CHAPTER 8
Polymers in Solution
8.1 Thermodynamics of polymer solutions
The interaction of long chain molecules with liquids is of considerable interest from both a practical and theoretical viewpoint.
For linear and branched polymers, liquids can usually be found which will dissolve the polymer completely to form a homogeneous solution, whereas cross-linked networks will only swell when in contact with compatible liquids.
In this chapter we shall deal with linear or branched polymers and treat the swelling of networks in chapter 14.
When an amorphous polymer is mixed with a suitable solvent, it disperses in the solvent and behaves as though it too is a liquid.
In a good solvent, classed as one which is highly compatible with the polymer, the liquid-polymer interactions expand the polymer coil, from its unperturbed dimensions, in proportion to the extent of these interactions.
In a ' poor ' solvent, the interactions are fewer and coil expansion or perturbation is restricted.
The fundamental thermodynamic equation used to describe these systems relates the Gibbs free energy function G to the enthalpy H and entropy S, i.e..
A homogeneous solution is obtained when the Gibbs free energy of mixing, i.e. when the Gibbs free energy of the solution G 12 is lower than the Gibbs functions of the components of the mixture G 1 and G 2.
8.2 Ideal mixtures of small molecules
To understand the behaviour of polymers in solution more fully, a knowledge of the enthalpic and entropic contributions to G M is essential, and it is instructive to consider first mixtures of small molecules, to establish some fundamental rules concerning ideal and non-ideal behaviour.
Raoult's law is a useful starting point and defines an ideal solution as one in which the activity of each component in a mixture a i is equal to its mole fraction x i.
This is valid only for components of comparable size, and where the intermolecular forces acting between both like and unlike molecules are equal.
The latter requirement means that component molecules of each species can interchange positions without altering the total energy of the system, i.e. and consequently it only remains for the entropy contribution S M to be calculated.
For a system in a given state, the entropy is related to the number of distinguishable arrangements the components in that state can adopt, and can be calculated from the Boltzmann law, where W is the number of statistical microstates available to the system.
We can begin by considering the mixing of N 1 molecules of component (1) with N 2 molecules of component (2) and this can be assumed to take place on a hypothetical lattice containing cells of equal size.
Although this formalism is not strictly necessary for the analysis, the arrangement of spherical molecules of equal size in the liquid state will, to the first near neighbour approximation, be similar to a regular lattice structure and so it is a useful structure to use as a framework for the mixing process.
The total number of possible ways in which the component molecules can be arranged on the lattice increases when mixing takes place and is equal to, but as the interchanging of a molecule of (1) with another molecule (1), or (2) with (2) will be an indistinguishable process, the net number of distinguishable arrangements will be
The configurational (or combinatorial) entropy S c can then be derived from the Boltzmann law and
For large values of N i, Stirling's approximation can be used to deal with the factorials viz. In, and equation (8.3) becomes which on dividing by N o gives If, the mole fraction of component i, then
For the pure components,, and as S M, the change in entropy on mixing, is given by then we can write so for a two component mixture
This expression is derived assuming
(a)
the volume change on mixing,
(b)
the molecules are all of equal size,
(c)
all possible arrangements have the same energy,, and
(d)
the motion of the components about their equilibrium positions remains unchanged on mixing
.
Thus the free energy of mixing, G M is which shows that mixing in ideal systems is an entropically driven, spontaneous process.
8.3 Non-ideal solutions
Any deviations from assumptions (a) to (d) will constitute a deviation from ideality (an ideal solution is a rare occurrence) and several more realistic types of solution can be identified:
(i)
Athermal solutions; where but S M is not ideal
(ii)
Regular solutions; where S M is ideal but,
(iii)
Irregular solutions; in which both S M and H M deviate from their ideal values.
Polymer solutions tend to fall into category (iii) and the non-ideal behaviour can be attributed not only to the existence of a finite heat of mixing but also to the large difference in size between the polymer and solvent molecules.
The polymer chain can be regarded as a series of small segments covalently bonded together and it is the effect of this chain connectivity which leads to deviations from an ideal entropy of mixing.
The effect of connectivity can be assessed by calculating the entropy change associated with the different number of ways of arranging polymer chains and solvent molecules on a lattice and, as it will be demonstrated, this differs from that calculated for the ideal solution.
This is embodied in the theory developed by Flory and Huggins, but still represents only the combinatorial contribution, whereas there are other (non-combinatorial) contributions to the entropy which come from the interaction of the polymer with the solvent and are much harder to quantify.
Nevertheless, the Flory-Huggins theory forms the cornerstone of polymer solution thermodynamics and is worth considering further.
8.4 Flory-Huggins theory
The dissolution of a polymer in a solvent can be regarded as a two stage process.
The polymer exists initially in the solid state where it is restricted to only one of the many conformations which are available to it as a free isolated molecule.
On passing into the liquid solution the chain achieves relative freedom and can now change rapidly among a multitude of possible equi-energetic conformations, dictated partly by the chain flexibility and partly by the interactions with the solvent.
Flory and Huggins considered that formation of the solution depends on
(a)
the transfer of the polymer chain from a pure, perfectly ordered state to a state of disorder which has the necessary freedom to allow the chain to be placed randomly on a lattice, and
(b)
the mixing process of the flexible chains with solvent molecules
.
The formalism of the lattice was used, for convenience, to calculate the combinatorial entropy of mixing following the method outlined in section 8.2 for small molecules, including the same starting assumptions and restrictions.
ENTROPY OF MIXING FOR ATHERMAL POLYMER SOLUTIONS
Consider a polymer chain consisting of r covalently bonded segments whose size is the same as the solvent molecules, i.e. where V 1 is the molar volume of component i.
To calculate the number of ways this chain can be added to a lattice, the necessary restriction imposed is that the segments must occupy r contiguous sites on the lattice because of the connectivity.
The problem is to examine the mixing of N 1 solvent molecules with N 2 monodisperse polymer molecules comprising r segments and we can begin by adding i polymer molecules to an empty lattice with a total number of cells N
Thus the number of vacant cells left which can accommodate the next molecule will be
The molecule can now be placed on the lattice, segment by segment, bearing in mind the restrictions imposed, viz. the connectivity of the segments, which requires the placing of each segment in a cell adjoining the preceding one.
This in turn will depend on the availability of a suitable vacancy.
The first segment can be placed in any empty cell but the second segment is restricted to the immediate near neighbours surrounding the first.
This can be given by the co-ordination number of the lattice z but we must also know if a cell in the co-ordination shell is empty.
If we let p i be the probability that an adjacent cell is vacant, then to a reasonable approximation this can be equated with the fraction of cells occupied by i polymer chains on the lattice i.e. which is valid for large values of z.
So the expected number of empty cells available for the second segment is zp i, and having removed one more vacant cell from the immediate vicinity, the third and each succeeding segment will have empty cells to choose from.
The total number of ways in which the molecule can be placed on the lattice is then
This gives the set of possible ways in which the molecule can be accommodated on the lattice.
The total number of ways for all N 2 molecules to be placed can then be obtained from the product of all possible ways, i.e.
The polymer molecules are all identical and so by analogy with equation (8.2) the total number of distinguishable ways of adding N 2 polymer molecules is
Substituting for W i gives
To evaluate the product term we can multiply and divide by r
This can be converted into the more convenient factorial form by remembering that the product is equivalent to and so equation (8.14) can be written as
The remaining empty cells on the lattice can now be filled by solvent molecules, but as there is only one distinguishable way in which this can be done,, there is no further contribution to W p and the entropy of the system.
The latter can now be calculated from the Boltzmann equation.
The factorials can again be approximated using Stirling's relation and while this requires considerable manipulation, which will be omitted here, it can eventually be shown that
To convert this into a form which will allow us to express this in the correct site fraction form we can add and subtract on the r.h.s. of equation (8.19) to give
For the pure solvent and the entropy.
Similarly the entropy of the pure polymer S 2 can be obtained for, which gives
Equation 8.21 then represents the entropy associated with the disordered or amorphous polymer on the lattice in the absence of solvent.
and so
It follows that the entropy change on mixing disordered polymer and solvent and so where  i the volume fraction can replace the site fraction if it is considered that the number of sites occupied by the polymer and solvent is proportional to their respective volumes.
Equation (8.22) is the expression for the combinatorial entropy of mixing of an athermal polymer solution and comparison with equation (8.7) shows that they are similar in form except for the fact that now the volume fraction is found to be the most convenient way of expressing the entropy change, rather than the mole fraction used for small molecules.
This change arises from the differences in size between the components which would normally mean mole fractions close to unity for the solvent especially when dilute solutions are being studied.
We can gain a further understanding of how the size of the polymer chain affects the magnitude of S M and why it differs from (equation 8.7), by recasting equation (8.22) in the following way.
The volume fraction  i can be expressed in terms of the number of moles n i, and the volume V i of component i, as where V is the total volume and.
If n i is converted to molar quantities then
As V i can conveniently be expressed as a function of a reference volume V o such that and assuming that, without introducing significant error, r can be equated with the degree of polymerization for the polymer then
If the volume fraction form is retained, then for a simple liquid mixture, but for a polymer solution and the last term in equation (8.24) will be smaller than the equivalent term calculated for small molecules.
Consequently S M per mole of lattice sites (or equivalent volume) will be very much less than and the contribution of the combinatorial entropy to the mixing process in a polymer solution is not as large as that for solutions of small molecules when calculated in terms of volume fractions and expressed as per mole of sites.
8.5 Enthalpy change on mixing
The derivation of S M from the lattice theory has been made on the assumption that no heat or energy change occurs on mixing.
This is an uncommon situation as experimental experience suggests that the energy change is finite.
We can make use of regular solution theory to obtain an expression for H M where this change in energy is assumed to arise from the formation of new solvent-polymer (1C2) contacts on mixing which replace some of the (1C1) and (2C2) contacts present in the pure solvent, and the pure polymer components respectively.
This can be represented by a quasi-chemical process where the formation of a solvent-polymer contact requires first the breaking of (1C1) and (2C2) contacts, and can be expressed as an interchange energy A 12 per contact, given by
Here  ii and  ij are the contact energies for each species.
The energy of mixing U M can be replaced by H M if no volume change takes place on mixing, and for q new contacts formed in solution
The number of contacts can be estimated from the lattice model by assuming that the probability of having a lattice cell occupied by a solvent molecule is simply the volume fraction  1.
This means that each polymer molecule will be surrounded by solvent molecules, and for N 2 polymer molecules
From the definition of  2 we obtain, hence which is the van Laar expression derived for regular solutions and shows that this approach can be applied to polymer systems.
To eliminate z, a dimensionless parameter per solvent molecule, is defined as which is the difference in energy between a solvent molecule when it is immersed in pure polymer and when in pure solvent.
It can also be expressed in the alternative form, where B is now an interaction density.
The final expression is and the interaction parameter  1 is an important feature of polymer solution theory which will be met with frequently.
8.6 Free energy of mixing
Having calculated the entropy and enthalpy contributions to mixing, these can now be combined to give the expression for the free energy of mixing, as
It is more useful to express equation (8.32) in terms of the chemical potentials of the pure solvent and the solvent in solution, by differentiating the expression with respect to the number of solvent molecules N 1 to obtain the partial molar Gibbs free energy of dilution (after multiplying by Avogadro's number),
This could also be carried out for the polymer, but as it makes no difference which one is taken (both having started from G M), equation (8.33) is more convenient to use.
While this expression is not strictly valid for the dilute solution regime it can be converted into a structure which is extremely informative about deviations from ideal solution behaviour encountered when measuring the molar mass by techniques such as osmotic pressure.
If the logarithmic term is expanded using a Taylor series: but truncated after the squared term, assuming  2 is small, then,
This can be modified by remembering that and, where v 2 is the partial specific volume of the polymer.
This can be related to the polymer molecular weight M 2 through so that and finally
Let us now anticipate the molar mass measurements to be described in chapter 9 and examine the osmotic pressure of a polymer solution in the light of equation (8.35).
8.7 Osmotic pressure
The osmotic pressure  of a solution can be regarded as the pressure which must be exerted on that solution to raise the chemical potential of the solvent in the solution back up to that of the pure solvent at a standard pressure P, i.e.
The compressibility of the solvent is equal to the molar volume of the solvent in solution, V 1, and can be assumed to be unchanged over a small range of pressures, thus giving
Substitution in (8.35) gives or
This is a limited virial expansion in which the first term is the classical van't Hoff expression for the osmotic pressure at infinite dilution.
The second term is related to the deviation from ideal behaviour and gives a relationship between the second virial coefficient B and the interaction parameter  1
Thus when then and the osmotic pressure is given by the ideal solution law.
8.8 Limitations of the Flory-Huggins theory
The simple lattice theory does not describe the behaviour of dilute polymer solutions particularly well because the following simplifications in the theoretical treatment are invalid:
(1)
it was assumed that the segment-locating process is purely statistical, but this would only be true if  12 was zero;
(2)
the treatment assumed that the flexibility of the chain is unaltered on passing into the solution from the solid state  this limits the calculation of S M to the combinatorial contribution only and neglects any contribution from continual flexing of the chain in solution which will contribute to the non-combinatorial or excess entropy of mixing;
(3)
any possible specific solvent-polymer interactions which might lead to orientation of the solvent molecules in the vicinity of the polymer chain are neglected i.e. polar solutions may be inadequately catered for by this theory;
(4)
a uniform density of lattice site occupation is assumed, but this will only apply to relatively concentrated solutions;
(5)
the parameter  1 is often concentration-dependent but this is ignored
.
It is now accepted that a non-combinatorial entropy contribution arises from the formation of new (1C2) contacts in the mixture which change the vibrational frequencies of the two components, i.e. assumption (d) in section 8.2 must be relaxed.
This can be allowed for by recognizing that  1 is actually a free energy parameter comprising entropic  H and enthalpic  S contributions, such that.
These are defined by
Experiments tend to show that the major contribution comes from the  s component, indicating that there is a large decrease in entropy (non-combinatorial) which is acting against the dissolution process of a polymer in a solvent.
In spite of much justifiable criticism, the Flory-Huggins theory can still generate considerable interest because of the limited amount of success which can be claimed for it in relation to phase equilibria studies.
8.9 Phase equilibria
Use can be made of the Flory-Huggins theory to predict the equilibrium behaviour of two liquid phases when both contain amorphous polymer and one or even two solvents.
Consider a two component system consisting of a liquid (1) which is a poor solvent for a polymer (2).
Complete miscibility occurs when the Gibbs free energy of mixing is less than the Gibbs free energies of the components, and the solution maintains its homogeneity only as long as G M remains less than the Gibbs free energy of any two possible co-existing phases.
The situation is represented by curve T 5 in figure 8.2.
The miscibility of this type of system is observed to be strongly temperature dependent and as T decreases the solution separates into two phases.
Thus at any temperature, say T 1,; the Gibbs free energy of any mixture, composition in the composition range to, is higher than either of the two co-existing phases whose compositions are and and phase separation takes place.
The compositions of the two phases and do not correspond to the two minima, but are measured from the points of contact of the double tangent AB with the Gibbs free energy curve.
The same is true for other temperatures lying below T c, and the inflexion points can be joined to bound an area representing the heterogeneous two phase system, where there is limited solubility of component 2 in 1 and vice-versa.
This is called a cloud-point curve.
As the temperature is increased the limits of this two phase co-existence contract, until eventually they coalesce to produce a homogeneous, one phase, mixture at T c, the critical solution temperature.
This is sometimes referred to as the critical consolute point.
In general, we can say that if the free energy-composition curve has a shape which allows a tangent to touch it at two points, phase separation will occur.
The critical solution temperature is an important quantity and can be accurately defined in terms of the chemical potential.
It represents the point at which the inflexion points on the curve merge, and so it is the temperature where the first, second, and third derivatives of the Gibbs free energy with respect to mole fraction are zero.
It is also true that the partial molar Gibbs free energies of each component are equal at this point and it emerges that the conditions for incipient phase separation are
By remembering that, application of these criteria for equilibrium to equation (8.33) leads to the first derivative of that equation while the second derivative is where the subscript c denotes critical conditions.
The critical composition at which phase separation is first detected is then and which indicates that at infinitely large chain length.
The interaction parameter  1 is a useful measure of the solvent power.
Poor solvents have values of  1 close to 0.5 while an improvement in solvent power lowers  1.
Generally, a variation from 0.5 to -1.0 can be observed although for many synthetic polymer solutions the range is 0.6 to 0.3.
A linear temperature dependence for  1 is also predicted of the general form, which suggests that as the temperature increases the solvating power of the liquid should increase.
This has implications for polymer fractionation.
8.10 Fractionation
The relations derived in this and other chapters normally assume that the polymer sample has a unique molar mass.
This situation is rarely achieved in practice and it is useful to know the form of the molar mass distribution in a sample, as this can have a significant bearing on the physical properties.
It is also advantageous to be able to prepare sample fractions, whose homogeneity is considerably better than the parent polymer, especially when testing dilute solution theory.
We have seen that the chain length can be related to the solvent power, expressed as  1, by equation (8.46) and this is illustrated in figure 8.3.
The implication is that if  1 can be carefully controlled, conditions could be attained which would allow a given molecular species to precipitate, while leaving larger or smaller molecules in solution.
This process is known as fractionation.
Experimentally, a polymer sample can be fractionated in a variety of ways and three in common use are:
(1)
addition of a non-solvent to a polymer solution;
(2)
lowering the temperature of the solution; and
(3)
column chromatography
.
In the first method the control of  1 is effected by adding a non-solvent to the polymer solution.
If the addition is slow,  1 increases gradually until the critical value for large molecules is reached.
This causes precipitation of the longest chains first and these can be separated from the shorter chains which remain in solution.
In practice the polymer solution is held at a constant temperature while precipitant is added to the stirred solution.
When the solution becomes turbid the mixture is warmed until the precipitate dissolves.
The solution is then returned to the original temperature and the precipitate which reforms is allowed to settle and then separated.
This ensures that the precipitated fraction is not broadened by local precipitation during addition of the non-solvent.
Successive additions of small quantities of non-solvent to the solution allow a series of fractions of steadily decreasing molar mass to be separated.
In the second method,  1 is varied by altering the temperature, with similar results.
For both techniques, it is useful to dissolve the polymer initially in a poor solvent with a large  1 value.
This ensures that only small quantities of non-solvent are required to precipitate the polymer in method 1, and that the temperature changes required in method 2 are small.
In column chromatography the polymer is precipitated on the inert support medium at the top of a column which has a temperature gradient imposed along its length.
The packing is usually glass beads of 0.1 to 0.3 mm diameter.
A solvent + non-solvent mixture is used to elute the sample and fractionation is achieved by using a solvent gradient.
This is generated in a mixing system, situated above the column, by constantly increasing the solvent to non-solvent ratio and as the mixture is initially a poor solvent which is gradually enriched by the good solvent the low molar mass fractions are eluted first.
Fractions of increasing molar mass are collected from the bottom of the column.
In each of the techniques, the mass and molar mass of the fractions are recorded and a distribution curve for the sample can be constructed from the results.
However, as the fractions themselves have a molar mass distribution, extensive overlapping of the fractions will occur as shown schematically in figure 8.4.
Consequently a simple histogram constructed from the mass and molar mass of each fraction will not provide a good representation of the distribution and a method must be used to compensate for the overlapping.
A useful approach was proposed by Schulz who suggested that a cumulative mass fraction be plotted against the molar mass.
The cumulative mass fraction can be calculated by adding half the mass fraction W i of the i th fraction to the total mass fraction of those fractions preceding it, i.e.
The values of are plotted against the corresponding M i and connected by a smooth curve as shown in figure 8.5, to give the integral distribution curve.
The differential curve can be obtained by determining the slope of this curve at selected molar masses and plotting this against the appropriate molar mass.
8.11 Flory-Krigbaum theory
To overcome the limitations of the lattice theory resulting from the discontinuous nature of a dilute polymer solution, Flory and Krigbaum discarded the idea of a uniform distribution of chain segments in the liquid.
Instead they considered the solution to be composed of areas containing polymer which were separated by the solvent.
In these areas the polymer segments were assumed to possess a Gaussian distribution about the centre of mass, but even with this distribution the chain segments still occupy a finite volume from which all other chain segments are excluded.
It is within this excluded volume that the long range interactions originate which are discussed more fully in chapter 10.
Flory and Krigbaum defined an enthalpy parameter and an entropy of dilution parameter such that the thermodynamic functions used to describe these long range effects are given in terms of the excess partial molar quantities
From equation (8.33) it can be seen that the excess free energy of dilution is
Combination of these non-ideal terms then yields
As we saw from equation (8.40), when and the solution appears to behave as though it were ideal.
The point at which this occurs is known as the FLORY or THETA point and is in some ways analogous to the Boyle point for a non-ideal gas.
Under these conditions
The temperature at which these conditions are obtained is the FLORY or THETA temperature , conveniently defined as.
This tells us that  will only have a meaningful value when  1 and  1 have the same sign.
Substitution in (8.50) followed by rearrangement gives and shows that deviations from ideal behaviour vanish when.
The theta temperature is a well defined state of the polymer solution at which the excluded volume effects are eliminated and the polymer coil is in an unperturbed condition (see chapter 10).
Above the theta temperature expansion of the coil takes place, caused by interactions with the solvent, whereas below * the polymer segments attract one another, the excluded volume is negative, the coils tend to collapse and eventual phase separation occurs.
8.12 Location of the theta temperature
The theta temperature of a polymer-solvent system can be measured from phase separation studies.
The value of  1.
c; at the critical concentration is related to the chain length of the polymer by equation (8.46), and substitution in (8.52) leads to where now we have replaced r with the equivalent degree of polymerization  n.
Rearrangement gives
Remembering that, where M and are the molar mass and partial specific volume of the polymer, and V 1 is the molar volume of the solvent, the equation states that the critical temperature is a function of M and the value of T c at infinite M is the theta temperature for the system.
Precipitation data for several systems have proved the validity of equation (8.54).
Linear plots are obtained with a positive slope from which the entropy parameter  1 can be calculated as shown in figure 8.6.
Typical values are shown in table 8.1, but  1 values measured for systems such as polystyrene + cyclohexane have been found to be almost ten times larger than those derived from other methods of measurement.
This appears to arise from the assumption in the Flory-Huggins theory that  1 is concentration independent and improved values of  1 are obtained when this is rectified.
The theta temperature, calculated from equation (8.54) for each system is in good agreement with that measured from the temperature variation of A 2 (=B/RT see chapter 9).
Curves of A 2, measured at various temperatures in the vicinity of , are constructed as a function of temperature for one or more molar masses as shown in figure 8.7.
Intersection of the curves with the T-axis occurs when and.
The curves for each molar mass of the same polymer should all intersect at.
8.13 Lower critical solution temperatures
So far we have been concerned with non-polar solutions of amorphous polymers, whose solubility is increased with rising temperature, because the additional thermal motion helps to decrease attractive forces between like molecules, and encourages energetically less favourable contacts.
The phase diagram for such systems, when the solvent is poor, is depicted by area A in figure 8.8, where the critical temperature T c occurs near the maximum of the cloud-point curve and is often referred to as the upper critical solution temperature (UCST).
This behaviour follows from that depicted in figure 8.2.
For non-polar systems S M is normally positive but weighted heavily by T and so solubility depends mainly on the magnitude of H M, which is normally endothermic (positive).
Consequently as T decreases G M eventually becomes positive and phase separation takes place.
Values of  and  1, in table 8.1, show that for systems 1 to 4 the entropy parameter is positive, as expected, but for poly (acrylic acid) in dioxan and polymethacrylonitrile in butanone,  is negative at the theta temperature.
As, when, the enthalpy is also negative for these systems.
This means that systems 5 and 6 exhibit an unusual decrease in solubility as the temperature rises, and the cloud-point curve is now inverted as in area B. The corresponding critical temperature is located at the minimum of the miscibility curve and is known as the lower critical solution temperature (LCST).
In systems 5 and 6 this phenomenon is a result of hydrogen-bond formation between the polymer and solvent, which enhances the solubility.
As hydrogen bonds are thermally labile a rise in T reduces the number of bonds and causes eventual phase separation.
In solutions, which are stabilized in this way by secondary bonding, the LCST usually appears below the boiling temperature of the solvent but it has been found experimentally that an LCST can be detected in non-polar systems when these are examined at temperatures approaching the critical temperature of the solvent.
Polyisobutylene in a series of n-alkanes, polystyrene in methyl acetate and cyclohexane, and cellulose acetate in acetone all exhibit LCSTs.
The LCST is located by heating the solutions, in sealed tubes, up to temperatures approaching the gas-liquid critical point of the solvent.
As the temperature rises, the liquid expands much more rapidly than the polymer, which is restrained by the covalent bonding between its segments.
At high temperatures, the spaces between the solvent molecules have to be reduced if mixing is to take place and when this eventually results in too great a loss of entropy, phase separation occurs.
The separation of polymer/solvent systems into two phases as the temperature increases is now recognized to be a characteristic feature of all polymer solutions.
This presents a problem of interpretation within the framework of regular solution theory, as the accepted form of  1 predicts a monotonic change with temperature and is incapable of dealing with two critical consolute points.
The problem of how to accommodate, in a theoretical framework, the existence of two miscibility gaps requires a new approach, and a more elaborate treatment by Prigogine and co-workers encompasses the difference in size between the components of a mixture, which can not be ignored for polymer solutions.
They replaced the rigid lattice model used by Flory and Huggins, which is valid only at absolute zero, with a flexible lattice whose cells change volume, with temperature and pressure.
This allowed them to include in their theory dissimilarities in free volume between polymer and solvent, together with the corresponding interactions.
The same approach was extended by both Patterson and Flory to deal specifically with polymer systems.
The most important of the new parameters is the so-called ' structural effect ' which is related to the number of degrees of freedom ' 3 c ' which a molecule possesses, divided by the number of external contacts q.
This structural factor is a measure of the number of external degrees of freedom per segment and changes with the length of the component.
Thus the ratio decreases as a liquid becomes increasingly polymeric.
The expansion and free volume can then be characterized by the ratio of the thermal energy arising from the external degrees of freedom available to the component, U thermal, and the interaction energy between neighbouring non-bonded segments, U cohesive which will oppose the thermal energy effects, i.e. where  * is the characteristic cohesive energy per contact.
For convenience q may be replaced by r, the number of chain segments, although q will actually be less than r because some of the external contacts are used in forming the covalent bonds in the chain.
Free volume dissimilarities become increasingly important as the size of one component increases with respect to the second, as in polymer solutions, and when these differences are sufficiently large, phase separation can be observed at the LCST.
The differences in expansivity can be accounted for if the interaction parameter is now expressed as where the first term reflects the interchange energy on forming contacts of unlike type and includes segment size differences, while the second term is the new ' structural ' contribution coming from free volume changes on mixing a dense polymer with an expanded solvent.
This can be represented schematically in figure 8.9.
The first term in (8.56), shown by curve 1, is merely an expression of the Flory-Huggins theory where X decreases constantly with rising temperature, but now inclusion of the new free volume term, shown by curve 2, modifies the behaviour of .
The second term gains in importance as the expansivities of the two components become increasingly divergent with temperature and the net effect is to increase  again until it once more attains its critical value at high temperature.
The LCST which results, is then a consequence of these free volume differences and is an entropically controlled phenomenon.
This can be illustrated in the following ways.
In terms of the flexible lattice model, one can imagine the polymer and liquid lattices expanding at different rates until a temperature is reached at which the highly expanded liquid lattice can no longer be distorted sufficiently to accommodate the less expanded polymer lattice and form a solution, i.e. the loss in entropy during the distortion becomes so large and unfavourable that phase separation (LCST) takes place.
Alternatively, a polymer solution can be thought of as a system formed by the condensation of solvent into a polymer.
As the temperature increases, the entropy loss incurred during condensation becomes greater until eventually it is so unfavourable that condensation in the polymer is impossible, and phase separation takes place.
Neither picture is particularly rigorous but they serve to emphasize the fact that the LCST is an entropically controlled phenomenon.
8.14 Solubility and the cohesive energy density
Solvent-polymer compatibility problems are repeatedly encountered in industry.
For example, in situations requiring the selection of elastomers for use as hose-pipes or gaskets, the correct choice of elastomer is of prime importance, as contact with highly compatible fluids may cause serious swelling and impair the operation of the system.
The wrong selection can have far reaching consequences; the initial choice of an elastomer for the seals in the landing gear of the DC-8 aircraft resulted in serious jamming because the seals become swollen when in contact with the hydraulic fluid.
This almost led to grounding of the plane but replacement with an incompatible elastomer made from ethylene-propylene copolymer rectified the fault.
To avoid such problems a technologist may wish to have at his disposal a rough guide to aid the selection of solvents for a polymer or to assess the extent of polymer-liquid interaction other than those already described.
Here use can be made of a semi-empirical approach suggested by Hildebrand and based on the premise that ' like dissolves like '.
The treatment involves relating the enthalpy of mixing to the cohesive energy density and defines a solubility parameter, where E is the molar energy of vaporization and V is the molar volume of the component.
The proposed relation for the heat of mixing of two non-polar components shows that H M is small for mixtures with similar solubility parameters and this indicates compatibility.
Values of the solubility parameter for simple liquids can be readily calculated from the enthalpy of vaporization.
The same method can not be used for a polymer and one must resort to comparative techniques.
Usually  for a polymer is established by finding the solvent which will produce maximum swelling of a network or the largest value of the limiting viscosity number, as both indicate maximum compatibility.
The polymer is then assigned a similar value of .
Alternatively, Small and Hoy have tabulated a series of group molar attraction constants from which a good estimate of  for most polymers can be made.
The suggested group contributions are shown in table 8.3 and the solubility parameter for a polymer can be estimated from the sum of the various molar attraction constants F for the groups which make up the repeat unit i.e.
Here V is the molar volume of the repeat unit whose molar mass is M o and p is the polymer density.
Thus for poly (methyl methacrylate) with and, we have using the Hoy values Therefore
For a more complex polyhydroxyether of Bisphenol A structure and with
Both estimates are within 10 per cent of experimentally determined values.
Attempts to correlate  with  1 from the Flory-Huggins equation have met with limited success because of the unjustifiable assumptions made in the derivation.
It is now believed, however, that  1 is not an enthalpy parameter but a free energy parameter and a relation of the form c.f. section 8.8 has improved the correlation.
Here is supposed to compensate for the lack of a non-combinatorial entropy contribution in the Flory-Huggins treatment.
Unfortunately, solubility is not a simple process and secondary bonding may play an important role in determining component interactions.
More detailed approaches have been suggested, which introduce a three-dimensional  composed of contributions from van der Waals dispersion forces, dipole-dipole interaction, and hydrogen bonding.
The overall solubility parameter is then the sum of the various contributions
Usually two dimensional plots are constructed first, before the three dimensional ' solubility volume ' is established, as shown in figure 8.10.
This is not a convenient construction and often a plot of versus  H is considered to be sufficiently accurate as  n and  p are usually similar and the main polar contribution comes from the hydrogen bonding factor  H.
8.15 Polymer-polymer mixtures
In the constant search for new materials with improved performance, the idea of mixing two or more different polymers to form new substances having a combination of all the attributes of the components, is deceptively attractive.
Deceptively, because in practice it is rarely accomplished and only in a few cases have polymer blends or mixtures achieved industrial importance.
The main reason is that most common polymers do not mix with one another to form homogeneous, one phase solutions or blends, and an explanation for this is to be found in the thermodynamics of solutions which have been outlined in the previous sections.
As we have seen, when two liquids, or a liquid and a polymer are mixed, the formation of a homogeneous, one phase solution is assisted mainly by the large favourable gain in combinatorial entropy.
This entropic contribution is progressively reduced when one or both components increase in size, and the reason for this becomes obvious on inspection of equation (8.24).
When r 1 and r 2 both increase, then S M becomes smaller; consequently attempts to mix two high molar mass polymer samples will receive little assistance from this function and must depend increasingly on a favourable (negative) heat of mixing embodied in the  parameter.
This loss of entropy can be conveniently illustrated using the simple lattice model shown in figure 8.11.
Here a 10 x 10 lattice, containing 50 white and 50 black units randomly mixed (a), will result in approximately 10 30 possible different arrangements of the units on mixing.
If these white units are now connected to other white units, and black to black (b), to form five equal chains of each colour with, then the number of possible arrangements of these chains decreases to about 10 3.
Thus as r 1 and r 2 approach infinity S M will become negligible and the free energy of mixing will become essentially dependent on H M which now has to be either very small or negative.
The heat of mixing for the majority of polymer (1)  polymer (2) pairs tends to be endothermic and can be approximated by reference to the solubility parameters using equation (8.57).
This can be written as where the reference volume normally assumes a value of.
The critical value for  12 can be estimated from where x i is the degree of polymerization, related to the actual degree of polymerization x n and the reference volume by with V R the molar volume of the repeat unit.
The critical values for  12 above which the two polymers will phase separate, calculated for various mixtures with, are shown in table 8.4 along with the corresponding differences in .
This shows that for mixing to take place between high molecular weight components the solubility parameters would have to be virtually identical.
This limits the number of possible combinations such that only a few examples exist in this category.
These include polystyrene/ poly (-methyl styrene) below M 70 000, and the polyacrylates mixed with the corresponding poly vinyl esters, e.g. and
The situation changes if H M is negative as this will encourage mixing, and the search for binary polymer blends which are miscible has focussed on combinations in which specific intermolecular interactions, such as hydrogen bonds, dipole-dipole interactions, ion-dipole interactions, or charge transfer complex formation, can exist between the component polymers.
A substantial number of miscible blends have now been discovered using this principle and it is possible to identify certain groups or repeat units, which when incorporated in polymer chains tend to enter into these intermolecular interactions and enhance the miscibility.
A short selection of some of these complementary groups is given in table 8.5, where a polymer containing groups or composed of units from column 1 will tend to form miscible blends with polymers containing groups or composed of units from column 2.
Thus it is believed that polystyrene forms miscible blends with poly (vinyl methylether), and with poly (phenylene oxide) s (examples 1 and 2, respectively) because of interactions between the -electrons of the phenyl rings and the lone pairs of the ether oxygens.
Similarly it has been suggested that a weak hydrogen bond, which is strong enough to induce miscibility can form between the carbonyl unit of poly (methyl methacrylate) and the -hydrogen of poly (vinyl chloride) (example 3 where).
i.e. Much stronger hydrogen bonding interactions can be obtained if units such as or sites for ion-dipole interactions such as can be built into chains, and even in relatively small amounts these can transform immiscible pairs into totally miscible blends.
Many of these blends undergo quite rapid demixing as the temperature is raised and an LCST phase boundary can be located above the glass transition temperature of the blend.
The origins of the lower critical phase separation phenomenon in polymer blends are not yet clearly understood and three possible causes have been proposed.
(i)
Free volume dissimilarities may become unfavourable to mixing on increasing the temperature.
(ii)
There may be unfavourable entropy contributions arising from non-random mixing.
(iii)
A temperature dependent heat of mixing may result if specific intermolecular interactions, which dissociate on heating, are responsible for miscibility at lower temperatures.
While the latter seems the most likely cause in many blends where specific interactions have been identified, miscible blends can also be obtained when certain statistical copolymers are mixed with either a homopolymer, or another copolymer, in which no such interactions have been located.
Thus poly (styrene-stat-acrylonitrile) will form miscible blends with poly (methyl methacrylate) if the composition of the copolymer lies in the range 10C39 wt% acrylonitrile.
This range of compositions is called the ' miscibility window ' and has been reported to be present in other systems.
The drive towards formation of a miscible solid solution in these cases is believed to arise when large repulsive interactions exist between the monomer units (A) and (B) comprising the copolymer; on mixing with a polymer (C), the number of these unfavourable (A B) contacts are reduced by forming less repulsive (A-C) or (B-C) contacts and a miscible blend results.
Many of these blends also exhibit an LCST.
Thus the driving force towards lower critical phase separation in polymer-polymer solutions may depend on the system or may be a combination of the effects (i)  (iii).
CHAPTER 9
Polymer Characterization Molar Masses
9.1 Introduction
Many of the distinctive properties of polymers are a consequence of the long chain lengths, which are reflected in the large molar masses of these substances.
While such large molar masses are now taken for granted, it was difficult in 1920 to believe and accept that these values were real and not just caused by the aggregation of much smaller molecules.
Values of the order of 10 6 g mol -1 are now accepted without question, but the accuracy of the measurements is much lower than for simple molecules.
This is not surprising, especially when polymer samples exhibit polydispersity, and the molar mass is, at best, an average dependent on the particular method of measurement used.
Estimation of the molar mass of a polymer is of considerable importance, as the chain length can be a controlling factor in determining solubility, elasticity, fibre forming capacity, tear strength, and impact strength in many polymers.
The methods used to determine the molar mass M are either relative or absolute.
Relative methods require calibration with samples of known M and include viscosity and vapour pressure osmometry.
The absolute methods are often classified by the type of average they yield, i.e. colligative techniques yield number averages, light scattering and the ultracentrifuge yield higher averages, e.g. weight and z-average.
9.2 Molar masses, molecular weights, and SI units
The dimensionless quantity ' the relative molecular mass' (molecular weight) defined as the average mass of the molecule divided by [formula] the mass of an atom of the nuclide C 12, is often used in polymer chemistry, and called the molecular weight.
In this book the quantity molar mass is used and appropriate SI units are given.
9.3 Number average molar mass M n
Determination of the number average molar mass M n involves counting the total number of molecules, regardless of their shape or size, present in a unit mass of the polymer.
The methods are conveniently grouped into three categories:
end-group assay,
thermodynamic, and
transport methods
.
9.4 End-group assay
The technique is of limited value and can only be used when the polymer has an end group amenable to analysis.
It can be used to follow the progress of linear condensation reactions when an end group, such as a carboxyl, is present which can be titrated.
It is used to detect amino end groups in nylons dissolved in m-cresol, by titration with methanolic perchloric acid solution, and can be applied to vinyl polymers if an initiator fragment, perhaps containing halogen, is attached to the end of the chain.
The sensitivity of the method decreases rapidly as the chain length increases and the number of end groups drops.
A practical upper limit might reach an M n of about 15000 g mol -1.
9.5 Colligative properties of solutions: thermodynamic considerations
Because chemical methods are rather limited, the most widely used techniques for measuring the molar mass of a polymer are physical.
Among the more common methods are those which depend on the colligative properties of dilute solutions.
These are
(a)
lowering of the vapour pressure
(b)
elevation of the boiling point
(c)
depression of the freezing point
(d)
osmotic pressure.
A colligative property is defined as one which is a function of the number of solute molecules present per unit volume of solution and is unaffected by the chemical nature of the solute.
Thus if Y represents any of the above colligative properties then where N i is the number of particles of each solute component i, and K is a proportionality constant.
The concentration of a solution per unit volume of solution V is where w i is the mass of the component and N A is the Avogadro constant.
The colligative property can be expressed in the reduced form Y/c so that
Hence any colligative method should yield the number average molar mass M n of a polydisperse polymer.
When a solute, such as a polymer (component 2), is dissolved in a solvent (component 1) to form a homogeneous solution, there is a change in the chemical potential which can be expressed in terms of the activity of the solvent a 1.
During the measurement of the molar mass of the polymer using a colligative method, an equilibrium is established when the chemical potential of the solvent in the solution is equal to that of the pure solvent, where the pure solvent is either in another phase or separated from the solution by a semi-permeable membrane.
The equilibration procedure can be achieved either by changing the temperature or the pressure of the system, and the amount of this change is then a measure of the activity of the solvent in solution.
This tells us nothing immediately about the solute but, if very dilute solutions are used, the following useful approximations can be made.
The activity of the solvent can be considered to be equal to the mole fraction of the solvent x 1.
By expanding the logarithmic term and assuming that in dilute solutions this can be restricted to the first expansion term, ln a 1 can be related to the mole fraction of the solute x 2.
We can now make use of these approximations to calculate M n.
9.6 Ebullioscopy and cryoscopy
In principle these two methods can be treated together and the relevant expressions are derived from the Clausius-Clapeyron equation describing the temperature dependence of the vapour pressure of a liquid where H 1 is the latent heat of vapourization.
If P is taken as the vapour pressure of a solution whose pure solvent vapour pressure is P o, then for solutions containing an involatile solute which gives
If the solution is very dilute the change in temperature T can be related to the solute mole fraction by and substituting for gives where c 2 is the solute concentration (mass per unit volume solution).
Polymer solutions do not behave in this ideal manner even in the dilute solution regime and for accurate molar mass measurements, deviations from ideality must be eliminated.
A more accurate representation of the behaviour of the polymer solution can be obtained using equation (8.35) where Substitution in equation (9.9) yields and rearrangement eliminating higher terms gives
The non-ideal behaviour can then be eliminated by extrapolating the experimental data to c 2 0 where equation (9.14) reduces to equation (9.11) and can be calculated.
For ebulliometry, T, H, and T are the boiling temperature of the solvent, the enthalpy of vaporization of the solvent, and the elevation of the boiling temperature respectively, while for cryoscopy they represent the freezing temperature of the solvent, the enthalpy of fusion of the solvent, and the depression of the freezing temperature.
The equation represents the limiting case at infinite dilution and it is necessary to extrapolate for a series of solutions to c = 0 in order to calculate M n.
The measurements are limited by the sensitivity of the thermometer used to obtain T.
At present this can rarely detect a T of less than with any precision, and the limit of accurate measurement of M n is in the region of 25 000 to 30 000 g mol -1.
9.7 Osmotic pressure
Measurement of the osmotic pressure  of a polymer solution can be carried out in the type of cell represented schematically in figure 9.1.
The polymer solution is separated from the pure solvent by a membrane, permeable only to solvent molecules.
Initially, the chemical potential  1, of the solvent in the solution, is lower than that of the pure solvent,  1  and solvent molecules tend to pass through the membrane into the solution in order to attain equilibrium.
This causes a build up of pressure in the solution compartment until, at equilibrium, the pressure exactly counteracts the tendency for further solvent flow.
This pressure is the osmotic pressure.
The expression for the reduced osmotic pressure has already been derived in section 8.7 and has the form where the limiting form, valid only at infinite dilution, is
Only under special conditions, when the polymer is dissolved in a theta-solvent, will be independent of concentration.
Experimentally, a series of concentrations is studied and the results treated according to one or other of the following virial expansions.
McMillan and Meyer suggested, while alternative forms are also used: and
The coefficients B, A 2,  2 and B 3, A 3,  3, are the second and third virial coefficients.
When solutions are sufficiently dilute a plot of against c is linear and the third virial coefficients (B 3, A 3,  3) can be neglected.
The various forms of the second virial coefficient are interrelated by
Although not normally detected, the third virial coefficient occasionally contributes to the non-ideal behaviour in dilute solutions and a curved plot is obtained (figure 9.2a.)
This increases the uncertainty of the extrapolation, but can be overcome by recasting equation (9.19) and introducing a polymer-solvent interaction parameter g
It has been found that g = 0.25 in good solvents so that equation (9.21) becomes
A plot of against c is now linear and this extrapolation is illustrated in figure 9.2b.
This example (figure 9.2) also shows the differing solubility of poly (methyl methacrylate) in the three solvents.
In a good solvent, toluene, the slope or A 2 is large, but as the solvent becomes poorer (acetone) A 2 decreases, until it is zero in the theta-solvent acetonitrile.
Thus A 2 provides a useful measure of the thermodynamic quality of the solvent and measures the deviation from ideality of the polymer solution.
The value of M n is calculated from the intercept using equation (9.16).
The corresponding values of the second virial coefficient are obtained from the slopes of the plots (table 9.1).
PRACTICAL OSMOMETRY
The static method of determining the osmotic pressure of a polymer solution, using volumes of 3 to 20 cm 3 of solution, is a relatively slow process which requires about 24 h to equilibrate at each concentration.
Several designs, suitable for this type of measurement, are typified by the Pinner Stabin instrument shown schematically in figure 9.1.
The osmometer is assembled, under a layer of solvent, by clamping two membranes (kept continually moist with solvent) on either side of the glass cell c.
These are retained in position by two metal plates perforated and grooved to allow contact between the membrane and solvent which is in the outer container.
The preparation of the membranes is very important and must be carefully carried out.
They are normally made of cellulose or a cellulose derivative and should be slowly conditioned from the storage liquid to the solvent in use.
This is done by transferring the membrane to mixtures progressively richer in the solvent, allowing them time to equilibrate with the mixture, then transferring again until pure solvent is reached.
Equilibration in each mixture usually takes a few hours.
When assembled, the osmometer is placed in a jacket containing enough solvent to cover the bottom part of the reference capillary s.
Solvent is then withdrawn from the cell c and a solution of polymer added by means of a syringe.
Care is taken during the filling stage to avoid trapping bubbles in the cell.
The level of the solution is then adjusted to a few centimetres above the level of solvent in s by means of a levelling rod 1.
Mercury is added to the cup t, to ensure a leak free system, and the osmometer is left undisturbed in a thermostat bath controlled to  0.01 K to reach equilibrium.
The osmotic pressure can be calculated from the difference in heights h between the solvent and solution in s and m respectively and  is measured from  = hpg, for each concentration where p is the density of the solution and g the acceleration of free fall.
Results are plotted as against c as described and M n is calculated from the intercept.
The method suffers from the disadvantage that it is slow and consequently diffusion of low molar mass material could be large enough to introduce serious error in the measurement.
Two or three high-speed automatic membrane-osmometers have now been designed to reduce these drawbacks and are commercially available.
The Mechrolab osmometer, shown schematically in figure 9.3, consists of a solution + solvent cell of volume approximately 1 cm 3, with the solvent side connected to a reservoir attached to a servo-driven elevator.
When solution is added to the top-half of the cell, solvent in the lower-half tends to flow into the upper section to equalize the chemical potentials.
The flow is detected optically by the movement of a bubble in a capillary below the cell.
The movement activates the servo-motor, which alters the hydrostatic head thereby counteracting the flow.
The movement of the solvent reservoir is then a measure of the osmotic pressure of the solution.
Equalization is rapid (5 to 30 min) and permeation is readily detected, if present, by following the change of head as a function of time on a recorder.
There is no actual flow of solvent in the Mechrolab instrument.
A slightly different principle, which allows solvent flow to take place, forms the basis of the Melab and Knauer models.
The Melab osmometer has a stainless steel cell (volume 0.5 cm 3), with solution and solvent compartments separated by the membrane.
One wall of the cell is a flexible stainless steel diaphragm connected through a strain gauge to a recorder.
As solvent diffuses through the membrane, the increase in volume causes the diaphragm to move.
The motion is detected by the gauge and translated into a pressure.
The design has the advantage that both solvent and solution compartments are easily rinsed out and the cell does not have to be dismantled if contamination by permeation of low molar mass solute occurs.
All osmotic pressure measurements are extremely sensitive to temperature and must be carried out under rigorously controlled temperature conditions.
This is allowed for in each instrument and in addition, measurements can be made over a range of temperatures (278 to 373 K).
Solvents should be chosen which are chemically stable and have a low to medium vapour pressure at the temperature of operation, as this avoids problems of bubble formation in the measuring chamber.
9.8 Transport methods  vapour pressure osmometer
In conventional osmometry, the membrane permeability imposes a lower limit of about M n = 15 000 g mol -1.
A technique, based on the lowering of the vapour pressure, called vapour pressure osmometry is a useful method of measuring values of M n from 50 to 20 000 g mol -1.
It is a relative method and is calibrated using such low molar mass standards as benzil, methyl stearate, or glucose penta-acetate.
The apparatus consists of a thermostatted chamber, saturated with solvent vapour at the temperature of measurement, and containing two differential matched thermistors which are capable of detecting temperature differences as low as 10 -4 K. Two syringes, one for solvent and one for solution, are used to apply a drop of solution to one thermistor, and a drop of solvent to the other.
As there is a difference in vapour pressure between the solution and the solvent drops, solvent from the vapour phase will condense on the solution drop causing its temperature to rise.
Because of the large excess of solvent present, evaporation, and hence cooling of the solvent drop, is negligible.
When equilibrium is attained, the temperature difference between the two drops T is a measure of the extent of the vapour pressure lowering by the solute.
The thermistors form part of a Wheatstone bridge, and T is recorded as a difference in resistance R.
The molar mass can then be calculated from where K * is the calibration constant.
As with other methods M n is obtained by extrapolating the data to.
The calibration constant is estimated by measuring R for solutions of known concentration prepared from standard compounds of known molar mass M k then
In some instances an additional correction for the dilution of the drop of solution may be necessary.
9.9 Light scattering
Light scattering is one of the most popular methods for determining the weight average molar mass M w.
The phenomenon of light scattering by small particles is familiar to us all; the blue colour of the sky or the varied colours of a sunset, the poor penetration of car headlights in a fog is caused by water droplets scattering the light, and the obvious presence of dust in a sunbeam or the Tyndall effect in an irradiated colloidal solution are further examples of this effect.
The fundamentals of light scattering were expounded by Lord Rayleigh in 1871 during his studies on gases, where the particle is small compared with the wavelength of the incident radiation.
Light is an electromagnetic wave, produced by the interaction of a magnetic and electric field, both oscillating at right angles to one another in the direction of propagation.
When a beam of light strikes the atoms or molecules of the medium, the electrons are perturbed or displaced and oscillate about their equilibrium positions with the same frequency as the exciting beam.
This induces transient dipoles in the atoms or molecules, which act as secondary scattering centres by re-emitting the absorbed energy in all directions, i.e. scattering takes place.
For gases, Rayleigh showed that the reduced intensity of the scattered light R  at any angle  to the incident beam, of wavelength  could be related to the molar mass of the gas M, its concentration c, and the refractive index increment by
The quantity R  is often referred to as the Rayleigh ratio and is equal to where I is the intensity of the incident beam, i, is the quantity of light scattered per unit volume by one centre at an angle  to the incident beam, and r is the distance of the centre from the observer.
This is valid for a gas, where all the particles are considered to be independent scattering centres and the addition of more centres, which increases n increases the scattering.
The situation changes when dealing with a liquid as remains unaffected by the addition of molecules and can be expected to be zero.
This conceptual difficulty was overcome in the fluctuation theories of Smoluchowski and Einstein; they postulated that optical discontinuities exist in the liquid arising from the creation and destruction of holes during Brownian motion.
Scattering emanates from these centres, created by local density fluctuations, which produce changes in in any volume element.
When a solute is dissolved in a liquid, scattering from a volume element again arises from liquid inhomogeneities, but now an additional contribution from fluctuations in the solute concentration is present and for polymer solutions the problem is to isolate and measure these additional effects.
This was achieved by Debye in 1944, who showed that for a solute whose molecules are small compared with the wavelength of the light used, the reduced angular scattering intensity of the solute is and that this is related to the change in Gibbs free energy with concentration of the solute.
As G is related to the osmotic pressure , we have
Here n and n are the refractive indices of solvent and solution respectively, and N is the number of polymer molecules.
Differentiation of the virial expansion for  with respect to c, followed by substitution in equation (9.27) and rearrangement leads to where
Alternatively, the scattering can be expressed as a turbidity  where and the equation becomes
The new constant is.
Both equations are valid for molecules smaller than when the angular scattering is symmetrical.
Here ˡ is the wavelength of light in solution.
For small particles, M w can be calculated from either equation (9.28) or (9.31).
The important experimental point to remember is that dust will also scatter light and contribute to the scattering intensity.
Great care must be taken to ensure that solutions are clean and free of extraneous matter.
Solutions of the polymer are prepared in a concentration series and clarified either by centrifugation for a few hours at about 25 000 g, or filtered through a grade 5 sinter glass filter.
Alternatively, a millipore filter, porosity 0.45 * can be used.
A number of instruments are available commercially; only one is described here and the schematic diagram 9.5 provides the main features of the model.
Light is obtained from a water-cooled mercury vapour lamp and one of three wavelengths 365, 436, or 546 nm can be selected by means of an appropriate filter.
As the scattering intensity is a function of  -4, use of a lower wavelength enhances the scattering, but the choice is left to the operator.
The light beam, which can be polarized, or left unpolarized, is collimated before passing through the cell.
The measuring cell is immersed in a vat of liquid, usually benzene or xylene which can be thermostatted at temperatures between 273 and 400 K. Scattering is detected by a photomultiplier, capable of revolving round the cell and the intensity is recorded on a galvanometer.
The 90 scattering is plotted as against c and linear extrapolation of the results leads to M w as the intercept at.
Typical results are shown in table 9.2 for a polystyrene sample dissolved in benzene.
The relevant constants are,, the intercept and.
SCATTERING FROM LARGE PARTICLES
When polymer dimensions are greater than  /20,; intraparticle interference causes the scattered light from two or more centres to arrive considerably out of phase at the observation point, and the scattering envelope becomes dependent on the molecular shape.
This attenuation, produced by destructive interference, is zero in the direction of the incident beam, but increases as 0 increases because the path length difference  f in the forward direction is less than  b in the backward (see figure 9.6).
This difference can be measured from the dissymmetry coefficient Z which is unity for small particles, but greater than unity for large particles.
The scattering envelope reflects the scattering attenuation and is compared with that for small particles in figure 9.7.
The angular attenuation of scattering is measured by the particle scattering factor which is simply the ratio of the scattering intensity to the intensity in the absence of interference, measured at the same angle .
Gunier showed that a characteristic shape-independent geometric function, called the radius of gyration can be measured from large particle scattering.
It is defined as an average distance from the centre of gravity of a polymer coil to the chain end.
The function is size dependent and can be related to the polymer coil size by where, for monodisperse randomly coiling polymers.
In the limit of small  the expansion can be used, and the coil size can be estimated from without assuming a particular model.
Specific shapes can be related to if desired, as shown in figure 9.8a and b.
Two methods can be used to calculate M w and the particle size for large molecules.
(i)
Dissymmetry method.
If is not too large, one need only measure the scattering intensity at 90 and two angles symmetrical about 90, usually 45 and 135.
As Z is normally concentration dependent, the value at is obtained by plotting against c.
From published tables can be related to, and M w is calculated from the 90 scattering then corrected by multiplication with.
Also available in table form is the ratio presented as a function of Z, where is the root mean square distance between the ends of the polymer coil.
The corresponding functions for a rod and a sphere have different forms (figure 9.8b).
Polymer dimensions can be calculated in this way if an assumption is made about the best model.
A much more satisfying treatment of the data uses the double extrapolation method proposed by Zimm, which leads to the shape independent parameter.
(ii)
Zimm plots.
This is based on the knowledge that, as the scattering at zero angle is independent of size, is unity when.
Experimentally this is difficult to measure, and an extrapolation procedure has been devised which makes use of a modified form of equation (9.28) for large particles,
Substituting for leads to
If the scattering intensity for each concentration in a dilution series is measured over an angular range 35 to 145, the data can be plotted as against, where k is an arbitrary constant chosen to provide a convenient spread of the data in the grid-like graph which is obtained.
A double extrapolation is then carried out, as shown in figure 9.9, by joining all points of equal concentration and extrapolating to zero angle, and then all points measured at equal angles and extrapolating these to zero concentration.
For example, on the diagram the points corresponding to concentration c 3 are joined and extrapolated to intersect with an imaginary line corresponding to the value of kc 3 on the abscissa; similarly all points measured at 90 are joined and extrapolated until the point corresponding to is reached.
This is done for each concentration and each angle and the extrapolated points are then lines of and.
Both lines, on extrapolation to the axis, should intersect at the same point.
The intercept is then, the slope of the line yields A 2, whereas is obtained from the initial slope s i of the line i.e.
The radius of gyration calculated in this way for a polydisperse sample is a z-average.
9.10 Refractive index increment
Before M w can be calculated from light scattering measurements, the specific refractive index increment must be known for the particular polymer + solvent system under examination.
It is defined as where n and n o are the refractive indices of the solution and the solvent and c is the concentration.
Measurements of are made using a differential refractometer employing the same wavelength of light as used in the light scattering.
The monochromatic beam (selected by filter) from a mercury vapour lamp is directed through a differential cell, consisting of a solution and solvent compartment separated by a diagonal glass wall.
The deflection of the light beam is measured, first with solvent in the forward compartment and solution in the rear, giving deflection d 1 the position is reversed and deflection d 2 measured.
If similar readings for solvent alone, and, are obtained, then the total displacement d is
If the instrument is calibrated with aqueous KCI solutions of known n a relation, can be obtained where c is the calibration constant.
By measuring d for a number of concentrations of polymer, n is obtained from a knowledge of c, and from the slope of the plot of n against c.
9.11 Small angle X-ray scattering
The theoretical outline presented for light scattering studies is valid for electromagnetic radiation of all wavelengths.
For X-rays,  is as low as 0.154 nm, and as this is much smaller than typical polymer dimensions structural information over small distances should be available from X-ray scattering.
The intensity of scattering is a function of the electron density and therefore of the refractive index.
The molar mass is then related to the excess electron density  c of solute over solvent for =0.154; nm by where R o is the Rayleigh ratio at.
Experimental techniques are difficult because of the weak scattering, but the method has provided useful information on macromolecules with dimensions in the range 1 to 100 nm and, as such, is complementary to light scattering.
9.12 Ultracentrifuge
When macroscopic particles are allowed to settle in a liquid under gravity it is possible to determine their size and weight.
Macromolecules in solution are usually much smaller and it would take years for them to overcome the Brownian motion and form a sediment.
This problem can be overcome by subjecting them to an external force, strong enough to alter their spatial distribution by a significant amount in a short time.
In 1925, Svedberg first achieved this by subjecting polymer solutions to large force fields, generated at high speeds of rotation.
The technique is now a well established method for measuring M w and M z for both synthetic and biological macromolecules and has the added advantage that measurements require only small quantities of material.
The dilute solution of polymer is placed in a cell with a sector shaped centre piece in the form of a truncated cone, whose peak is located at the centre of the rotation.
The shape ensures that convective disturbances are minimized during the transportation of molecules to the cell bottom.
The cells are supported in a rotor of either titanium or aluminium alloy, which is attached to the drive motor by a fine steel wire, thereby allowing limited self-balancing to take place.
The rotor is spun in a vacuum chamber to minimize frictional heating during high speed rotations, as speeds of up to 68 000 r.p.m., capable of producing 372 000 g can be generated.
During rotation the cell passes through a collimated beam of light from a high pressure mercury lamp and the emergent beam then travels through the optical system to be recorded photographically.
Three types of optical system are available, schlieren, interference, and UV absorption.
Solvents, having densities and refractive indices sufficiently different from the polymer, are chosen to ensure movement of the polymer chains in the medium and the optical detection of this motion.
Most commercial instruments are extremely versatile, with an extensive choice of rotor speeds and a temperature control system.
Molar masses from 10 2 to 10 6 g mol -1 can be measured and this range is much wider than any other existing technique.
Two general methods are used to measure M,
(1)
sedimentation velocity and
(2)
sedimentation equilibrium
.
SEDIMENTATION VELOCITY
The centrifuge is operated at high speeds to transport the polymer molecules through the solvent to the cell bottom if the solvent density is less than the polymer, or to the top (flotation) if the reverse is true.
The rate of movement can be measured by following the change in refractive index n through the boundary region.
As the molecules sediment, a layer of pure solvent is left whose refractive index differs from the solution.
The boundary is located by the sharp change in n and its movement is followed as a function of time using one or other of the optical methods available.
When moving through the solution, the polymer will experience a centrifugal force which is, but as the molecule displaces a mass of solution it will be subject to a buoyancy effect and an opposing force.
The net force is then where r is the distance between the boundary and the centre of rotation, is the partial specific volume of the polymer,  is the angular velocity, the mass of the molecule and  is the density of the solution.
This force will be balanced by the frictional resistance of the medium F for a particular velocity and where R s is the spherical radius of the polymer particle and  is the viscosity of the medium.
These two forces are in equilibrium when a uniform particle velocity is attained and
The steady-state velocity in a unit gravitational field can then be defined as the sedimentation constant S, and where f is the frictional coefficient of the molecule and is related to the diffusion constant D by Substitution gives the Svedberg equation,
From this a molar mass M SD, is calculated if both S and D are known.
This average is close to M w but is usually smaller and depends on the method used to measure D.
The term is called the buoyancy factor and determines the direction of macromolecular transport in the cell.
If the factor is positive, the polymer chains sediment away from the centre of rotation to the cell bottom, if negative, they move in the opposite direction and float to the top.
The determination of M is absolute when S and D are known, but more commonly a relation of the form is established, using polymer fractions of known M, for a given solvent + polymer system.
This approach is similar to that used for the limiting viscosity number, which is a non-absolute method.
SEDIMENTATION EQUILIBRIUM
In the sedimentation equilibrium experiments the condition for equilibrium requires that the total potential, ̡, must be constant in all parts of the system.
For a polymer of molar mass M 2 dissolved in a solvent (1) and placed in a centrifugal field of angular velocity , its potential energy at a distance r from the centre of rotation is and its chemical potential is .
The total potential ̡ then becomes and the conditions for equilibrium are
Consider now the transportation of J moles of polymer across unit cross sectional area in unit time; the transport equation for this is where, is the mass conductivity which is proportional to the concentration of the substance and inversely proportional to the resistance offered by the medium to transport i.e. the frictional coefficient per mole f.
Now  is a function of T, P and c 2 but if T is constant then and as where  2 is the activity coefficient.
We have for an ideal solution
Substitution in equation (9.49) yields or
Using the equations (9.45), (9.46) and the definition of L, equation (9.50) can be written as
This shows that the flux J is then a net result of the sedimentation rate and the back diffusion of the molecules.
At equilibrium these balance, and the flow vanishes so that.
It follows that
This describes the concentration gradient at equilibrium for a single solute under ideal solution conditions, and integration between the meniscus r m and any point ' r ' in the cell gives
A graph of against r 2 can be constructed by measuring the concentrations at different points in the cell and M 2 can be calculated from the slope.
The main experimental problem, however, is being able to calculate the concentration at each point in the cell which is not always easy.
A more widely used method is to calculate the difference in concentration between that at the meniscus (cm) and the cell bottom (cb).
Rearranging and integrating equation (9.51) gives and the integral on the r h s. can be evaluated by considering mass conservation in a sector shaped cell, so that where c o is the initial concentration.
For a polydisperse polymer this gives a weight average molar mass M w and the z average M z can be calculated from the concentration gradients at the top and bottom of the cell.
The main disadvantage of the method lies in the long periods of time required to reach equilibrium.
Several variations exist which reduce this time scale such as studying the approach to equilibrium, using short columns, or meniscus depletion techniques can be employed but all are outside the scope of this text.
The value of M w calculated from equation (9.53) is, of course, an apparent value relating to the initial concentration of the solution, and extrapolation to zero concentration is necessary.
9.13 Viscosity
When a polymer dissolves in a liquid, the interaction of the two components stimulates an increase in polymer dimensions over that in the unsolvated state.
Because of the vast difference in size between solvent and solute, the frictional properties of the solvent in the mixture are drastically altered, and an increase in viscosity occurs which should reflect the size and shape of the dissolved solute, even in dilute solutions.
This was first recognized in 1930 by Staudinger, who found that an empirical relation existed between the relative magnitude of the increase in viscosity and the molar mass of the polymer.
One of the simplest methods of examining this effect is by capillary viscometry.
It has been shown that the ratio of the flow time of a polymer solution t to that of the pure solvent t o is effectively equal to the ratio of their viscosity if the densities are equal.
This latter approximation is reasonable for dilute solutions and provides a measure of the relative viscosity  r
As this has a limiting value of unity, a more useful quantity is the specific viscosity
Even in dilute solutions molecular interference is likely to occur and  sp is extrapolated to zero concentration to obtain a measure of the influence of an isolated polymer coil.
This is accomplished in either of two ways;  sp can be expressed as a reduced quantity and extrapolated to according to the relation and the intercept is the limiting viscosity number [  ] which is a characteristic parameter for the polymer in a particular solvent, k is a shape dependent factor called the Huggins constant and has values between 0.3 and 0.9 for randomly coiling vinyl polymers.
The alternative extrapolation method uses the inherent viscosity as where k is another shape dependent factor.
The dimensions of [  ] are the same as the reciprocal of the concentration.
When measuring [  ] solutions are filtered to remove spurious particles, then flow times for solvent and solutions are recorded in U-tube viscometers such as the ' Cannon-Fenske ' or the ' Ubbelohde suspended level dilution ' models.
Dilution viscometers are most convenient when a concentration series is to be measured.
In these the concentration can be changed in situ, whereas fresh solution concentrations of exactly the same volume must be introduced for each measurement in the non dilution Cannon-Fenske.
In the Ubbelohde viscometer an aliquot of solution of known volume is pipetted into bulb D through A. The solution is then pumped into E, by applying a pressure down A with C closed off; the pressure is released and C is opened to allow the excess solution to drain back into D. This leaves the end of the capillary open or suspended.
Solution then flows down the capillary and drains round the sides of the bulb back into D, but as no back pressure from the excess solution exists, the volume in D plays no part in determining the flow time t.
This suspended level is the feature which allows dilution to be carried out in D without affecting t.
Thus addition of a known amount of solvent to the solution in D, followed by mixing, gives the next concentration in the series.
The flow time t, is the time taken for the solution meniscus to pass from x to y in bulb E.
For a given polymer + solvent system at a specified temperature, [  ] can be related to M through the Mark-Houwink equation
K v and v can be established by calibrating with polymer fractions of known molar mass, and once this has been established for a system, [  ] alone will give M for an unknown fraction.
This is normally achieved by plotting log [  ] against log M and interpolation is then quite straightforward.
Values of v lie between 0.5 for a polymer dissolved in a theta-solvent to about 0.8 in very good solvents for linear randomly coiling vinyl polymers, and typical values for systems studied by viscosity and sedimentation are given in table 9.3.
The exponents v and b are indicative of solvent quality.
When the solvent is ideal, i.e. a theta-solvent, both v and b are 0.5, but as the solvent becomes thermodynamically better, and deviations from ideality larger, then v increases and b decreases.
VISCOSITY AVERAGE MOLECULAR WEIGHT
Polymer samples are normally polydisperse and it is of interest to examine the type of average molecular weight that might be expected from a measurement of *lsqb;  ].
As the specific viscosity will depend on the contributions from each of the polymer molecules in the sample we can write
If we now divide by the total concentration and substitute for then or
Comparison with equation (9.59) shows that the viscosity average M v is then and that this lies between M n and M w in magnitude, but will be usually closer to M w.
9.14 Gel permeation chromatography
The molar mass distribution (MMD) of a polymer sample has a significant influence on its properties and a knowledge of the shape of this distribution is fundamental to the full characterization of a polymer.
The determination of the MMD by conventional fractionation techniques is time consuming, and a rapid, efficient and reliable method which can provide a measure of the MMD in a matter of hours has been developed.
This is gel permeation chromatography (GPC).
Known alternatively by its more descriptive name ' size exclusion chromatography ' (SEC), the method depends on the use of mechanically stable, highly crosslinked gels which have a distribution of different pore sizes and can, by means of a sieving action, effect separation of a polymer sample into fractions, dictated by their molecular volume.
The non-ionic gel stationary phase is commonly composed of crosslinked polystyrene or macroporous silica particles, which do not swell significantly in the carrier solvents.
A range of pore sizes is fundamental to the success of this size fractionation procedure which depends on two processes.
These are
(a)
separation by size exclusion alone, which is the more important feature, and
(b)
a dispersion process, controlled by molecular diffusion which may lead to an artificial broadening of the MMD
.
Looking first at the mechanism of the separation process (a); in simple terms, the large molecules, which occupy the greatest effective volume in solution, are excluded from the smaller pore sizes in the gel and pass quickly through the larger channels between the gel particles.
This results in their being eluted first from the column.
As the molecular size of the polymer decreases there is an increasing probability that the molecules can diffuse into the smaller pores and channels in the gel which slows their time of passage through the column by providing a potentially longer path length before being eluted.
By choosing a series of gel columns with an appropriate range of pore sizes, an effective size separation can be obtained.
The efficiency of the separation process is then a function of the dependence of the retention (or elution) volume V R on the molar mass M, and a reliable relationship between the two parameters must be established.
The value of V R depends on the interstitial void volume V o and the accessible part of the pore volume in the gel, where V i is the total internal pore volume and KD is the partition coefficient between V and the portion accessible to a given solute.
Thus for very large molecules, and rapid elution takes place, whereas for very small molecules which can penetrate all the available pore volume.
This is shown schematically in figure 9.12 and clearly the technique can not discriminate amongst molecular sizes with or.
For samples which fall within the appropriate range it has been suggested that a universal calibration curve can be constructed to relate V R and M, by assuming that the hydrodynamic volume of a macromolecule is related to the product, where [  ] is the intrinsic viscosity of the polymer in the carrier solvent used, at the temperature of measurement.
A universal calibration curve is then obtained by plotting against V R for a given carrier solvent and a fixed temperature.
Experimental verification of this is shown in figure 9.13 for a variety of different polymers and can be utilized in the following way.
To obtain the MMD, the mass of the polymer being eluted must be measured.
This can be achieved continuously using refractive index, UV or IR detectors, which will give a mass distribution as a function of VR.
It is still necessary to estimate the molar mass of each fraction before the MMD curve can be constructed.
If the universal calibration curve is valid for the system then where the subscripts s and u denote the standard calibration and the polymer under study, respectively.
As the chains of the polymer under examination may swell in the carrier solvent to a different extent compared to an equal molar mass sample of the standard, the hydrodynamic volumes will not necessarily be equivalent.
This can be compensated for by applying a correction based on the knowledge of the appropriate Mark-Houwink relations for each, the standard and the unknown, measured in the solvent used for elution.
The molar mass M u can then be obtained from
Thus a calibration curve constructed for standard samples of polystyrene can be used to determine M for other polymers if the Mark-Houwink relations are also known.
This can be avoided if a ' Viscotek ', which is a combined differential refractometer and viscometer, is attached to the end of the column, as this measures both concentration and the  sp for the fraction.
By assuming in dilute solutions the molar mass can be obtained from the Mark-Houwink relation.
Alternatively, a low angle laser light scattering (LALLS) instrument can be attached in series with a concentration detector.
This gives a direct measurement of M w, using equation (9.28), if all the parameters in the equation are known.
When using SEC, care must be taken not to overload the columns with too large a polymer sample as this results in a non-linear response, characterized by losses in resolution and column efficiency.
Also, although the band broadening referred to earlier can be minimized by using long, efficient columns, it may never be entirely eliminated.
The MMD curves may then be broadened by this phenomenon and appropriate corrections must be applied.
Unfortunately these are often difficult to calculate accurately, although it has been shown by Tung that the error introduced by broadening is negligible if for the sample.
CHAPTER 10
Polymer Characterization  Chain Dimensions and Structures
As the size and shape of a polymer chain are of considerable interest to the polymer scientist it is useful to know how these factors can be assessed.
Much of the information can be derived from studies of dilute solutions; an absolute measurement of polymer chain size can be obtained from light scattering, when the polymer is large compared with the wavelength of the incident light.
Sometimes the absolute measurement can not be used but the size can be deduced indirectly from viscosity measurements, which are related to the volume occupied by the chain in solution.
Armed with this information we must now determine how meaningful it is and to do this a clearer understanding of the factors governing the shape of the polymer is required.
We can confine ourselves to models of the random coil, as this is usually believed to be most appropriate for synthetic polymers; other models  rods, discs, spheres, spheroids  are also postulated, but need not concern us at this level.
10.1 Average chain dimensions
A polymer chain in dilute solution can be pictured as a coil, continuously changing its shape under the action of random thermal motions.
This means, that at any time, the volume occupied by a chain in solution, could differ from that occupied by its neighbours, and these size differences are further accentuated by the fact that each sample will contain a variety of chain lengths.
Taking these two points into consideration leads us to the conclusion that meaningful chain dimensions can only be values averaged over the many conformations assumed.
Two such averages have been defined:
(a)
the average root mean square distance between the chain ends; and
(b)
the average root mean square radius of gyration which is a measure of the average distance of a chain element from the centre of gravity of the coil.
The angular brackets denote averaging due to chain polydispersity in the sample and the bar indicates averaging for the many conformational sizes available to chains of the same molar mass.
The two quantities are related, in the absence of excluded volume effects, for simple chains by but as the actual dimensions obtained can depend on the conditions of the measurement, other factors must also be considered.
10.2 Freely jointed chain model
The initial attempts to arrive at a theoretical representation of the dimensions of a linear chain, treated the molecule as a number n of chain elements, joined by bonds of length l.
By assuming the bonds act like universal joints, complete freedom of rotation about the chain bonds can be postulated.
This model allows the chain to be pictured as in figure 10.1(a) which resembles the path of a diffusing gas molecule and as random flight statistics have proved useful in describing gases, a similar approach is used here.
In two dimensions the diagram is more picturesquely called the ' drunkard's walk ' and r f is estimated by considering first the simplest case of two links.
The end-to-end distance r f follows from the cosine law that see figure 10.1(b), or
When the number of bonds, n is large, the angle  will vary over all possible values so that the sum of all these terms will be zero, and as equation(10.3) will reduce to
This shows that the distance between the chain ends, for this model is proportional to the square root of the number of bonds and so, is considerably shorter than a fully extended chain.
The result is the same if the molecule is thought to occupy three-dimensional space, but if it is centred on a co-ordinate system both positive and negative contributions occur with equal probability.
To overcome this the dimension is expressed always as the square which eliminates negative signs.
This model is, however, unrealistic.
Polymer chains occupy a volume in space, and the dimensions of any macromolecule are influenced by the bond angles and by interactions between the chain elements.
These interactions can be classified into two groups:
(i)
Short range interactions which occur between neighbouring atoms or groups, and are usually forces of steric repulsion caused by the overlapping of electron clouds;
(ii)
Long range interactions which are comprised of attractive and repulsive forces between segments, widely separated in a chain, that occasionally approach one another during molecular flexing, and between segments and solvent molecules
.
These are often termed excluded volume effects.
10.3 Short range effects
The expansion of a covalently bonded polymer chain will be restricted by the valence angles between each chain atom.
In general this angle is  for a homoatomic chain and equation (10.4) can be modified to allow for these short range interactions.
For the simplest case of an all carbon backbone chain such as polyethylene, and so that equation (10.5) reduces to
This indicates that the polyethylene chain is twice as extended as the freely jointed chain model when short range interactions are considered.
10.4 Chain stiffness
As we have already seen in chapter 1 for butane and polyethylene, steric repulsions impose restrictions to bond rotation.
This means that equation (10.5) has to be modified further and now becomes where is the average cosine of the angle of rotation of the bonds in the backbone chain.
The parameter is the average mean square of the unperturbed dimension, which is a characteristic parameter for a given polymer chain.
The freely jointed dimensions are now more realistic when restricted by the factor  the skeletal factor  composed of the two terms where  is known as the steric parameter and is for simple chains.
For more complex chains, containing rings or heteroatomic chains, e.g. polydienes, polyethers, polysaccharides, and proteins, an estimate of  is obtained from
Values of the unperturbed dimension can be obtained experimentally from dilute solution measurements made either directly in a theta-solvent (see section 9.9) or by using indirect measurements in non-ideal solvents and employing an extrapolation procedure.
The geometry of each chain allows the calculation of, and results are expressed either as  or as the characteristic ratio.
Both provide a measure of chain stiffness in dilute solution.
The range of values normally found for  is from about 1.5 to 2.5 as shown in table 10.1.
10.5 Treatment of dilute solution data
We can now examine some of the ways of calculating the polymer dimensions from experimental data.
THE SECOND VIRIAL COEFFICIENT
An investigation of the dilute solution behaviour of a polymer can provide useful information about the size and shape of the coil, the extent of polymer-solvent interaction and the molar mass.
Deviations from ideality, as we have seen in section 9.7, are conveniently expressed in terms of virial expansions, and when solutions are sufficiently dilute, the results can be adequately described by the terms up to the second virial coefficient A 2 while neglecting higher terms.
The value of A 2 is a measure of solvent-polymer compatibility, as the parameter reflects the tendency of a polymer segment to exclude its neighbours from the volume it occupies.
Thus a large positive A 2 indicates a good solvent for the polymer while a low value (sometimes even negative) shows that the solvent is relatively poor.
The virial coefficient can be related to the Flory dilute solution parameters by where is a molar mass dependent function of the excluded volume.
The exact form of can be defined explicitly by one of several theories, and while each leads to a slightly different form, all predict that is unity when theta conditions are attained and the excluded volume effect vanishes.
Equation (10.10) can be used to analyse data such as that in figure 8.7.
Once  has been located, the entropy parameter  1 can be calculated by replotting the data as against T. Extrapolation to, where, allows  1 to be estimated for the system under theta conditions.
This method of measuring  and  1, is only accurate when the solvent is poor, and extrapolations are short.
The dependence of A 2 on M can often be predicted, for good solvents, by a simple equation where  varies from 0.15 to 0.4, depending on the system and k is a constant.
EXPANSION FACTOR 
The value of A 2 will tell us whether or not the size of the polymer coil, which is dissolved in a particular solvent, will be perturbed or expanded over that of the unperturbed state, but the extent of this expansion is best estimated by calculating the expansion factor .
If the temperature of a system, containing a polymer of finite M, drops much below  the number of polymer-polymer contacts increases until precipitation of the polymer occurs.
Above this temperature, the chains are expanded, or perturbed, from the equilibrium size attained under pseudo-ideal conditions, by long range interactions.
The extent of this coil expansion is determined by two long range effects.
The first results from the physical exclusion of one polymer segment by another from a hypothetical lattice site which reduces the number of possible conformations available to the chain.
This serves to lower the probability that tightly coiled conformations will be favoured.
The second is observed in very good solvents, where the tendency is for polymer-solvent interactions to predominate, and leads to a preference for even more extended conformations.
In a given solvent an equilibrium conformation is eventually achieved when the forces of expansion are balanced by forces of contraction in the molecule.
The tendency to contract arises from the both the polymer-polymer interactions and the resistance to expansion of the chain into over extended and energetically less favoured conformations.
The extent of this coil perturbation by long range effects is measured by an expansion factor , introduced by Flory.
This relates the perturbed and unperturbed dimensions by
In good solvents (large, positive A 2) the coil is more extended than in poor solvents (low A 2) and  is correspondingly larger.
Since  is solvent and temperature dependent a more characteristic dimension to measure for the polymer is which can be calculated from light scattering in a theta-solvent, or indirectly as next described.
FLORY-FOX THEORY
The molecular dimensions of a polymer chain in any solvent can be calculated directly from light scattering measurements, using equation (9.36), if the coil is large enough to scatter light in an asymmetric manner, but when the chain is too short to be measured accurately in this way an alternative technique has to be used.
Flory and Fox suggested that as the viscosity of a polymer solution will depend on the volume occupied by the polymer chain, it should be feasible to relate coil size and [  ].
They assumed that if the unperturbed polymer is approximated by a hydrodynamic sphere, then [  ], the limiting viscosity number in a theta solvent, could be related to the square root of the molar mass by where
Equations (10.13) and (10.14) are actually derived for monodisperse samples, and when measurements are performed with heterodisperse polymers, the appropriate averages to use are M n and.
The parameter  was originally considered to be a universal constant, but experimental work suggests that it is a function of the solvent, molar mass, and heterogeneity.
Values can vary from an experimental one of 2.1  10 23 to a theoretical limit of about 2.84  10 23 when [  ] is expressed in cm 3 g -1.
A most probable value of 2.5  10 23 has been found to be acceptable for most flexible heterodisperse polymers in good solvents.
For non-ideal solvents equation (10.13) can be expanded to give where a is the linear expansion factor, pertaining to viscosity measurements, and is a measure of long range interactions.
As the derivation is based on an unrealistic Gaussian distribution of segments in good solvents, it has been suggested that   is related to the more direct measurement of  in equation (10.12) by
Considerable experimental evidence exists to support this conclusion.
INDIRECT ESTIMATES OF
It is not always possible to find a suitable theta-solvent for a polymer and methods have been developed which allow unperturbed dimensions to be estimated in non-ideal (good) solvents.
Several methods of extrapolating data for [  ] have been suggested.
The most useful of these was proposed by Stockmayer and Fixman, using the equation: when  is assumed to adopt its limiting theoretical value, B is related to the thermodynamic interaction parameter  1 by and examination of equation (10.10) shows that B is also proportional to A 2.
The unperturbed dimension can be estimated by plotting against M ?; K , is obtained from the intercept and is calculated from equation (10.14).
A similar procedure has been proposed by Cowie and Bywater, in which the intrinsic frictional coefficient [ f ] measured from sedimentation or diffusion experiments, will provide the same information using where and P o is a ' constant ' with a limiting value of 5.2.
These extrapolation procedures all depend on the validity of the theoretical treatment and reliability must be judged in this light.
Fortunately, it has been demonstrated that most non-polar polymers can be treated in this way and results agree well with direct measurements of.
For more polar polymers, specific solvent effects become more pronounced and extrapolations have to be regarded with corresponding caution.
INFLUENCE OF TACTICITY ON CHAIN DIMENSIONS
Studies of the dilute solution behaviour of polymers with a specific stereostructure have revealed that the unperturbed dimensions may depend on the chain configuration.
This can be seen from the data in table 10.1 where isotactic, syndiotactic, and atactic poly (methyl methacrylate) have different  values.
If the size of a polymer chain can be affected by its configuration, the microstructure must be well characterized before an accurate assessment of experimental data can be made.
This can be achieved using n.m.r. and infrared techniques.
10.6 Nuclear magnetic resonance (n.m.r.)
High resolution n.m.r. has proved to be a particularly useful tool in the study of the microstructure of polymers in solution, where the extensive molecular motion reduces the effect of long range interactions and allows the short range effects to dominate.
Interpretation of chain tacticity, based on the work of Bovey and Tiers, can be illustrated using poly (methyl methacrylate).
The three possible steric configurations are shown in figure 10.2 where R is the group.
For the purposes of n.m.r. measurements three consecutive monomer units in a chain are considered to define a configuration and called a triad.
The term heterotactic is used now to define a triad which is neither isotatic nor syndiotactic.
In the structures shown, the three equivalent protons of the -methyl group absorb radiation at a single frequency, but this frequency will be different for each of the three kinds of triad, because the environment of the -methyl groups in each is different.
For poly (methyl methacrylate) samples, which were prepared under different conditions to give the three forms, resonances at  = 8.78, 8.95 and 9.09 were observed, which were assigned to the isotatic, heterotactic, and syndiotactic triads respectively.
Thus in a sample with a mixture of configurations a triple peak will be observed and the area under each of these peaks will correspond to the amount of each triad present in the polymer chain.
This is illustrated in figure 10.3, where one sample is predominantly isotactic, but also contains smaller percentages of the heterotactic and syndiotactic configurations.
The analysis can be carried further.
The fraction of each configuration, P i, P h, and P s, measured from the respective peak areas, can be related to  m the probability that a monomer adding on to the end of a growing chain will have the same configuration as the unit it is joining.
This leads to the relations
Curves plotted according to this simple analysis are shown in figure 10.4 where they are compared with experimental data obtained for various tactic forms of poly (-methyl styrene).
Differences in the microstructure of polydienes and copolymers can also be made using n.m.r.
In the polydienes the difference between 1,2- and 1,4-addition can be distinguished on examination of the resonance peaks corresponding to terminal olefinic protons, found at, and non-terminal olefinic protons observed at.
Not only is the local field acting on the nucleus altered by environment, it is also sensitive to molecular motion, and it has been observed that as the molecular motion within a sample increases, the resonance lines become narrower.
Determination of the width, or second moment, of an n.m.r. resonance line, then provides a sensitive measure of low frequency internal motions in solid polymers and can be used to study transitions and segmental rotations in the polymer sample.
Line widths are also altered by the polymer crystallinity.
Partially crystalline polymers present complex spectra as they are multi-phase materials, in which the molecular motions are more restricted in the crystalline phase than in the amorphous phase.
However, attempts to estimate percentage crystallinity in a sample using n.m.r. have not been particularly successful.
The method is illustrated in figure 10.5 for poly(tetrafluoroethylene) where glass and other transitions are readily detected.
Below 200 K the chains are virtually immobile, but above 200 K the lines sharpen as rotation begins.
This is associated with the glass transition, but the way the line width increases in this region is governed by sample crystallinity.
10.7 Infrared spectroscopy
Infrared spectroscopy can be used to characterize long chain polymers because the infrared active groups, present along the chain, absorb as if each was a localized group in a simple molecule.
Identification of polymer samples can be made by making use of the ' finger-print ' region, where it is least likely for one polymer to exhibit exactly the same spectrum as another.
This region lies within the range 6.67 to 12.50 m.
In addition to identification, the technique has been used to elucidate certain aspects of polymer microstructure, such as branching, crystallinity, tacticity, and cis-trans isomerism.
The relative proportions of, and; addition in polybutadienes can be ascertained by making use of the differences in absorption between (CH) out of plane bending vibrations, which depend on the type of substitution at the olefinic bond.
Terminal and internal groups can also be distinguished, as an absorption band at about 11.0 m is characteristic of a vinyl group and indicates.
The is characterized by an absorption band at about 13.6 m, whereas the configuration exhibits a band at about 10.4 m.
An estimate of cis-trans isomerism can be made by measuring the absorbance A of each band, where and I o and I are the intensities of the incident and transmitted radiation respectively.
This is calculated by locating a base line across the minima on either side of the absorption band and the vertical height to the top of the band from the base line is converted into a composition using the equation where P cis is the fraction of cis configuration, A cis is the absorbance at 13.6 m, A trans the absorbance at 10.4 m, and if we assume that the 1,2 content is negligible.
Polyisoprenes can also be analyzed in this way, only now the bands at 11.0 and 11.25, m are used to estimate the and, while a band at 8.7 m corresponds to the linkage.
The infrared spectra of highly stereoregular polymers are distinguishable from those of their less regular counterparts, but many of the differences can be attributed to crystallinity rather than tacticity as such.
The application of infrared to stereostructure determination in polymers is less reliable than n.m.r., but has achieved moderate success for poly (methyl methacrylate) and polypropylene.
In poly (methyl methacrylate), a methyl deformation at 7.25 m is unaffected by microstructure, and comparison of this with a band at 9.40 m, which is present only in atactic or syndiotactic polymers allows an estimate of the syndiotacticity to be made from the ratio.
Similarly provides a measure of the isotactic content.
An alternative method is to calculate the quantity J as an average of the two equations where the absorption band at 10.10 m is now used.
If J lies between 100 and 115 a highly syndiotactic polymer is indicated, if between 25 and 30 the polymer is highly isotactic.
For polypropylene, the characteristic band for the syndiotactic polymer appears at 11.53 m, and the syndiotactic index I s is.
Values of I s about 0.8 indicate highly syndiotactic samples.
Spectra can be measured in a number of ways; for soluble polymers a film can be cast, perhaps even on the N a Cl plate to be used and examined directly.
Measurements can also be made in solution, if the solvent absorption in any important region is low, or by a differential method.
10.8 X-ray diffraction
The extent of sample crystallinity can influence the behaviour of a polymer sample greatly.
A particularly effective way of examining partially crystalline polymers is by X-ray diffraction.
The crystallites present in a powdered or unoriented polymer sample diffract X-ray beams from parallel planes for incident angles  which are determined by the Bragg equation where  is the wavelength of the radiation, d is the distance between the parallel planes in the crystallites, and n is an integer.
The reinforced waves reflected by all the small crystallites produce diffraction rings, or haloes, which are sharply defined for highly crystalline materials and become increasingly diffuse when the amorphous content is high.
If the polymer sample is oriented, by drawing a fibre, or by applying tension to a film, the crystallites tend to become aligned in the direction of the stress and the X-ray pattern is improved.
In some samples of stereoregular or symmetrical polymers, the degree of three-dimensional ordering of the chains may be sufficiently high to allow a structural analysis of the polymer to be accomplished.
Sample crystallinity can be estimated from the X-ray patterns by plotting the density of the scattered beam against the angle of incidence.
If this can be done for an amorphous sample and a corresponding sample which is highly crystalline, a relative measure of crystallinity for other samples of the same polymer can be obtained.
In figure 10.6 the shaded portion is the amorphous polypropylene, while the maxima arise from the crystallites.
10.9 Thermal analysis
When a substance undergoes a physical or chemical change a corresponding change in enthalpy is observed.
This forms the basis of the technique known as differential thermal analysis (DTA) in which the change is detected by measuring the enthalpy difference between the material under study and an inert standard.
The sample is placed in a heating block and warmed at a uniform rate.
The sample temperature is then monitored by means of a thermocouple and compared with the temperature of an inert reference such as powdered alumina, or simply an empty sample pan, which is subjected to the same linear heating programme.
As the temperature of the block is raised at a constant rate the sample temperature T s and that of the reference, T r will keep pace until a change in the sample takes place.
If the change is exothermic T s will exceed T r for a short period, but if it is endothermic T s will temporarily lag behind T r.
This temperature difference T is recorded and transmitted to a chart recorder where changes such as melting or crystallization are recorded as peaks.
A third type of change can be detected.
Since the heat capacities of sample and reference are different T is never actually zero, and a change in heat capacity, such as that associated with a glass transition, will cause a shift in the base line.
All three possibilities are shown in figure 10.7 for quenched terylene.
Other changes such as sample decomposition, crosslinking, and the existence of polymorphic forms can also be detected.
As T measured in DTA is a function of the thermal conductivity and bulk density of the sample, it is non-quantitative and relatively uninformative.
To overcome these drawbacks an alternative procedure known as differential scanning calorimetry (DSC) is used.
This technique retains the constant mean heat input but instead of measuring the temperature difference during a change a servo-system immediately increases the energy input to either sample or reference to maintain both at the same temperature.
The thermograms obtained are similar to DTA, but actually represent the amount of electrical energy supplied to the system, not T, and so the areas under the peaks will be proportional to the change in enthalpy which occurred.
An actual reference sample can be dispensed with in practice and an empty sample pan used instead.
Calibration of the instrument will allow the heat capacity of a sample to be calculated in a quantitative manner.
This information is additional to that gained on crystallization, melting, glass transitions, and decompositions.
CHAPTER 11
The Crystalline State
11.1 Introduction
When polymers are irradiated by a beam of X-rays, scattering produces diffuse haloes on the photographic plate for some polymers, while for others a series of sharply defined rings superimposed on a diffuse background is recorded.
The former are characteristic of amorphous polymers, and illustrate that a limited amount of short range order exists in most polymeric solids.
The latter patterns are indicative of considerable three-dimensional order and are typical of polycrystalline samples containing a large number of unoriented crystallites associated with amorphous regions.
The rings are observed to sharpen into arcs, or discrete spots, if the polymer is drawn or stretched, a process which orients the axes of the crystallites in one direction.
The occurrence of significant crystallinity in a polymer sample is of considerable consequence to a materials scientist.
The properties of the sample  the density, optical clarity, modulus, and general mechanical response all change dramatically when crystallites are present and the polymer is no longer subject to the rules of linear visco-elasticity, which apply to amorphous polymers as outlined in Chapter 13.
However, a polymer sample is rarely completely crystalline and the properties also depend on the amount of crystalline order.
It is important then to examine crystallinity in polymers and determine the factors which control the extent of crystallinity.
11.2 Mechanism of crystallization
A polymer in very dilute solution can be effectively regarded as an isolated chain whose shape is governed by short and long range inter- and intra-molecular interactions.
In the aggregated state this is no longer true, the behaviour of the chain is now influenced largely by the proximity of the neighbouring chains and the secondary valence forces which act between them.
These factors determine the orientation of chains relative to each other in the undiluted state, and this is essentially an interplay between the entropy and internal energy of the system which is expressed in the usual thermodynamic form
In the melt, polymers normally attain a state of maximum entropy consistent with a stable state of minimum free energy.
Crystallization is a process involving the orderly arrangement of chains and is consequently associated with a large negative entropy of activation.
If a favourable free energy change is to be obtained for crystallite formation, the entropy term has to be offset by a large negative energy contribution.
The alignment of polymer chains at specific distances from one another to form crystalline nuclei will be assisted when intermolecular forces are strong.
The greater this interaction between chains the more favourable will be the energy parameter and this provides some indication of the type of chain which might be expected to crystallize from the melt, viz.
(1)
Symmetrical chains which allow the regular close packing required for crystallite formation.
(2)
Chains possessing groups which encourage strong intermolecular attraction thereby stabilizing the alignment.
In addition to the thermodynamic requirements, kinetic factors relating to the flexibility and mobility of a chain in the melt must also be considered.
Thus polyisobutylene might be expected to crystallize because the chain is symmetrical, but it will only do so if maintained at an optimum temperature for several months.
This is presumably a result of the flexibility of the chain which allows extensive convolution thereby impeding stabilization of the required long range alignment.
The creation of a three-dimensional ordered phase from a disordered state is a two stage process.
Just above its melting temperature a polymer behaves like a highly viscous liquid in which the chains are all tangled up with their neighbours.
Each chain pervades a given volume in the sample, but as the temperature decreases the volume available to the molecule also decreases.
This in turn restricts the number of disordered conformational states available to the chain due to the constraining influence of intramolecular interactions among chains in juxtaposition.
As a result there is an increasing tendency for the polymer to assume an ordered conformation in which the chain bonds are in the rotational states of lowest energy.
However, various other factors will tend to oppose crystallization; chain entanglements will hinder the diffusion of chains into suitable orientations and if the temperature is above the melting temperature, thermal motions will be sufficient to disrupt the potential nuclei before significant growth can take place.
This restricts crystallization to a range of temperatures between T g and T m.
The first step in crystallite formation is the creation of a stable nucleus brought about by the ordering of chains in a parallel array, stimulated by intramolecular forces, followed by the stabilization of long range order by the secondary valence forces which aid the packing of molecules into a three-dimensional ordered structure.
The second stage is the growth of the crystalline region, the size of which is governed by the rate of addition of other chains to the nucleus.
As this growth is counteracted by thermal redispersion of the chains at the crystal-melt interface, the temperature must be low enough to ensure that this disordering process is minimal.
11.3 Temperature and growth rate
Measurable rates of crystallization occur between and, a range in which the thermal motion of the polymer chains is conducive to the formation of stable ordered regions.
The growth rate of crystalline areas passes through a maximum in this range as illustrated in figure 11.1 for isotactic polystyrene.
Close to T m the segmental motion is too great to allow many stable nuclei to form, while near T g the melt is so viscous that molecular motion is extremely slow.
As the temperature drops from T m, the melt viscosity, which is a function of the molar mass, increases and the diffusion rate decreases, thereby giving the chains greater opportunity to rearrange themselves to form a nucleus.
This means that there will exist an optimum temperature of crystallization, which depends largely on the interval T m to T g, but also on the molar mass of the sample.
The melt usually has to be supercooled by about 5 to 20 K before a significant number of nuclei appear which possess the critical dimensions required for stability and further growth.
If a nucleating agent is added to the system, crystallization can be induced at higher temperatures.
This is known as heterogeneous nucleation and only affects the crystallization rate, not the spherulitic growth rate, at a given temperature.
11.4 Melting
The melting of a perfectly crystalline substance is an equilibrium process characterized by a marked volume change and a well-defined melting temperature.
Polymers are never perfectly crystalline, but contain disordered regions and crystallites of varying size.
The process is normally incomplete because crystallization takes place when the polymer is a viscous liquid.
In this state, the chains are highly entangled, and as sufficient time must be allowed for the chains to diffuse into the three-dimensional order required for crystallite formation, the crystalline perfection of the sample is affected by the thermal history.
Thus, rapid cooling from the melt usually prevents the development of significant crystallinity.
The result is that melting takes place over a range of temperatures, and this range is a useful indication of sample crystallinity.
Effect of crystallite size on melting.
The range of temperature, which covers the melting of a polymer, is indicative of the size and perfection of the crystallites in the sample.
This is illustrated in a study of the melting of natural rubber samples, which has shown that the melting range is a function of the temperature of crystallization.
At low crystallization temperatures the nucleation density in the rubber melt is high, segmental diffusion rates are low, and small imperfect crystalline regions are formed.
Thus broad melting ranges are measured for samples crystallized at these lower temperatures, and these become narrower as the crystallization temperature increases.
This suggests that careful annealing at the appropriate temperature could produce samples with a high degree of crystallinity.
These samples might then exhibit almost perfect first order phase changes at the melting temperature.
A close approximation to these conditions has been attained by Mandelkern, who annealed a linear polyethylene for 40 days.
The improvement in the crystalline organization is obvious from examination of the resulting fusion curves in figure 11.2, where the variation of specific volume with temperature for this sample is compared with that for a branched polyethylene of low crystallinity.
The effect of branching is to decrease the percentage crystallinity, broaden the melting range, and reduce the average melting temperature.
The points A and B in the diagram represent the temperatures at which the largest crystallites disappear and are regarded as the respective melting temperatures T m for the samples.
The effect of crystal size on T m is shown more clearly in figure 11.3.
The small crystals melt about 30 K lower than the large ones due to the greater contribution from the interfacial free energy in the smaller crystallites, i.e. there is an excess of free energy associated with the disordered chains emerging from the ends of ordered crystallites and this is relatively greater for the small crystallites, resulting in lower melting temperatures.
11.5 Thermodynamic parameters
Even with carefully annealed specimens, it is thought that the equilibrium melting temperature of the completely crystalline polymer T m is never actually attained.
The temperature T m is related to the change in enthalpy H u and the entropy change S u, for the first order melting transition of pure crystalline polymer to pure amorphous melt, by
The enthalpy change can be estimated by adding varying quantities of a diluent to the polymer, which serves to depress the observed melting temperature, and measuring T m for each polymer + diluent mixture.
The results are then plotted according to the Flory equation where is the ratio of the molar volume of the repeating unit in the chain to that of the diluent, and  1, is the volume fraction of the diluent.
The factor is equivalent to the Flory interaction parameter  1, indicating that equation (11.2) is dependent on the polymer-diluent interaction.
For practical purposes T m is taken to be the melting temperature of the undiluted polymer irrespective of the crystalline content.
Typical values obtained in this way are shown in table 11.1
In many cases the entropy change is the most important influence on the magnitude of the melting temperature of a polymer.
A large part of this entropy is due to the additional freedom which allows the chain conformational changes to occur in the melt, after the restrictions of the crystalline lattice.
In the crystalline phase the chain bonds are in their lowest energy state.
If the energy difference between the rotational states  is low, the population of the higher energy states will increase in the melt and considerable flexing of the chain is achieved.
The contribution of S u is then high.
When  is large, the tendency to populate the high energy states is not too great, consequently the chain is less flexible and S u is lower.
Two polymers which exist in the all trans state in the crystal are polyethylene and poly(tetrafluoroethylene).
For polyethylene  is about 3.0 kJ mol -, but it is as high as 18.0 kJ mol - for poly(tetrafluoroethylene).
Hence the polyethylene chain is much more flexible in the melt and gains considerably more entropy on melting, so that T m is correspondingly lower.
11.6 Crystalline arrangement of polymers
The formation of stable crystalline regions in a polymer requires that,
(i)
an economical close packed arrangement of the chains can be achieved in three dimensions, and that
(ii)
a favourable change in internal energy is obtained during this process
.
This imposes restrictions on the type of chain which can be crystallized with ease and, as mentioned earlier, one would expect symmetrical linear chains such as polyesters, polyamides, and polyethylene to crystallize most readily.
FACTORS AFFECTING CRYSTALLINITY AND T m
These can be dealt with under the general headings, symmetry, intermolecular bonding, tacticity, branching and molar mass.
Symmetry.
The symmetry of the chain shape influences both T m and the ability to form crystallites.
Polyethylene and poly(tetrafluoroethylene) are both sufficiently symmetrical to be considered as smooth stiff cylindrical rods.
In the crystal these rods tend to roll over each other and change position when thermally agitated.
This motion within the crystal lattice, called premelting, increases the entropy of the crystal and effectively stabilizes it.
Consequently, more thermal energy is required before the crystal becomes unstable, and T m is raised.
Flat or irregularly shaped polymers, with bends and bumps in the chain, can not move in this way without disrupting the crystal lattice, and so have lower T m values.
This is only one aspect.
For crystallite formation in a polymer, easy close-packing of the chains in a regular three-dimensional fashion is required.
Again linear symmetrical molecules are best.
Polyethylene, poly(tetrafluoroethylene) and other chains with more complex backbones containing,, and groups all possess a suitable symmetry for crystallite formation and usually assume extended zig-zag conformations when aligned in the lattice.
Chains containing irregular units, which detract from the linear geometry, reduce the ability of a polymer to crystallize.
Thus cis-double bonds (I), o- and m-phenylene groups (11), or cis-oriented puckered rings (III), all encourage bending and twisting in the chains and make regular close-packing very difficult.
If, however, the phenylene rings are para-oriented, the chains retain their axial symmetry and can crystallize more readily.
Similarly, incorporation of a trans-double bond maintains the chain symmetry.
This is highlighted when comparing the amorphous elastomeric cis-polyisoprene with the highly crystalline trans-polyisoprene which has no virtue as an elastomer, or cis-poly(1,3-butadiene), with trans-poly(1,3-butadiene),.
Intermolecular bonding.
In polyethylene crystallites, the close packing achieved by the chains allows the van der Waals forces to act co-operatively and provide additional stability to the crystallite.
Any interaction between chains in the crystal lattice will help to hold the structure together more firmly and raise the melting temperature.
Polymers containing polar groups, e.g. Cl, CN, or OH, can be held rigid, and aligned, in a polymer matrix by the strong dipole-dipole interactions between the substituents, but the effect is most obvious in the symmetrical polyamides.
These polymers can form intermolecular hydrogen bonds which greatly enhance crystallite stability.
This is illustrated in figure 11.4 for nylon-6,6, where the extended zig-zag conformation is ideally suited to allow regular intermolecular hydrogen bonding.
The increased stability is reflected in T m which for nylon-6,6 is 540 K compared with 410 K for polyethylene.
The structures of related polyamides do not always lead to this neat arrangement of intermolecular bond formation; for example the geometry of an extended nylon-7,7 chain allows the formation of only every second possible hydrogen bond when the chains are aligned and fully extended.
However, the process is so favourable energetically, that sufficient deformation of the chain takes place to enable formation of all possible hydrogen bonds.
The added stability that this imparts to the crystallite far outweighs the limited loss of energy caused by chain flexing.
Secondary bonds can therefore lead to a stimulation of the crystallization process in the appropriate polymers.
Tacticity.
Chain symmetry and flexibility both affect the crystallinity of a polymer sample.
If a chain possesses large pendant groups, these will increase the rigidity but also increase the difficulty of close packing to form a crystalline array.
This latter problem can be overcome if the groups are arranged in a regular fashion along the chain.
Isotactic polymers tend to form helices to accommodate the substituents in the most stable steric positions; these helices are regular forms capable of regular alignment.
Thus atactic polystyrene is amorphous but isotactic polystyrene is semi-crystalline.
Syndiotactic polymers are also sufficiently regular to crystallize, but not necessarily as a helix, rather in glide planes.
Branching in the side group tends to stiffen the chain and raise T m as shown in the series poly(but-1-ene),; poly (3-methyl but-1-ene),; poly (3,3-dimethyl but-1-ene),.
If the side group is flexible and non-polar, T m is lowered.
Branching and molar mass.
If the chain is substantially branched, the packing efficiency deteriorates and the crystalline content is lowered.
Polyethylene provides a good example of this (figure 11.2) where extensive branching lowers the density and T m of the polymer.
Molar mass can also alter T m.
Chain ends are relatively free to move and if the number of chain ends is increased by reducing the molar mass, then T m is lowered because of the decrease in energy required to stimulate chain motion and melting.
For example, polypropylene, with M = 2000 g mol -1, has, whereas a sample with M = 30 000 g mol -1, has.
11.7 Morphology and kinetics
Having once established that certain polymeric materials are capable of crystallizing, fundamental studies are directed along two main channels of interest centred on
(a)
the mode and kinetics of crystallization, and
(b)
the morphology of the sample on completion of the process
.
Although the morphology depends largely on the crystallizing conditions, we shall consider the macro- and microscopic structure first before dealing with the kinetics of formation.
11.8 Morphology
A number of distinct morphological units have been identified during the crystallization of polymers from the melt, which have helped to clarify the mechanism.
We shall now discuss the ordered forms which have been identified.
Crystallites.
In an X-ray pattern produced by a semicrystalline polymer, the discrete maxima observed arise from the scattering by small regions of three-dimensional order, which are called crystallites.
They are formed in the melt by diffusion of molecules, or sections of molecules, into close packed ordered arrays; these then crystallize.
The sizes of these crystallites are small relative to the length of a fully extended polymer chain, but they are also found to be independent of the molar mass and rarely exceed 1 to 100 nm.
As a result, various portions of one chain may become incorporated in more than one crystallite during growth, thereby imposing a strain on the polymer which retards the process of crystallite formation.
This will also introduce imperfections in the crystallites which continue growing until the strains imposed by the surrounding crystallites eventually stop further enlargement.
Thus a matrix of ordered regions with disordered interfacial areas is formed, but, unlike materials with small molar masses, the ordered and disordered regions are not discrete entities and can not be separated by differential solution techniques unless the solvent causes selective degradation of the primary bonds in the amorphous regions.
Crystallites of cellulose have been isolated from wood pulp in this way by treatment with acid to hydrolyse and remove the amorphous regions.
Typical dimensions of the remaining crystallites were 46 nm long by 7.3 nm wide corresponding to bundles of about 100 to 150 chains in each crystallite.
The first attempts to explain the crystalline structure of a polymer sample produced a model called the fringe-micelle structure.
The chain was envisaged as meandering throughout the system, entering and leaving several ordered regions along its length.
The whole structure was thus made up of crystalline regions imbedded randomly in a continuous amorphous matrix.
This model has now been virtually discarded in the light of more recent research which has revealed features incompatible with this picture.
Single crystals.
When a polymer is crystallized from the melt, imperfect polycrystalline aggregations are formed in association with a substantial amorphous content.
This is a consequence of chain entanglement and the high viscosity of the melt combining to hinder the diffusion of chains into the ordered arrays necessary for crystallite formation.
If these restrictions to free movement are reduced and a polymer is allowed to crystallize from a dilute solution, it is possible to obtain well-defined single crystals.
By working with solutions in which the amount of polymer is considerably less than 0.1 per cent the chance of a chain being incorporated in more than one crystal is greatly reduced, thereby increasing the possibility of isolated single crystals being formed.
These crystals are usually very small, but they have been detected for a range of polymers including polyesters, polyamides, polyethylene, cellulose acetate, and poly (4-methyl pentene-1).
Although small, these single crystals can be studied using an electron microscope.
This reveals that they are made up of thin lamellae, often lozenge shaped, sometimes oval, about 10 to 20 nm thick, depending on the temperature of crystallization.
The most surprising feature of these lamellae is that while the molecular chains may be as long as 1000 nm, the direction of the chain axis is across the thickness of the platelet.
This means that the chain must be folded many times like a concertina to be accommodated in the crystal.
For a polymer such as polyethylene, the fold in the chain is completed using only 3 or 4 monomer units with bonds in the gauche conformation.
The extended portions in between have about 40 monomers units all in the trans conformation.
The crystals, thus formed, have a hollow pyramid shape, because of the requirement that the chain folding must involve a staggering of the chains if the most efficient packing is to be achieved.
There is also a remarkable constancy of lamellar thickness, but this increases as the temperature increases.
While opinions vary between kinetic and thermodynamic reasons for this constancy of fold distance, it is suggested that the fold structure allows the maximum amount of crystallization of the molecule at a length which produces a free energy minimum in the crystal.
One suggestion is that the folding maintains the appropriate kinetic unit of the chain at any given temperature; as this would be expected to lengthen with increasing temperature, it would account for the observed thickening of the lamellae.
Hedrites.
If the concentration of the polymer solution is increased a crystalline polyhedral structure emerges composed of lamellae joined together along a common plane.
These have also been detected growing from a melt which suggests that lamellar growth can take place in the melt and may be a sub-unit of the spherulite.
Crystallization from the melt.
Whereas crystallization from dilute solutions may result in the formation of single polymer crystals, this perfection is not achieved when dealing with polymers cooled from the melt.
The basic characteristic feature is still the lamellar-like crystallite with amorphous surfaces or interfaces, but the way these are formed may be different based on the careful investigation of melt crystallized polymers using neutron scattering techniques.
The two models that have been proposed to describe the fine structure of these lamellae and their surface characteristics in semicrystalline polymers, differ mainly in the way the chains are thought to enter and leave the ordered lamellae regions.
These are:
(a)
the regular folded array with adjacent re-entry of the chains, but with some loose folding and emergent chain ends or cilia that contribute to the disordered surface, or
(b)
the switchboard model, where there is some folding of the chains but re-entry is now quite random.
Both are represented schematically in figure 11.5 but the exact nature of the structure has been the subject of considerable controversy.
While the morphology of the single crystals grown from dilute solutions may be more regular and resemble the first model, for polymers that are crystallized from the melt (and this is by far the more important procedure technically) the mass of evidence tends to favour a form of the switchboard model.
Measurements of the densities of several semicrystalline polymers points to the fact that a significant fraction of the chain units are in a non crystalline environment.
This is not consistent with the regular folded form of the crystallites where the amorphous part is associated only with loose folding of the chains and cilia.
Even more persuasive are small angle neutron scattering studies.
These have demonstrated that the radii of gyration of several semicrystalline polymers remain essentially unchanged on moving from the melt phase to the semicrystalline phase (table 11.2).
This means that there is no significant reordering of the chain conformation when crystallization takes place after cooling from the melt, which would be required if a regularly folded chain structure was to be constructed in the lamellae.
To explain these observations Fischer has proposed the solidification model in which crystallization is believed to take place by the straightening of sections of the polymer coil followed by alignment of these sequences in regular arrays forming the lamellar structure.
This precludes the need for the extensive, long range, diffusion of the chain through a highly viscous medium that would be necessary if a regular chain folded structure was to be constructed.
The process is shown schematically in figure 11.6 and the resulting structure is a variation of the switchboard model.
This hypothesis can account for the fact that on cooling, rapid crystal growth is seen to occur which is inconsistent with the need for long range diffusion if the regularly folding lamellae were forming.
The solidification model shows that the chains can be incorporated into the basic lamellar form with the minimum amount of movement and that there will be extensive meandering of chains between the lamellae forming the interfacial amorphous regions.
Spherulites.
Examination of thin sections of semicrystalline polymers reveals that the crystallites themselves are not arranged randomly, but form regular birefringent structures with circular symmetry.
These structures, which exhibit a characteristic Maltese cross optical extinction pattern, are called spherulites.
While spherulites are characteristic of crystalline polymers, they have also been observed to form in low molar mass compounds which are crystallized from highly viscous media.
Each spherulite grows radially from a nucleus formed either by the density fluctuations which result in the initial chain ordering process or from an impurity in the system.
As the structure is not a single crystal, the sizes found vary from somewhat greater than a crystallite to diameters of a few millimetres.
The number, size, and fine structure depend on the temperature of crystallization, which determines the critical size of the nucleating centre.
This means that large fibrous structures form near T m, whereas greater numbers of small spherulites grow at lower temperatures.
When the nucleation density is high, the spherical symmetry tends to be lost as the spherulite edges impinge on their neighbours to form a mass such as shown in figure 11.7.
A study of the fine structure of a spherulite shows that it is built up of fibrous sub-units, growth takes place by the formation of fibrils which spread outwards from the nucleus in bundles, into the surrounding amorphous phase.
As this fibrillar growth advances, branching takes place, and at some intermediate stage in the development, the spherulite often resembles a sheaf of grain.
This forms as the fibrils fan out and begin to create the spherical outline.
Although the fibrils are arranged radially, the molecular chains lie at right angles to the fibril axis.
This has led to the suggestion that the fine structure is created from a series of lamellar crystals winding helically along the spherulite radius.
Growth proceeds from a small crystal nucleus which develops into a fibril.
Low branching and twisting then produces bundles of diverging and spreading fibrils which eventually fill out into the characteristic spherical structure.
In between the branches of the fibrils are amorphous areas and these, along with the amorphous interfacial regions between the lamellae, make up the disordered content of the semi-crystalline polymer (figure 11.8).
Spherulites are classified as positive when the refractive index of the polymer chain is greater across the chain than along the axis, and negative when the greater refractive index is in the axial direction.
They also show various other features such as zig-zag patterns, concentric rings, and dendritic structures.
11.9 Kinetics of crystallization
The crystalline content of a polymer has a profound effect on its properties and it is important to know how the rate of crystallization will vary with the temperature, especially during the processing and manufacturing of polymeric articles.
The chemical structure of the polymer is also an important feature in the crystallization; for example, polyethylene crystallizes readily and can not be quenched rapidly enough to give a largely amorphous sample whereas this is readily accomplished for isotactic polystyrene.
However, this aspect will be discussed more fully later.
Isothermal crystallization.
Two main factors influence the rate of crystallization at any given temperature:
(i)
the rate of nucleation; and
(ii)
the subsequent rate of growth of these nuclei to macroscopic dimensions.
The kinetic treatment of crystallization from the melt is based on the radial growth of a front through space and can be likened to someone scattering a handful of gravel onto the surface of a pond.
Each stone is a nucleus which, when it strikes the surface, generates expanding circles (similar to spherulites in two dimensions).
These grow unimpeded for a while but the leading edges eventually collide with others and growth rates are altered.
When a similar picture is adopted for the crystallization of a polymer certain basic assumptions are made first.
The formation of ordered growth centres by the alignment of chains from the melt is called spontaneous nucleation.
When the temperature of crystallization is close to the melting temperature, nucleation is sporadic and only a few large spherulites will grow.
At lower temperatures, nucleation is rapid and a large number of small spherulites are formed.
The growth of the spherulites may occur in one, two, or three dimensions and the rate of radial growth is taken to be linear at any temperature.
Finally the density  c of the crystalline phase is considered to be uniform throughout but different from that of the melt  L.
A kinetic treatment has been developed taking account of these points.
The Avrami equation.
The kinetic approach relies on the establishment of a relation between the density of the crystalline and melt phases and the time.
This provides a measure of the overall crystallization rate.
It is assumed that the spherulites grow from nuclei whose relative positions in the melt remain unaltered, and the analysis allows for the eventual impingement of the growing discs on one another.
The final relation describing the process is known as the Avrami equation expressed as where k is the rate constant, w o and w L are the masses of the melt at zero time and that left after time t.
The exponent n is the Avrami exponent and is an integer which can provide information on the geometric form of the growth.
Sporadic nucleation is assumed to be a first-order mechanism and if we consider that a two-dimensional disc is formed, then.
Rapid nucleation is a zeroth-order process in which the growth centres are formed at the same time, and for each growth unit listed in table 11.3, the corresponding values of the exponent would be.
Thus the Avrami exponent is the sum of the order of the rate process and the number of dimensions the morphological unit possesses.
Dilatometry.
As crystallization involves the close packing of chains in regular three-dimensional structures, the economical use of space is accompanied by an increase in density.
Thus the rate of crystallization can be followed by recording the density changes which are readily detected in a dilatometer.
This is achieved by placing the polymer in a dilatometer with a confining liquid, such as mercury, so that any volume change can be recorded as a movement of the liquid meniscus in a capillary.
A typical design is shown in figure 11.9.
The polymer is introduced into the dilatometer between the point A and the capillary.
The apparatus is then pumped out and sealed under vacuum at the point A. Sufficient mercury is then added to enclose the polymer and extend into the capillary, after which the tube is sealed at B, and placed in a thermostat at a temperature somewhat higher than the melting temperature of the polymer.
When the sample is completely molten the dilatometer is transferred to a second thermostat set at the temperature selected for crystallization to take place and allowed to equilibrate.
The initial period of temperature adjustment to the second temperature may make the initial height h o rather difficult to locate, but usually a plot such as shown in figure 10.9(b) is recorded.
If secondary crystallization takes place the final portion of the curve may tail away making h more difficulty to measure.
The mass fraction of the uncrystallized polymer can be related to the volume changes and to the heights measured in the dilatometer by where h t, h o, and h  are the heights at time t, the beginning, and the end of the process respectively, with V t, V o, and V  the corresponding volumes.
The slope of a plot of against t allows evaluation of the Avrami exponent n while k can be calculated from the intercept.
Deviations from Avrami equation.
The Avrami equation can describe some but not all systems investigated.
The crystallization isotherms of poly (ethylene terephthalate) can be fitted by equation (11.3) using n = 4 above 473 K and n = 2 at 383 K. The equation should be used with caution, however, as non-integer values have been reported and the geometric shape of the morphological unit is not always that predicted by the value of n calculated from the experimental data.
Secondary crystallization.
Deviations from the Avrami treatment may also be observed towards the end of the crystallization process and values of h are often difficult to determine accurately, as shown in the curve derived from dilatometric data.
The tailing of the curve is a result of a secondary crystallization process which is a slower reorganization of the crystalline regions to produce more perfectly formed crystallites.
CHAPTER 12
The Amorphous State
12.1 Molecular motion
A linear polymer chain can be treated as a ' one-dimensional co-operative system ' in which the rotation of a chain segment is restricted or aided by the neighbouring segments.
For long chains, co-operative motion can not be expected to extend along the entire length, and the polymer tends to act as if it were composed of a series of interconnected, but independent, kinetic units.
Any significant movement of such a chain is generated by rotation about the single bonds connecting the atoms in the chain, and depends on the ease of interchange of any element from one rotational state to another.
The height of the potential energy barrier E (c.f. figure 1.3) will determine the rapidity of conformational change at any temperature, and when the temperature of the polymer increases, the additional thermal energy allows E to be overcome more often.
This encourages increasing molecular motion until eventually the polymer behaves like a viscous liquid (assuming that no thermal degradation takes place).
In the amorphous state the distribution of polymer chains in the matrix is completely random, with none of the strictures imposed by the ordering encountered in the crystallites of partially crystalline polymers.
This allows the onset of molecular motion in amorphous polymers to take place at temperatures below the melting temperature of such crystallites.
Consequently, as the molecular motion in an amorphous polymer increases, the sample passes from a glass, through a rubber-like state, until finally it becomes molten.
These transitions lead to changes in the physical properties and material application of a polymer, and it is important to examine physical changes wrought in an amorphous polymer as a result of variations in the molecular motion.
12.2 The five regions of viscoelastic behaviour
The physical nature of an amorphous polymer is related to the extent of the molecular motion in the sample, which in turn is governed by the chain flexibility and the temperature of the system.
Examination of the mechanical behaviour shows that there are five distinguishable states in which a linear amorphous polymer can exist and these are readily displayed if a parameter such as the elastic modulus is measured over a range of temperatures.
The general behaviour of a polymer can be typified by results obtained for an amorphous atactic polystyrene sample.
The relaxation modulus E r was measured at a standard time interval of 10 s and, is shown as a function of temperature in figure 12.1.
Five distinct regions can be identified on this curve.
(i)
The glassy state.
This is section A to B lying below 363 K and it is characterized by a modulus between 10 9.5; and.
Here co-operative molecular motion along the chain is frozen, causing the material to respond like an elastic solid to a stress, and the strain time curve is of the form shown in figure 12.1(a).
(ii)
Leathery or retarded highly elastic state.
This is the transition region B to C where the modulus drops sharply from about 10 9 to about over the temperature range 363 to 393 K. The glass transition temperature T g is located in this area and the rapid change in modulus reflects the constant increase in molecular motion as the temperature rises from T g to about.
Just above T g the movement of the chain segments is still rather slow, imparting what can best be described as leathery properties to the material.
The strain-time curve is that shown in figure 12.1(b).
(iii)
The rubbery state.
At approximately 30 K above the glass transition the modulus curve begins to flatten out into the plateau region C to D in the modulus interval 10 5.7; to and extends up to about 420 K.
(iv)
Rubbery flow.
After the rubbery plateau the modulus again decreases from 10 5.4 to in the section D to E. The effect of applied stress to a polymer in states (iii) and (iv) is shown in figure 12.1(c) where there is instantaneous elastic response followed by a region of flow.
(v)
Viscous state.
Above a temperature of 450 K, in the section E to F, there is little evidence of any elastic recovery in the polymer and all the characteristics of a viscous liquid become evident (figure 12.1(d)).
Here there is a steady decrease of the modulus from as the temperature increases.
The overall shape of the curve shown in figure 12.1 is typical for linear amorphous polymers in general, although the temperatures quoted are specific to polystyrene and will differ for other polymers.
Variations in shape are found for different molar masses and when the sample is crosslinked or partly crystalline.
The value of the modulus provides a good indication of the state of the polymer and can be obtained from the curve.
12.3 The viscous region
Before considering the flow in polymer melts, the viscous behaviour of simple liquids will be examined.
The application of a force to a simple liquid of low molar mass is relieved by the flow of molecules past one another into new positions in the system.
A liquid, forced to flow in this way by a shearing force , experiences a viscous resistance expressed by where v is the velocity of flow along a tube of radius x, so that is the velocity gradient or shear rate , and  is the viscosity coefficient of the liquid.
A liquid is said to exhibit Newtonian flow if  is independent of  but substances which show deviations from this flow pattern, with either decreasing or increasing ratios, are termed non-Newtonian.
(See figure 12.2.)
Most polymers fall into this latter category, with  decreasing as the shear rate increases.
The temperature dependence of  can normally be expressed in the form where A is a constant and E D represents the activation energy required to create a hole big enough for a molecule to translate or ' jump ' into during flow.
In liquids with larger or irregularly shaped molecules, the deformation is slower as the molecules restrict the easy translation of one past the other.
This results in a high value of 
12.4 Kinetic units in polymer chains
Resistance to flow in polymer systems is even greater, because now the molecules are covalently bonded into long chains which are coiled and entangled and translational motion must, of necessity, be a co-operative process.
It would be unreasonable to expect easy co-operative motion along the entire polymer chain, but as there is normally some degree of flexibility in the chain, local segmental motion can take place more readily.
The polymer can then be considered as a series of kinetic units; each of these moves in an independent manner and involves the co-operative movement of a number of consecutive chain atoms.
Crankshaft motion.
If we now consider an arbitrary kinetic unit which involves the movement of six atoms by rotation about two chain bonds, the movement can be visualized as shown diagrammatically in figure 12.3.
The amorphous or molten polymer is a conglomeration of badly packed interlacing chains and the extra empty space caused by this random molecular arrangement is called the free volume which essentially consists of all the holes in the matrix.
When sufficient thermal energy is present in the system the vibrations can cause a segment to jump into a hole by co-operative bond rotation and a series of such jumps will enable the complete polymer chain eventually to change its position.
Heating will cause a polymer sample to expand thereby creating more room for movement of each kinetic unit and the application of a stress in a particular direction will encourage flow by segmental motion in the direction of the stress.
The segmental transposition involving six carbon atoms is called crankshaft motion and is believed to require an activation energy of about 25 kJ mol -1.
12.5 Effect of chain length
Although it is thought that translation of a polymer chain proceeds by means of a series of segmental jumps involving short kinetic units, which may each consist of between 15 and 30 chain atoms, the complete movement of a chain can not remain unaffected by the surrounding chains.
As stated previously, considerable entanglement exists in the melt and any motion will be retarded by other chains.
According to Bueche the polymer molecule may drag along several others during flow and the energy dissipation is then a combination of the friction between the chain plus those which are entangled and the neighbouring chains as they slip past each other.
It would seem reasonable to assume from this, that the length of the chains in the sample must play a significant role in determining the resistance to flow and the effect of chain length on log , measured at low shear rates to ensure Newtonian flow, is illustrated in figure 12.4.
The plot comprises two linear portions meeting at a critical chain length [formula] c.
Above [formula] c the relation describing the flow behaviour is and K 1 is proportional to the 3.4 power of [formula].
Below [formula] c,  is directly proportional to and the expression becomes where K 1 and K 2 are temperature dependent constants.
The critical chain length [formula] c is interpreted as representing the dividing point between chains which are too short to provide a significant contribution to  from entanglement effects and those large enough to cause retardation of flow by intertwining with their neighbours.
If [formula] is defined as the number of atoms in the backbone chain of a polymer then typical values for [formula] c are 610 for polyisobutylene, 730 for polystyrene, and 208 for poly (methyl methacrylate).
In general [formula] c is lower for polar polymers than for non-polar polymers.
12.6 The reptation model
The theory proposed by Bueche tends to suggest a very clear-cut distinction between the movement of chains of length less than [formula] c and the relative immobility of the entangled chains with lengths greater than [formula] c.
As independent chain mobility can not be discounted for these longer chains after the onset of entanglement, a modified model is required to account for the ability of long chains to translate and diffuse through the polymer matrix, i.e. the entanglement network must be considered as being transient.
Such a concept is embodied in the ' reptation ' model proposed by de Gennes.
In this approach the chain is assumed to be contained in a hypothetical tube which is placed initially in a three dimensional network formed from the other entangled chains.
Although for simplicity, these network ' knots' are regarded as a set of fixed obstacles round which the isolated chain under consideration must wriggle during translation, in practice the network ' knots' would also be in motion.
The contours of the tube are then defined by the position of the entanglement points in the network.
Two types of chain motion can be envisaged, a conformational change taking place within the confines of the tube, and more importantly, reptation.
The latter is imagined to be a snake-like movement that translates the chain through the tube and allows it to escape at the tube ends.
Mechanistically it can be regarded as the movement of a kink in the chain along its length (see figure 12.5) until this reaches the end of the chain and leaves it.
Motion of this kind translates the chain through the tube, like a snake moving through grass, and successive defects moving the chain in this way will eventually carry it completely out of the hypothetical tube.
The motion can be characterized by a reptation time, or more accurately by a relaxation time, , that is a measure of the time required for a chain to escape completely from its tube.
If the tube is defined as having the same length as the unperturbed chain, nl o, where l o is the bond length under  conditions (corrected for short range interactions), then the time required for the chain to reptate out of the tube is proportional to the square of the distance travelled, i.e.
Here D 1 is the diffusion constant within the tube, and is distinguished from translation outside the tube which will be slower and more difficult.
This can be expressed as the frictional coefficient for the chain, again within the tube confines.
However, because the reptation is assumed to occur by migration of a segmental kink along the chain, the force needed to do this is applied one segment at a time and so it is more appropriate to use the frictional factor per segment .
Thus or
Equation (12.6b) shows that the relaxation time is proportional to the cube of the chain length.
This is the fundamental result of the reptation model.
The cube dependence is not a precise match with the 3.4 exponent obtained from viscosity measurements of long chains, but it is acceptable, particularly as the model gives a satisfactory picture of how a polymer chain can overcome the restraining influence of entanglements and move within the matrix.
Typically  o is of the order of 10 -10 seconds for and so the relaxation time for a polymer chain with would be about 100 seconds.
Reptation theory has been developed further by Doi and Edwards and is being applied to both viscoelastic and solution behaviour.
It has been shown that for a chain moving in the melt, over time-scales that greatly exceed the lifetime of the tube , a reptation self-diffusion coefficient D rept, can be measured which is inversely proportional to n 2, i.e. the diffusion law is
This law holds for the ' welding ' of polymers at an interface which can be explained by reptation.
When two blocks of the same polymer are brought together and held at a temperature just above the T g for a time t, interdiffusion of the chains takes place from each block across the interface (see figure 12.6) thereby joining the blocks together.
The strength of the junction formed will depend on t which should be smaller than the reptation time , i.e. the mixed layer ought to be smaller than the size of the coil if an interfacial link is to be formed.
The situation changes if the blocks are composed of two different polymers which, as a pair, can form a miscible blend.
Although welding can again take place, the diffusion law (equation 12.7) is now altered.
It has been found that if a block of poly(vinylchloride) is brought in contact with a block of polycaprolactone, at temperatures above T g, then D rept is higher than expected and is proportional to.
This has been interpreted as being a consequence of the negative enthalpy of mixing in the system which acts as an additional driving force for the chains on either side of the boundary to cross into the other matrix.
This driving force will be proportional to the number of monomers in a chain, hence the change in the diffusion law.
Reptation theory can also be applied to polymer dissolution processes.
12.7 Temperature dependence of 
When a polymer is transformed into a melt without degradation and is stable at even higher temperatures,  is observed to decrease rapidly as the temperature increases.
If it is still stable at temperatures in excess of 100 K above T g, the temperature dependence has an exponential form where according to the Eyring rate theory H is the activation enthalpy of viscous flow and is a more representative parameter than the energy.
Values of H vary slowly over a range from 20 to 120 kJ mol -1.
When the temperature is lowered towards T g, H changes dramatically and a simple equation such as (12.8) is no longer valid.
The increase in H, observed with temperature lowering, can be equated with a rapid loss of free volume as T g is approached.
Hence H becomes dependent on the availability of a suitable hole for a segment to move into, rather than being representative of the potential energy barrier to rotation.
This approach suggests that the jump frequency decreases when there is an increasing co-operative motion among the chains needed to produce holes.
12.8 Rubbery state
With a decrease in temperature, the flow of a polymer melt becomes increasingly sluggish as the chain motion becomes too slow to effect complete untangling of the polymer coils.
The viscosity increases rapidly to a value of about as T g is approached, but on passing from the melt to the glass a region of rubbery flow and elasticity is traversed.
In this state the polymer exhibits several unique properties which are dealt with in chapter 14 and only a brief description of the chain behaviour in this region is given here.
Long range elasticity.
The rubber-like region, which lies above T g, appears when the rotation about the segment links is free enough to enable the chains to assume any of the immense number of equi-energetic conformations available, without significant chain untangling taking place.
The majority of these shapes will be compact coils because the possibility of their occurrence is much greater than for the more extended forms.
When a polymer, which is not too crystalline and has a reasonably high molar mass (&lt;20 000 g mol -1), is in this elastic state it will elongate quite readily in the direction of an applied stress, e.g. natural rubber will stretch easily when pulled.
If the stress is applied for a short time, then removed, the sample snaps back to its original length suggesting that some ' memory ' of its initial unstretched condition is retained.
The ability of an elastomer to regain its former size, when extensions of up to 400 per cent have been experienced, is associated with the long chain character of the material.
This retractive action of linear uncrosslinked polymers can be observed if the time interval between extension and release is short, but if the stress is maintained for some time, then a relaxation process takes place allowing the tension to decay eventually to zero.
This can be explained quite simply.
The molecules are initially in highly coiled shapes but application of a force causes rotation about the chain bonds resulting in an elongation of the molecules in the direction of the stress.
This produces a distribution of chain conformations which differs significantly from the most probable distribution, and as this is an unstable state the chains will rapidly recoil when the stress is released in an attempt to regain their original shape distribution.
For short periods of stress in an amorphous elastomer, the entanglement and intertwining of chains with their neighbours acts as a physical restraint to excessive chain movement and the elastomer regains its original length when the stress is removed.
If however, the stress is maintained for a sufficient time, there is a general tendency for chains to unravel and slip past one another into new positions where the segments can relax and regain a stable coiled form.
The resultant flow relieves the tension and produces the observed stress decay.
The process is shown schematically in figure 12.7.
When the molar mass is too low to produce sufficient entanglement, the material will flow more readily and behave like a viscous liquid.
Similarly, as the temperature increases further and further above the glass transition, the enhanced segmental movement facilitates stress decay because of the greater ease of chain disentanglement.
12.9 Glass transition region
When the polymer is at a temperature below its glass temperature, chain motion is frozen.
The polymer then behaves like a stiff spring storing all the available energy in stretching as potential energy, when work is performed on it.
If sufficient thermal energy is supplied to the system to allow the chain segments to move co-operatively, a transition from the glass to the rubber-like state begins to take place.
Motion is still restricted at this stage, but as the temperature increases further a larger number of chains begin to move with greater freedom.
In mechanical terms the transition can be likened to the transformation of a stiff spring to a weak spring.
As weak springs can only stored a fraction of the potential energy that a strong spring can hold, the remainder is lost as heat and if the change from a strong to a weak spring takes place over a period of time, equivalent to the observation time, then the energy loss is detected as mechanical damping.
Finally when molecular motion increases to a sufficiently high level, all the chains behave like weak springs the whole time.
This means that the modulus is much lower, but so too is the damping, which passes through a maximum in the vicinity of T g.
The maximum appears because the polymer is passing from the low-damping glassy state, through the high-damping transition region, to the lower-damping rubber-like state.
Treloar has described a very apt demonstration of the transition.
A thin rubber rod is wound round a cylinder to create the shape of a spring and then frozen in this shape using liquid nitrogen.
The cylinder (possibly of paper) is then removed leaving the rubber spring.
The rubber is now in the glassy state and it acts like a stiff metal spring by regaining its shape rapidly after an extension.
As the temperature is raised a gradual loss in the elastic recovery is observed after each applied stress, until a stage is reached when there is no recovery and the rubber remains in the deformed shape.
With a further increase in temperature the rod straightens under its own weight and eventually regains its rubber-like elasticity at slightly higher temperatures.
THE GLASS TRANSITION TEMPERATURE, T g
The transition from the glass to the rubber-like state is an important feature of polymer behaviour, marking as it does a region where dramatic changes in the physical properties, such as hardness and elasticity, are observed.
The changes are completely reversible, however, and the transition from a glass to a rubber is a function of molecular motion, not polymer structure.
In the rubber-like state or in the melt the chains are in relatively rapid motion, but as the temperature is lowered the movement becomes progressively slower until eventually the available thermal energy is insufficient to overcome the rotational energy barriers in the chain.
At this temperature, which is known as the glass transition temperature T g, the chains become locked in whichever conformation they possessed when T g was reached.
Below T g the polymer is in the glassy state and is, in effect, a frozen liquid with a completely random structure.
Although the glass-rubber transition itself does not depend on polymer structure, the temperature at which T g is observed depends largely on the chemical nature of the polymer chain and for most common synthetic polymers lies between 170 and 500 K. It is quite obvious that T g is an important characteristic property of any polymer as it has an important bearing on the potential application of a polymer.
Thus for a polymer with a flexible chain, such as polyisoprene, the thermal energy available at about 300 K is sufficient to cause the chain to change shape many thousands of times in a second.
This polymer has.
On the other hand, virtually no motion can be detected in atactic poly (methyl methacrylate) at 300 K, but at 450 K, the chains are in rapid motion.
In this case.
This means that at 300 K polyisoprene is likely to exhibit rubber-like behaviour and be useful as an elastomer, whereas poly (methyl methacrylate) will be a glassy material.
If the operating temperature was lowered to 100 K, both polymers would be glasses.
EXPERIMENTAL DEMONSTRATION OF T g
The glass transition is not specific to long chain polymers.
Any substance, which can be cooled to a sufficient degree below its melting temperature without crystallizing, will form a glass.
The phenomenon can be conveniently demonstrated using glucose penta-acetate (GPA).
A crystalline sample of GPA is melted, then chilled rapidly in ice-water to form a brittle amorphous mass.
By working the hard material between one's fingers, the transition from glass to rubber will be felt as the sample warms up.
A little perseverance, with further rubbing and pulling, will eventually result in the recrystallization of the rubbery phase, which then crumbles to a powder.
DETECTION OF T g
The transition from a glass to a rubber-like state is accompanied by marked changes in the specific volume, the modulus, the heat capacity, the refractive index, and other physical properties of the polymer.
The glass transition is not a first-order transition, in the thermodynamic sense, as no discontinuities are observed when the entropy or volume of the polymer are measured as a function of temperature.
If the first derivative of the property-temperature curve is measured, a change in the vicinity of T g is found; for this reason it is sometimes called a second-order transition.
Thus while the change in a physical property can be used to locate T g, the transition bears many of the characteristics of a relaxation process and the precise value of T g can depend on the method used and the rate of the measurement.
Techniques for locating T g can be divided into two categories, dynamic and static.
In the static methods, changes in the temperature dependence of an intensive property, such as density or heat capacity are followed and measurements are carried out slowly, to allow the sample to equilibrate and relax at each observation temperature.
In dynamic mechanical methods a rapid change in modulus is indicative of the glass transition, but now the transition region is dependent on the frequency of the applied force.
If we assume that, in the transition region, the restrictions to motion still present in the sample, allow only a few segments to move in some time interval, say 10 s, then considerably fewer will have moved if the observation time is less than 10 s.
This means that the location of the transition region and T g will depend on the experimental approach used, and T g is found to increase 5 to 7 K for every tenfold increase in the frequency of the measuring techniques.
This time dependence of segmental motion corresponds to the strong-weak transformation of a hypothetical spring and results in the high damping which imparts the lifeless leathery consistency to the polymer in this region.
The temperature of maximum damping is usually associated with T g, and at low frequencies the value assigned to T g is within a few kelvins of that obtained from the static methods.
As the static methods lead to more consistent values some of these can be described.
Measurement of T g from V-T curves.
One of the most frequently used methods of locating T g is to follow the change in the volume of the polymer as a function of the temperature.
The polymer sample is placed in the bulb of a dilatometer, degassed, and a confining liquid such as mercury added.
If the bulb is attached to a capillary the change in polymer volume can be traced by noting the overall change in volume registered by the movement of the mercury level in the capillary.
A variation of this method makes use of a density gradient column.
A small sample of polymer suspended in this column provides a direct measure of the polymer density which can be measured easily as the temperature is varied.
Typical specific-volume-temperature curves are shown in figure 12.8 for poly (vinyl acetate).
These consist of two linear portions whose slopes differ and closer inspection reveals that over a narrow range of temperature of between 2 and 5 K the slope changes continuously.
To locate T g, the linear portions are extrapolated and intersect at the point which is taken to be the characteristic transition temperature of the material.
Each point on the curve is normally recorded after allowing the polymer time to equilibrate at the chosen temperature and as the rate of measurement affects the magnitude of T g quite noticeably the equilibration time should be several hours at least.
The effect of the measuring rate on T g was demonstrated by Kovacs who recorded the volume of a polymer at each temperature, over a range including the transition, using two rates of cooling.
If the sample was cooled rapidly (0.02 h) to each temperature the value of T g derived from the resulting curve was some 8 K higher than that measured from results obtained using a slow cooling rate (100 h).
Refractive index measurements.
The change in refractive index of the polymer with temperature has been used by several workers to establish T g.
A linear decrease in refractive index is observed as the temperature increases, and as the transition is passed, the rate of decrease becomes greater; T g is again taken as the intersection of the linear extrapolation.
Heat capacity and other methods.
The glass transition temperature can be detected calorimetrically by following the change in heat capacity with change in temperature.
The curve for atactic polypropylene is shown in figure 12.9 where the abrupt increase in c p at about 260 K, corresponds to the glass transition.
Among other reported techniques the most useful include differential thermal analysis, dielectric loss measurements, X- and -ray absorption, and gas permeability studies.
All indicate the existence of the phenomenon which we call the glass transition.
12.10 Factors affecting T g
We have seen that the magnitude of T g varies over a wide temperature range for different polymers.
As T g depends largely on the amount of thermal energy required to keep the polymer chains moving, a number of factors which affect rotation about chain links, will also influence T g.
These include
(1)
chain flexibility,
(2)
molecular structure (steric effects),
(3)
molar mass (see section 12.12),
(4)
branching and crosslinking
Chain flexibility.
The flexibility of the chain is undoubtedly the most important factor influencing T g.
It is a measure of the ability of a chain to rotate about the constituent chain bonds, hence a flexible chain has a low T g whereas a rigid chain has a high T g.
For symmetrical polymers, the chemical nature of the chain backbone is all important.
Flexibility is obtained when the chains are made up of bond sequences which are able to rotate easily, and polymers containing,, or links will have correspondingly low values of T g.
The value of T g is raised markedly by inserting groups which stiffen the chain by impeding rotation, so that more thermal energy is required to set the chain in motion.
The p-phenylene ring is particularly effective in this respect, but when carried to extremes, produces a highly intractable, rigid structure, poly(p-phenylene) with no softening point.
The basic structure can be modified by introducing flexible groups in the chain and some examples are given in table 12.1.
Steric effects.
When the polymer chains are unsymmetrical, with repeat units of the type, an additional restriction to rotation is imposed by steric effects.
These arise when bulky pendant groups hinder the rotation about the backbone and cause T g to increase.
The effect is accentuated by increasing the size of the side group and there is some evidence of a correlation between T g and the molar volume V  of the pendant group.
It can be seen in table 12.2, that T g increases with increasing V  in the progressive series, polyethylene, polypropylene, polystyrene, and poly (vinyl naphthalene).
Superimposed on this group size factor are the effects of polarity and the intrinsic flexibility of the pendant group itself.
An increase in the lateral forces in the bulk state will hinder molecular motion and increase T g.
Thus polar groups tend to encourage a higher T g than non-polar groups of similar size, as seen when comparing polypropylene, poly (vinyl chloride) and polyacrylonitrile.
The influence of side chain flexibility is evident on examination of the polyacrylate series from methyl through butyl, and also in the polypropylene to poly(hex-1-ene) series.
A further increase in steric hindrance is imposed by substituting an -methyl group, which restricts rotation even further and leads to higher T g.
For the pair polystyrene poly (-methyl styrene), the increase in T g is 70 K, while the difference between poly (methyl methacrylate) and poly (methyl acrylate) is 100 K.
These steric factors all affect the chain flexibility and are simply additional contributions to the main chain effects.
Configurational effects.
Cis-trans isomerism in polydienes and tacticity variations in certain -methyl substituted polymers alter chain flexibility and affect T g.
Some examples are shown in table 12.3.
It is interesting to note that when no -methyl group is present in a polymer, tacticity has little influence on T g.
Effect of crosslinks on T g
When crosslinks are introduced into a polymer, the density of the sample is increased proportionally.
As the density increases, the molecular motion in the sample is restricted and T g rises.
For a high crosslink density the transition is broad and ill-defined, but at lower values, T g is found to increase linearly with the number of crosslinks.
12.11 Theoretical treatments
Before embarking on a rather brief description of the theoretical interpretations of the glass transition a word of caution should be given.
In the foregoing sections several features of the results point to the fact that, in the vicinity of T g, rate effects are closely associated with changes in certain thermodynamic properties.
This has engendered two schools of thought on the origins of this phenomenon, together with variations on each theme.
The elementary level of this text precludes detailed critical discussion of the relative merits of any particular treatment, and to avoid prejudicing the issue with personal comment the main ideas of each are outlined together with a more recent and possibly unifying approach to complete the picture.
THE FREE VOLUME THEORY
The free volume concept has been touched on in previous sections but it is instructive now to consider this idea more closely and to draw together the various points alluded to earlier.
The free volume, V f, is defined as the unoccupied space in a sample, arising from the inefficient packing of disordered chains in the amorphous regions of a polymer sample.
The presence of these empty spaces can be inferred from the fact that when a polystyrene glass is dissolved in benzene there is a contraction in the total volume.
This and similar observations indicate that the polymer can occupy less volume when surrounded by benzene molecules and that there must have been unused space in the glassy matrix to allow this increase in packing efficiency to occur.
On that basis, the observed specific volume of a sample, V, will be composed of the volume actually occupied by the polymer molecules, V o, and the free volume in the system.
i.e.
Each term will, of course, be temperature dependent.
The free volume is a measure of the space available for the polymer to undergo rotation and translation, and when the polymer is in the liquid or rubber-like states the amount of free volume will increase with temperature as the molecular motion increases.
If the temperature is decreased, this free volume will contract and eventually reach a critical value when there is insufficient free space to allow large scale segmental motion to take place.
The temperature at which this critical value is reached is the glass transition temperature.
Below T g the free volume will remain essentially constant as the temperature decreases further since the chains have now been immobilized and frozen in position.
In contrast, the occupied volume will alter because of the changing amplitude of thermal vibrations in the chains and, to the first approximation, will be a linear function of temperature irrespective of whether the polymer is in the liquid or glassy state.
The glass transition can then be visualized as the onset of co-ordinated segmental motion made possible by an increase of the holes in the polymer matrix to a size sufficient to allow this type of motion to occur.
This is manifest as a change in the specific volume due solely to an increase in the free volume and is shown schematically as the cross hatched area in figure 12.10, where the broken line indicates the temperature dependence of V o.
The precise definition of the average amount of free volume present in a totally amorphous polymer remains unclear, but it must also depend to some extent on the thermal history of the sample.
A number of suggestions have been made.
Simha and Boyer observed that a general empirical relationship exists between the T g and the difference in expansion coefficients of the liquid and glass states.
From the examination of a wide range of polymers they concluded that where K 1 is a constant with a value of 0.113.
This implies that the free volume fraction is the same for all polymers, i.e. 11.3 per cent of the total volume in the glassy state.
The definition of the S-B free volume can be seen from figure 12.10 to be where V o  is the hypothetical liquid volume at absolute zero.
This definition is perhaps too rigid and discounts differing chain flexibilities, so a more accurate representation is thought to be given by
The values obtained are still much higher than the estimates from the Williams, Landel, Ferry (WLF) equation.
This is an empirical equation but it can also be derived from free volume considerations by starting with a description of the viscosity of the system.
In section 12.7 the Arrhenius equation was used to describe the temperature dependence of viscous flow, but an empirical equation proposed by Doolittle gives a much better description of viscous flow, and has a similar form where A and B are constants and.
On a molecular level, the ratio is then a measure of the average volume of the polymer relative to that of the holes.
Thus when, i.e. the polymer chain is larger than the average hole size, the viscosity will be correspondingly high, whereas when, the viscosity will be low.
We can now introduce a free volume fraction f and substitute in equation (12.13)
Next, a comparison can be made between the viscosity of a polymer melt at a temperature, and that at a reference temperature such as and so
Here f  and f g are the fractional free volumes at T and T g respectively.
From figure 12.10 it can be seen that V f is assumed to remain constant during expansion of the polymer in the glassy state but that above T g there is a steady increase with rising temperature.
If  f is the expansion coefficient of the free volume above T g, then the temperature dependence of f  can be written Substitution of equation (12.17) in equation (12.16) gives Rearranging and dividing by  f
Equation (12.19) is one form of the WLF equation, but as viscosity is a time dependent quantity and is proportional to the flow time, t, and density, , then and where the small differences in density have been neglected.
This can be compared with the form of the WLF equation where  is the reduced variables shift factor, C 1 and C 2 are constants that can be evaluated from experimental data, and are found to be and when T g is the reference temperature.
A more general description can be used where T s is an arbitrary reference temperature usually located 50 K above T g.
C 1 and C 2 now have different values, and the shift factor is expressed as a ratio of relaxation times, , at T and T s.
As we shall see in chapter 13, the relaxation time is a function of the viscosity and modulus (G) of the polymer and, according to the Maxwell model,.
The modulus will be much less temperature dependent than the viscosity so we can write which demonstrates the equivalence of the empirical equation (12.22) with that derived from the free volume theory, equations (12.19) and 12.21).
The WLF equation can be used to describe the temperature dependence of dynamic mechanical, and dielectric relaxation behaviour of polymers near the glass transition where the response is no longer described by an Arrhenius relation.
This will be dealt with in chapter 13.
The equations (12.19) and (12.22) can be used to evaluate f g as we see that and.
On the basis of viscosity data, B can be assigned a value of unity, leading to and.
If  f is assumed to be equivalent to, this value compares well with the average value of determined for 18 polymers covering a wide range of T g.
The free volume fraction of 2.5 per cent is low compared with the S-B estimation but is comparable to that derived from the Gibbs-Di Marzio theory.
Other values of 8 per cent from the ' hole ' theory of Hirai and Eyring, and 12 per cent calculated by Miller from heats of vaporization and liquid compressibilities, illustrate the uncertainty surrounding the magnitude of this free volume parameter.
The free volume theory deals with the need for space to be available before co-operative motion, characteristic of the glass transition, can be initiated, but it tells us little about the molecular motion itself.
Other approaches have chosen to base their description of the glass transition on a thermodynamic analysis.
GIBBS-DI MARZIO THERMODYNAMIC THEORY
Comments on the thermodynamic theories will be restricted to the proposals of Gibbs and Di Marzio (G-D) who, while acknowledging that kinetic effects are inevitably encountered when measuring T g, consider the fundamental transition to be a true equilibrium.
The data reported by Kovacs in section 12.9 imply that the observed T g would decrease further if a sufficiently long time for measurement was allowed.
This aspect is considered in the G-D theory by defining a new transition temperature T 2 at which the configurational entropy of the system is zero.
This temperature can be considered in effect to be the limiting value T g would reach in a hypothetical experiment taking an infinitely long time.
On this basis the experimentally detectable T g is a time dependent relaxation process and the observed value is a function of the time scale of the measuring technique.
The theoretical derivation is based on a lattice treatment.
The configurational entropy is found by calculating the number of ways that n x linear chains each x segments long can be placed on a diamond lattice, for which the coordination number z = 4, together with n o holes.
The restrictions imposed on the placing of a chain on the lattice are embodied in the hindered rotation which is expressed as the ' flex energy '  and  h which is the energy of formation of a hole.
The flex energy is the energy difference between the potential energy minimum of the located bond and the potential minima of the remaining possible orientations which may be used on the lattice.
Thus for polyethylene the trans position is considered most stable and the gauche positions are the flexed ones with  the energy difference between the ground and flexed states.
This of course varies with the nature of the polymer.
The quantity  h is a measure of the cohesive energy.
The configurational entropy S conf is derived from the partition function describing the location of holes and polymer molecules.
As the temperature drops towards T 2 the number of available configurational states in the system decreases until at the temperature T 2 the system possesses only one degree of freedom.
This leads to where and
The fractions of unoccupied and occupied sites are f o and f x respectively while S o is a function of f o, f x, and z.
The main weaknesses of this theory are
(a)
that a chain of zero stiffness would have a T g of 0 K and
(b)
that the T g would be essentially independent of any intermolecular interactions
.
In spite of these limitations, various aspects of the behaviour of copolymers, plasticized polymers, and the chain length dependence of T g, can be predicted in a reasonably satisfactory manner.
The temperature T 2 is not of course an experimentally measurable quantity but is calculated to lie approximately 50 K below the experimental T g and can be related to T g on this basis.
ADAM-GIBBS THEORY
While the kinetic approach embodied in the WLF equation and the equilibrium treatment of the G-D theory have both been successful in their way, the one-sided aspect of each probably masks the fact that they are not entirely incompatible with one another.
An attempt to reunite both channels of thought has been made by Adam and Gibbs who have outlined a molecular kinetic theory.
In this they relate the temperature dependence of the relaxation process to the temperature dependence of the size of a region, which is defined as a volume large enough to allow co-operative rearrangement to take place without affecting a neighbouring region.
This' co-operatively rearranging region ' is large enough to allow a transition to a new conformation, hence is determined by the chain conformation and by definition will equal the sample size at T 2 where only one conformation is available to each molecule.
Evaluation of the temperature dependence of the size of such a region leads to an expression for the co-operative transition probability,, which is simply the reciprocal of the relaxation time.
The polymer sample is described as an ensemble of co-operative regions, or subsystems, each containing  monomeric segments.
The transition probability of such a co-operative region is then calculated as a function of its size, to be where  is the activation energy for a co-operative rearrangement per monomer segment.
A lower limit to  is then defined; this is the smallest size *; capable of having two configurations available to it, with a critical configurational entropy which, from the definition, can be approximated by.
Thus where S c is the macroscopic configurational entropy for the ensemble.
Substitution gives and expressing this in the WLF form
The following approximations for S c can now be used and, remembering that, then Substitution in equation (12.29b) gives a WLF equation where
Results plotted according to the WLF equation could be predicted also from the molecular kinetic equation and show that the two approaches are compatible.
The Adam-Gibbs equations also lead to a value of, so the theory appears to resolve most of the differences between the kinetic and thermodynamic interpretations of the glass transition.
These theories point to the fundamental importance of T 2 as a true second-order transition temperature and to the experimental T g as the temperature governed by the time scale of the measuring technique.
The latter value has great practical significance, however, and is a parameter which is essential to the understanding of the physical behaviour of a polymer.
12.12 Dependence of T g on molar mass
The value of T g depends on the way in which it is measured but it is also found to be a function of the polymer chain length.
At high molar masses the glass temperature is essentially constant when measured by any given method, but decreases as the molar mass of the sample is lowered.
In terms of the simple free volume concept each chain end requires more free volume in which to move about than a segment in the chain interior.
With increasing thermal energy the chain ends will be able to rotate more readily than the rest of the chain and the more chain ends a sample has the greater the contribution to the free volume when these begin moving, consequently the glass transition temperature is lowered.
Bueche expressed this as where is the glass temperature of a polymer with a very large molar mass,  is the free volume contribution of one chain end and is 2 for a linear polymer,  is the polymer density, N A is Avogradro's constant, and  f is the free volume expansivity defined as The linear expression in equation (12.33) has been widely used and describes the behaviour of many polymer systems over a reasonable range of molar mass (&gt; 5000).
For short chains the relationship is no longer valid and it has been shown that if T g is plotted against log x, where x is the number of atoms or bonds in the polymer backbone, then three distinct regions can be identified for a number of common amorphous polymers (figure 12.11).
Region I denotes the range of chain lengths at which T g reaches its asymptotic value and the critical value x c at which this occurs increases as the chain becomes more rigid.
Thus x c is approximately 90 for a flexible polymer poly (dimethyl siloxane) but nearer 600 for the more rigid poly (-methyl styrene).
The relationship between and x c is then
In region II, T g is dependent on the molar mass and can be described by equation (12.35), but, on entering region III where the decrease in T g accelerates, this is no longer true.
The latter region incorporates the material that is oligomeric, and the line separating II and III represents the oligomer-polymer transition where the chains begin to become long enough to be considered capable of adopting a gaussian coil conformation.
12.13 The glassy state
When a linear amorphous polymer is in the glassy state, the material is rigid and brittle because the flow units of the chain are co-operatively immobile and effectively frozen in position.
The polymer sample is also optically transparent, as the chains are distributed in a random fashion and present no definite boundaries or discontinuities from which light can be reflected.
An amorphous polymer in this state has been likened to a plate of frozen spaghetti.
If a small stress is applied to a polymer glass, it exhibits a rapid elastic response resulting from purely local, bond angle, deformation.
Consequently, although the modulus is high, specimen deformation is limited to about 1 per cent, due to the lack of glide planes in the disordered mass.
This means that the sample has no way of dissipating a large applied stress, other than by bond rupture, and so a polymer glass is prone to brittle fracture.
12.14 Relaxation processes in the glassy state
Polymers do not form perfectly elastic solids, as a limited amount of bond rotation can occur in the glass which allows slight plastic deformation; this makes them somewhat tougher than an inorganic glass.
There is now ample evidence to support the suggestion that relaxation processes can be active in polymer glasses at temperature well below T g.
While the co-operative, long-range, chain motion which is released on passing from the glass to the rubber-like state is not possible at, other relaxations can take place.
Many of these processes can be identified as secondary loss peaks in dynamic mechanical, or dielectric measurements, as will be seen in chapter 13.
They often find their origin in the movement of groups that are attached pendant to the main chain, but relaxation of limited sections of the main chain can also be identified.
The molecular mechanisms for a number of these sub-glass transition relaxations have now been established, and by way of illustration some examples of group motions that have been found to be active in a series of poly (alkyl methacrylate) s will be described.
For the methyl, ethyl, and propyl derivatives a broad, mechanically active, damping peak is observed at 280 K (1 Hz), which is below the T g of each polymer.
Rotation of the oxycarbonyl group about has been identified as the cause.
However, if the group R is an alkyl or cycloalkyl unit then these can relax at even lower temperatures.
Thus when R is a methyl unit, rotation is possible in the glass at temperatures below 100 K, and the -methyl unit will also be capable of rotation at low temperatures.
When R is larger,, another relaxation process is seen at around 120 K, which is common to all of these polymers with.
This is believed to involve the relaxation of a four atom unit (-O-C-C-C-) or (-C-C-C-C-) which has been variously described by mechanisms including, and similar to, the Schatzki or Boyer crankshaft motions shown schematically in figures 12.3 and 12.12.
The latter mechanisms have also been used to account for limited segmental relaxations in the backbone of all carbon chain, single strand, polymers.
Even larger units can relax, and Heijboer has demonstrated that if R is a cyclohexyl ring, a relaxation at 180 K (1 Hz) can be located in the glass.
This can be attributed to an intramolecular chair-chair transition in the ring.
As these relaxations require energy, and are associated with a characteristic activation energy, it has been suggested that they may improve the impact resistance of some materials.
This point still requires confirmation as a general phenomenon, but there is little doubt that polymer molecules are not totally frozen or immobile when in the glassy state and that small sub units in the chain can remain mechanically and dielectrically active below T g.
CHAPTER 13
Mechanical Properties
13.1 Viscoelastic state
The fabrication of an article from a polymeric material in the bulk state, whether it be the moulding of a thermosetting plastic or the spinning of a fibre from the melt, involves deformation of the material by applied forces.
Afterwards, the finished article is inevitably subjected to stresses, hence it is important to be aware of the mechanical and rheological properties of each material and understand the basic principles underlying their response to such forces.
In classical terms the mechanical properties of elastic solids can be described by Hooke's law, which states that an applied stress is proportional to the resultant strain, but is independent of the rate of strain.
For liquids the corresponding statement is known as Newton's law, with the stress now independent of the strain, but proportional to the rate of strain.
Both are limiting laws, valid only for small strains or rates of strain, and while it is essential that conditions involving large stresses, leading to eventual mechanical failure, be studied, it is also important to examine the response to small mechanical stresses.
Both laws can prove useful under these circumstances.
In many cases, a material may exhibit the characteristics of both a liquid and a solid and neither of the limiting laws will adequately describe its behaviour.
The system is then said to be in a viscoelastic state.
A particularly good illustration of a viscoelastic material is provided by a silicone polymer known as' bouncing putty '.
If a sample is rolled into the shape of a sphere it can be bounced like a rubber ball, i.e. the rapid application and removal of a stress causes the material to behave like an elastic body.
If on the other hand, a stress is applied slowly over a longer period the material flows like a viscous liquid so that the spherical shape is soon lost if left to stand for some time.
Pitch behaves in a similar, if less spectacular, manner.
Before examining the viscoelastic behaviour of amorphous polymeric substances in more detail, some of the fundamental terms used will be defined.
13.2 Mechanical properties
Homogeneous, isotropic, elastic materials possess the simplest mechanical properties and three elementary types of elastic deformation can be observed when such a body is subjected to
(i)
simple tension,
(ii)
simple shear, and
(iii)
uniform compression.
Simple tension.
Consider a parallelepiped of length x o and cross-sectional area.
If this is subjected to a balanced pair of tensile forces F, its length changes by an increment dx so that.
When dx is small, Hooke's law is obeyed, and the tensile stress  is proportional to the tensile strain .
The constant of proportionality is known as the modulus, and for elastic solids where E is Young's modulus
The stress  is a measure of the force per unit area, and the strain or elongation is defined as the extension per unit length, i.e..
It should be pointed out, however, that other definitions of strain will be met with in the literature, most notably, is often called the true strain, while an expression arising from the kinetic theory of elasticity has the form
Of course, the extension dx will be accompanied by lateral contractions dy and dz, but although normally negative and equal, they can usually be assumed to be zero.
For an isotropic body, the change in length per unit length is related to the change in width per unit of length, such that where  p is known as Poisson's ratio and varies from 0.5, when no volume change occurs, to about 0.2.
Simple shear.
In simple shear the shape change is not accompanied by any change in volume.
If the base of the body, shown shaded in the diagram, figure 13.1(b) is firmly fixed, a transverse force F applied to the opposite face is sufficient to cause a deformation dx through an angle .
The shear modulus C is then given by the quotient of the shearing force per unit area and the shear per unit distance between shearing surfaces; and so For very small shearing strains tan   and
Both E and G depend on the shape of the specimen and it is usually necessary to define the shape carefully for any measurement.
Uniform compression.
When a hydrostatic pressure  p is applied to a body of volume V o, causing a change in volume V, a bulk modulus B can be defined as
The quantity B is often expressed in terms of the compressibility which is the reciprocal of the bulk modulus.
Similarly and are known as the tensile and shear compliances and given the symbols D and J respectively.
13.3 Interrelation of moduli
The relations given above pertain to isotropic bodies and for non-isotropic bodies the equations are considerably more complex.
Polymeric materials are normally either amorphous, or partially crystalline with randomly oriented crystallites embedded in a disordered matrix.
However, any symmetry possessed by an individual crystallite can be disregarded and the body as a whole is treated as being isotropic.
The various moduli can be related to each other in a simple manner, because an isotropic body is considered to possess only two independent elastic constants and so
This indicates, that for an incompressible elastic solid, i.e. one having a Poisson ratio of 0.5, Young's modulus is three times larger than the shear modulus.
These moduli have dimensions of pressure and typical values for several polymeric and non-polymeric materials can be compared at ambient temperatures in table 13.1.
The response of polymers to mechanical stresses can vary widely, and depends on the particular state the polymer is in at any given temperature.
13.4 Mechanical models describing viscoelasticity
A perfectly elastic material obeying Hooke's law behaves like a perfect spring.
The stress-strain diagram is shown in figure 13.2(a), and can be represented in mechanical terms by the model of a weightless spring whose modulus of extension represents the modulus of the material.
The application of a shear stress to a viscous liquid on the other hand, is relieved by viscous flow, and for small values of  s can be described by Newton's law where  is the coefficient of viscosity and is the rate of shear sometimes denoted by .
As stress is now independent of the strain the form of the diagram changes and can be represented by a dashpot which is a loose fitting piston in a cylinder containing a liquid of viscosity .
(Figure 13.2(b).)
Comparison of the two models shows that the spring represents a system storing energy which is recoverable, whereas the dashpot represents the dissipation of energy in the form of heat by a viscous material subjected to a deforming force.
The dashpot is used to denote the retarded nature of the response of a material to any applied stress.
Because of their chain-like structure, polymers are not perfectly elastic bodies and deformation is accompanied by a complex series of long and short range co-operative molecular rearrangements.
Consequently, the mechanical behaviour is dominated by viscoelastic phenomena, in contrast to materials such as metal and glass where atomic adjustments under stress are more localized and limited.
The Maxwell model.
One of the first attempts to explain the mechanical behaviour of materials such as pitch and tar was made by James Clark Maxwell.
He argued that when a material can undergo viscous flow and also respond elastically to a stress it should be described by a combination of both the Newton and Hooke laws.
This assumes that both contributions to the strain are additive so that.
Expressing this as the differential equation leads to the equation of motion of a Maxwell unit
Under conditions of constant shear strain the relation becomes and if the boundary condition is assumed that at zero time, the solution to this equation is where  o is the initial stress immediately after stretching the polymer.
This shows that when a Maxwell element is held at a fixed shear strain, the shearing stress will relax exponentially with time.
At a time the stress is reduced to times the original value and this characteristic time is known as the relaxation time .
The equations can be generalized for both shear and tension and G can be replaced by E. The mechanical analogue for the Maxwell unit can be represented by a combination of a spring and a dashpot arranged in series so that the stress is the same on both elements.
This means that the total strain is the sum of the strains on each element as expressed by equation (13.7).
A typical stress-strain curve predicted by the Maxwell model, is shown in figure 13.3(a).
Under conditions of constant stress, a Maxwell body shows instantaneous elastic deformation first, followed by a viscous flow.
Voigt-Kelvin model.
A second simple mechanical model can be constructed from the ideal elements by placing a spring and dashpot in parallel.
This is known as a Voigt Kelvin model.
Any applied stress is now shared between the elements and each is subjected to the same deformation.
The corresponding expression for strain is
Here is known as the retardation time and is a measure of the time delay in the strain after imposition of the stress.
For high values of the viscosity, the retardation time is long and this represents the length of time the model takes to attain or 0.632 of the equilibrium elongation.
Such models are much too simple to describe the complex viscoelastic behaviour of a polymer, nor do they provide any real insight into the molecular mechanism of the process, but in certain instances they can prove useful in assisting the understanding of the viscoelastic process.
13.5 Linear viscoelastic behaviour of amorphous polymers
A polymer can possess a wide range of material properties and of these the hardness, deformability, toughness, and ultimate strength, are amongst the most significant.
Certain features, such as high rigidity (modulus) and impact strength, combined with low creep characteristics are desirable in a polymer if eventually it is to be subjected to loading.
Unfortunately, these are conflicting properties, as a polymer with a high modulus and low creep response does not absorb energy by deforming easily, hence has poor impact strength.
This means a compromise must be sought depending on the use to which the polymer will be put, and this requires a knowledge of the mechanical response in detail.
The early work on viscoelasticity was performed on silk, rubber, and glass, and it was concluded that these materials exhibited a ' delayed elasticity ' manifest in the observation, that the imposition of a stress resulted in an instantaneous strain which continued to increase more slowly, with time.
It is this delay between cause and effect that is fundamental to the observed viscoelastic response and the three major examples of this hysteresis effect are
(1)
Creep, where there is a delayed strain response after the rapid application of a stress,
(2)
Stress-relaxation (section 13.7) in which the material is quickly subjected to a strain and a subsequent decay of stress is observed, and
(3)
Dynamic response (section 13.9) of a body to the imposition of a steady sinusoidal stress.
This produces a strain oscillating with the same frequency as, but out of phase with, the stress.
For maximum usefulness, these measurements must be carried out over a wide range of temperature.
CREEP
To be of any practical use, an object made from a polymeric material must be able to retain its shape when subjected to even small tensions or compressions over long periods of time.
This dimensional stability is an important consideration in choosing a polymer to use in the manufacture of an item.
No one wants a plastic telephone receiver which sags after sitting in its cradle for several weeks, or a car tyre that develops a flat spot if parked in one position for too long, or clothes made from synthetic fibres which become baggy and deformed after short periods of wear.
Creep tests provide a measure of this tendency to deform and are relatively easy to carry out.
Creep can be defined as a progressive increase in strain, observed over an extended time period, in a polymer subjected to a constant stress.
Measurements are carried out on a sample clamped in a thermostat.
A constant load is firmly fixed to one end and the elongation is followed by measuring the relative movement of two fiducial marks, made initially on the polymer, as a function of time.
To avoid excessive changes in the sample cross section, elongations are limited to a few per cent and are followed over approximately three decades of time.
The initial, almost instantaneous, elongation produced by the application of the tensile stress is inversely proportional to the rigidity or modulus of the material, i.e. an elastomer with a low modulus stretches considerably more than a material in the glassy state with a high modulus.
The initial deformation corresponds to portion OA of the curve (figure 13.4), increment a.
This rapid response is followed by a region of creep, A to B, initially fast but eventually slowing down to a constant rate represented by the section B to C. When the stress is removed the instantaneous elastic response OA is completely recovered and the curve drops from C to D, i.e. the distance.
There follows a slower recovery in the region D to E which is never complete, falling short of the initial state by an increment.
This is a measure of the viscous flow experienced by the sample and is a completely non-recoverable response.
If the tensile load is enlarged, both the elongation and the creep rate increase, so results are usually reported in terms of the creep compliance J(t), defined as the ratio of the relative elongation y at time t to the stress so that
At low loads J(t) is independent of the load.
This idealized picture of creep behaviour in a polymer has its mechanical equivalent constructed from the springs and dashpots described earlier.
The changes a and a correspond to the elastic response of the polymer and so we can begin with a Hookean spring.
The Voigt-Kelvin model is embodied in equation (13.11) and this reproduces the changes b and b.
The final changes c and c represent viscous flow and can be represented by a dashpot so that the whole model is a four element model  figure 13.5.
The behaviour can be explained in the following series of steps.
In diagram (i) the system is at rest.
The stress  is applied to spring E 1 and dashpot  3; it is also shared by E 2 and  2 but in a manner which varies with time.
In diagram (ii), representing zero time, the spring E 1 extends by an amount.
This is followed by a decreasing rate of creep with a progressively increasing amount of stress being carried by E 2 until eventually none is carried by  2 and E 2 is fully extended  diagram (iii).
Such behaviour is described by where the retardation time  R provides a measure of the time required for E 2 and  2 to reach 0.632 of their total deformation.
A considerably longer time is required for complete deformation to occur.
When spring E 2 is fully extended the creep attains a constant rate corresponding to movement in the dashpot  3.
Viscous flow continues and the dashpot  3 is deformed until the stress is removed.
At that time, E 1 retracts quickly along section a and a period of recovery ensues (b).
During this time spring E 2 forces the dashpot plunger in  2 back to its original position.
As no force acts on  3 it remains in the extended state, and corresponds to the non-recoverable viscous flow; region.
The system is then as shown in diagram (v).
In practice, a substance possesses a large number of retardation times which can be expressed as a distribution function where
To the first approximation, this is estimated from a plot of creep compliance against, and is the contribution from viscous flow.
STRESS-STRAIN MEASUREMENTS
The data derived from stress-strain measurements on thermoplastics are important from a practical viewpoint, providing as they do, information on the modulus, the brittleness, and the ultimate and yield strengths of the polymer.
By subjecting the specimen to a tensile force applied at a uniform rate and measuring the resulting deformation, a curve of the type shown in figure 13.6 can be constructed.
The shape of such a curve is dependent on the rate of testing, consequently, this must be specified if a meaningful comparison of data is to be made.
The initial portion of the curve is linear and the tensile modulus E is obtained from its slope.
The point L represents the stress beyond which a brittle material will fracture, and the area under the curve to this point is proportional to the energy required for brittle fracture.
If the material is tough no fracture occurs, and the curve then passes through a maximum or inflection point Y, known as the yield point.
Beyond this, the ultimate elongation is eventually reached and the polymer breaks at B. The area under this part of the curve is the energy required for tough fracture to take place.
EFFECT OF TEMPERATURE ON STRESS-STRAIN RESPONSE
Polymers such as polystyrene and poly (methyl methacrylate) with a high E at ambient temperatures fall into the category of hard brittle materials which break before point Y is reached.
Hard tough polymers can be typified by cellulose acetate and several curves measured at different temperatures are shown in figure 13.7(a).
Stress-strain curves for poly (methyl methacrylate) are also shown for comparison (figure 13.7(b)).
It can be seen that the effect of temperature on the characteristic shape of the curve is significant.
As the temperature increases both the rigidity and the yield strength decrease while the elongation generally increases.
For cellulose acetate there is a transformation from a hard brittle state below 273 K to a softer but tougher type of polymer at temperatures above 273 K. For poly (methyl methacrylate) the hard brittle characteristics are retained to a much higher temperature, but it eventually reaches a soft tough state at about 320 K. Thus if the requirements of high rigidity and toughness are to be met, the temperature is important.
Cellulose acetate meets these requirements if used at 298 K more satisfactorily than when used at 350 K where the modulus is smaller and the ability to absorb energy, represented by the area under the curve, is also lower.
13.6 Boltzmann superposition principle
If a Hookean spring is subjected to a series of incremental stresses at various times, the resulting extensions will be independent of the loading or past history of the spring.
A Newtonian dashpot also behaves in a predictable manner.
For viscoelastic materials the response to mechanical testing is time dependent, but the behaviour at any time can be predicted by applying a superposition principle proposed by Boltzmann.
This can be illustrated by a creep test using a simple Voigt-Kelvin model with a single retardation time  R, placed initially under a stress  o at time t o.
If after times t 1, t 2, t 3,... the system is subjected to additional stresses  1,  2,  3,... then the principle states that the creep response of the system can be predicted simply by summing the individual responses from each stress increment.
Thus if the stress alters continually, the summation can be replaced by an integral, and  n by a continually varying function, so that at time t * when the stress existed, the strain is given by
The principle has been applied successfully to the tensile creep of amorphous and rubber-like polymers, but it is not too successful if appreciable crystallinity exists in the sample.
Graphical representation of the principle is shown in figure 13.8.
13.7 Stress-relaxation
Stress-relaxation experiments involve the measurement of the force required to maintain the deformation produced initially by an applied stress as a function of time.
Stress-relaxation tests are not performed as often as creep tests because many investigators believe they are less readily understood.
The latter point is debatable and it may only be that the practical aspects of creep measurements are simpler.
As will be shown later, all the mechanical parameters are in theory interchangeable, and so all such measurements will contribute to the understanding of viscoelastic theory.
While stress-relaxation measurements are useful in a general study of polymeric behaviour, they are particularly useful in the evaluation of antioxidants in polymers, especially elastomers, because measurements on such systems are relatively easy to perform and are sensitive to bond rupture in the network.
Experimental stress-relaxation technique.
In a stress-relaxation experiment, the sample under study is deformed by a rapidly applied stress.
As the stress is normally observed to reach a maximum as soon as the material deforms and then decreases thereafter, it is necessary to alter this continually in order to maintain a constant deformation or measure the stress that would be required to accomplish this operation.
The apparatus used varies in complexity with the physical nature of the sample, being simplest for an elastomer and becoming more sophisticated when the polymer is more rigid.
One type of experimental set up is shown in figure 13.9.
The sample is fixed in position by means of clamps, one being attached to a spring beam above and the other to an adjustable rod R below.
A stress is applied to the sample by rapidly pulling rod R downwards and clamping it in position.
This causes the beam to bend and the displacement is measured by means of a strain gauge or a differential transformer.
The beam deflection is then fed to a recorder and a trace of stress against time is obtained.
The results are expressed as a relaxation modulus which is a function of the time of observation.
Typical data for polyisobutylene are shown in section 13.14, figure 13.21, where the logarithm of the relaxation modulus log is plotted against log t.
From the curves it can be seen that there is a rapid change in log over a narrow range of temperature corresponding to the glass transition.
Again a simple model with a single relaxation time is too crude, and the stress relaxation modulus is better represented by where is the distribution function of relaxation times.
This is suitable for a linear polymer but requires the additional term E, if the material is crosslinked.
13.8 Dynamic mechanical and dielectric thermal analysis
Non-destructive testing methods are particularly useful for assessing the physical properties of polymeric materials when an understanding of the performance at a molecular level is important.
The foregoing techniques for measuring mechanical properties are transient or non periodic methods and typically cover time intervals of up to 10 6 s.
For information relating to short times, two approaches that have been widely used are dynamic mechanical thermal analysis (DMTA) and dielectric thermal analysis (DETA).
These are both particular kinds of relaxation spectroscopy in which the sample is perturbed by a sinusoidal force (either mechanical or electrical) and the response of the material is measured over a range of temperatures and at different frequencies of the applied force.
From an analysis of the material response it is possible to derive information about the molecular motions in the sample, and how these can affect the modulus, damping characteristics and structural transitions.
Both techniques can be used to probe molecular motions in liquid or solid polymers, but when dielectric spectroscopy is used the relaxation or transition must involve movement of a dipole or a charge displacement if it is to be detected.
Thus while both DMTA and DETA can provide similar information about a sample, they can also be used in a complementary fashion, particularly when trying to identify the molecular mechanism of a particular process and in ascertaining whether or not the group is polar.
13.9 Dynamic mechanical thermal analysis (DMTA)
In DMTA a small sinusoidal stress is imparted to the sample in the form of a torque, push-pull, or a flexing mode, of angular frequency .
If the polymer is treated as a classical damped harmonic oscillator, both the elastic modulus and the damping characteristics can be obtained.
Elastic materials convert mechanical work into potential energy which is recoverable; for example an ideal spring, if deformed by a stress, stores the energy and uses it to recover its original shape after removal of the stress.
No energy is converted into heat during the cycle and so no damping is experienced.
Liquids on the other hand flow if subjected to a stress; they do not store the energy but dissipate it almost entirely as heat and thus possess high damping characteristics.
Viscoelastic polymers exhibit both elastic and damping behaviour.
Hence if a sinusoidal stress is applied to a linear viscoelastic material, the resulting stress will also be sinusoidal, but will be out of phase when there is energy dissipation or damping in the polymer.
Harmonic motion of a Maxwell element.
The application of a sinusoidal stress to a Maxwell element produces a strain with the same frequency as, but out of phase with, the stress.
This can be represented schematically in figure 13.10 where  is the phase angle between the stress and the strain.
The resulting strain can be described in the terms of its angular frequency  and the maximum amplitude  o using complex notation, by where, the frequency is v and.
The relation between the alternating stress and strain is written as where is the frequency dependent complex dynamic modulus defined by
This shows that is composed of two frequency dependent components; is the real part in phase with the strain called the storage modulus, and is the loss modulus defined as the ratio of the component 90 out of phase with the stress to the stress itself.
Hence measures the amount of stored energy and, sometimes called the imaginary part, is actually a real quantity measuring the amount of energy dissipated by the material.
The response is often expressed as a complex dynamic compliance especially if a generalized Voigt model is used.
For a Maxwell model
In more realistic terms, there is a distribution of relaxation times and a continuous distribution function can be derived, if required.
The damping in the system or the energy loss per cycle can be measured from the ' loss tangent ' tan .
This is a measure of the internal friction and is related to the complex moduli by
The onset of molecular motion in a polymer sample is reflected in the behaviour of E and E.
A schematic diagram (figure 13.11) of the variation of E and E as a function of , assuming only a single value for  in the model, shows that a maximum in the loss angle is observed where.
This represents a transition point such as T g, T m or some other region where significant molecular motion occurs in the sample.
The maximum is characteristic of the dynamic method as the creep and relaxation techniques merely show a change in the modulus level.
13.10 Experimental methods
There are three main experimental approaches for measuring the dynamic mechanical properties of a sample,
(a)
free vibration,
(b)
forced vibration  resonance,
(c)
forced vibration  non-resonance
.
The mechanical response is usually determined at low frequencies and over as wide a temperature range as possible and examples of each are described in the following section.
TORSIONAL PENDULUM  FREE VIBRATION
A study of the mechanical damping and shear modulus under free vibration can be made using a torsional pendulum.
The specimen is firmly fixed at one end and the other end is clamped to a disc, with a large moment of inertia, which can move freely.
As the polymer sample should not be under a tensile stress, the suspension wire supporting the disc is passed over a pulley and the weight of the disc and sample are counterbalanced by loading the end.
If the disc is subjected to an angular displacement and then released, the sample will twist backwards and forwards about the vertical axis.
The oscillations stimulated in the sample are picked up by an arm attached to the rigidly fixed end held in torsion bars, and transmitted to a recorder by a linear variable differential transformer.
The sample movements are traced as a series of oscillations whose frequency is a function of the physical state of the sample.
The period of oscillation P is taken as the distance between adjacent maxima or minima and the amplitude A is a measure of the height from one minimum to the preceding maximum.
The exponential decay of the amplitude along the axis provides an indication of the mechanical damping.
At a temperature the sample absorbs most of the energy and damping is high whereas at a much lower temperature the material tends to store the energy and mechanical damping is considerably lower.
A quantitative measure of the damping is provided by the logarithmic decrement defined as the logarithmic decrease in amplitude per cycle.
It is calculated from the ratio of amplitudes of any two successive oscillations using the relation
The shear modulus can also be derived from the data, being inversely proportional to the square of the period, where K is a factor depending on the shape and the size of the sample and I is the polar moment of inertia.
The method can cover the complete range of moduli encountered in polymeric systems but is confined to a relatively narrow frequency range of 0.01 to 10 Hz.
VIBRATING REED RESONANCE
For resonance forced vibration measurements a sample in the form of a thin strip is clamped firmly at one end leaving the other end free.
The clamped end of the system is then vibrated laterally at a given frequency v and the amplitude of the vibration induced at the free end of the sample is recorded.
A range of frequencies wide enough to ensure that it encompasses the resonant frequency of the sample v r is then examined.
The resonant frequency is detected as the maximum of a graph of amplitude against frequency.
The results provide information on the elastic modulus E since it is related to the square of the resonance frequency by where c is a numerical constant, L is the free length of the sample, D is its thickness, and  is the sample density.
If the amplitudes are expressed as ratios of the amplitude to the maximum amplitude, then damping is measured from the half-width h of the curve, i.e.
This technique is not as useful as the torsional pendulum but covers the higher frequency range 10 to 103 Hz.
FORCED VIBRATION NON-RESONANCE
Several types of instrument can be used for this type of test, and these are usually limited to measurements on rigid polymers or rubbers.
One such instrument is shown in the block diagram 13.14.
The sample C is attached firmly at each end to a strain gauge; one of these is a force transducer measuring the applied sinusoidal force and the other records the sample deformation.
A sinusoidal tensile stress of a given frequency can be generated in the vibrator A and if the electrical vectors from the force and displacement are represented by and then by satisfying the condition the tangent of the phase angle  between the stress and the strain may be calculated from This operation of adjustment followed by subtraction of the electrical vectors is performed directly in the recording circuit.
The complex elastic modulus E * is given by where F is the amplitude of the tensile force, A is the sample cross-sectional area, L is the sample length, and L is the amplitude of elongation.
Tensile storage and loss moduli E and E follow from and.
A second version now widely used for these measurements is the Polymer Laboratories DMTA instrument and a schematic diagram of the working head is shown in figure 13.15.
Several damping arrangements are available for the sample so that measurements may be made in the bending, shear, or tensile, modes.
In the bending mode the sample, in the form of a small bar, is clamped firmly at both ends and the central point is vibrated by means of a ceramic drive shaft.
This can be driven at frequencies selected from the range 0.01 to 200 Hz.
The applied stress is proportional to the a.c. current fed to the drive shaft and the strain is detected using a transducer that measures the displacement of the drive clamp.
Temperature can be controlled over the range 120 to 770 K, either isothermally or more normally by ramping up and down at various fixed rates.
13.11 Correlation of mechanical damping terms
The several practical methods described express the damping and moduli in slightly different forms but these can all be interrelated quite simply.
In general, one can select a dissipation factor or loss tangent derived from the ratio or to represent the energy conversion per cycle.
This leads to the equivalent forms and
To a first approximation it is also possible to write thereby allowing use of the data from either type of measurement to characterize the sample.
It should also be noted that if complex moduli are used the corresponding complex compliances are given by.
Moduli can also be related to the viscosity, and where  is known as the dynamic viscosity.
The approximations and can be made when damping is low, and the absolute value for the modulus |G| or |E| can be related to the complex components by.
A similar expression holds for |G|.
13.12 Dielectric thermal analysis (DETA)
Dry polymers are very poor conductors of electricity and can be regarded as insulators.
Application of an electric field to a polymer can lead to polarization of the sample, which is a surface effect, but if the polymer contains groups that can act as permanent dipoles then the applied field will cause them to align in the direction of the field.
When the electric field is released, the dipoles can relax back into a random orientation, but, due to the frictional resistance experienced by the groups in the bulk polymer this will not be instantaneous.
The process of disordering can be characterized by a relaxation time, but may not be easily measured.
It is more convenient to apply a sinusoidally varying voltage to the sample and to study the dipole polarization under steady state conditions.
In DETA a small alternating electric field is applied to the sample and the electric charge displacement Q is measured by following the current.
The complex dielectric permittivity  * can be measured from the change in amplitude and, if the phase lag between the applied voltage and the outcoming current is determined (see figure 13.16), then  * can be resolved into the two components š, the storage (dielectric constant) and š, the loss (dielectric loss).
The frequencies used in the measurements must now be in the range where orientational polarization of the dipoles in the polymer is active.
These frequencies are much higher than used normally for DMTA and typically lie in the range 20 Hz to 100 kHz.
While the main variable is temperature, the factors š and š can be studied as a function of the angular frequency , and in the frequency region where there is a relaxation, š decreases as shown in figure 13.17.
The magnitude of this decrease is a measure of the strength of the molecular dipole involved in the relaxation, where  o is the static dielectric constant related to the actual dipole moment of the polymer and š is the dielectric constant measured at high frequencies.
When the dielectric loss factor is measured at a characteristic frequency  max and a given temperature, it passes through a maximum when a relaxation occurs, and the dipole relaxation time, can be obtained.
At frequencies above  max the dipoles can not move fast enough to follow the alternating field so both š and š are low.
When the frequency is lower than  max the permanent dipoles can follow the field quite closely and so š is high because the dipoles align easily with each change in polarity; š on the other hand is low again because now the voltage and the current are approximately 90 out of phase.
Dielectric relaxation processes can be described formally by the following relations: and
A useful way of examining the data is to measure the ratio of the two factors, which gives the dielectric loss tangent
Dipolar groups in a polymer coil may not all be able to relax at the same speed because of the variable steric restrictions they may experience, imposed by their environment.
This can be caused by the disordered packing of chains in the amorphous glassy phase, and a random distribution of the available free volume, or perhaps even by the random coil structure of the chain itself causing local environmental changes.
The result is that a distribution of relaxation times is to be expected for a given process and this results in a broadening of the dielectric loss peak.
Thus, the more mobile a dipolar group, the easier it is for it to follow the electric field up to higher frequencies, whereas the less mobile groups can only orient at lower frequencies.
13.13 Comparison between DMTA and DETA
Data from mechanical and dielectric measurements can be related, certainly in a qualitative, if not always in a quantitative way.
Formally, the dielectric constant can be regarded as the equivalent of the mechanical compliance, rather than the modulus, and this highlights the fact that mechanical techniques measure the ability of the system to resist movement, whereas the dielectric approach is a measurement of the ability of the system to move, given that the groups involved must also be dipolar.
Interestingly, the dielectric loss appears to match the loss modulus more closely than the loss compliance when data are compared for the same system.
Both techniques respond in a similar fashion to a change in the frequency of the measurement.
When the frequency is increased, the transitions and relaxations that are observed in a sample appear at higher temperatures.
This is illustrated from work on poly (ethylene terephthalate) where the loss peak representing the glass transition has been measured by both DMTA and DETA at several frequencies between 0.01 Hz and 100 kHz (figure 13.18).
The maximum of this loss peak is seen to move from a temperature of about 360 K (0.01 Hz) to about 400 K (100 kHz), which is an increase of 40 K over a frequency change of seven orders of magnitude.
This is close to the rule of thumb that the temperature for the maximum of a loss peak (or a relaxation process) will change by approximately 7 K for each decade of change in frequency.
This type of measurement can be used to estimate the activation energy for a transition or relaxation process, if the frequency v, at T max is expressed as a function of reciprocal temperature according to the relation
Data plotted using equation (13.33) for the -relaxation process in a series of poly(alkylmethacrylate)s are shown in figure 13.19.
Both techniques have been used and separately give good straight lines with the same slope, but the fact that the lines do not overlap precisely indicates that the measurements may not be exactly equivalent.
The results from DMTA and DETA can be used in a complementary manner to distinguish between relaxations involving polar and non polar units relaxing in the polymeric system.
Thus if the response of poly (ethylene terephthalate) to both DMTA and DETA is examined, two major loss peaks can be identified in each, as seen in figure 13.20.
The high temperature loss peak (-peak) can be assigned to the glass transition and this can be confirmed by d.s.c.
measurements.
There is a second loss (-relaxation) which appears at lower temperatures and suggests that there is a relaxation process active in the glassy state.
It is not immediately obvious which group is responsible for this process, but it is active both mechanically and dielectrically.
Examination of the polymer structures suggests that the relaxation in the glass could involve libration of the phenyl ring, motion of the oxycarbonyl unit or rearrangement of the (-O-C-C-O-) unit.
From the spectra it can be seen that the intensity of the -peak relative to the -peak is much stronger in the dielectric response compared with the mechanical measurements.
This suggests that the group undergoing relaxation is associated with a dipole moment and thus rules out the phenyl ring libration as a likely process.
This does not give irrefutable evidence of the participation of the oxycarbonyl unit but it does point in this direction.
13.14 Time-temperature superposition principle
A curve of the logarithm of the modulus against time and temperature is shown in figure 13.21.
This provides a particularly useful description of the behaviour of a polymer and allows one to estimate, among other things, either the relaxation or retardation spectrum.
The practical time scale for most stress-relaxation measurements ranges from 10 1 to 10 6 s but a wider range of temperature is desirable.
Such a range can be covered relatively easily by making use of the observation, first made by Leaderman, that for viscoelastic materials time is equivalent to temperature.
A composite isothermal curve covering the required extensive time scale can then be constructed from data collected at different temperatures.
This is accomplished by translation of the small curves along the log t axis until they are all superimposed to form a large composite curve.
The technique can be illustrated using data for polyisobutylene at several temperatures.
An arbitrary temperature T o is first chosen to serve as a reference which in the present case is 298 K. As values of the relaxation modulus have been measured at widely differing temperatures, they must be corrected for changes in the sample density with temperature to give a reduced modulus, where  and  o are the polymer densities at T and T o respectively.
This correction is small and can often be neglected.
Each curve of reduced modulus is shifted with respect to the curve at T o until all fit together forming one master curve.
The curve obtained at each temperature is shifted by an amount
The parameter a  is the shift factor and is positive if the movement of the curve is to the left of the reference and negative for a move to the right.
The shift factor is a function of temperature only and decreases with increasing temperature, it is, of course, unity at T o.
The superposition principle can also be applied to creep data.
Curves exhibiting the creep behaviour of polymers at different temperatures can be compared by plotting against log t.
This reduces all the curves at various temperatures to the same shape but displaced along the log t axis.
Superposition to form a master curve is readily achieved by movement along the log t axis, where the shift factor a  has the same characteristics as for the relaxation data.
This shift factor has also been defined as the ratio of relaxation or retardation times at the temperatures T and T o i.e. and is related to the viscosities.
If the viscosities obey the Arrhenius equation, then by neglecting the correction factor, we can express a  in an exponential form as or where b is a constant.
This equation is very similar in form to the WLF equation,
For polyisobutylene, the shift factor a  can be predicted if is used with and.
As outlined in chapter 12, the reference temperature is often chosen to be T g with and, from which a  can be calculated for various amorphous polymers.
The superposition principle can be used to predict the creep and relaxation behaviour at any temperature if some results are already available, with the proviso that the most reliable predictions can be made for interpolated temperatures rather than long extrapolations.
The principle can also be applied to dielectric data which can be shifted either along the temperature or the frequency axis.
An example of the latter type of shift is shown in figure 13.22, where instead of time dependence measurements the frequency dependence of the -relaxation in poly (vinyl acetate) has been studied at fixed temperatures in the range 212 to 266 K. A master curve can be constructed for this relaxation region by plotting against, where the ' max ' subscript refers to the peak maximum at each experimental temperature.
13.15 A molecular theory for viscoelasticity
So far the interpretation of viscoelastic behaviour has been largely phenomenological, relying on the application of mechanical models to aid the elucidation of the observed phenomena.
These are, at best, no more than useful physical aids to illustrate the mechanical response and suffer from the disadvantage that a given process may be described in this way using more than one arrangement of springs and dashpots.
In an attempt to gain a deeper understanding on a molecular level, Rouse, Zimm, Bueche, and others have attempted to formulate a theory of polymer viscoelasticity based on a chain model consisting of a series of sub-units.
Each sub-unit is assumed to behave like an entropy spring and is expected to be large enough to realize a Gaussian distribution of segments (i.e.&gt; 50 carbon atoms).
This approach, although still somewhat restrictive has led to reasonable predictions of relaxation and retardation spectra.
One starts with a single isolated chain and the assumption that it exhibits both viscous and elastic behaviour.
If the chain is left undisturbed it will also adopt the most notable conformation or segmental distribution, so that, with the exception of high frequencies, the observed elasticity is predominantly entropic.
Thus the application of a stress to the molecule will cause distortion, by altering the equilibrium conformation to a less probable one, resulting in a decrease in the entropy and a corresponding increase in the free energy of the system.
When the stress is removed the chain segments will diffuse back to their unstressed positions even though the whole molecule may have changed its spatial position in the meantime.
If on the other hand, the stress is maintained, strain relief is sought by converting the excess free energy into heat, thereby stimulating the thermal motion of the segments back to their original positions.
Stress relaxation is then said to have occurred.
For a chain molecule composed of a large number of segments, movement of the complete molecule depends on the co-operative movement of all the segments, and as stress-relaxation depends on the number of ways the molecule can regain its most probable conformation, each possible co-ordinated movement is treated as a mode of motion with a characteristic relaxation time.
For simplicity we can represent the polymer as in figure 13.23.
The first mode p = 1 represents translation of the molecule as a whole and has the longest relaxation time  1 because the maximum number of co-ordinated segmental movements are involved.
The second mode p = 2 corresponds to the movement of the chain ends in opposite directions; for p = 3, both chain ends move in the same direction, but the centre moves in the opposite direction.
Higher modes 4, 5... m follow involving a progressively decreasing degree of co-operation for each succeeding mode and correspondingly lower relaxation times  p.
This means that a single polymer chain possesses a wide distribution of relaxation times.
Using this concept, Rouse considered a molecule in dilute solution under sinusoidal shear and derived the relations where  and  s are the viscosities of the solution and the solvent respectively, n is the number of molecules per unit volume, k is the Boltzmann constant, and  is the angular frequency of the applied stress which is zero for steady flow.
These equations are strictly applicable only to dilute solutions of non-draining monodisperse coils, but can be extended to undiluted polymers above their glass temperature if suitably modified.
This becomes necessary when chain entanglements begin to have a significant effect on the relaxation times.
The undiluted system is represented as a collection of polymer segments dissolved in a liquid matrix composed of other polymer segments and  s can be replaced by a monomeric frictional coefficient  o.
This provides a measure of the viscous resistance experienced by a chain and is characteristic of a given polymer at a particular temperature.
The continuous relaxation and retardation spectra calculated from the Rouse theory are and where is the unperturbed mean square end-to-end distance of a chain of molar mass M and density  containing N monomer units.
The equations predict linearity in the plots and against with slopes of-? and +? respectively.
Comparison with experimental results for poly (methyl acrylate) shows validity only for longer values of the relaxation and retardation times.
The Rouse model only pertains to the region covering intermediate  values.
The reason for this lies in the response of a polymer to an alternating stress.
At low frequencies Brownian motion can relieve the deformation caused by the stress before the next cycle takes place, but as the frequency increases the conformational change begins to lag behind the stress and energy is not only dissipated but stored as well.
Finally at very high frequencies only enough time exists for bond deformation to occur.
As it was stipulated that each segment be long enough to obey Gaussian statistics, short relaxation times may not allow a segment sufficient time to rearrange and regain this distribution.
Thus the contribution from short segments to the distribution functions tends to be lost and deviations from the theoretical represent departure from ideal Gaussian behaviour.
This approach to viscoelastic theory is reasonably successful in the low modulus regions but it requires considerable modification if the high modulus and rubbery plateau regions are to be described.
