Rheological constitutive equation for a model of soft glassy materials

Abstract

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. H\'ebraud, 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 $x=1$ (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 ${G}^{\ensuremath{'}}$ and the loss modulus ${G}^{\ensuremath{''}}$ vary with frequency as ${\ensuremath{\omega}}^{x\ensuremath{-}1}$ for $1<x<2$, 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 $x<2$, 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 ${G}^{\ensuremath{'}}$ and ${G}^{\ensuremath{''}}$ at finite strain amplitude $\ensuremath{\gamma}$. Near the glass transition, ${G}^{\ensuremath{''}}$ exhibits a maximum as $\ensuremath{\gamma}$ 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 ${\ensuremath{\gamma}}_{c}$ at which measurements still probe the linear response is predicted to be roughly frequency independent.