# Spatial cooperativity in soft glassy flows

## Reference

Goyon, J., Colin, A., Ovarlez, G., Ajdari, A., Bocquet, L., Nature 454 (2008).

## Keywords

spatial cooperativity, glass, velocity profile, shear stress, shear strain, Couette cell, Herschel-Bulkley fluid

## Summary

A general feature of glassy materials is a strong nonlinear flow rule relating stress and strain. This feature is no well-documented and poorly understood. Many have tried to understand the glass transition by studying the dynamical heterogeneities in glass-forming materials, but how these heterogeneities affect flow remains unclear. Using a local velocity measurement technique, the authors study the local flow of a film of confined glassy material.

The authors test flow in two main geometries: shear planar flow in a wide gap Couette cell, and pressure driven planar flow in a narrow microchannel (tens of hundreds micrometers in width). The substance tested was an emulsion of silicone droplets (6.5um in diameter) in a glycerine-water mixture. The local flow curves, which relate the local shear stress <math>\sigma</math> to the local shear rate <math>\dot{\gamma}</math>, are obtained from the measured velocity profiles of both geometries. Figure 1 shows the results for the wide-gap Couette cell, and Figure 2 shows the results for the narrow microchannel. Evidently, the flow curve is highly dependent on the geometry. In the wide-gap case, the curve follows the Herschel-Bulkley model. However, in the microchannel setup, the data does not follow a single rheological curve. The finite-size effect in the microchannel setup which did not appear in the wide-gap setup suggests that there are extended spatial correlations in the system.

The authors checked that the change in rheology is not caused by a structural change in the emulsion, nor a change in density, nor boundary effects. In order to explain this finite-size effect, they developed a model considering the plastic rearrangements that occur in concentrated emulsions. In Conceptually, localized plastic events induce a non-local, long-range elastic relaxation of the stress over the system. The rate of plastic rearrangements <math>v</math> is equivalent to a fluidity <math>f</math> defined as <math>\sigma = (1/f)\dot{\gamma}</math>. In the absence of non-local effects, the fluidity reduces to the bulk fluidity <math>f_{bulk} = \dot{\gamma}/\sigma_{bulk}</math>, but to take into account the non-local effects, the fluidity is assumed to be of the form

<math>f(z)=f_{bulk}+\xi^2 \frac{\partial^2f(z)}{\partial^2 z}</math>.

In the above, <math>\xi</math> denotes a "flow cooperativity length." This equation was solved numerically using appropriate boundary conditions. Surprisingly, the authors found that a unique flow cooperativity length accurately describes all the microchannel data, even across different shear velocities. The predictions are shown as dashed lines in Figure 2. Furthermore, the authors applied the model at different volume fractions, finding a unique cooperativity length for each that matched the data similarly well. It is noted that the cooperativity length is on the same order as dynamical heterogeneities, and that the two might be related. However, whereas dynamical heterogeneities have a maximum at the glass transition, the cooperativity length has a very different behavior, only being nonzero above the transition and increasing from there. Instead of the size of mobile regions, perhaps the the cooperativity length is the characteristic length for plastic events.