Rheological constitutive equation for a model of soft glassy materials

We solve exactly and describe in detail a simplified scalar model for the low frequency shear rheology of foams, emulsions, slurries, etc. [P. Sollich, F. Lequeux, P. Hébraud, and M. E. Cates, Phys. Rev. Lett. 78, 2020 (1997)]. The model attributes similarities in the rheology of such “soft glassy materials” to the shared features of structural disorder and metastability. By focusing on the dynamics of mesoscopic elements, it retains a generic character. Interactions are represented by a mean-field noise temperature $x$, with a glass transition occurring at $x=1\mathrm{}$ (in appropriate units). The exact solution of the model takes the form of a constitutive equation relating stress to strain history, from which all rheological properties can be derived. For the linear response, we find that both the storage modulus $G^{\prime }$ and the loss modulus $G^{″}$ vary with frequency as ${\omega }^{x-1}$ for $1<x<\mathrm{}2\mathrm{}$, becoming flat near the glass transition. In the glass phase, aging of the moduli is predicted. The steady shear flow curves show power-law fluid behavior for $x<\mathrm{}2\mathrm{}$, with a nonzero yield stress in the glass phase; the Cox-Merz rule does not hold in this non-Newtonian regime. Single and double step strains further probe the nonlinear behavior of the model, which is not well represented by the Bernstein-Kearseley-Zapas relation. Finally, we consider measurements of $G^{\prime }$ and $G^{″}$ at finite strain amplitude $\gamma$. Near the glass transition, $G^{″}$ exhibits a maximum as $\gamma$ is increased in a strain sweep. Its value can be strongly overestimated due to nonlinear effects, which can be present even when the stress response is very nearly harmonic. The largest strain ${\gamma }_c$ at which measurements still probe the linear response is predicted to be roughly frequency independent.