Supercooled liquids under shear: A mode-coupling theory approach

We generalize the mode-coupling theory of supercooled fluids to systems under stationary shear flow. Our starting point is the generalized fluctuating hydrodynamic equations with a convection term. The method is applied to a two dimensional colloidal suspension. The shear rate dependence of the intermediate scattering function and shear viscosity is analyzed. The results show a drastic reduction of the structural relaxation time due to shear and strong shear thinning behavior of the viscosity which are in qualitative agreement with recent simulations. The microscopic theory with minimal assumptions can explain the behavior far beyond the linear response regime. Many complex fluids such as suspensions, polymer solutions, and granular fluids exhibit very diverse rheological behavior. Shear thinning is among the most-known phenomena. Recently, it was found by experiments[1] and simulations[2] that supercooled liquids near the glass-transition also show strong shear thinning behavior. They have observed that near the transition temperature, the structural relaxation time and the shear viscosity both decrease as γ̇−ν , where γ̇ is the shear rate and ν is an exponent which is less than but close to 1. For such systems driven far from equilibrium, the nonequilibrium parameter γ̇ is not a small perturbation parameter but plays a role more like an intensive parameter which characterizes the “thermodynamic state” of the system[3]. Such rheological behavior is interesting in its own right, but understanding the dynamics of supercooled liquids in a nonequilibrium state is more important because it has possibility to shed light on an another typical and perhaps more important nonequilibrium problem, non-stationary aging. Aging is the slow relaxation after a sudden quench of temperature below the glass transition temperature. In this case, the waiting time plays a similar role to (the inverse of) the shear rate. Aging behavior has been extensively studied for spin glasses (see Ref.[4] and references therein). The relationship between aging and a system driven far away from the equilibrium was considered using a schematic model based on the exactly solvable p-spin spin glass by Berthier, Barrat and Kurchan[5] and its validity was tested numerically for supercooled liquids[6]. There are attempts to study the aging of structural glasses theoretically[7] but it has not been analyzed and compared with the simulation results[8]. In this paper, we investigate the dynamics of supercooled liquids under shear theoretically, by extending the standard mode-coupling theory (MCT). We start with generalized fluctuating hydrodynamic equations with a convection term. Using several approximations, we obtain a closed nonlinear equation for the intermediate scattering function for the sheared system. The theory is applicable to both normal liquids and colloidal suspensions in the absence of hydrodynamic interactions. Numerical results will be presented only for the colloidal suspensions, but generalization to liquids are straightforward. Some of the preliminary results have already been published in Ref.[9]. We shall consider a two dimensional colloidal suspension under a stationary simple shear flow given by v0(r) = Γ · r = (γ̇y,0), (1) where (Γ)αβ = γ̇δαxδβ y is the velocity gradient matrix. The hydrodynamic fluctuations for density ρ(r,t) and the velocity field v(r,t) obey the following set of Langevin equations[10].