# Shear Melting of a Colloidal Glass

Entry: Chia Wei Hsu, AP 225, Fall 2010

Christoph Eisenmann, Chanjoong Kim, Johan Mattsson, and David Weitz, Phys. Rev. Lett. **104**, 035502 (2010)

## Background

Colloidal suspensions serve as a model system for glass transition. A colloidal glass exhibit cooperativity and dynamic heterogeneity that are typically seen in glass transitions, as well as its own unique properties such as shear melting. Because of the large particle size, colloids form soft solids and can be fluidized through shear. In order to understand the relationship of shear-induced melting to others (such as melting by increasing temperature or reducing volume fraction), we need to understand the microscopic behavior of shear-melting.

## Experiment

The authors study sterically stabilized poly(methylmethacrylate) particles (average radius <math>R=0.6\mu m</math>) suspended in a mixture of cis-decalin and cycloheptylbromide (which matches the particle density and index of refraction) at a volume fraction of <math>\phi=0.61\pm0.03</math>. The colloid particles are fluorescently labeled, and the suspension is contained between two parallel glass plates <math>40 \mu m</math> apart in a shear cell.

A symmetrical triangular time-dependent strain is applied in the <math>y</math> direction with strain amplitude up to <math>\gamma \approx 0.5</math> and strain periods between 25 and 100 <math>s</math>. At such shear rate, particle motions are dominated by the imposed shear (as opposed to diffusion). The particle positions <math>x(t)</math> and <math> y(t)</math> are tracked with confocal microscopy during shear, with data collected 2 frames per second. Then the mean displacement due to shear is subtracted.

## Results

The authors first calculate an effective mean-squared-displacement (MSD), <math>\langle\Delta x^2(\Delta t)\rangle_0=\langle ( x ( t_0 + \Delta t)-x(t_0))^2\rangle</math>, where<math>t_0</math> is the initial time of a half strain-cycle. MSD is compared for different values of <math>\gamma</math> (fig 1a, b). The MSD crosses over from subdiffusive to a steady state diffusive behavior at a transition at <math>\gamma \approx 0.08</math> is observed. At the transition, the distribution of particle displacements <math>P(\Delta x (\Delta t))</math> becomes non-Gaussian, reflecting dynamic heterogeneities. This is measured by the non-Gaussian parameter <math>\alpha_2</math> (fig 1c, d).

To study the cooperative motions in the sheared colloids, the authors look at the dynamic clusters in the sample, and compare it to an unsheared colloidal liquid with comparable diffusion constant (fig 2 a,b). To be more quantitative, they compare the dynamic cluster size distribution of the two (fig 2 c). The sheared glass has a longer tail of larger dynamic clusters. Integrating the data, they obtain an average dynamic cluster size about 2-3 particles, setting the scale of cooperative motion.

Lastly, they measure the diffusion constant<math>D</math> with a linear fit of the MSD data: <math>\langle \Delta x^2(t)\rangle_y=2Dt</math> (fig 3). Interestingly, <math>D</math> is linear with the shear rate <math>\dot{\gamma}</math>. They propose to explain this relation with a modified version of the Stokes-Einstein relation. Replace the thermal energy <math>k_B T</math> with the shear energy <math>(4\pi R_s^3/3)\eta \dot{\gamma}</math>, the Stokes-Einstein relation becomes <math>D=(2/9)R_s^2 \dot{\gamma}</math>. Here <math>R_s</math> is a characteristic radius. In the data <math>R_s=1.8R</math>, agreeing with the observation that a cluster contains about 3 particles.

## Soft Matter discussion

Explanation of the glass transition is one of the biggest mysteries in soft matter. Shear melting of a colloidal glass provides another way to investigate this phenomenon. Here strain plays a role similar to temperature, and the critical strain <math>\gamma \approx 0.08</math> is analogous to the glass transition temperature <math>T_g</math>. Indeed, under external shear, the particle dynamics is not determined by thermal fluctuation but by the imposed shear. That is also why we can simply swap the thermal energy with shear energy, and modify the Stokes-Einstein relation (which is just an example of fluctuation-dissipation ) to explain the diffusion constant of the sheared system. Further, we note that in this example, shearing enhances the mobility of the particles, and that in similar to the role of active transport in biological systems (see Intracellular transport by active diffusion).