,_, while_CS an_AT1 expression_NN1 arising_VVG from_II the_AT kinetic_JJ theory_NN1 of_IO elasticity_NN1 has_VHZ the_AT form_NN1 Of_RR21 course_RR22 ,_, the_AT extension_NN1 dx_MC will_VM be_VBI accompanied_VVN by_II lateral_JJ contractions_NN2 dy_NN1 and_CC dz_NNU ,_, but_CCB although_CS normally_RR negative_JJ and_CC equal_JJ ,_, they_PPHS2 can_VM usually_RR be_VBI assumed_VVN to_TO be_VBI zero_MC ._. 
For_IF an_AT1 isotropic_JJ body_NN1 ,_, the_AT change_NN1 in_II length_NN1 per_II unit_NN1 length_NN1 is_VBZ related_VVN to_II the_AT change_NN1 in_II width_NN1 per_II unit_NN1 of_IO length_NN1 ,_, such_CS21 that_CS22 where_CS p_ZZ1 is_VBZ known_VVN as_II Poisson_NP1 's_GE ratio_NN1 and_CC varies_VVZ from_II 0.5_MC ,_, when_CS no_AT volume_NN1 change_NN1 occurs_VVZ ,_, to_II about_RG 0.2_MC ._. 
Simple_JJ shear_VV0 ._. 
In_II simple_JJ shear_VV0 the_AT shape_NN1 change_NN1 is_VBZ not_XX accompanied_VVN by_II any_DD change_NN1 in_II volume_NN1 ._. 
If_CS the_AT base_NN1 of_IO the_AT body_NN1 ,_, shown_VVN shaded_JJ in_II the_AT diagram_NN1 ,_, figure_NN1 13.1(b)_FO is_VBZ firmly_RR fixed_VVN ,_, a_AT1 transverse_JJ force_NN1 F_ZZ1 applied_VVN to_II the_AT opposite_JJ face_NN1 is_VBZ sufficient_JJ to_TO cause_VVI a_AT1 deformation_NN1 dx_MC through_II an_AT1 angle_NN1 ._. 
The_AT shear_VV0 modulus_NN1 C_ZZ1 is_VBZ then_RT given_VVN by_II the_AT quotient_NN1 of_IO the_AT shearing_JJ force_NN1 per_II unit_NN1 area_NN1 and_CC the_AT shear_VV0 per_II unit_NN1 distance_NN1 between_II shearing_VVG surfaces_NN2 ;_; and_CC so_RR For_IF very_RG small_JJ shearing_NN1 strains_VVZ tan_JJ and_CC Both_RR E_ZZ1 and_CC G_ZZ1 depend_VV0 on_II the_AT shape_NN1 of_IO the_AT specimen_NN1 and_CC it_PPH1 is_VBZ usually_RR necessary_JJ to_TO define_VVI the_AT shape_NN1 carefully_RR for_IF any_DD measurement_NN1 ._. 
Uniform_JJ compression_NN1 ._. 
When_CS a_AT1 hydrostatic_JJ pressure_NN1 p_ZZ1 is_VBZ applied_VVN to_II a_AT1 body_NN1 of_IO volume_NN1 V_ZZ1 o_ZZ1 ,_, causing_VVG a_AT1 change_NN1 in_II volume_NN1 V_ZZ1 ,_, a_AT1 bulk_NN1 modulus_NN1 B_ZZ1 can_VM be_VBI defined_VVN as_II The_AT quantity_NN1 B_ZZ1 is_VBZ often_RR expressed_VVN in_II31 terms_II32 of_II33 the_AT compressibility_NN1 which_DDQ is_VBZ the_AT reciprocal_JJ of_IO the_AT bulk_NN1 modulus_NN1 ._. 
Similarly_RR and_CC are_VBR known_VVN as_II the_AT tensile_JJ and_CC shear_VV0 compliances_NN2 and_CC given_VVN the_AT symbols_NN2 D_ZZ1 and_CC J_ZZ1 respectively._NNU 13.3_MC Interrelation_NN1 of_IO moduli_NN2 The_AT relations_NN2 given_VVN above_II pertain_NN1 to_II isotropic_JJ bodies_NN2 and_CC for_IF non-isotropic_JJ bodies_NN2 the_AT equations_NN2 are_VBR considerably_RR more_RGR complex_JJ ._. 
Polymeric_JJ materials_NN2 are_VBR normally_RR either_RR amorphous_JJ ,_, or_CC partially_RR crystalline_JJ with_IW randomly_RR oriented_VVD crystallites_NN2 embedded_VVN in_II a_AT1 disordered_JJ matrix_NN1 ._. 
However_RR ,_, any_DD symmetry_NN1 possessed_VVN by_II an_AT1 individual_JJ crystallite_NN1 can_VM be_VBI disregarded_VVN and_CC the_AT body_NN1 as_II a_AT1 whole_NN1 is_VBZ treated_VVN as_CSA being_VBG isotropic_JJ ._. 
The_AT various_JJ moduli_NN2 can_VM be_VBI related_VVN to_II each_PPX221 other_PPX222 in_II a_AT1 simple_JJ manner_NN1 ,_, because_CS an_AT1 isotropic_JJ body_NN1 is_VBZ considered_VVN to_TO possess_VVI only_RR two_MC independent_JJ elastic_JJ constants_NN2 and_CC so_RR This_DD1 indicates_VVZ ,_, that_DD1 for_IF an_AT1 incompressible_JJ elastic_JJ solid_JJ ,_, i.e._REX one_PN1 having_VHG a_AT1 Poisson_NP1 ratio_NN1 of_IO 0.5_MC ,_, Young_NP1 's_GE modulus_NN1 is_VBZ three_MC times_NNT2 larger_JJR than_CSN the_AT shear_VV0 modulus_NN1 ._. 
These_DD2 moduli_NN2 have_VH0 dimensions_NN2 of_IO pressure_NN1 and_CC typical_JJ values_NN2 for_IF several_DA2 polymeric_JJ and_CC non-polymeric_JJ materials_NN2 can_VM be_VBI compared_VVN at_II ambient_JJ temperatures_NN2 in_II table_NN1 13.1_MC ._. 
The_AT response_NN1 of_IO polymers_NN2 to_II mechanical_JJ stresses_NN2 can_VM vary_VVI widely_RR ,_, and_CC depends_VVZ on_II the_AT particular_JJ state_NN1 the_AT polymer_NN1 is_VBZ in_RP at_II any_DD given_JJ temperature._NNU 13.4_MC Mechanical_JJ models_NN2 describing_VVG viscoelasticity_NN1 A_ZZ1 perfectly_RR elastic_JJ material_NN1 obeying_VVG Hooke_NP1 's_GE law_NN1 behaves_VVZ like_II a_AT1 perfect_JJ spring_NN1 ._. 
The_AT stress-strain_JJ diagram_NN1 is_VBZ shown_VVN in_II figure_NN1 13.2(a)_FO ,_, and_CC can_VM be_VBI represented_VVN in_II mechanical_JJ terms_NN2 by_II the_AT model_NN1 of_IO a_AT1 weightless_JJ spring_NN1 whose_DDQGE modulus_NN1 of_IO extension_NN1 represents_VVZ the_AT modulus_NN1 of_IO the_AT material_NN1 ._. 
The_AT application_NN1 of_IO a_AT1 shear_VV0 stress_NN1 to_II a_AT1 viscous_JJ liquid_NN1 on_II the_AT other_JJ hand_NN1 ,_, is_VBZ relieved_VVN by_II viscous_JJ flow_NN1 ,_, and_CC for_IF small_JJ values_NN2 of_IO s_ZZ1 can_VM be_VBI described_VVN by_II Newton_NP1 's_GE law_NN1 where_RRQ is_VBZ the_AT coefficient_NN1 of_IO viscosity_NN1 and_CC is_VBZ the_AT rate_NN1 of_IO shear_VV0 sometimes_RT denoted_VVD by_RP ._. 
As_CSA stress_NN1 is_VBZ now_RT independent_JJ of_IO the_AT strain_NN1 the_AT form_NN1 of_IO the_AT diagram_NN1 changes_NN2 and_CC can_VM be_VBI represented_VVN by_II a_AT1 dashpot_NN1 which_DDQ is_VBZ a_AT1 loose_JJ fitting_JJ piston_NN1 in_II a_AT1 cylinder_NN1 containing_VVG a_AT1 liquid_NN1 of_IO viscosity_NN1 ._. 
(_( Figure_NN1 13.2(b)_FO ._. )_) 
Comparison_NN1 of_IO the_AT two_MC models_NN2 shows_VVZ that_CST the_AT spring_NN1 represents_VVZ a_AT1 system_NN1 storing_VVG energy_NN1 which_DDQ is_VBZ recoverable_JJ ,_, whereas_CS the_AT dashpot_NN1 represents_VVZ the_AT dissipation_NN1 of_IO energy_NN1 in_II the_AT form_NN1 of_IO heat_NN1 by_II a_AT1 viscous_JJ material_NN1 subjected_VVN to_II a_AT1 deforming_JJ force_NN1 ._. 
The_AT dashpot_NN1 is_VBZ used_VVN to_TO denote_VVI the_AT retarded_JJ nature_NN1 of_IO the_AT response_NN1 of_IO a_AT1 material_NN1 to_II any_DD applied_JJ stress_NN1 ._. 
Because_II21 of_II22 their_APPGE chain-like_JJ structure_NN1 ,_, polymers_NN2 are_VBR not_XX perfectly_RR elastic_JJ bodies_NN2 and_CC deformation_NN1 is_VBZ accompanied_VVN by_II a_AT1 complex_JJ series_NN of_IO long_JJ and_CC short_JJ range_NN1 co-operative_JJ molecular_JJ rearrangements_NN2 ._. 
Consequently_RR ,_, the_AT mechanical_JJ behaviour_NN1 is_VBZ dominated_VVN by_II viscoelastic_JJ phenomena_NN2 ,_, in_II contrast_NN1 to_II materials_NN2 such_II21 as_II22 metal_NN1 and_CC glass_NN1 where_CS atomic_JJ adjustments_NN2 under_II stress_NN1 are_VBR more_RGR localized_JJ and_CC limited_JJ ._. 
The_AT Maxwell_NP1 model_NN1 ._. 
One_MC1 of_IO the_AT first_MD attempts_NN2 to_TO explain_VVI the_AT mechanical_JJ behaviour_NN1 of_IO materials_NN2 such_II21 as_II22 pitch_NN1 and_CC tar_NN1 was_VBDZ made_VVN by_II James_NP1 Clark_NP1 Maxwell_NP1 ._. 
He_PPHS1 argued_VVD that_CST when_CS a_AT1 material_NN1 can_VM undergo_VVI viscous_JJ flow_NN1 and_CC also_RR respond_VV0 elastically_RR to_II a_AT1 stress_NN1 it_PPH1 should_VM be_VBI described_VVN by_II a_AT1 combination_NN1 of_IO both_DB2 the_AT Newton_NP1 and_CC Hooke_NP1 laws_NN2 ._. 
This_DD1 assumes_VVZ that_CST both_DB2 contributions_NN2 to_II the_AT strain_NN1 are_VBR additive_JJ so_CS that_DD1 ._. 
Expressing_VVG this_DD1 as_II the_AT differential_JJ equation_NN1 leads_VVZ to_II the_AT equation_NN1 of_IO motion_NN1 of_IO a_AT1 Maxwell_NP1 unit_NN1 Under_II conditions_NN2 of_IO constant_JJ shear_VV0 strain_VV0 the_AT relation_NN1 becomes_VVZ and_CC if_CS the_AT boundary_NN1 condition_NN1 is_VBZ assumed_VVN that_CST at_II zero_MC time_NNT1 ,_, the_AT solution_NN1 to_II this_DD1 equation_NN1 is_VBZ where_RRQ o_ZZ1 is_VBZ the_AT initial_JJ stress_NN1 immediately_RR after_II stretching_VVG the_AT polymer_NN1 ._. 
This_DD1 shows_VVZ that_CST when_CS a_AT1 Maxwell_NP1 element_NN1 is_VBZ held_VVN at_II a_AT1 fixed_JJ shear_VV0 strain_NN1 ,_, the_AT shearing_JJ stress_NN1 will_VM relax_VVI exponentially_RR with_IW time_NNT1 ._. 
At_II a_AT1 time_NNT1 the_AT stress_NN1 is_VBZ reduced_VVN to_II times_NNT2 the_AT original_JJ value_NN1 and_CC this_DD1 characteristic_JJ time_NNT1 is_VBZ known_VVN as_II the_AT relaxation_NN1 time_NNT1 ._. 
The_AT equations_NN2 can_VM be_VBI generalized_VVN for_IF both_DB2 shear_VV0 and_CC tension_NN1 and_CC G_ZZ1 can_VM be_VBI replaced_VVN by_II E._NP1 The_AT mechanical_JJ analogue_NN1 for_IF the_AT Maxwell_NP1 unit_NN1 can_VM be_VBI represented_VVN by_II a_AT1 combination_NN1 of_IO a_AT1 spring_NN1 and_CC a_AT1 dashpot_NN1 arranged_VVN in_II series_NN so_CS21 that_CS22 the_AT stress_NN1 is_VBZ the_AT same_DA on_II both_DB2 elements_NN2 ._. 
This_DD1 means_VVZ that_CST the_AT total_JJ strain_NN1 is_VBZ the_AT sum_NN1 of_IO the_AT strains_NN2 on_II each_DD1 element_NN1 as_CSA expressed_VVN by_II equation_NN1 (_( 13.7_MC )_) ._. 
A_AT1 typical_JJ stress-strain_JJ curve_NN1 predicted_VVN by_II the_AT Maxwell_NP1 model_NN1 ,_, is_VBZ shown_VVN in_II figure_NN1 13.3(a)_FO ._. 
Under_II conditions_NN2 of_IO constant_JJ stress_NN1 ,_, a_AT1 Maxwell_NP1 body_NN1 shows_VVZ instantaneous_JJ elastic_JJ deformation_NN1 first_MD ,_, followed_VVN by_II a_AT1 viscous_JJ flow_NN1 ._. 
Voigt-Kelvin_JJ model_NN1 ._. 
A_AT1 second_MD simple_JJ mechanical_JJ model_NN1 can_VM be_VBI constructed_VVN from_II the_AT ideal_JJ elements_NN2 by_II placing_VVG a_AT1 spring_NN1 and_CC dashpot_VV0 in_II parallel_NN1 ._. 
This_DD1 is_VBZ known_VVN as_II a_AT1 Voigt_NP1 Kelvin_NP1 model_NN1 ._. 
Any_DD applied_JJ stress_NN1 is_VBZ now_RT shared_VVN between_II the_AT elements_NN2 and_CC each_DD1 is_VBZ subjected_VVN to_II the_AT same_DA deformation_NN1 ._. 
The_AT corresponding_JJ expression_NN1 for_IF strain_NN1 is_VBZ Here_RL is_VBZ known_VVN as_II the_AT retardation_NN1 time_NNT1 and_CC is_VBZ a_AT1 measure_NN1 of_IO the_AT time_NNT1 delay_NN1 in_II the_AT strain_NN1 after_II imposition_NN1 of_IO the_AT stress_NN1 ._. 
For_IF high_JJ values_NN2 of_IO the_AT viscosity_NN1 ,_, the_AT retardation_NN1 time_NNT1 is_VBZ long_JJ and_CC this_DD1 represents_VVZ the_AT length_NN1 of_IO time_NNT1 the_AT model_NN1 takes_VVZ to_TO attain_VVI or_CC 0.632_MC of_IO the_AT equilibrium_NN1 elongation_NN1 ._. 
Such_DA models_NN2 are_VBR much_RR too_RG simple_JJ to_TO describe_VVI the_AT complex_JJ viscoelastic_JJ behaviour_NN1 of_IO a_AT1 polymer_NN1 ,_, nor_CC do_VD0 they_PPHS2 provide_VVI any_DD real_JJ insight_NN1 into_II the_AT molecular_JJ mechanism_NN1 of_IO the_AT process_NN1 ,_, but_CCB in_II certain_JJ instances_NN2 they_PPHS2 can_VM prove_VVI useful_JJ in_II assisting_VVG the_AT understanding_NN1 of_IO the_AT viscoelastic_JJ process._NNU 13.5_MC Linear_JJ viscoelastic_JJ behaviour_NN1 of_IO amorphous_JJ polymers_NN2 A_ZZ1 polymer_NN1 can_VM possess_VVI a_AT1 wide_JJ range_NN1 of_IO material_NN1 properties_NN2 and_CC of_IO these_DD2 the_AT hardness_NN1 ,_, deformability_NN1 ,_, toughness_NN1 ,_, and_CC ultimate_JJ strength_NN1 ,_, are_VBR amongst_II the_AT most_RGT significant_JJ ._. 
Certain_JJ features_NN2 ,_, such_II21 as_II22 high_JJ rigidity_NN1 (_( modulus_NN1 )_) and_CC impact_NN1 strength_NN1 ,_, combined_VVN with_IW low_JJ creep_NN1 characteristics_NN2 are_VBR desirable_JJ in_II a_AT1 polymer_NN1 if_CS eventually_RR it_PPH1 is_VBZ to_TO be_VBI subjected_VVN to_II loading_NN1 ._. 
Unfortunately_RR ,_, these_DD2 are_VBR conflicting_JJ properties_NN2 ,_, as_CSA a_AT1 polymer_NN1 with_IW a_AT1 high_JJ modulus_NN1 and_CC low_JJ creep_NN1 response_NN1 does_VDZ not_XX absorb_VVI energy_NN1 by_II deforming_VVG easily_RR ,_, hence_RR has_VHZ poor_JJ impact_NN1 strength_NN1 ._. 
This_DD1 means_VVZ a_AT1 compromise_NN1 must_VM be_VBI sought_VVN depending_II21 on_II22 the_AT use_NN1 to_II which_DDQ the_AT polymer_NN1 will_VM be_VBI put_VVN ,_, and_CC this_DD1 requires_VVZ a_AT1 knowledge_NN1 of_IO the_AT mechanical_JJ response_NN1 in_II detail_NN1 ._. 
The_AT early_JJ work_NN1 on_II viscoelasticity_NN1 was_VBDZ performed_VVN on_II silk_NN1 ,_, rubber_NN1 ,_, and_CC glass_NN1 ,_, and_CC it_PPH1 was_VBDZ concluded_VVN that_CST these_DD2 materials_NN2 exhibited_VVD a_AT1 '_GE delayed_JJ elasticity_NN1 '_GE manifest_JJ in_II the_AT observation_NN1 ,_, that_CST the_AT imposition_NN1 of_IO a_AT1 stress_NN1 resulted_VVN in_II an_AT1 instantaneous_JJ strain_NN1 which_DDQ continued_VVD to_TO increase_VVI more_RGR slowly_RR ,_, with_IW time_NNT1 ._. 
It_PPH1 is_VBZ this_DD1 delay_NN1 between_II cause_NN1 and_CC effect_VV0 that_DD1 is_VBZ fundamental_JJ to_II the_AT observed_JJ viscoelastic_JJ response_NN1 and_CC the_AT three_MC major_JJ examples_NN2 of_IO this_DD1 hysteresis_NN1 effect_NN1 are_VBR (_( 1_MC1 )_) Creep_VV0 ,_, where_CS there_EX is_VBZ a_AT1 delayed_JJ strain_NN1 response_NN1 after_II the_AT rapid_JJ application_NN1 of_IO a_AT1 stress_NN1 ,_, (_( 2_MC )_) Stress-relaxation_NN1 (_( section_NN1 13.7_MC )_) in_II which_DDQ the_AT material_NN1 is_VBZ quickly_RR subjected_VVN to_II a_AT1 strain_NN1 and_CC a_AT1 subsequent_JJ decay_NN1 of_IO stress_NN1 is_VBZ observed_VVN ,_, and_CC (_( 3_MC )_) Dynamic_JJ response_NN1 (_( section_NN1 13.9_MC )_) of_IO a_AT1 body_NN1 to_II the_AT imposition_NN1 of_IO a_AT1 steady_JJ sinusoidal_JJ stress_NN1 ._. 
This_DD1 produces_VVZ a_AT1 strain_NN1 oscillating_VVG with_IW the_AT same_DA frequency_NN1 as_CSA ,_, but_CCB out_II21 of_II22 phase_NN1 with_IW ,_, the_AT stress_NN1 ._. 
For_IF maximum_JJ usefulness_NN1 ,_, these_DD2 measurements_NN2 must_VM be_VBI carried_VVN out_RP over_II a_AT1 wide_JJ range_NN1 of_IO temperature_NN1 ._. 
CREEP_VV0 To_TO be_VBI of_IO any_DD practical_JJ use_NN1 ,_, an_AT1 object_NN1 made_VVN from_II a_AT1 polymeric_JJ material_NN1 must_VM be_VBI able_JK to_TO retain_VVI its_APPGE shape_NN1 when_CS subjected_VVN to_II even_RR small_JJ tensions_NN2 or_CC compressions_NN2 over_II long_JJ periods_NN2 of_IO time_NNT1 ._. 
This_DD1 dimensional_JJ stability_NN1 is_VBZ an_AT1 important_JJ consideration_NN1 in_II choosing_VVG a_AT1 polymer_NN1 to_TO use_VVI in_II the_AT manufacture_NN1 of_IO an_AT1 item_NN1 ._. 
No_PN121 one_PN122 wants_VVZ a_AT1 plastic_NN1 telephone_NN1 receiver_NN1 which_DDQ sags_VVZ after_II sitting_VVG in_II its_APPGE cradle_NN1 for_IF several_DA2 weeks_NNT2 ,_, or_CC a_AT1 car_NN1 tyre_NN1 that_CST develops_VVZ a_AT1 flat_JJ spot_NN1 if_CS parked_VVN in_II one_MC1 position_NN1 for_IF too_RG long_RR ,_, or_CC clothes_NN2 made_VVN from_II synthetic_JJ fibres_NN2 which_DDQ become_VV0 baggy_JJ and_CC deformed_JJ after_II short_JJ periods_NN2 of_IO wear_NN1 ._. 
Creep_VV0 tests_NN2 provide_VV0 a_AT1 measure_NN1 of_IO this_DD1 tendency_NN1 to_TO deform_VVI and_CC are_VBR relatively_RR easy_JJ to_TO carry_VVI out_RP ._. 
Creep_NN1 can_VM be_VBI defined_VVN as_II a_AT1 progressive_JJ increase_NN1 in_II strain_NN1 ,_, observed_VVD over_RP an_AT1 extended_JJ time_NNT1 period_NN1 ,_, in_II a_AT1 polymer_NN1 subjected_VVN to_II a_AT1 constant_JJ stress_NN1 ._. 
Measurements_NN2 are_VBR carried_VVN out_RP on_II a_AT1 sample_NN1 clamped_VVN in_II a_AT1 thermostat_NN1 ._. 
A_AT1 constant_JJ load_NN1 is_VBZ firmly_RR fixed_VVN to_II one_MC1 end_NN1 and_CC the_AT elongation_NN1 is_VBZ followed_VVN by_II measuring_VVG the_AT relative_JJ movement_NN1 of_IO two_MC fiducial_JJ marks_NN2 ,_, made_VVD initially_RR on_II the_AT polymer_NN1 ,_, as_CSA a_AT1 function_NN1 of_IO time_NNT1 ._. 
To_TO avoid_VVI excessive_JJ changes_NN2 in_II the_AT sample_NN1 cross_NN1 section_NN1 ,_, elongations_NN2 are_VBR limited_VVN to_II a_AT1 few_DA2 per_NNU21 cent_NNU22 and_CC are_VBR followed_VVN over_RP approximately_RR three_MC decades_NNT2 of_IO time_NNT1 ._. 
The_AT initial_NN1 ,_, almost_RR instantaneous_JJ ,_, elongation_NN1 produced_VVN by_II the_AT application_NN1 of_IO the_AT tensile_JJ stress_NN1 is_VBZ inversely_RR proportional_JJ to_II the_AT rigidity_NN1 or_CC modulus_NN1 of_IO the_AT material_NN1 ,_, i.e._REX an_AT1 elastomer_NN1 with_IW a_AT1 low_JJ modulus_NN1 stretches_VVZ considerably_RR more_DAR than_CSN a_AT1 material_NN1 in_II the_AT glassy_JJ state_NN1 with_IW a_AT1 high_JJ modulus_NN1 ._. 
The_AT initial_JJ deformation_NN1 corresponds_VVZ to_II portion_NN1 OA_NP1 of_IO the_AT curve_NN1 (_( figure_NN1 13.4_MC )_) ,_, increment_VV0 a_AT1 ._. 
This_DD1 rapid_JJ response_NN1 is_VBZ followed_VVN by_II a_AT1 region_NN1 of_IO creep_NN1 ,_, A_ZZ1 to_II B_ZZ1 ,_, initially_RR fast_JJ but_CCB eventually_RR slowing_VVG down_RP to_II a_AT1 constant_JJ rate_NN1 represented_VVN by_II the_AT section_NN1 B_ZZ1 to_II C._NP1 When_CS the_AT stress_NN1 is_VBZ removed_VVN the_AT instantaneous_JJ elastic_JJ response_NN1 OA_NP1 is_VBZ completely_RR recovered_VVN and_CC the_AT curve_NN1 drops_VVZ from_II C_NP1 to_II D_NP1 ,_, i.e._REX the_AT distance_NN1 ._. 
There_EX follows_VVZ a_AT1 slower_JJR recovery_NN1 in_II the_AT region_NN1 D_ZZ1 to_II E_ZZ1 which_DDQ is_VBZ never_RR complete_VV0 ,_, falling_VVG short_JJ of_IO the_AT initial_JJ state_NN1 by_II an_AT1 increment_NN1 ._. 
This_DD1 is_VBZ a_AT1 measure_NN1 of_IO the_AT viscous_JJ flow_NN1 experienced_VVN by_II the_AT sample_NN1 and_CC is_VBZ a_AT1 completely_RR non-recoverable_JJ response_NN1 ._. 
If_CS the_AT tensile_JJ load_NN1 is_VBZ enlarged_VVN ,_, both_DB2 the_AT elongation_NN1 and_CC the_AT creep_NN1 rate_NN1 increase_NN1 ,_, so_CS results_NN2 are_VBR usually_RR reported_VVN in_II31 terms_II32 of_II33 the_AT creep_NN1 compliance_NN1 J(t)_NP1 ,_, defined_VVN as_II the_AT ratio_NN1 of_IO the_AT relative_JJ elongation_NN1 y_ZZ1 at_II time_NNT1 t_ZZ1 to_II the_AT stress_NN1 so_CS21 that_CS22 At_II low_JJ loads_NN2 J(t)_NP1 is_VBZ independent_JJ of_IO the_AT load_NN1 ._. 
This_DD1 idealized_JJ picture_NN1 of_IO creep_NN1 behaviour_NN1 in_II a_AT1 polymer_NN1 has_VHZ its_APPGE mechanical_JJ equivalent_NN1 constructed_VVN from_II the_AT springs_NN2 and_CC dashpots_NN2 described_VVN earlier_RRR ._. 
The_AT changes_NN2 a_AT1 and_CC a_AT1 correspond_VV0 to_II the_AT elastic_JJ response_NN1 of_IO the_AT polymer_NN1 and_CC so_RR we_PPIS2 can_VM begin_VVI with_IW a_AT1 Hookean_JJ spring_NN1 ._. 
The_AT Voigt-Kelvin_JJ model_NN1 is_VBZ embodied_VVN in_II equation_NN1 (_( 13.11_MC )_) and_CC this_DD1 reproduces_VVZ the_AT changes_NN2 b_ZZ1 and_CC b_ZZ1 ._. 
The_AT final_JJ changes_NN2 c_ZZ1 and_CC c_ZZ1 represent_VV0 viscous_JJ flow_NN1 and_CC can_VM be_VBI represented_VVN by_II a_AT1 dashpot_NN1 so_CS21 that_CS22 the_AT whole_JJ model_NN1 is_VBZ a_AT1 four_MC element_NN1 model_NN1 figure_NN1 13.5_MC ._. 
The_AT behaviour_NN1 can_VM be_VBI explained_VVN in_II the_AT following_JJ series_NN of_IO steps_NN2 ._. 
In_II diagram_NN1 (_( i_ZZ1 )_) the_AT system_NN1 is_VBZ at_II rest_NN1 ._. 
The_AT stress_NN1 is_VBZ applied_VVN to_TO spring_VVI E_ZZ1 1_MC1 and_CC dashpot_VVI 3_MC ;_; it_PPH1 is_VBZ also_RR shared_VVN by_II E_ZZ1 2_MC and_CC 2_MC but_CCB in_II a_AT1 manner_NN1 which_DDQ varies_VVZ with_IW time_NNT1 ._. 
In_II diagram_NN1 (_( ii_MC )_) ,_, representing_VVG zero_MC time_NNT1 ,_, the_AT spring_NN1 E_ZZ1 1_MC1 extends_VVZ by_II an_AT1 amount_NN1 ._. 
This_DD1 is_VBZ followed_VVN by_II a_AT1 decreasing_JJ rate_NN1 of_IO creep_NN1 with_IW a_AT1 progressively_RR increasing_JJ amount_NN1 of_IO stress_NN1 being_VBG carried_VVN by_II E_ZZ1 2_MC until_CS eventually_RR none_PN is_VBZ carried_VVN by_II 2_MC and_CC E_ZZ1 2_MC is_VBZ fully_RR extended_VVN diagram_NN1 (_( iii_MC )_) ._. 
Such_DA behaviour_NN1 is_VBZ described_VVN by_II where_RRQ the_AT retardation_NN1 time_NNT1 R_ZZ1 provides_VVZ a_AT1 measure_NN1 of_IO the_AT time_NNT1 required_VVN for_IF E_ZZ1 2_MC and_CC 2_MC to_TO reach_VVI 0.632_MC of_IO their_APPGE total_JJ deformation_NN1 ._. 
A_AT1 considerably_RR longer_JJR time_NNT1 is_VBZ required_VVN for_IF complete_JJ deformation_NN1 to_TO occur_VVI ._. 
When_CS spring_NN1 E_ZZ1 2_MC is_VBZ fully_RR extended_VVN the_AT creep_NN1 attains_VVZ a_AT1 constant_JJ rate_NN1 corresponding_VVG to_II movement_NN1 in_II the_AT dashpot_NN1 3_MC ._. 
Viscous_JJ flow_NN1 continues_VVZ and_CC the_AT dashpot_NN1 3_MC is_VBZ deformed_JJ until_CS the_AT stress_NN1 is_VBZ removed_VVN ._. 
At_II that_DD1 time_NNT1 ,_, E_ZZ1 1_MC1 retracts_VVZ quickly_RR along_II section_NN1 a_AT1 and_CC a_AT1 period_NN1 of_IO recovery_NN1 ensues_VVZ (_( b_ZZ1 )_) ._. 
During_II this_DD1 time_NNT1 spring_NN1 E_ZZ1 2_MC forces_NN2 the_AT dashpot_NN1 plunger_VV0 in_II 2_MC back_NN1 to_II its_APPGE original_JJ position_NN1 ._. 
As_II no_AT force_NN1 acts_VVZ on_II 3_MC it_PPH1 remains_VVZ in_II the_AT extended_JJ state_NN1 ,_, and_CC corresponds_VVZ to_II the_AT non-recoverable_JJ viscous_JJ flow_NN1 ;_; region_NN1 ._. 
The_AT system_NN1 is_VBZ then_RT as_CSA shown_VVN in_II diagram_NN1 (_( v_ZZ1 )_) ._. 
In_II practice_NN1 ,_, a_AT1 substance_NN1 possesses_VVZ a_AT1 large_JJ number_NN1 of_IO retardation_NN1 times_NNT2 which_DDQ can_VM be_VBI expressed_VVN as_II a_AT1 distribution_NN1 function_NN1 where_RRQ To_II the_AT first_MD approximation_NN1 ,_, this_DD1 is_VBZ estimated_VVN from_II a_AT1 plot_NN1 of_IO creep_NN1 compliance_NN1 against_II ,_, and_CC is_VBZ the_AT contribution_NN1 from_II viscous_JJ flow_NN1 ._. 
STRESS-STRAIN_JJ MEASUREMENTS_NN2 The_AT data_NN derived_VVN from_II stress-strain_JJ measurements_NN2 on_II thermoplastics_NN2 are_VBR important_JJ from_II a_AT1 practical_JJ viewpoint_NN1 ,_, providing_VVG as_CSA they_PPHS2 do_VD0 ,_, information_NN1 on_II the_AT modulus_NN1 ,_, the_AT brittleness_NN1 ,_, and_CC the_AT ultimate_JJ and_CC yield_VV0 strengths_NN2 of_IO the_AT polymer_NN1 ._. 
By_II subjecting_VVG the_AT specimen_NN1 to_II a_AT1 tensile_JJ force_NN1 applied_VVN at_II a_AT1 uniform_JJ rate_NN1 and_CC measuring_VVG the_AT resulting_JJ deformation_NN1 ,_, a_AT1 curve_NN1 of_IO the_AT type_NN1 shown_VVN in_II figure_NN1 13.6_MC can_VM be_VBI constructed_VVN ._. 
The_AT shape_NN1 of_IO such_DA a_AT1 curve_NN1 is_VBZ dependent_JJ on_II the_AT rate_NN1 of_IO testing_NN1 ,_, consequently_RR ,_, this_DD1 must_VM be_VBI specified_VVN if_CS a_AT1 meaningful_JJ comparison_NN1 of_IO data_NN is_VBZ to_TO be_VBI made_VVN ._. 
The_AT initial_JJ portion_NN1 of_IO the_AT curve_NN1 is_VBZ linear_JJ and_CC the_AT tensile_JJ modulus_NN1 E_ZZ1 is_VBZ obtained_VVN from_II its_APPGE slope_NN1 ._. 
The_AT point_NN1 L_ZZ1 represents_VVZ the_AT stress_NN1 beyond_II which_DDQ a_AT1 brittle_JJ material_NN1 will_NN1 fracture_NN1 ,_, and_CC the_AT area_NN1 under_II the_AT curve_NN1 to_II this_DD1 point_NN1 is_VBZ proportional_JJ to_II the_AT energy_NN1 required_VVN for_IF brittle_JJ fracture_NN1 ._. 
If_CS the_AT material_NN1 is_VBZ tough_JJ no_AT fracture_NN1 occurs_VVZ ,_, and_CC the_AT curve_NN1 then_RT passes_VVZ through_II a_AT1 maximum_NN1 or_CC inflection_NN1 point_NN1 Y_ZZ1 ,_, known_VVN as_II the_AT yield_NN1 point_NN1 ._. 
Beyond_II this_DD1 ,_, the_AT ultimate_JJ elongation_NN1 is_VBZ eventually_RR reached_VVN and_CC the_AT polymer_NN1 breaks_VVZ at_II B._NP1 The_AT area_NN1 under_II this_DD1 part_NN1 of_IO the_AT curve_NN1 is_VBZ the_AT energy_NN1 required_VVN for_IF tough_JJ fracture_NN1 to_TO take_VVI place_NN1 ._. 
EFFECT_NN1 OF_IO TEMPERATURE_NN1 ON_II STRESS-STRAIN_JJ RESPONSE_NN1 Polymers_NN2 such_II21 as_II22 polystyrene_NN1 and_CC poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) with_IW a_AT1 high_JJ E_ZZ1 at_II ambient_JJ temperatures_NN2 fall_VV0 into_II the_AT category_NN1 of_IO hard_JJ brittle_JJ materials_NN2 which_DDQ break_VV0 before_II point_NN1 Y_ZZ1 is_VBZ reached_VVN ._. 
Hard_RR tough_JJ polymers_NN2 can_VM be_VBI typified_VVN by_II cellulose_NN1 acetate_NN1 and_CC several_DA2 curves_NN2 measured_VVN at_II different_JJ temperatures_NN2 are_VBR shown_VVN in_II figure_NN1 13.7(a)_FO ._. 
Stress-strain_JJ curves_NN2 for_IF poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) are_VBR also_RR shown_VVN for_IF comparison_NN1 (_( figure_NN1 13.7(b)_FO )_) ._. 
It_PPH1 can_VM be_VBI seen_VVN that_CST the_AT effect_NN1 of_IO temperature_NN1 on_II the_AT characteristic_JJ shape_NN1 of_IO the_AT curve_NN1 is_VBZ significant_JJ ._. 
As_II the_AT temperature_NN1 increases_VVZ both_RR the_AT rigidity_NN1 and_CC the_AT yield_NN1 strength_NN1 decrease_NN1 while_CS the_AT elongation_NN1 generally_RR increases_VVZ ._. 
For_IF cellulose_NN1 acetate_NN1 there_EX is_VBZ a_AT1 transformation_NN1 from_II a_AT1 hard_JJ brittle_JJ state_NN1 below_RG 273_MC K_ZZ1 to_II a_AT1 softer_JJR but_CCB tougher_JJR type_NN1 of_IO polymer_NN1 at_II temperatures_NN2 above_II 273_MC K._NP1 For_IF poly_NN1 (_( methyl_NN1 methacrylate_NN1 )_) the_AT hard_JJ brittle_JJ characteristics_NN2 are_VBR retained_VVN to_II a_AT1 much_RR higher_JJR temperature_NN1 ,_, but_CCB it_PPH1 eventually_RR reaches_VVZ a_AT1 soft_JJ tough_JJ state_NN1 at_II about_RG 320_MC K._NP1 Thus_RR if_CS the_AT requirements_NN2 of_IO high_JJ rigidity_NN1 and_CC toughness_NN1 are_VBR to_TO be_VBI met_VVN ,_, the_AT temperature_NN1 is_VBZ important_JJ ._. 
Cellulose_VV0 acetate_NN1 meets_VVZ these_DD2 requirements_NN2 if_CS used_VVN at_II 298_MC K_ZZ1 more_RRR satisfactorily_RR than_CSN when_CS used_VVN at_II 350_MC K_ZZ1 where_RRQ the_AT modulus_NN1 is_VBZ smaller_JJR and_CC the_AT ability_NN1 to_TO absorb_VVI energy_NN1 ,_, represented_VVN by_II the_AT area_NN1 under_II the_AT curve_NN1 ,_, is_VBZ also_RR lower._NNU 13.6_MC Boltzmann_NP1 superposition_NN1 principle_NN1 If_CS a_AT1 Hookean_JJ spring_NN1 is_VBZ subjected_VVN to_II a_AT1 series_NN of_IO incremental_JJ stresses_NN2 at_II various_JJ times_NNT2 ,_, the_AT resulting_JJ extensions_NN2 will_VM be_VBI independent_JJ of_IO the_AT loading_NN1 or_CC past_JJ history_NN1 of_IO the_AT spring_NN1 ._. 
A_AT1 Newtonian_JJ dashpot_NN1 also_RR behaves_VVZ in_II a_AT1 predictable_JJ manner_NN1 ._. 
For_IF viscoelastic_JJ materials_NN2 the_AT response_NN1 to_II mechanical_JJ testing_NN1 is_VBZ time_NNT1 dependent_NN1 ,_, but_CCB the_AT behaviour_NN1 at_II any_DD time_NNT1 can_VM be_VBI predicted_VVN by_II applying_VVG a_AT1 superposition_NN1 principle_NN1 proposed_VVN by_II Boltzmann_NP1 ._. 
This_DD1 can_VM be_VBI illustrated_VVN by_II a_AT1 creep_NN1 test_NN1 using_VVG a_AT1 simple_JJ Voigt-Kelvin_JJ model_NN1 with_IW a_AT1 single_JJ retardation_NN1 time_NNT1 R_ZZ1 ,_, placed_VVN initially_RR under_II a_AT1 stress_NN1 o_ZZ1 at_II time_NNT1 t_ZZ1 o_ZZ1 ._. 
If_CS after_CS times_II t_ZZ1 1_MC1 ,_, t_ZZ1 2_MC ,_, t_ZZ1 3_MC ,_, ..._... the_AT system_NN1 is_VBZ subjected_VVN to_II additional_JJ stresses_NN2 1_MC1 ,_, 2_MC ,_, 3_MC ,_, ..._... then_RT the_AT principle_NN1 states_VVZ that_CST the_AT creep_NN1 response_NN1 of_IO the_AT system_NN1 can_VM be_VBI predicted_VVN simply_RR by_II summing_VVG the_AT individual_JJ responses_NN2 from_II each_DD1 stress_NN1 increment_NN1 ._. 
Thus_RR if_CS the_AT stress_NN1 alters_VVZ continually_RR ,_, the_AT summation_NN1 can_VM be_VBI replaced_VVN by_II an_AT1 integral_JJ ,_, and_CC n_ZZ1 by_II a_AT1 continually_RR varying_JJ function_NN1 ,_, so_CS21 that_CS22 at_II time_NNT1 t_ZZ1 *_FU when_RRQ the_AT stress_NN1 existed_VVD ,_, the_AT strain_NN1 is_VBZ given_VVN by_II The_AT principle_NN1 has_VHZ been_VBN applied_VVN successfully_RR to_II the_AT tensile_JJ creep_NN1 of_IO amorphous_JJ and_CC rubber-like_JJ polymers_NN2 ,_, but_CCB it_PPH1 is_VBZ not_XX too_RG successful_JJ if_CS appreciable_JJ crystallinity_NN1 exists_VVZ in_II the_AT sample_NN1 ._. 
Graphical_JJ representation_NN1 of_IO the_AT principle_NN1 is_VBZ shown_VVN in_II figure_NN1 13.8._MC 13.7_MC Stress-relaxation_JJ Stress-relaxation_JJ experiments_NN2 involve_VV0 the_AT measurement_NN1 of_IO the_AT force_NN1 required_VVN to_TO maintain_VVI the_AT deformation_NN1 produced_VVD initially_RR by_II an_AT1 applied_JJ stress_NN1 as_II a_AT1 function_NN1 of_IO time_NNT1 ._. 
Stress-relaxation_JJ tests_NN2 are_VBR not_XX performed_VVN as_RG often_RR as_CSA creep_NN1 tests_NN2 because_CS many_DA2 investigators_NN2 believe_VV0 they_PPHS2 are_VBR less_RGR readily_RR understood_VVN ._. 
The_AT latter_DA point_NN1 is_VBZ debatable_JJ and_CC it_PPH1 may_VM only_RR be_VBI that_CST the_AT practical_JJ aspects_NN2 of_IO creep_NN1 measurements_NN2 are_VBR simpler_JJR ._. 
As_CSA will_VM be_VBI shown_VVN later_RRR ,_, all_DB the_AT mechanical_JJ parameters_NN2 are_VBR in_II theory_NN1 interchangeable_JJ ,_, and_CC so_RR all_DB such_DA measurements_NN2 will_VM contribute_VVI to_II the_AT understanding_NN1 of_IO viscoelastic_JJ theory_NN1 ._. 
While_CS stress-relaxation_JJ measurements_NN2 are_VBR useful_JJ in_II a_AT1 general_JJ study_NN1 of_IO polymeric_JJ behaviour_NN1 ,_, they_PPHS2 are_VBR particularly_RR useful_JJ in_II the_AT evaluation_NN1 of_IO antioxidants_NN2 in_II polymers_NN2 ,_, especially_RR elastomers_NN2 ,_, because_CS measurements_NN2 on_II such_DA systems_NN2 are_VBR relatively_RR easy_JJ to_TO perform_VVI and_CC are_VBR sensitive_JJ to_TO bond_VVI rupture_NN1 in_II the_AT network_NN1 ._. 
Experimental_JJ stress-relaxation_JJ technique_NN1 ._. 
In_II a_AT1 stress-relaxation_JJ experiment_NN1 ,_, the_AT sample_NN1 under_II study_NN1 is_VBZ deformed_JJ by_II a_AT1 rapidly_RR applied_JJ stress_NN1 ._. 
As_II the_AT stress_NN1 is_VBZ normally_RR observed_VVN to_TO reach_VVI a_AT1 maximum_NN1 as_CS31 soon_CS32 as_CS33 the_AT material_NN1 deforms_NN2 and_CC then_RT decreases_VVZ thereafter_RT ,_, it_PPH1 is_VBZ necessary_JJ to_TO alter_VVI this_DD1 continually_RR in_BCL21 order_BCL22 to_TO maintain_VVI a_AT1 constant_JJ deformation_NN1 or_CC measure_VV0 the_AT stress_NN1 that_CST would_VM be_VBI required_VVN to_TO accomplish_VVI this_DD1 operation_NN1 ._. 
The_AT apparatus_NN1 used_VVN varies_VVZ in_II complexity_NN1 with_IW the_AT physical_JJ nature_NN1 of_IO the_AT sample_NN1 ,_, being_VBG simplest_JJT for_IF an_AT1 elastomer_NN1 and_CC becoming_VVG more_RGR sophisticated_JJ when_CS the_AT polymer_NN1 is_VBZ more_RGR rigid_JJ ._. 
One_MC1 type_NN1 of_IO experimental_JJ set_NN1 up_RP is_VBZ shown_VVN in_II figure_NN1 13.9_MC ._. 
The_AT sample_NN1 is_VBZ fixed_VVN in_II position_NN1 by_II31 means_II32 of_II33 clamps_NN2 ,_, one_PN1 being_VBG attached_VVN to_II a_AT1 spring_NN1 beam_VV0 above_RL and_CC the_AT other_JJ to_II an_AT1 adjustable_JJ rod_NN1 R_ZZ1 below_RL ._. 
A_AT1 stress_NN1 is_VBZ applied_VVN to_II the_AT sample_NN1 by_II rapidly_RR pulling_VVG rod_NN1 R_ZZ1 downwards_RL and_CC clamping_VVG it_PPH1 in_II position_NN1 ._. 
This_DD1 causes_VVZ the_AT beam_NN1 to_TO bend_VVI and_CC the_AT displacement_NN1 is_VBZ measured_VVN by_II31 means_II32 of_II33 a_AT1 strain_NN1 gauge_NN1 or_CC a_AT1 differential_JJ transformer_NN1 ._. 
The_AT beam_NN1 deflection_NN1 is_VBZ then_RT fed_VVN to_II a_AT1 recorder_NN1 and_CC a_AT1 trace_NN1 of_IO stress_NN1 against_II time_NNT1 is_VBZ obtained_VVN ._. 
The_AT results_NN2 are_VBR expressed_VVN as_II a_AT1 relaxation_NN1 modulus_NN1 which_DDQ is_VBZ a_AT1 function_NN1 of_IO the_AT time_NNT1 of_IO observation_NN1 ._. 
Typical_JJ data_NN for_IF polyisobutylene_NN1 are_VBR shown_VVN in_II section_NN1 13.14_MC ,_, figure_NN1 13.21_MC ,_, where_CS the_AT logarithm_NN1 of_IO the_AT relaxation_NN1 modulus_NN1 log_NN1 is_VBZ plotted_VVN against_II log_NN1 t_ZZ1 ._. 
From_II the_AT curves_NN2 it_PPH1 can_VM be_VBI seen_VVN that_CST there_EX is_VBZ a_AT1 rapid_JJ change_NN1 in_II log_NN1 over_II a_AT1 narrow_JJ range_NN1 of_IO temperature_NN1 corresponding_VVG to_II the_AT glass_NN1 transition_NN1 ._. 
Again_RT a_AT1 simple_JJ model_NN1 with_IW a_AT1 single_JJ relaxation_NN1 time_NNT1 is_VBZ too_RG crude_JJ ,_, and_CC the_AT stress_NN1 relaxation_NN1 modulus_NN1 is_VBZ better_RRR represented_VVN by_II where_RRQ is_VBZ the_AT distribution_NN1 function_NN1 of_IO relaxation_NN1 times_NNT2 ._. 
This_DD1 is_VBZ suitable_JJ for_IF a_AT1 linear_JJ polymer_NN1 but_CCB requires_VVZ the_AT additional_JJ term_NN1 E_ZZ1 ,_, if_CS the_AT material_NN1 is_VBZ crosslinked._NNU 13.8_MC Dynamic_JJ mechanical_JJ and_CC dielectric_JJ thermal_JJ analysis_NN1 Non-destructive_JJ testing_NN1 methods_NN2 are_VBR particularly_RR useful_JJ for_IF assessing_VVG the_AT physical_JJ properties_NN2 of_IO polymeric_JJ materials_NN2 when_RRQ an_AT1 understanding_NN1 of_IO the_AT performance_NN1 at_II a_AT1 molecular_JJ level_NN1 is_VBZ important_JJ ._. 
The_AT foregoing_JJ techniques_NN2 for_IF measuring_VVG mechanical_JJ properties_NN2 are_VBR transient_JJ or_CC non_FU periodic_JJ methods_NN2 and_CC typically_RR cover_VV0 time_NNT1 intervals_NN2 of_IO up_RG21 to_RG22 10_MC 6_MC s_ZZ1 ._. 
For_IF information_NN1 relating_VVG to_II short_JJ times_NNT2 ,_, two_MC approaches_VVZ that_CST have_VH0 been_VBN widely_RR used_VVN are_VBR dynamic_JJ mechanical_JJ thermal_JJ analysis_NN1 (_( DMTA_NP1 )_) and_CC dielectric_JJ thermal_JJ analysis_NN1 (_( DETA_NP1 )_) ._. 
These_DD2 are_VBR both_RR particular_JJ kinds_NN2 of_IO relaxation_NN1 spectroscopy_NN1 in_II which_DDQ the_AT sample_NN1 is_VBZ perturbed_VVN by_II a_AT1 sinusoidal_JJ force_NN1 (_( either_RR mechanical_JJ or_CC electrical_JJ )_) and_CC the_AT response_NN1 of_IO the_AT material_NN1 is_VBZ measured_VVN over_II a_AT1 range_NN1 of_IO temperatures_NN2 and_CC at_II different_JJ frequencies_NN2 of_IO the_AT applied_JJ force_NN1 ._. 
From_II an_AT1 analysis_NN1 of_IO the_AT material_NN1 response_NN1 it_PPH1 is_VBZ possible_JJ to_TO derive_VVI information_NN1 about_II the_AT molecular_JJ motions_NN2 in_II the_AT sample_NN1 ,_, and_CC how_RRQ these_DD2 can_VM affect_VVI the_AT modulus_NN1 ,_, damping_VVG characteristics_NN2 and_CC structural_JJ transitions_NN2 ._. 
Both_DB2 techniques_NN2 can_VM be_VBI used_VVN to_TO probe_VVI molecular_JJ motions_NN2 in_II liquid_NN1 or_CC solid_JJ polymers_NN2 ,_, but_CCB when_RRQ dielectric_JJ spectroscopy_NN1 is_VBZ used_VVN the_AT relaxation_NN1 or_CC transition_NN1 must_VM involve_VVI movement_NN1 of_IO a_AT1 dipole_NN1 or_CC a_AT1 charge_NN1 displacement_NN1 if_CS it_PPH1 is_VBZ to_TO be_VBI detected_VVN ._. 
Thus_RR while_CS both_RR DMTA_NP1 and_CC DETA_NN1 can_VM provide_VVI similar_JJ information_NN1 about_II a_AT1 sample_NN1 ,_, they_PPHS2 can_VM also_RR be_VBI used_VVN in_II a_AT1 complementary_JJ fashion_NN1 ,_, particularly_RR when_CS trying_VVG to_TO identify_VVI the_AT molecular_JJ mechanism_NN1 of_IO a_AT1 particular_JJ process_NN1 and_CC in_II ascertaining_VVG whether_CSW31 or_CSW32 not_CSW33 the_AT group_NN1 is_VBZ polar._NNU 13.9_MC Dynamic_JJ mechanical_JJ thermal_JJ analysis_NN1 (_( DMTA_NP1 )_) In_II DMTA_NP1 a_AT1 small_JJ sinusoidal_JJ stress_NN1 is_VBZ imparted_VVN to_II the_AT sample_NN1 in_II the_AT form_NN1 of_IO a_AT1 torque_NN1 ,_, push-pull_JJ ,_, or_CC a_AT1 flexing_JJ mode_NN1 ,_, of_IO angular_JJ frequency_NN1 ._. 
If_CS the_AT polymer_NN1 is_VBZ treated_VVN as_II a_AT1 classical_JJ damped_JJ harmonic_JJ oscillator_NN1 ,_, both_DB2 the_AT elastic_JJ modulus_NN1 and_CC the_AT damping_NN1 characteristics_NN2 can_VM be_VBI obtained_VVN ._. 
Elastic_JJ materials_NN2 convert_VV0 mechanical_JJ work_NN1 into_II potential_JJ energy_NN1 which_DDQ is_VBZ recoverable_JJ ;_; for_REX21 example_REX22 an_AT1 ideal_JJ spring_NN1 ,_, if_CS deformed_JJ by_II a_AT1 stress_NN1 ,_, stores_NN2 the_AT energy_NN1 and_CC uses_VVZ it_PPH1 to_TO recover_VVI its_APPGE original_JJ shape_NN1 after_II removal_NN1 of_IO the_AT stress_NN1 ._. 
No_AT energy_NN1 is_VBZ converted_VVN into_II heat_NN1 during_II the_AT cycle_NN1 and_CC so_RR no_AT damping_NN1 is_VBZ experienced_VVN ._. 
Liquids_NN2 on_II the_AT other_JJ hand_NN1 flow_NN1 if_CS subjected_VVN to_II a_AT1 stress_NN1 ;_; they_PPHS2 do_VD0 not_XX store_VVI the_AT energy_NN1 but_CCB dissipate_VV0 it_PPH1 almost_RR entirely_RR as_CSA heat_NN1 and_CC thus_RR possess_VV0 high_JJ damping_NN1 characteristics_NN2 ._. 
Viscoelastic_JJ polymers_NN2 exhibit_VV0 both_RR elastic_JJ and_CC damping_VVG behaviour_NN1 ._. 
Hence_RR if_CS a_AT1 sinusoidal_JJ stress_NN1 is_VBZ applied_VVN to_II a_AT1 linear_JJ viscoelastic_JJ material_NN1 ,_, the_AT resulting_JJ stress_NN1 will_VM also_RR be_VBI sinusoidal_JJ ,_, but_CCB will_VM be_VBI out_II21 of_II22 phase_NN1 when_CS there_EX is_VBZ energy_NN1 dissipation_NN1 or_CC damping_VVG in_II the_AT polymer_NN1 ._. 
Harmonic_JJ motion_NN1 of_IO a_AT1 Maxwell_NP1 element_NN1 ._. 
The_AT application_NN1 of_IO a_AT1 sinusoidal_JJ stress_NN1 to_II a_AT1 Maxwell_NP1 element_NN1 produces_VVZ a_AT1 strain_NN1 with_IW the_AT same_DA frequency_NN1 as_CSA ,_, but_CCB out_II21 of_II22 phase_NN1 with_IW ,_, the_AT stress_NN1 ._. 
This_DD1 can_VM be_VBI represented_VVN schematically_RR in_II figure_NN1 13.10_MC where_RRQ is_VBZ the_AT phase_NN1 angle_NN1 between_II the_AT stress_NN1 and_CC the_AT strain_NN1 ._. 
The_AT resulting_JJ strain_NN1 can_VM be_VBI described_VVN in_II the_AT terms_NN2 of_IO its_APPGE angular_JJ frequency_NN1 and_CC the_AT maximum_JJ amplitude_NN1 o_ZZ1 using_VVG complex_JJ notation_NN1 ,_, by_II where_RRQ ,_, the_AT frequency_NN1 is_VBZ v_ZZ1 and_CC ._. 
The_AT relation_NN1 between_II the_AT alternating_JJ stress_NN1 and_CC strain_NN1 is_VBZ written_VVN as_II where_RRQ is_VBZ the_AT frequency_NN1 dependent_JJ complex_JJ dynamic_JJ modulus_NN1 defined_VVN by_II This_DD1 shows_VVZ that_CST is_VBZ composed_VVN of_IO two_MC frequency_NN1 dependent_JJ components_NN2 ;_; is_VBZ the_AT real_JJ part_NN1 in_II phase_NN1 with_IW the_AT strain_NN1 called_VVN the_AT storage_NN1 modulus_NN1 ,_, and_CC is_VBZ the_AT loss_NN1 modulus_NN1 defined_VVN as_II the_AT ratio_NN1 of_IO the_AT component_NN1 90_MC out_II21 of_II22 phase_NN1 with_IW the_AT stress_NN1 to_II the_AT stress_NN1 itself_PPX1 ._. 
Hence_RR measures_VVZ the_AT amount_NN1 of_IO stored_JJ energy_NN1 and_CC ,_, sometimes_RT called_VVN the_AT imaginary_JJ part_NN1 ,_, is_VBZ actually_RR a_AT1 real_JJ quantity_NN1 measuring_VVG the_AT amount_NN1 of_IO energy_NN1 dissipated_VVD by_II the_AT material_NN1 ._. 
The_AT response_NN1 is_VBZ often_RR expressed_VVN as_II a_AT1 complex_JJ dynamic_JJ compliance_NN1 especially_RR if_CS a_AT1 generalized_JJ Voigt_NN1 model_NN1 is_VBZ used_VVN ._. 
For_IF a_AT1 Maxwell_NP1 model_NN1 In_II more_RGR realistic_JJ terms_NN2 ,_, there_EX is_VBZ a_AT1 distribution_NN1 of_IO relaxation_NN1 times_NNT2 and_CC a_AT1 continuous_JJ distribution_NN1 function_NN1 can_VM be_VBI derived_VVN ,_, if_CS required_VVN ._. 
The_AT damping_NN1 in_II the_AT system_NN1 or_CC the_AT energy_NN1 loss_NN1 per_II cycle_NN1 can_VM be_VBI measured_VVN from_II the_AT '_GE loss_NN1 tangent_NN1 '_GE tan_NN1 ._. 
This_DD1 is_VBZ a_AT1 measure_NN1 of_IO the_AT internal_JJ friction_NN1 and_CC is_VBZ related_VVN to_II the_AT complex_JJ moduli_NN2 by_II The_AT onset_NN1 of_IO molecular_JJ motion_NN1 in_II a_AT1 polymer_NN1 sample_NN1 is_VBZ reflected_VVN in_II the_AT behaviour_NN1 of_IO E_ZZ1 and_CC E._NP1 A_ZZ1 schematic_JJ diagram_NN1 (_( figure_NN1 13.11_MC )_) of_IO the_AT variation_NN1 of_IO E_ZZ1 and_CC E_ZZ1 as_II a_AT1 function_NN1 of_IO ,_, assuming_VVG only_RR a_AT1 single_JJ value_NN1 for_IF in_II the_AT model_NN1 ,_, shows_VVZ that_CST a_AT1 maximum_NN1 in_II the_AT loss_NN1 angle_NN1 is_VBZ observed_VVN where_RRQ ._. 
This_DD1 represents_VVZ a_AT1 transition_NN1 point_NN1 such_II21 as_II22 T_ZZ1 g_ZZ1 ,_, T_ZZ1 m_ZZ1 or_CC some_DD other_JJ region_NN1 where_CS significant_JJ molecular_JJ motion_NN1 occurs_VVZ in_II the_AT sample_NN1 ._. 
The_AT maximum_NN1 is_VBZ characteristic_JJ of_IO the_AT dynamic_JJ method_NN1 as_II the_AT creep_NN1 and_CC relaxation_NN1 techniques_NN2 merely_RR show_VV0 a_AT1 change_NN1 in_II the_AT modulus_NN1 level._NNU 13.10_MC Experimental_JJ methods_NN2 There_EX are_VBR three_MC main_JJ experimental_JJ approaches_NN2 for_IF measuring_VVG the_AT dynamic_JJ mechanical_JJ properties_NN2 of_IO a_AT1 sample_NN1 ,_, (_( a_ZZ1 )_) free_JJ vibration_NN1 ,_, (_( b_ZZ1 )_) forced_JJ vibration_NN1 resonance_NN1 ,_, (_( c_ZZ1 )_) forced_JJ vibration_NN1 non-resonance_NN1 ._. 
The_AT mechanical_JJ response_NN1 is_VBZ usually_RR determined_VVN at_II low_JJ frequencies_NN2 and_CC over_RP as_RG wide_RR a_AT1 temperature_NN1 range_NN1 as_CSA possible_JJ and_CC examples_NN2 of_IO each_DD1 are_VBR described_VVN in_II the_AT following_JJ section_NN1 ._. 
TORSIONAL_JJ PENDULUM_NN1 FREE_JJ VIBRATION_NN1 A_ZZ1 study_NN1 of_IO the_AT mechanical_JJ damping_NN1 and_CC shear_VV0 modulus_NN1 under_II free_JJ vibration_NN1 can_VM be_VBI made_VVN using_VVG a_AT1 torsional_JJ pendulum_NN1 ._. 
The_AT specimen_NN1 is_VBZ firmly_RR fixed_VVN at_II one_MC1 end_NN1 and_CC the_AT other_JJ end_NN1 is_VBZ clamped_VVN to_II a_AT1 disc_NN1 ,_, with_IW a_AT1 large_JJ moment_NN1 of_IO inertia_NN1 ,_, which_DDQ can_VM move_VVI freely_RR ._. 
As_II the_AT polymer_NN1 sample_NN1 should_VM not_XX be_VBI under_II a_AT1 tensile_JJ stress_NN1 ,_, the_AT suspension_NN1 wire_NN1 supporting_VVG the_AT disc_NN1 is_VBZ passed_VVN over_II a_AT1 pulley_NN1 and_CC the_AT weight_NN1 of_IO the_AT disc_NN1 and_CC sample_NN1 are_VBR counterbalanced_VVN by_II loading_VVG the_AT end_NN1 ._. 
If_CS the_AT disc_NN1 is_VBZ subjected_VVN to_II an_AT1 angular_JJ displacement_NN1 and_CC then_RT released_VVN ,_, the_AT sample_NN1 will_VM twist_VVI backwards_RL and_CC forwards_RL about_II the_AT vertical_JJ axis_NN1 ._. 
The_AT oscillations_NN2 stimulated_VVN in_II the_AT sample_NN1 are_VBR picked_VVN up_RP by_II an_AT1 arm_NN1 attached_VVN to_II the_AT rigidly_RR fixed_JJ end_NN1 held_VVN in_II torsion_NN1 bars_NN2 ,_, and_CC transmitted_VVN to_II a_AT1 recorder_NN1 by_II a_AT1 linear_JJ variable_JJ differential_JJ transformer_NN1 ._. 
The_AT sample_NN1 movements_NN2 are_VBR traced_VVN as_II a_AT1 series_NN of_IO oscillations_NN2 whose_DDQGE frequency_NN1 is_VBZ a_AT1 function_NN1 of_IO the_AT physical_JJ state_NN1 of_IO the_AT sample_NN1 ._. 
The_AT period_NN1 of_IO oscillation_NN1 P_ZZ1 is_VBZ taken_VVN as_II the_AT distance_NN1 between_II adjacent_JJ maxima_NN2 or_CC minima_NN2 and_CC the_AT amplitude_NN1 A_ZZ1 is_VBZ a_AT1 measure_NN1 of_IO the_AT height_NN1 from_II one_MC1 minimum_NN1 to_II the_AT preceding_JJ maximum_NN1 ._. 
The_AT exponential_NN1 decay_NN1 of_IO the_AT amplitude_NN1 along_II the_AT axis_NN1 provides_VVZ an_AT1 indication_NN1 of_IO the_AT mechanical_JJ damping_NN1 ._. 
At_II a_AT1 temperature_NN1 the_AT sample_NN1 absorbs_VVZ most_DAT of_IO the_AT energy_NN1 and_CC damping_NN1 is_VBZ high_JJ whereas_CS at_II a_AT1 much_RR lower_JJR temperature_NN1 the_AT material_NN1 tends_VVZ to_TO store_VVI the_AT energy_NN1 and_CC mechanical_JJ damping_NN1 is_VBZ considerably_RR lower_JJR ._. 
A_AT1 quantitative_JJ measure_NN1 of_IO the_AT damping_NN1 is_VBZ provided_VVN by_II the_AT logarithmic_JJ decrement_NN1 defined_VVN as_II the_AT logarithmic_JJ decrease_NN1 in_II amplitude_NN1 per_II cycle_NN1 ._. 
It_PPH1 is_VBZ calculated_VVN from_II the_AT ratio_NN1 of_IO amplitudes_NN2 of_IO any_DD two_MC successive_JJ oscillations_NN2 using_VVG the_AT relation_NN1 The_AT shear_VV0 modulus_NN1 can_VM also_RR be_VBI derived_VVN from_II the_AT data_NN ,_, being_VBG inversely_RR proportional_JJ to_II the_AT square_NN1 of_IO the_AT period_NN1 ,_, where_RRQ K_ZZ1 is_VBZ a_AT1 factor_NN1 depending_II21 on_II22 the_AT shape_NN1 and_CC the_AT size_NN1 of_IO the_AT sample_NN1 and_CC I_ZZ1 is_VBZ the_AT polar_JJ moment_NN1 of_IO inertia_NN1 ._. 
The_AT method_NN1 can_VM cover_VVI the_AT complete_JJ range_NN1 of_IO moduli_NN2 encountered_VVN in_II polymeric_JJ systems_NN2 but_CCB is_VBZ confined_VVN to_II a_AT1 relatively_RR narrow_JJ frequency_NN1 range_NN1 of_IO 0.01_MC to_II 10_MC Hz_NNU ._. 
VIBRATING_NP1 REED_NP1 RESONANCE_NN1 For_IF resonance_NN1 forced_JJ vibration_NN1 measurements_NN2 a_AT1 sample_NN1 in_II the_AT form_NN1 of_IO a_AT1 thin_JJ strip_NN1 is_VBZ clamped_VVN firmly_RR at_II one_MC1 end_NN1 leaving_VVG the_AT other_JJ end_NN1 free_JJ ._. 
The_AT clamped_JJ end_NN1 of_IO the_AT system_NN1 is_VBZ then_RT vibrated_VVN laterally_RR at_II a_AT1 given_JJ frequency_NN1 v_ZZ1 and_CC the_AT amplitude_NN1 of_IO the_AT vibration_NN1 induced_VVN at_II the_AT free_JJ end_NN1 of_IO the_AT sample_NN1 is_VBZ recorded_VVN ._. 
A_AT1 range_NN1 of_IO frequencies_NN2 wide_RR enough_RR to_TO ensure_VVI that_CST it_PPH1 encompasses_VVZ the_AT resonant_JJ frequency_NN1 of_IO the_AT sample_NN1 v_ZZ1 r_ZZ1 is_VBZ then_RT examined_VVN ._. 
The_AT resonant_JJ frequency_NN1 is_VBZ detected_VVN as_II the_AT maximum_NN1 of_IO a_AT1 graph_NN1 of_IO amplitude_NN1 against_II frequency_NN1 ._. 
The_AT results_NN2 provide_VV0 information_NN1 on_II the_AT elastic_JJ modulus_NN1 E_ZZ1 since_CS it_PPH1 is_VBZ related_VVN to_II the_AT square_NN1 of_IO the_AT resonance_NN1 frequency_NN1 by_II where_RRQ c_ZZ1 is_VBZ a_AT1 numerical_JJ constant_JJ ,_, L_ZZ1 is_VBZ the_AT free_JJ length_NN1 of_IO the_AT sample_NN1 ,_, D_ZZ1 is_VBZ its_APPGE thickness_NN1 ,_, and_CC is_VBZ the_AT sample_NN1 density_NN1 ._. 
If_CS the_AT amplitudes_NN2 are_VBR expressed_VVN as_CSA ratios_NN2 of_IO the_AT amplitude_NN1 to_II the_AT maximum_JJ amplitude_NN1 ,_, then_RT damping_VVG is_VBZ measured_VVN from_II the_AT half-width_NN1 h_ZZ1 of_IO the_AT curve_NN1 ,_, i.e._REX This_DD1 technique_NN1 is_VBZ not_XX as_RG useful_JJ as_CSA the_AT torsional_JJ pendulum_NN1 but_CCB covers_VVZ the_AT higher_JJR frequency_NN1 range_NN1 10_MC to_II 103_MC Hz_NNU ._. 
FORCED_JJ VIBRATION_NN1 NON-RESONANCE_NN1 Several_DA2 types_NN2 of_IO instrument_NN1 can_VM be_VBI used_VVN for_IF this_DD1 type_NN1 of_IO test_NN1 ,_, and_CC these_DD2 are_VBR usually_RR limited_VVN to_II measurements_NN2 on_II rigid_JJ polymers_NN2 or_CC rubbers_NN2 ._. 
One_MC1 such_DA instrument_NN1 is_VBZ shown_VVN in_II the_AT block_NN1 diagram_NN1 13.14_MC ._. 
The_AT sample_NN1 C_ZZ1 is_VBZ attached_VVN firmly_RR at_II each_DD1 end_NN1 to_II a_AT1 strain_NN1 gauge_NN1 ;_; one_MC1 of_IO these_DD2 is_VBZ a_AT1 force_NN1 transducer_NN1 measuring_VVG the_AT applied_JJ sinusoidal_JJ force_NN1 and_CC the_AT other_JJ records_NN2 the_AT sample_NN1 deformation_NN1 ._. 
A_AT1 sinusoidal_JJ tensile_JJ stress_NN1 of_IO a_AT1 given_JJ frequency_NN1 can_VM be_VBI generated_VVN in_II the_AT vibrator_NN1 A_ZZ1 and_CC if_CS the_AT electrical_JJ vectors_NN2 from_II the_AT force_NN1 and_CC displacement_NN1 are_VBR represented_VVN by_II and_CC then_RT by_II satisfying_VVG the_AT condition_NN1 the_AT tangent_NN1 of_IO the_AT phase_NN1 angle_NN1 between_II the_AT stress_NN1 and_CC the_AT strain_NN1 may_VM be_VBI calculated_VVN from_II This_DD1 operation_NN1 of_IO adjustment_NN1 followed_VVN by_II subtraction_NN1 of_IO the_AT electrical_JJ vectors_NN2 is_VBZ performed_VVN directly_RR in_II the_AT recording_NN1 circuit_NN1 ._. 
The_AT complex_JJ elastic_JJ modulus_NN1 E_ZZ1 *_FU is_VBZ given_VVN by_II where_RRQ F_ZZ1 is_VBZ the_AT amplitude_NN1 of_IO the_AT tensile_JJ force_NN1 ,_, A_ZZ1 is_VBZ the_AT sample_NN1 cross-sectional_JJ area_NN1 ,_, L_ZZ1 is_VBZ the_AT sample_NN1 length_NN1 ,_, and_CC L_ZZ1 is_VBZ the_AT amplitude_NN1 of_IO elongation_NN1 ._. 
Tensile_JJ storage_NN1 and_CC loss_NN1 moduli_NN2 E_ZZ1 and_CC E_ZZ1 follow_VV0 from_II and_CC ._. 
A_AT1 second_MD version_NN1 now_RT widely_RR used_VVN for_IF these_DD2 measurements_NN2 is_VBZ the_AT Polymer_NN1 Laboratories_NN2 DMTA_NN1 instrument_NN1 and_CC a_AT1 schematic_JJ diagram_NN1 of_IO the_AT working_JJ head_NN1 is_VBZ shown_VVN in_II figure_NN1 13.15_MC ._. 
Several_DA2 damping_NN1 arrangements_NN2 are_VBR available_JJ for_IF the_AT sample_NN1 so_CS21 that_CS22 measurements_NN2 may_VM be_VBI made_VVN in_II the_AT bending_NN1 ,_, shear_VV0 ,_, or_CC tensile_JJ ,_, modes_NN2 ._. 
In_II the_AT bending_NN1 mode_NN1 the_AT sample_NN1 ,_, in_II the_AT form_NN1 of_IO a_AT1 small_JJ bar_NN1 ,_, is_VBZ clamped_VVN firmly_RR at_II both_DB2 ends_NN2 and_CC the_AT central_JJ point_NN1 is_VBZ vibrated_VVN by_II31 means_II32 of_II33 a_AT1 ceramic_JJ drive_NN1 shaft_NN1 ._. 
This_DD1 can_VM be_VBI driven_VVN at_II frequencies_NN2 selected_VVN from_II the_AT range_NN1 0.01_MC to_II 200_MC Hz_NNU ._. 
The_AT applied_JJ stress_NN1 is_VBZ proportional_JJ to_II the_AT a.c._NN1 current_NN1 fed_VVN to_II the_AT drive_NN1 shaft_NN1 and_CC the_AT strain_NN1 is_VBZ detected_VVN using_VVG a_AT1 transducer_NN1 that_CST measures_VVZ the_AT displacement_NN1 of_IO the_AT drive_NN1 clamp_NN1 ._. 
Temperature_NN1 can_VM be_VBI controlled_VVN over_II the_AT range_NN1 120_MC to_II 770_MC K_ZZ1 ,_, either_RR isothermally_RR or_CC more_RGR normally_RR by_II ramping_VVG up_RP and_CC down_RP at_II various_JJ fixed_JJ rates._NNU 13.11_MC Correlation_NN1 of_IO mechanical_JJ damping_NN1 terms_NN2 The_AT several_DA2 practical_JJ methods_NN2 described_VVN express_VV0 the_AT damping_NN1 and_CC moduli_NN2 in_II slightly_RR different_JJ forms_NN2 but_CCB these_DD2 can_VM all_DB be_VBI interrelated_VVN quite_RG simply_RR ._. 
In_RR21 general_RR22 ,_, one_PN1 can_VM select_VVI a_AT1 dissipation_NN1 factor_NN1 or_CC loss_NN1 tangent_NN1 derived_VVN from_II the_AT ratio_NN1 or_CC to_TO represent_VVI the_AT energy_NN1 conversion_NN1 per_II cycle_NN1 ._. 
This_DD1 leads_VVZ to_II the_AT equivalent_JJ forms_NN2 and_CC To_II a_AT1 first_MD approximation_NN1 it_PPH1 is_VBZ also_RR possible_JJ to_TO write_VVI thereby_RR allowing_VVG use_NN1 of_IO the_AT data_NN from_II either_DD1 type_NN1 of_IO measurement_NN1 to_TO characterize_VVI the_AT sample_NN1 ._. 
It_PPH1 should_VM also_RR be_VBI noted_VVN that_CST if_CS complex_JJ moduli_NN2 are_VBR used_VVN the_AT corresponding_JJ complex_JJ compliances_NN2 are_VBR given_VVN by_II ._. 
Moduli_NN2 can_VM also_RR be_VBI related_VVN to_II the_AT viscosity_NN1 ,_, and_CC where_RRQ is_VBZ known_VVN as_II the_AT dynamic_JJ viscosity_NN1 ._. 
The_AT approximations_NN2 and_CC can_VM be_VBI made_VVN when_CS damping_NN1 is_VBZ low_JJ ,_, and_CC the_AT absolute_JJ value_NN1 for_IF the_AT modulus_NN1 G_ZZ1 or_CC E_ZZ1 can_VM be_VBI related_VVN to_II the_AT complex_JJ components_NN2 by_II ._. 
A_AT1 similar_JJ expression_NN1 holds_VVZ for_IF G._NP1 13.12_MC Dielectric_JJ thermal_JJ analysis_NN1 (_( DETA_NP1 )_) Dry_JJ polymers_NN2 are_VBR very_RG poor_JJ conductors_NN2 of_IO electricity_NN1 and_CC can_VM be_VBI regarded_VVN as_II insulators_NN2 ._. 
Application_NN1 of_IO an_AT1 electric_JJ field_NN1 to_II a_AT1 polymer_NN1 can_VM lead_VVI to_II polarization_NN1 of_IO the_AT sample_NN1 ,_, which_DDQ is_VBZ a_AT1 surface_NN1 effect_NN1 ,_, but_CCB if_CS the_AT polymer_NN1 contains_VVZ groups_NN2 that_CST can_VM act_VVI as_RG permanent_JJ dipoles_NN2 then_RT the_AT applied_JJ field_NN1 will_VM cause_VVI them_PPHO2 to_TO align_VVI in_II the_AT direction_NN1 of_IO the_AT field_NN1 ._. 
When_CS the_AT electric_JJ field_NN1 is_VBZ released_VVN ,_, the_AT dipoles_NN2 can_VM relax_VVI back_RP into_II a_AT1 random_JJ orientation_NN1 ,_, but_CCB ,_, due_II21 to_II22 the_AT frictional_JJ resistance_NN1 experienced_VVN by_II the_AT groups_NN2 in_II the_AT bulk_NN1 polymer_NN1 this_DD1 will_VM not_XX be_VBI instantaneous_JJ ._. 
The_AT process_NN1 of_IO disordering_VVG can_VM be_VBI characterized_VVN by_II a_AT1 relaxation_NN1 time_NNT1 ,_, but_CCB may_VM not_XX be_VBI easily_RR measured_VVN ._. 
It_PPH1 is_VBZ more_RGR convenient_JJ to_TO apply_VVI a_AT1 sinusoidally_RR varying_JJ voltage_NN1 to_II the_AT sample_NN1 and_CC to_TO study_VVI the_AT dipole_NN1 polarization_NN1 under_II steady_JJ state_NN1 conditions_NN2 ._. 
In_II DETA_NP1 a_AT1 small_JJ alternating_JJ electric_JJ field_NN1 is_VBZ applied_VVN to_II the_AT sample_NN1 and_CC the_AT electric_JJ charge_NN1 displacement_NN1 Q_ZZ1 is_VBZ measured_VVN by_II following_VVG the_AT current_JJ ._. 
The_AT complex_JJ dielectric_JJ permittivity_NN1 *_FU can_VM be_VBI measured_VVN from_II the_AT change_NN1 in_II amplitude_NN1 and_CC ,_, if_CS the_AT phase_NN1 lag_NN1 between_II the_AT applied_JJ voltage_NN1 and_CC the_AT outcoming_JJ current_NN1 is_VBZ determined_VVN (_( see_VV0 figure_NN1 13.16_MC )_) ,_, then_RT *_FU can_VM be_VBI resolved_VVN into_II the_AT two_MC components_NN2 ,_, the_AT storage_NN1 (_( dielectric_JJ constant_JJ )_) and_CC ,_, the_AT loss_NN1 (_( dielectric_JJ loss_NN1 )_) ._. 
The_AT frequencies_NN2 used_VVN in_II the_AT measurements_NN2 must_VM now_RT be_VBI in_II the_AT range_NN1 where_CS orientational_JJ polarization_NN1 of_IO the_AT dipoles_NN2 in_II the_AT polymer_NN1 is_VBZ active_JJ ._. 
These_DD2 frequencies_NN2 are_VBR much_RR higher_JJR than_CSN used_VVN normally_RR for_IF DMTA_NP1 and_CC typically_RR lie_VV0 in_II the_AT range_NN1 20_MC Hz_NNU to_II 100_MC kHz_NNU ._. 
While_CS the_AT main_JJ variable_NN1 is_VBZ temperature_NN1 ,_, the_AT factors_NN2 and_CC can_VM be_VBI studied_VVN as_II a_AT1 function_NN1 of_IO the_AT angular_JJ frequency_NN1 ,_, and_CC in_II the_AT frequency_NN1 region_NN1 where_CS there_EX is_VBZ a_AT1 relaxation_NN1 ,_, decreases_VVZ as_CSA shown_VVN in_II figure_NN1 13.17_MC ._. 
The_AT magnitude_NN1 of_IO this_DD1 decrease_NN1 is_VBZ a_AT1 measure_NN1 of_IO the_AT strength_NN1 of_IO the_AT molecular_JJ dipole_NN1 involved_JJ in_II the_AT relaxation_NN1 ,_, where_CS o_ZZ1 is_VBZ the_AT static_JJ dielectric_JJ constant_NN1 related_VVN to_II the_AT actual_JJ dipole_NN1 moment_NN1 of_IO the_AT polymer_NN1 and_CC is_VBZ the_AT dielectric_JJ constant_NN1 measured_VVN at_II high_JJ frequencies_NN2 ._. 
When_CS the_AT dielectric_JJ loss_NN1 factor_NN1 is_VBZ measured_VVN at_II a_AT1 characteristic_JJ frequency_NN1 max_NN1 and_CC a_AT1 given_JJ temperature_NN1 ,_, it_PPH1 passes_VVZ through_II a_AT1 maximum_NN1 when_RRQ a_AT1 relaxation_NN1 occurs_VVZ ,_, and_CC the_AT dipole_NN1 relaxation_NN1 time_NNT1 ,_, can_VM be_VBI obtained_VVN ._. 
At_II frequencies_NN2 above_II max_NN1 the_AT dipoles_NN2 can_VM not_XX move_VVI fast_RR enough_RR to_TO follow_VVI the_AT alternating_JJ field_NN1 so_RG both_RR and_CC are_VBR low_JJ ._. 
When_CS the_AT frequency_NN1 is_VBZ lower_JJR than_CSN max_NN1 the_AT permanent_JJ dipoles_NN2 can_VM follow_VVI the_AT field_NN1 quite_RG closely_RR and_CC so_RR is_VBZ high_JJ because_CS the_AT dipoles_NN2 align_VV0 easily_RR with_IW each_DD1 change_NN1 in_II polarity_NN1 ;_; on_II the_AT other_JJ hand_NN1 is_VBZ low_RR again_RT because_CS now_RT the_AT voltage_NN1 and_CC the_AT current_JJ are_VBR approximately_RR 90_MC out_II21 of_II22 phase_NN1 ._. 
Dielectric_JJ relaxation_NN1 processes_NN2 can_VM be_VBI described_VVN formally_RR by_II the_AT following_JJ relations_NN2 :_: and_CC A_ZZ1 useful_JJ way_NN1 of_IO examining_VVG the_AT data_NN is_VBZ to_TO measure_VVI the_AT ratio_NN1 of_IO the_AT two_MC factors_NN2 ,_, which_DDQ gives_VVZ the_AT dielectric_JJ loss_NN1 tangent_JJ Dipolar_JJ groups_NN2 in_II a_AT1 polymer_NN1 coil_NN1 may_VM not_XX all_DB be_VBI able_JK to_TO relax_VVI at_II the_AT same_DA speed_NN1 because_II21 of_II22 the_AT variable_JJ steric_JJ restrictions_NN2 they_PPHS2 may_VM experience_VVI ,_, imposed_VVN by_II their_APPGE environment_NN1 ._. 
This_DD1 can_VM be_VBI caused_VVN by_II the_AT disordered_JJ packing_NN1 of_IO chains_NN2 in_II the_AT amorphous_JJ glassy_JJ phase_NN1 ,_, and_CC a_AT1 random_JJ distribution_NN1 of_IO the_AT available_JJ free_JJ volume_NN1 ,_, or_CC perhaps_RR even_RR by_II the_AT random_JJ coil_NN1 structure_NN1 of_IO the_AT chain_NN1 itself_PPX1 causing_VVG local_JJ environmental_JJ changes_NN2 ._. 
The_AT result_NN1 is_VBZ that_CST a_AT1 distribution_NN1 of_IO relaxation_NN1 times_NNT2 is_VBZ to_TO be_VBI expected_VVN for_IF a_AT1 given_JJ process_NN1 and_CC this_DD1 results_VVZ in_II a_AT1 broadening_NN1 of_IO the_AT dielectric_JJ loss_NN1 peak_NN1 ._. 
Thus_RR ,_, the_AT more_RGR mobile_JJ a_AT1 dipolar_JJ group_NN1 ,_, the_AT easier_JJR it_PPH1 is_VBZ for_IF it_PPH1 to_TO follow_VVI the_AT electric_JJ field_NN1 up_II21 to_II22 higher_JJR frequencies_NN2 ,_, whereas_CS the_AT less_RGR mobile_JJ groups_NN2 can_VM only_RR orient_VVI at_II lower_JJR frequencies._NNU 13.13_MC Comparison_NN1 between_II DMTA_NP1 and_CC DETA_NP1 Data_NN from_II mechanical_JJ and_CC dielectric_JJ measurements_NN2 can_VM be_VBI related_VVN ,_, certainly_RR in_II a_AT1 qualitative_JJ ,_, if_CS not_XX always_RR in_II a_AT1 quantitative_JJ way_NN1 ._. 
Formally_RR ,_, the_AT dielectric_JJ constant_NN1 can_VM be_VBI regarded_VVN as_II the_AT equivalent_NN1 of_IO the_AT mechanical_JJ compliance_NN1 ,_, rather_II21 than_II22 the_AT modulus_NN1 ,_, and_CC this_DD1 highlights_VVZ the_AT fact_NN1 that_CST mechanical_JJ techniques_NN2 measure_VV0 the_AT ability_NN1 of_IO the_AT system_NN1 to_TO resist_VVI movement_NN1 ,_, whereas_CS the_AT dielectric_JJ approach_NN1 is_VBZ a_AT1 measurement_NN1 of_IO the_AT ability_NN1 of_IO the_AT system_NN1 to_TO move_VVI ,_, given_CS21 that_CS22 the_AT groups_NN2 involved_VVN must_VM also_RR be_VBI dipolar_JJ ._. 
Interestingly_RR ,_, the_AT dielectric_JJ loss_NN1 appears_VVZ to_TO match_VVI the_AT loss_NN1 modulus_NN1 more_RGR closely_RR than_CSN the_AT loss_NN1 compliance_NN1 when_RRQ data_NN are_VBR compared_VVN for_IF the_AT same_DA system_NN1 ._. 
Both_DB2 techniques_NN2 respond_VV0 in_II a_AT1 similar_JJ fashion_NN1 to_II a_AT1 change_NN1 in_II the_AT frequency_NN1 of_IO the_AT measurement_NN1 ._. 
When_CS the_AT frequency_NN1 is_VBZ increased_VVN ,_, the_AT transitions_NN2 and_CC relaxations_NN2 that_CST are_VBR observed_VVN in_II a_AT1 sample_NN1 appear_VV0 at_II higher_JJR temperatures_NN2 ._. 
This_DD1 is_VBZ illustrated_VVN from_II work_NN1 on_II poly_NN1 (_( ethylene_NN1 terephthalate_NN1 )_) where_RRQ the_AT loss_NN1 peak_NN1 representing_VVG the_AT glass_NN1 transition_NN1 has_VHZ been_VBN measured_VVN by_II both_RR DMTA_NP1 and_CC DETA_NN1 at_II several_DA2 frequencies_NN2 between_II 0.01_MC Hz_NNU and_CC 100_MC kHz_NNU (_( figure_NN1 13.18_MC )_) ._. 
The_AT maximum_NN1 of_IO this_DD1 loss_NN1 peak_NN1 is_VBZ seen_VVN to_TO move_VVI from_II a_AT1 temperature_NN1 of_IO about_RG 360_MC K_ZZ1 (_( 0.01_MC Hz_NNU )_) to_II about_RG 400_MC K_ZZ1 (_( 100_MC kHz_NNU )_) ,_, which_DDQ is_VBZ an_AT1 increase_NN1 of_IO 40_MC K_ZZ1 over_II a_AT1 frequency_NN1 change_NN1 of_IO seven_MC orders_NN2 of_IO magnitude_NN1 ._. 
This_DD1 is_VBZ close_JJ to_II the_AT rule_NN1 of_IO thumb_NN1 that_CST the_AT temperature_NN1 for_IF the_AT maximum_NN1 of_IO a_AT1 loss_NN1 peak_NN1 (_( or_CC a_AT1 relaxation_NN1 process_NN1 )_) will_VM change_VVI by_II approximately_RR 7_MC K_ZZ1 for_IF each_DD1 decade_NNT1 of_IO change_NN1 in_II frequency_NN1 ._. 
This_DD1 type_NN1 of_IO measurement_NN1 can_VM be_VBI used_VVN to_TO estimate_VVI the_AT activation_NN1 energy_NN1 for_IF a_AT1 transition_NN1 or_CC relaxation_NN1 process_NN1 ,_, if_CS the_AT frequency_NN1 v_ZZ1 ,_, at_II T_ZZ1 max_NN1 is_VBZ expressed_VVN as_II a_AT1 function_NN1 of_IO reciprocal_JJ temperature_NN1 according_II21 to_II22 the_AT relation_NN1 Data_NN plotted_VVD using_VVG equation_NN1 (_( 13.33_MC )_) for_IF the_AT -relaxation_JJ process_NN1 in_II a_AT1 series_NN of_IO poly(alkylmethacrylate)s_NN2 are_VBR shown_VVN in_II figure_NN1 13.19_MC ._. 
Both_DB2 techniques_NN2 have_VH0 been_VBN used_VVN and_CC separately_RR give_VV0 good_JJ straight_JJ lines_NN2 with_IW the_AT same_DA slope_NN1 ,_, but_CCB the_AT fact_NN1 that_CST the_AT lines_NN2 do_VD0 not_XX overlap_VVI precisely_RR indicates_VVZ that_CST the_AT measurements_NN2 may_VM not_XX be_VBI exactly_RR equivalent_JJ ._. 
The_AT results_NN2 from_II DMTA_NP1 and_CC DETA_NN1 can_VM be_VBI used_VVN in_II a_AT1 complementary_JJ manner_NN1 to_TO distinguish_VVI between_II relaxations_NN2 involving_VVG polar_JJ and_CC non_FU polar_JJ units_NN2 relaxing_VVG in_II the_AT polymeric_JJ system_NN1 ._. 
Thus_RR if_CS the_AT response_NN1 of_IO poly_NN1 (_( ethylene_NN1 terephthalate_NN1 )_) to_II both_RR DMTA_NP1 and_CC DETA_NN1 is_VBZ examined_VVN ,_, two_MC major_JJ loss_NN1 peaks_NN2 can_VM be_VBI identified_VVN in_II each_DD1 ,_, as_CSA seen_VVN in_II figure_NN1 13.20_MC ._. 
The_AT high_JJ temperature_NN1 loss_NN1 peak_NN1 (_( -peak_NN1 )_) can_VM be_VBI assigned_VVN to_II the_AT glass_NN1 transition_NN1 and_CC this_DD1 can_VM be_VBI confirmed_VVN by_II d.s.c._NNU measurements_NN2 ._. 
There_EX is_VBZ a_AT1 second_MD loss_NN1 (_( -relaxation_NN1 )_) which_DDQ appears_VVZ at_II lower_JJR temperatures_NN2 and_CC suggests_VVZ that_CST there_EX is_VBZ a_AT1 relaxation_NN1 process_VV0 active_JJ in_II the_AT glassy_JJ state_NN1 ._. 
It_PPH1 is_VBZ not_XX immediately_RR obvious_JJ which_DDQ group_NN1 is_VBZ responsible_JJ for_IF this_DD1 process_NN1 ,_, but_CCB it_PPH1 is_VBZ active_JJ both_RR mechanically_RR and_CC dielectrically_RR ._. 
Examination_NN1 of_IO the_AT polymer_NN1 structures_NN2 suggests_VVZ that_CST the_AT relaxation_NN1 in_II the_AT glass_NN1 could_VM involve_VVI libration_NN1 of_IO the_AT phenyl_NN1 ring_NN1 ,_, motion_NN1 of_IO the_AT oxycarbonyl_NN1 unit_NN1 or_CC rearrangement_NN1 of_IO the_AT (_( -O-C-C-O-_NN1 )_) unit_NN1 ._. 
From_II the_AT spectra_NN2 it_PPH1 can_VM be_VBI seen_VVN that_CST the_AT intensity_NN1 of_IO the_AT -peak_NN1 relative_II21 to_II22 the_AT -peak_NN1 is_VBZ much_RR stronger_JJR in_II the_AT dielectric_JJ response_NN1 compared_VVN with_IW the_AT mechanical_JJ measurements_NN2 ._. 
This_DD1 suggests_VVZ that_CST the_AT group_NN1 undergoing_VVG relaxation_NN1 is_VBZ associated_VVN with_IW a_AT1 dipole_NN1 moment_NN1 and_CC thus_RR rules_VVZ out_RP the_AT phenyl_NN1 ring_NN1 libration_NN1 as_II a_AT1 likely_JJ process_NN1 ._. 
This_DD1 does_VDZ not_XX give_VVI irrefutable_JJ evidence_NN1 of_IO the_AT participation_NN1 of_IO the_AT oxycarbonyl_NN1 unit_NN1 but_CCB it_PPH1 does_VDZ point_VVI in_II this_DD1 direction._NNU 13.14_MC Time-temperature_JJ superposition_NN1 principle_NN1 A_ZZ1 curve_NN1 of_IO the_AT logarithm_NN1 of_IO the_AT modulus_NN1 against_II time_NNT1 and_CC temperature_NN1 is_VBZ shown_VVN in_II figure_NN1 13.21_MC ._. 
This_DD1 provides_VVZ a_AT1 particularly_RR useful_JJ description_NN1 of_IO the_AT behaviour_NN1 of_IO a_AT1 polymer_NN1 and_CC allows_VVZ one_PN1 to_TO estimate_VVI ,_, among_II other_JJ things_NN2 ,_, either_RR the_AT relaxation_NN1 or_CC retardation_NN1 spectrum_NN1 ._. 
The_AT practical_JJ time_NNT1 scale_NN1 for_IF most_DAT stress-relaxation_JJ measurements_NN2 ranges_VVZ from_II 10_MC 1_MC1 to_II 10_MC 6_MC s_ZZ1 but_CCB a_AT1 wider_JJR range_NN1 of_IO temperature_NN1 is_VBZ desirable_JJ ._. 
Such_DA a_AT1 range_NN1 can_VM be_VBI covered_VVN relatively_RR easily_RR by_II making_VVG use_NN1 of_IO the_AT observation_NN1 ,_, first_MD made_VVN by_II Leaderman_NP1 ,_, that_DD1 for_IF viscoelastic_JJ materials_NN2 time_NNT1 is_VBZ equivalent_JJ to_II temperature_NN1 ._. 
A_AT1 composite_JJ isothermal_JJ curve_NN1 covering_VVG the_AT required_JJ extensive_JJ time_NNT1 scale_NN1 can_VM then_RT be_VBI constructed_VVN from_II data_NN collected_VVN at_II different_JJ temperatures_NN2 ._. 
This_DD1 is_VBZ accomplished_VVN by_II translation_NN1 of_IO the_AT small_JJ curves_NN2 along_II the_AT log_NN1 t_ZZ1 axis_NN1 until_CS they_PPHS2 are_VBR all_DB superimposed_VVN to_TO form_VVI a_AT1 large_JJ composite_JJ curve_NN1 ._. 
The_AT technique_NN1 can_VM be_VBI illustrated_VVN using_VVG data_NN for_IF polyisobutylene_NN1 at_II several_DA2 temperatures_NN2 ._. 
An_AT1 arbitrary_JJ temperature_NN1 T_ZZ1 o_ZZ1 is_VBZ first_MD chosen_VVN to_TO serve_VVI as_II a_AT1 reference_NN1 which_DDQ in_II the_AT present_JJ case_NN1 is_VBZ 298_MC K._NP1 As_CSA values_NN2 of_IO the_AT relaxation_NN1 modulus_NN1 have_VH0 been_VBN measured_VVN at_II widely_RR differing_JJ temperatures_NN2 ,_, they_PPHS2 must_VM be_VBI corrected_VVN for_IF changes_NN2 in_II the_AT sample_NN1 density_NN1 with_IW temperature_NN1 to_TO give_VVI a_AT1 reduced_JJ modulus_NN1 ,_, where_RRQ and_CC o_ZZ1 are_VBR the_AT polymer_NN1 densities_NN2 at_II T_ZZ1 and_CC T_ZZ1 o_ZZ1 respectively_RR ._. 
This_DD1 correction_NN1 is_VBZ small_JJ and_CC can_VM often_RR be_VBI neglected_VVN ._. 
Each_DD1 curve_NN1 of_IO reduced_JJ modulus_NN1 is_VBZ shifted_VVN with_II31 respect_II32 to_II33 the_AT curve_NN1 at_II T_ZZ1 o_ZZ1 until_CS all_DB fit_VV0 together_RL forming_VVG one_MC1 master_NN1 curve_NN1 ._. 
The_AT curve_NN1 obtained_VVN at_II each_DD1 temperature_NN1 is_VBZ shifted_VVN by_II an_AT1 amount_NN1 The_AT parameter_NN1 a_AT1 is_VBZ the_AT shift_NN1 factor_NN1 and_CC is_VBZ positive_JJ if_CS the_AT movement_NN1 of_IO the_AT curve_NN1 is_VBZ to_II the_AT left_NN1 of_IO the_AT reference_NN1 and_CC negative_JJ for_IF a_AT1 move_NN1 to_II the_AT right_NN1 ._. 
The_AT shift_NN1 factor_NN1 is_VBZ a_AT1 function_NN1 of_IO temperature_NN1 only_RR and_CC decreases_VVZ with_IW increasing_JJ temperature_NN1 ,_, it_PPH1 is_VBZ ,_, of_RR21 course_RR22 ,_, unity_NN1 at_II T_ZZ1 o_ZZ1 ._. 
The_AT superposition_NN1 principle_NN1 can_VM also_RR be_VBI applied_VVN to_TO creep_VVI data_NN ._. 
Curves_NN2 exhibiting_VVG the_AT creep_NN1 behaviour_NN1 of_IO polymers_NN2 at_II different_JJ temperatures_NN2 can_VM be_VBI compared_VVN by_II plotting_VVG against_II log_NN1 t_ZZ1 ._. 
This_DD1 reduces_VVZ all_DB the_AT curves_NN2 at_II various_JJ temperatures_NN2 to_II the_AT same_DA shape_NN1 but_CCB displaced_VVD along_II the_AT log_NN1 t_ZZ1 axis_NN1 ._. 
Superposition_NN1 to_TO form_VVI a_AT1 master_NN1 curve_NN1 is_VBZ readily_RR achieved_VVN by_II movement_NN1 along_II the_AT log_NN1 t_ZZ1 axis_NN1 ,_, where_CS the_AT shift_NN1 factor_NN1 a_AT1 has_VHZ the_AT same_DA characteristics_NN2 as_CSA for_IF the_AT relaxation_NN1 data_NN ._. 
This_DD1 shift_NN1 factor_NN1 has_VHZ also_RR been_VBN defined_VVN as_II the_AT ratio_NN1 of_IO relaxation_NN1 or_CC retardation_NN1 times_NNT2 at_II the_AT temperatures_NN2 T_ZZ1 and_CC T_ZZ1 o_ZZ1 i.e._REX and_CC is_VBZ related_VVN to_II the_AT viscosities_NN2 ._. 
If_CS the_AT viscosities_NN2 obey_VV0 the_AT Arrhenius_NP1 equation_NN1 ,_, then_RT by_II neglecting_VVG the_AT correction_NN1 factor_NN1 ,_, we_PPIS2 can_VM express_VVI a_AT1 in_II an_AT1 exponential_NN1 form_NN1 as_CSA or_CC where_RRQ b_ZZ1 is_VBZ a_AT1 constant_JJ ._. 
This_DD1 equation_NN1 is_VBZ very_RG similar_JJ in_II form_NN1 to_II the_AT WLF_NP1 equation_NN1 ,_, For_IF polyisobutylene_NN1 ,_, the_AT shift_NN1 factor_NN1 a_AT1 can_VM be_VBI predicted_VVN if_CS is_VBZ used_VVN with_IW and_CC ._. 
As_CSA outlined_VVN in_II chapter_NN1 12_MC ,_, the_AT reference_NN1 temperature_NN1 is_VBZ often_RR chosen_VVN to_TO be_VBI T_ZZ1 g_ZZ1 with_IW and_CC ,_, from_II which_DDQ a_AT1 can_VM be_VBI calculated_VVN for_IF various_JJ amorphous_JJ polymers_NN2 ._. 
The_AT superposition_NN1 principle_NN1 can_VM be_VBI used_VVN to_TO predict_VVI the_AT creep_NN1 and_CC relaxation_NN1 behaviour_NN1 at_II any_DD temperature_NN1 if_CS some_DD results_NN2 are_VBR already_RR available_JJ ,_, with_IW the_AT proviso_NN1 that_CST the_AT most_RGT reliable_JJ predictions_NN2 can_VM be_VBI made_VVN for_IF interpolated_JJ temperatures_NN2 rather_CS21 than_CS22 long_JJ extrapolations_NN2 ._. 
The_AT principle_NN1 can_VM also_RR be_VBI applied_VVN to_II dielectric_JJ data_NN which_DDQ can_VM be_VBI shifted_VVN either_RR along_II the_AT temperature_NN1 or_CC the_AT frequency_NN1 axis_NN1 ._. 
An_AT1 example_NN1 of_IO the_AT latter_DA type_NN1 of_IO shift_NN1 is_VBZ shown_VVN in_II figure_NN1 13.22_MC ,_, where_CS instead_II21 of_II22 time_NNT1 dependence_NN1 measurements_NN2 the_AT frequency_NN1 dependence_NN1 of_IO the_AT -relaxation_NN1 in_II poly_NN1 (_( vinyl_NN1 acetate_NN1 )_) has_VHZ been_VBN studied_VVN at_II fixed_JJ temperatures_NN2 in_II the_AT range_NN1 212_MC to_II 266_MC K._NP1 A_ZZ1 master_NN1 curve_NN1 can_VM be_VBI constructed_VVN for_IF this_DD1 relaxation_NN1 region_NN1 by_II plotting_VVG against_II ,_, where_CS the_AT '_GE max_NN1 '_GE subscript_NN1 refers_VVZ to_II the_AT peak_NN1 maximum_NN1 at_II each_DD1 experimental_JJ temperature._NNU 13.15_MC A_ZZ1 molecular_JJ theory_NN1 for_IF viscoelasticity_NN1 So_RG far_RR the_AT interpretation_NN1 of_IO viscoelastic_JJ behaviour_NN1 has_VHZ been_VBN largely_RR phenomenological_JJ ,_, relying_VVG on_II the_AT application_NN1 of_IO mechanical_JJ models_NN2 to_TO aid_VVI the_AT elucidation_NN1 of_IO the_AT observed_JJ phenomena_NN2 ._. 
These_DD2 are_VBR ,_, at_RR21 best_RR22 ,_, no_AT more_DAR than_CSN useful_JJ physical_JJ aids_NN2 to_TO illustrate_VVI the_AT mechanical_JJ response_NN1 and_CC suffer_VVI from_II the_AT disadvantage_NN1 that_CST a_AT1 given_JJ process_NN1 may_VM be_VBI described_VVN in_II this_DD1 way_NN1 using_VVG more_DAR than_CSN one_MC1 arrangement_NN1 of_IO springs_NN2 and_CC dashpots_NN2 ._. 
In_II an_AT1 attempt_NN1 to_TO gain_VVI a_AT1 deeper_JJR understanding_NN1 on_II a_AT1 molecular_JJ level_NN1 ,_, Rouse_NP1 ,_, Zimm_NP1 ,_, Bueche_NP1 ,_, and_CC others_NN2 have_VH0 attempted_VVN to_TO formulate_VVI a_AT1 theory_NN1 of_IO polymer_NN1 viscoelasticity_NN1 based_VVN on_II a_AT1 chain_NN1 model_NN1 consisting_VVG of_IO a_AT1 series_NN of_IO sub-units_NN2 ._. 
Each_DD1 sub-unit_NN1 is_VBZ assumed_VVN to_TO behave_VVI like_II an_AT1 entropy_NN1 spring_NN1 and_CC is_VBZ expected_VVN to_TO be_VBI large_JJ enough_RR to_TO realize_VVI a_AT1 Gaussian_JJ distribution_NN1 of_IO segments_NN2 (_( i.e.&gt;_FO 50_MC carbon_NN1 atoms_NN2 )_) ._. 
This_DD1 approach_NN1 ,_, although_CS still_RR somewhat_RR restrictive_JJ has_VHZ led_VVN to_II reasonable_JJ predictions_NN2 of_IO relaxation_NN1 and_CC retardation_NN1 spectra_NN2 ._. 
One_PN1 starts_VVZ with_IW a_AT1 single_JJ isolated_JJ chain_NN1 and_CC the_AT assumption_NN1 that_CST it_PPH1 exhibits_VVZ both_RR viscous_JJ and_CC elastic_JJ behaviour_NN1 ._. 
If_CS the_AT chain_NN1 is_VBZ left_JJ undisturbed_JJ it_PPH1 will_VM also_RR adopt_VVI the_AT most_RGT notable_JJ conformation_NN1 or_CC segmental_JJ distribution_NN1 ,_, so_CS21 that_CS22 ,_, with_IW the_AT exception_NN1 of_IO high_JJ frequencies_NN2 ,_, the_AT observed_JJ elasticity_NN1 is_VBZ predominantly_RR entropic_JJ ._. 
Thus_RR the_AT application_NN1 of_IO a_AT1 stress_NN1 to_II the_AT molecule_NN1 will_VM cause_VVI distortion_NN1 ,_, by_II altering_VVG the_AT equilibrium_NN1 conformation_NN1 to_II a_AT1 less_RGR probable_JJ one_PN1 ,_, resulting_VVG in_II a_AT1 decrease_NN1 in_II the_AT entropy_NN1 and_CC a_AT1 corresponding_JJ increase_NN1 in_II the_AT free_JJ energy_NN1 of_IO the_AT system_NN1 ._. 
When_CS the_AT stress_NN1 is_VBZ removed_VVN the_AT chain_NN1 segments_NN2 will_VM diffuse_VVI back_RP to_II their_APPGE unstressed_JJ positions_NN2 even_CS21 though_CS22 the_AT whole_JJ molecule_NN1 may_VM have_VHI changed_VVN its_APPGE spatial_JJ position_NN1 in_II the_AT meantime_NNT1 ._. 
If_CS on_II the_AT other_JJ hand_NN1 ,_, the_AT stress_NN1 is_VBZ maintained_VVN ,_, strain_VV0 relief_NN1 is_VBZ sought_VVN by_II converting_VVG the_AT excess_JJ free_JJ energy_NN1 into_II heat_NN1 ,_, thereby_RR stimulating_VVG the_AT thermal_JJ motion_NN1 of_IO the_AT segments_NN2 back_RP to_II their_APPGE original_JJ positions_NN2 ._. 
Stress_NN1 relaxation_NN1 is_VBZ then_RT said_VVN to_TO have_VHI occurred_VVN ._. 
For_IF a_AT1 chain_NN1 molecule_NN1 composed_VVN of_IO a_AT1 large_JJ number_NN1 of_IO segments_NN2 ,_, movement_NN1 of_IO the_AT complete_JJ molecule_NN1 depends_VVZ on_II the_AT co-operative_JJ movement_NN1 of_IO all_DB the_AT segments_NN2 ,_, and_CC as_CSA stress-relaxation_NN1 depends_VVZ on_II the_AT number_NN1 of_IO ways_NN2 the_AT molecule_NN1 can_VM regain_VVI its_APPGE most_RGT probable_JJ conformation_NN1 ,_, each_DD1 possible_JJ co-ordinated_JJ movement_NN1 is_VBZ treated_VVN as_II a_AT1 mode_NN1 of_IO motion_NN1 with_IW a_AT1 characteristic_JJ relaxation_NN1 time_NNT1 ._. 
For_IF simplicity_NN1 we_PPIS2 can_VM represent_VVI the_AT polymer_NN1 as_CSA in_II figure_NN1 13.23_MC ._. 
The_AT first_MD mode_NN1 p_ZZ1 =_FO 1_MC1 represents_VVZ translation_NN1 of_IO the_AT molecule_NN1 as_II a_AT1 whole_NN1 and_CC has_VHZ the_AT longest_JJT relaxation_NN1 time_NNT1 1_MC1 because_CS the_AT maximum_JJ number_NN1 of_IO co-ordinated_JJ segmental_JJ movements_NN2 are_VBR involved_VVN ._. 
The_AT second_MD mode_NN1 p_ZZ1 =_FO 2_MC corresponds_VVZ to_II the_AT movement_NN1 of_IO the_AT chain_NN1 ends_VVZ in_II opposite_JJ directions_NN2 ;_; for_IF p_ZZ1 =_FO 3_MC ,_, both_DB2 chain_VV0 ends_NN2 move_VV0 in_II the_AT same_DA direction_NN1 ,_, but_CCB the_AT centre_NN1 moves_VVZ in_II the_AT opposite_JJ direction_NN1 ._. 
Higher_JJR modes_NN2 4_MC ,_, 5_MC ..._... m_ZZ1 follow_VV0 involving_VVG a_AT1 progressively_RR decreasing_JJ degree_NN1 of_IO co-operation_NN1 for_IF each_DD1 succeeding_JJ mode_NN1 and_CC correspondingly_RR lower_JJR relaxation_NN1 times_II p_ZZ1 ._. 
This_DD1 means_VVZ that_CST a_AT1 single_JJ polymer_NN1 chain_NN1 possesses_VVZ a_AT1 wide_JJ distribution_NN1 of_IO relaxation_NN1 times_NNT2 ._. 
Using_VVG this_DD1 concept_NN1 ,_, Rouse_NP1 considered_VVD a_AT1 molecule_NN1 in_II dilute_JJ solution_NN1 under_II sinusoidal_JJ shear_VV0 and_CC derived_VVD the_AT relations_NN2 where_RRQ and_CC s_ZZ1 are_VBR the_AT viscosities_NN2 of_IO the_AT solution_NN1 and_CC the_AT solvent_NN1 respectively_RR ,_, n_ZZ1 is_VBZ the_AT number_NN1 of_IO molecules_NN2 per_II unit_NN1 volume_NN1 ,_, k_ZZ1 is_VBZ the_AT Boltzmann_NP1 constant_JJ ,_, and_CC is_VBZ the_AT angular_JJ frequency_NN1 of_IO the_AT applied_JJ stress_NN1 which_DDQ is_VBZ zero_NN1 for_IF steady_JJ flow_NN1 ._. 
These_DD2 equations_NN2 are_VBR strictly_RR applicable_JJ only_RR to_II dilute_JJ solutions_NN2 of_IO non-draining_JJ monodisperse_NN1 coils_NN2 ,_, but_CCB can_VM be_VBI extended_VVN to_II undiluted_JJ polymers_NN2 above_II their_APPGE glass_NN1 temperature_NN1 if_CS suitably_RR modified_VVN ._. 
This_DD1 becomes_VVZ necessary_JJ when_CS chain_NN1 entanglements_NN2 begin_VV0 to_TO have_VHI a_AT1 significant_JJ effect_NN1 on_II the_AT relaxation_NN1 times_NNT2 ._. 
The_AT undiluted_JJ system_NN1 is_VBZ represented_VVN as_II a_AT1 collection_NN1 of_IO polymer_NN1 segments_NN2 dissolved_VVN in_II a_AT1 liquid_JJ matrix_NN1 composed_VVN of_IO other_JJ polymer_NN1 segments_NN2 and_CC s_ZZ1 can_VM be_VBI replaced_VVN by_II a_AT1 monomeric_JJ frictional_JJ coefficient_NN1 o_ZZ1 ._. 
This_DD1 provides_VVZ a_AT1 measure_NN1 of_IO the_AT viscous_JJ resistance_NN1 experienced_VVN by_II a_AT1 chain_NN1 and_CC is_VBZ characteristic_JJ of_IO a_AT1 given_JJ polymer_NN1 at_II a_AT1 particular_JJ temperature_NN1 ._. 
The_AT continuous_JJ relaxation_NN1 and_CC retardation_NN1 spectra_NN2 calculated_VVN from_II the_AT Rouse_NP1 theory_NN1 are_VBR and_CC where_RRQ is_VBZ the_AT unperturbed_JJ mean_JJ square_JJ end-to-end_JJ distance_NN1 of_IO a_AT1 chain_NN1 of_IO molar_JJ mass_JJ M_NN1 and_CC density_NN1 containing_VVG N_ZZ1 monomer_NN1 units_NN2 ._. 
The_AT equations_NN2 predict_VV0 linearity_NN1 in_II the_AT plots_NN2 and_CC against_II with_IW slopes_NN2 of-_NN1 and_CC +_FO respectively_RR ._. 
Comparison_NN1 with_IW experimental_JJ results_NN2 for_IF poly_NN1 (_( methyl_NN1 acrylate_NN1 )_) shows_VVZ validity_NN1 only_RR for_IF longer_JJR values_NN2 of_IO the_AT relaxation_NN1 and_CC retardation_NN1 times_NNT2 ._. 
The_AT Rouse_NP1 model_NN1 only_RR pertains_VVZ to_II the_AT region_NN1 covering_VVG intermediate_JJ values_NN2 ._. 
The_AT reason_NN1 for_IF this_DD1 lies_VVZ in_II the_AT response_NN1 of_IO a_AT1 polymer_NN1 to_II an_AT1 alternating_JJ stress_NN1 ._. 
At_II low_JJ frequencies_NN2 Brownian_JJ motion_NN1 can_VM relieve_VVI the_AT deformation_NN1 caused_VVN by_II the_AT stress_NN1 before_II the_AT next_MD cycle_NN1 takes_VVZ place_NN1 ,_, but_CCB as_CSA the_AT frequency_NN1 increases_VVZ the_AT conformational_JJ change_NN1 begins_VVZ to_TO lag_VVI behind_II the_AT stress_NN1 and_CC energy_NN1 is_VBZ not_XX only_RR dissipated_VVN but_CCB stored_VVD as_RR21 well_RR22 ._. 
Finally_RR at_II very_RG high_JJ frequencies_NN2 only_RR enough_DD time_NNT1 exists_VVZ for_IF bond_NN1 deformation_NN1 to_TO occur_VVI ._. 
As_CSA it_PPH1 was_VBDZ stipulated_VVN that_CST each_DD1 segment_NN1 be_VBI long_RR enough_RR to_TO obey_VVI Gaussian_JJ statistics_NN ,_, short_JJ relaxation_NN1 times_NNT2 may_VM not_XX allow_VVI a_AT1 segment_NN1 sufficient_JJ time_NNT1 to_TO rearrange_VVI and_CC regain_VVI this_DD1 distribution_NN1 ._. 
Thus_RR the_AT contribution_NN1 from_II short_JJ segments_NN2 to_II the_AT distribution_NN1 functions_NN2 tends_VVZ to_TO be_VBI lost_VVN and_CC deviations_NN2 from_II the_AT theoretical_JJ represent_VV0 departure_NN1 from_II ideal_JJ Gaussian_JJ behaviour_NN1 ._. 
This_DD1 approach_NN1 to_II viscoelastic_JJ theory_NN1 is_VBZ reasonably_RR successful_JJ in_II the_AT low_JJ modulus_NN1 regions_NN2 but_CCB it_PPH1 requires_VVZ considerable_JJ modification_NN1 if_CS the_AT high_JJ modulus_NN1 and_CC rubbery_JJ plateau_NN1 regions_NN2 are_VBR to_TO be_VBI described_VVN ._. 
