# 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 model

## 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,