Slip, yield, and bands in colloidal crystals under oscillatory shear
Original entry by Joerg Fritz, AP225 Fall 2009
Itai Cohen, Benny Davidovitch, Andrew B. Schofield, Michael P. Brenner, and David A. Weitz: Physical Review Letters, 2006, 97, pp 215502-1 to 215502-4
Rheological measurments for complex fluids are notoriously difficult. One reason for this is the formation of shear bands where the material separates into bands with significantly different strain and flow rates. The paper describes a new experimental approach to this problem where confocal microscopy is used to investigate the behavior of colloidal crystals under oscillatory shear. For large shear rates the suspension forms shear bands that exhibit a harmonic response to the applied forcing. This creates an interesting theoretical problems. Nonlinear models of rheology, which are usually used to describe shear bands are ruled out by this observation. Instead the authors present a linear model that accounts for all the phenomena seen in the experiments.
The study uses a dense suspension of polymethyl methacrylate particles marked with rhodamine dye and stabilized by a hydrostearic acid suspended in a mixture of cyclohexyl bromide and decalin. The mixture can be chosen in such a way as to match particle density and index of refraction, which makes 3-dimensional imaging of the dyed colloids possible. This suspension is contained in a shear cell which simple consists of a movable microscope slide and a fixed glass plate. This setup allows to control the suspension thickness, the shear frequency and the strain amplitude.
After preparing the samples to create a uniform random crystal (rhcp) by shearing it at a selected frequency for over an hour measurments were started. The maximal particle displacement u was measured at different heights z above the oscillating plate, for a large range of frequencies and strain rates. Figure 1 shows a plot for two different strain rates where frequency is used as a parameter and varied from 0.02 to 60 Hz.
The plots show quite surprisingly that for higher frequencies a region close to the upper static plate exhibits a significantly larger strain than that near the lower oscillating plate. We have a very nice visual representation of shear banding. A comparison of both plots shows that for increased strain rate the upper region gets bigger until the high strain region stretches the full domain.
Another interesting behavior can be observed at the top plate (<math>z=0</math>). The displacement has been normalized by the amplitude of the cover slip, but th measured displacements never reach 1, which indicates slip at this surface for all frequencies and strain rates.
Since normal explanations for the formation of shear banding rely on non-linear characteristics of the materials the author also test for any aharmonic contributions for the variation of the displacement field over time for one selected position z. Figure 2 shows the results at three different positions between the two plates. Surprisingly, the displacements in all three regions can be fitted perfectly to a harmonic oscillation, thus making an explanation along the lines of non-linear material characteristics impossible.
A linear model
These observations motivate the development of a model that is linear in its response to forcing. We can note that at rest the suspension forms a rhcp crystal. This phase occupies the entire gap for very small strains. Previous studies have shown that near equilibrium the crystal stress can be modeled as a sum of viscous and elastic components with effective viscosity and shear modulus. The authors extend this idea to still be valid in the initially larger area when the shear band begin to form. In the second phase, that only appears for larger strains, the colloidal sheets flow freely over each other, which can be interpreted as a vanishing shear modulus. This allows to model this phase as a Newtonian fluid with effective viscosity. For a given frequency the stress in both phases is linearly proportional to the strain. This automatically creates a harmonic response to oscillatory displacements, which is consistent with the flow patters that can be observed in figure 2.
This model can be used to calculate a general non-dimensional displacement field as a function of position. By using realistic parameters the solid lines in figure 1 can be generated. The agreement with the experimental results is quite striking, indicating that the model used by the authors in fact captures all of the colloid behavior exhibited in this experiment.