Theory of Nonlinear Rheology and Yielding of Dense Colloidal Suspensions

Abstract

A first principles approach to the nonlinear flow of dense suspensions is presented which captures shear thinning of colloidal fluids and dynamical yielding of colloidal glasses. The advection of density fluctuations plays a central role, suppressing the caging of particles and speeding up structural relaxation. A mode coupling approach is developed to explore these effects.PACS numbers: 82.70. Dd, 83.60.Df, 83.50.Ax, 64.70.Pf, The properties of dispersions under flow are central to their processing and technological use [1,2]. But especially the non-linear rheology is not yet well understood. For the simplest case of steady shearing, the low density behavior is known [3], but upon increasing the density the growing importance of particle interactions requires theoretical approximation [4,5], hinders simulations [6], and calls for studies of model systems, e.g. [7,8]. Of major interest is the arrest of the structural relaxation when approaching solidification for higher densities, which raises the question of how the imposition of steady shearing might interfere with glass formation. The linear phenomenology is familiar: a colloidal fluid possesses a viscosity and flows, while a colloidal glass characterized by elastic constants, only distorts under strain [1,2]. But the nonlinear rheology of glassy colloids, which exhibit a continuous slowing down of the structural relaxation due to particle blocking (the "cage effect") [1], is less clear. While the mode coupling theory (MCT) recovers the linear phenomenology of this fluid-to-glass transition from microscopic starting points [9], a nonlinear external driving introduces new time scales whose influence on (non-)equilibration is not understood, and has been addressed only in minimal models [10] or mean-field approaches [11]. Moreover, as the true nature of the glass is still uncertain, its behavior under shearing may provide broader new insights (as suggested by recent simulation studies [12,13]).Here we develop a first-principles approach for the simplest case of a disordered colloidal suspension under steady imposed shear, neglecting both many-body hydrodynamics and the resulting velocity fluctuations. We first identify some generic features in the yield properties of glass; approximations suggested by the MCT are then introduced in order to derive quantitative predictions.The system consists of N spherical particles (diameter d) dispersed in a volume V of solvent with imposed flow profile v(r) = κ r, where for simple shear with velocity along the x-axis and its gradient along the y-axis, the shear rate tensor is κ ij =γ δ ix δ jy . The effect of the shear rateγ on the particle dynamics is measured by the Peclet number The system is taken to be in quiescent equilibrium (γ = 0) at t ≤ 0 when averages . . . (γ=0) are the canonical equilibrium ones. Then at t = 0 + , the velocity profile is switched on instantaneously, so that the steady state distribution function Ψ s , which satisfies Ω (γ) Ψ s = 0, will be approached at long times, t → ∞. If Ψ s was known the st...

A statistical mechanical approach to the Payne effect in filled rubbers

“…a power law dependence for large deformation amplitudes according to Equation (19). Experimental data fitting with Equations (15)-(18) generated & = n 3 1/2 [24].…”

“…The physical origin of selecting ! in the phase diagram was not really clear, though recent mode-coupling theories have attempted to include the shear stress [19]. In the studies of Wang and Robertson Figure 1.…”

Arrest and Flow of Colloidal Glasses

“…through a time-dependent vertex V , too long to write down here [18]. Using a nonequilibrium Kubo-type relationship, one can also obtain an expression for the steady state viscosity η = σ(γ)/γ where σ(γ) is the shear stress as a function of shear rate.…”

“…(In practice also, the following results were obtained in Refs. [18] using a projection operator formalism which differs in detail from the version of MCT outlined above.) Despite all this, the structure of the resulting equations is remarkably similar to Eqs.…”

“…[18], M. Fuchs and the author developed a theory, along MCT lines, of colloidal suspensions under flow. The work was intended mainly to address the case of repulsion-driven glasses, and to study the effect of imposed shear flow either on a glass, or on a fluid phase very near the glass transition.…”