Shear melting of a colloidal glass

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Particle motion diagram from Eisenmann paper.
Figure 1. Dynamic clusters for a low volume fraction liquid (a) and a sheared glass (b). Full circles represent particle initial positions and open circles are final positions after a shear strain <math>\gamma = 0.15</math>. Red/blue: moving in +/- y-direction. Green/orange: moving in +/- x-direction. Shear is in the y-direction.

Confocal microscopy is used to investigate shear melting of colloidal glasses at the microscopic level. A phase transition occurs at a strain of about 8%, as this is when shear melting begins. For larger shear strain than this, cooperative motions of groups of particles can be found, where the average group size is about 3 particles. It is also evident that diffusive behavior of the particles is driven by shear for large strains. In particular, it is found that the effective diffusion coefficient is linearly proportional to shear rate, and this can be explained mathematically with a modified form of the Stokes-Einstein model of diffusion in which thermal energy is replaced by shear energy.

General Information

Keywords: colloid, glass, shear melting, phase transition

Authors: Christoph Eisenmann, Chanjoong Kim, Johan Mattsson, and David Weitz.

Date: February 4, 2009.

Departments of Physics and HSEAS, Harvard University, Cambridge, Massachusetts 02138, USA

Preprint (2009). [1]


A colloidal glass of poly(methyl methacrylate) particles with an average diameter of 1.2 microns is used to study the effects of shear melting. The particles are sandwiched between two parallel glass plates in a specially designed air-tight shear cell. The particles are fluorescently labeled so that their positions can be tracked using a confocal microscope when shear is applied. The position of each particle is tracked in two dimensions and its position is determined after subtracting the mean displacement due to the shear applied.

First data set from Eisenmann paper.
Figure 2. (a) Mean squared displacement (MSD) versus <math>\gamma</math> for different strain rates <math>\dot{\gamma}</math>. The inset shows the distribution function of particle displacement for <math>\gamma = 0.08</math> (non-Gaussian, red squares) and <math>\gamma = 1</math> (Gaussian, blue circles). (b) MSD versus <math>\gamma</math>, calculated over time intervals after the onset of diffusive behavior. The similarity to (a) indicates that there is no critical strain for this system. (c) Normalized non-Gaussian parameter <math>\alpha_2</math> as a function of strain. (d) Normalized non-Gaussian parameter <math>\alpha_2</math> as a function of strain, computed using the data in (b).

For strains <math>\gamma < 0.08</math>, subdiffusive behavior is observed, consistent with the normal behavior of a colloidal glass. In this regime, the particle motion is highly heterogeneous. The system transitions to the diffusive regime for larger shears, however, suggesting that for strains larger than 0.08, shear melting occurs, and so this quantity corresponds to a glass transition for the colloid system. The particle motion is more homogeneous for these larger strains (see Figure 2 a, b). The fact that the non-Gaussian parameter <math>\alpha_2</math> is maximum at <math>\gamma \approx 0.08</math> (see Figure 2c, d) provides further evidence of a phase transition at this quantity of shear.

To quantify the range of cooperative motions in the sheared sample, the authors examine particles that move farther than some threshold distance <math>\delta</math> within a time interval <math>\Delta t</math> for a given strain <math>\gamma</math>. They discover that there are clusters of particles that move together in the shear melted glass, and these clusters are larger than those corresponding to the unsheared low volume fraction liquid. On average, the dynamic cluster size is about 2-3 particles, and this size does not appear to vary with shear-rate at all. Furthermore, some clusters move parallel to the applied strain while others move perpendicular to it.

These measurements allow the authors to measure the shear-induced effective diffusion coefficient for the particle motion at a given shear rate <math>\dot{\gamma}</math>. From linear fits to the data, the long-time diffusion constant can be extracted, as

<math>\langle \Delta x^2 (t) \rangle = 2Dt</math> .

To explain why the long-time diffusion constant may be linearly proportional to shear rate, one can consider the Stokes-Einstein equation for the diffusion coefficient of an equilibrium fluid,

<math>D = \frac{k_B T}{6 \pi \eta R}</math> .

If the thermal energy is replaced by the shear energy,

<math>E_{\gamma} = \frac{4 \pi}{3} R_s^3 \eta \dot{\gamma}</math> ,

where <math>R_s</math> is the radius of the characteristic volume element or cluster size of 3 particles, then the equation for the diffusion coefficient becomes:

<math>D = \frac{2}{9} R_s^2 \dot{\gamma}</math> ,

and one can see that D should be proportional to shear rate. This raises an interesting possibility: "shear can be regarded as an 'effective temperature'" in a shear melted colloid system. Additional research will be necessary to understand to what extent this notion can be generalized.

Connection to soft matter

This research examines a fundamental problem in soft matter: When can structured fluids be characterized as having properties of a solid (non-diffusive) and when do they have properties of a melt, with a diffusion coefficient? It is shown that various metrics can be used which all indicate that a shear strain of about 0.08 is the critical value for this phase transition in colloidal glasses. However, in some ways it is not a well-defined phase transition because the 0.08 strain can be measured relative to any given start time, including those for which sufficient strain has already occurred such that the system should be in the diffusive regime.

Moreover, the authors show that the particles in the shear melted glass tend to flow in clusters with a characteristic average size. By accounting for this cluster size, they demonstrate that an effective diffusion coefficient can be calculated. However, to calculate it, the thermal energy in the Stokes-Einstein equation must be replaced with the shear energy, expressed as a function of the cluster size. After this substitution, it can be seen that, as one might expect, the diffusion coefficient is proportional to the shear-rate <math>\dot{\gamma}</math>. This has fascinating potential implications for other problems in soft matter; if shear energy can be thought of as a thermal energy, then perhaps shear plays the role of an effective temperature in a shear melting system. Additional research will be necessary to understand the implications of this assertion.