Rheological constitutive equation for 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. Hebraud, M.E. Cates, Phys. Rev. Lett. 78, 2020 (1997)]. The model attributes similarities in the rheology of such ``soft glassy materials&#39;&#39; 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&#39; and the loss modulus G&#39;&#39; vary with frequency as ω^{x-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 BKZ relation. Finally, we consider measurements of G&#39; and G&#39;&#39; at finite strain amplitude γ. Near the glass transition, G&#39;&#39; exhibits a maximum as γis 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 γ_c at which measurements still probe the linear response is predicted to be roughly frequency-independent.