Viscoelastic properties of oxide-coated liquid metals
I. INTRODUCTION
Small droplets of simple liquids tend to be spherical because this shape minimizes their surface area and surface free energy. Liquid droplets, and even bubbles, can deviate from this behavior when solids are incorporated onto their surface, thereby "solidifying" the liquid surface Subramaniam et al. 2005; Xu et al. 2005. A similar realization of this principle occurs when the outer surface of a low-viscosity liquid metal is solidified, in atmosphere, by an oxide?
To quantify the effect of oxidation on material properties, previous researchers have focused on surface tension measurements Eustathopoulos and Drevet 1998; Ricci et al. 2005. Such measurements have limited usefulness for understanding the mechanical properties of the oxide skins in air, as demonstrated by recent experiments of flow of liquid metal in microchannels Dickey et al. 2008. The liquid-like characteristics of eutectic gallium indium EGaInwere most apparent when the material was injected at constant pressure into a micro fluidic channel that became progressively narrower. The narrowing of the channel caused the meniscus of the EGaIn to stop at a position that depended on the applied pressure. Advancing the meniscus further required higher applied pressure. A force balance between the pressure and the surface stress in the static meniscus revealed the maximum surface stress DY that the oxide skin could support without flowing. For EGaIn, DY =0.630 N/m, a value that is in excellent agreement with a separate measurement of an apparent surface tension, in atmosphere, obtained using the pendant drop method 0.624 N/mZrnic and Swatik 1969. This result implies that for this particular case, EGaIn can be characterized in terms of surface tension, a liquid-like characteristic. In this respect, EGaIn is not unlike mercury Hg, a low-viscosity liquid metal that does not oxidize in atmosphere: measurements of DY obtained with Hg in the micro fluidic device also agree with literature values of its surface tension 0.450 and 0.480 N/m, respectively Dickey et al. 2008. However, despite the similarity in how these two materials flow into microchannels, EGaIn and Hg exhibited qualitatively different behavior when the pressure was removed.
II. EXPERIMENTAL
Our experiments were conducted using three types of stress-controlled rheometers: the AR-G2 TA Instruments, the C-VOR, and the Bohlin Gemini HRnano rheometer both from Malvern Instruments. The rheometers were equipped with parallel-plate geometries of different size. In this configuration, the bottom plate is stationary and the upper plate is rigidly attached to a low-friction bearing. The upper plate is rotated by a motor that applies a torque to the bearing. When the sample is loaded between the two plates, the stress in the sample is assumed to be proportional to, and the strain is assumed to be proportional to the bearing displacement. In most of the experiments with the AR-G2, the upper plate material was made of hard anodized aluminum, and a homemade stainless steel plate covered the lower plate. In the case of the Malvern rheometers, both the upper and lower plates consisted of stainless steel. Our results did not appear to depend significantly on the choice of the materials used.
RESULTS
These measurements were performed on EGaIn using the HRnano equipped with parallel plates of 20 and 40 mm diameter. The rheometer performed strain oscillations at =1 rad/s of first ascending and then descending strain amplitude 0 and calculated apparent linear viscoelastic moduli G and G. We performed these experiments four times for each plate radius, with each test preceded by a strain-rate-controlled pre-shear (&plusmn;1.3 or &plusmn;13 s-1), applied for 5 min with no equilibration time before the oscillatory strain measurements.
B. Steady and oscillatory shear
The elastic characteristics of the oxide skin are highly dependent on the shear history. Because the shear-history dependence can change GS by more than an order of magnitude, it can potentially lead to large discrepancies between different measurements of the mechanical properties of the oxide skin. These strain history effects might be present in other configurations that are spherically designed to measure surface rheology. To control these effects during rheological measurements, it is important to understand the mechanisms that cause them. We seek to better understand the difference between the stiff and soft states, by first considering the conditions under which both states were measured. Analysis of the strain oscillations reveals that when the material is in the stiff state, significant internal stresses from the pre-shear are stored in the oxide film. These stresses vanish during the large oscillations and are not present in the soft state, suggesting that the internal stress plays an important role in determining GS.
The string will minimize its stretching by tracing the path of shortest distance between the attachment points of the string. If this path is projected onto one of the plates, it will not follow the arc of the circles but will be a chord see Fig.7. If the same parallel plates are now connected by a thin elastic membrane attached to the outer rims of the plates, then relative rotation of the plates will cause the membrane to stretch. If the stretched elastic elements of the membrane are modeled as a series of elastic strings regularly spaced around the circumference of the plates, then upon shearing, the strings will trace the path of shortest distance between their end points and the membrane will bow inward.
The tendency of the membrane to bow inward when sheared tends to decrease the volume enclosed by the membrane and the parallel plates. However, if this space is occupied by an incompressible fluid, then the fluid can accommodate the bowing of the outer membrane by slightly displacing the parallel plates away from each other. In our experiments, this displacement is detected by the strain gauges. The relaxation of the tension in the elastic strings is accompanied by a decrease in the measured pressure p of the bulk.
The stress relaxation experiment is similar to the strain-controlled oscillations that we performed using two different plate size with the HRnano. The oscillations were performed immediately following a pre-shear and without allowing the material to relax. In the oscillatory experiments, the steady stress SE vanished when 0 became sufficiently large, which occurred after about 15 min of oscillations. However, in the stress relaxation experiment in which no oscillations were performed, the steady stress persisted for several hours, in both Ga and EGaIn. This result indicates that the large oscillations do indeed play an essential role in causing SE to vanish during the oscillatory tests.
E. Time-dependence
Our analysis is performed on the raw angular displacement and normal force data that are provided by the rheometer during the equilibration period at a rate of 250 data points per second. We convert these values to and p, respectively, and observe that immediately following the cessation of the pre-shear, the rheometer reverses direction and exhibits damped ringing about a new equilibrium position. The ringing is due to a combination of sample elasticity and rheometer inertia, and the damping occurs because of viscous dissipation in the sample. The slow elastic recoil that follows the pre-shear persists for the entire equilibration period, even when the equilibration is as long as 1 h. The recoil indicates that the elastic stresses that were stored during the preceding pre-shear are slowly relaxing. Because p is an increasing function of S, we expect that the relaxation of S should correspond to a decrease in p. We test this hypothesis by binning the raw measurements of p obtained from the rheometer into increments that are linearly spaced in time and averaging all the p values within each bin. We then plot the average p value from each bin as a function of the average time of each bin.
Free oscillations typically damp out until the material effectively arrests at an equilibrium strain. However, a static equilibrium strain was not observed in our samples. At times greater than about 5 s, the oscillations did not damp out completely but rather continued to vibrate throughout the equilibration time at a small amplitude that varied in time. At times, the amplitude of the vibrations gradually increased, and at other times it gradually decreased, as is apparent in Fig. 10a, and in detail in Fig. 13a. These small oscillations occur because the sample exhibits a combination of low viscosity and high elasticity that makes it sensitive to minor perturbations from either the environment or the instrument itself.
The time-dependence of GSduring the equilibrium period indicates that a sample in the stiff state does not evolve to the soft state when the applied stress is zero. This observation, however, does not apply to relaxation that occurs under constant strain conditions. At the conclusion of the stress relaxation experiment, p had descended to 120 Pa, not far from the value of about 150 Pa that is associated with soft state. Although we did not perform controlled oscillations following the stress relaxation, we were able to collect about 9 s of passive strain vibrations from raw data that was taken less than 2 min after the conclusion of the stress relaxation, and immediately preceding the onset of the pre-shear associated with the following oscillatory tests. A Fourier transform of these vibrations indicates that G'S=1.70&plusmn;0.1 N/m and p=-146 Pa. These values are representative of the soft state and indicate that the soft state can be obtained by allowing stress relaxation to occur under constant strain conditions.
IV. DISCUSSION
Our results suggest the possibility of rationalizing the mechanism responsible for the difference between the stiff and soft states in terms of our simple visualization of Fig. 7 that treats the membrane as a series of elastic strings. When the plates are rotated and the strings are stretched to their maximum extension about =0.02, they exert a positive pressure on the bulk liquid. If the plates are maintained at this angular displacement, the strings can only relax their tension by increasing their equilibrium length. In physical terms, this may correspond to additional oxidation that replaces strained oxide elements with unstrained elements. Such oxidation would likely increase the surface area of the oxide film. When the torque that was stretching the film is subsequently set to zero, the film will be less taut because it has a larger surface area. The more "baggy" or wrinkled film will exert less pressure on the bulk liquid. The wrinkled film might also exhibit a lower apparent elastic modulus because it can accommodate deformation simply by changing its shape rather than straining the oxide skin. Because of this, we expect that apparent elasticity measured in the soft state does not represent the intrinsic properties of the material but is dependent on the particular shape that the material has achieved due to the combination of both stress and oxidation in its strain history.
V. CONCLUSIONS
The dependence of our measurements on both time and strain history have the potential to significantly complicate elasticity measurements. However, we have shown that, at least for the parallel plate geometry, the state of the material can be reset through a shear rejuvenation step that returns GS to an average value that is fairly independent of time. The mechanism by which shear can bring about these effects is not clear and will hopefully be elucidated by future measurements.
We have rationalized both the time-dependant increase and decrease in GS in terms of additional oxidation of the oxide film. Oxidation may decrease the apparent elasticity by increasing the surface area of the oxide film thereby increasing its "bagginess." However, oxidation might also reinforce the oxide film, thereby increasing the apparent elasticity. If oxidation truly is responsible for these effects, then we expect that these effects may also be present in other geometries that are typically used to characterize surface rheology.
Although our experiments were performed only on liquid metals, similar behavior may extend to a wider range of complex fluids consisting of liquids coated by solid-like shells. These results show that even in a simple geometry, under simple shear, solid-coated liquids can display complex mechanical behavior. A better understanding of the mechanisms governing the properties of solid-coated liquids will facilitate their rheological characterization, as well as the use of these materials in advanced technologies.