The rheological behavior of concentrated colloidal dispersions

A simple model for the rheological behavior of concentrated colloidal dispersions is developed. For a suspension of Brownian hard spheres there are two contributions to the macroscopic stress: a hydrodynamic and a Brownian stress. For small departures from equilibrium, the hydrodynamic contribution is purely dissipative and gives the high-frequency dynamic viscosity. The Brownian contribution has both dissipative and elastic parts and is responsible for the viscoelastic behavior of colloidal dispersions. An evolution equation for the pair-distribution function is developed and from it a simple scaling relation is derived for the viscoelastic response. The Brownian stress is shown to be proportional to the equilibrium radial-distribution function at contact, g(2;φ), divided by the short-time self-diffusivity, D0s(φ), both evaluated at the volume fraction φ of interest. This scaling predicts that the Brownian stress diverges at random close packing, φm, with an exponent of −2, that is, η′0 ∼ η(1 − φ/φm)−2, where η′0 is the steady shear viscosity of the dispersion and η is the viscosity of the suspending fluid. Both the scaling law and the predicted magnitude are in excellent accord with experiment. For viscoelastic response, the theory predicts that the proper time scale is a2/D0s, where a is the particle radius, and, when appropriately scaled, the form of the viscoelastic response is a universal function for all volume fractions, again in agreement with experiment. In the presence of interparticle forces there is an additional contribution to the stress analogous to the Brownian stress. When the length scale characterizing the interparticle forces is comparable to the particle size, the theory predicts that there is only a quantitative contribution from the interparticle forces to the stress; the qualitative behavior, particularly the singular scaling of the viscosity and the form of the viscoelastic response, remains unchanged from the Brownian case. For strongly repulsive interparticle forces characterized by a length scale b (≫a), however, the theory predicts that the viscosity diverges at the random close packing volume fraction, φbm, based on the length scale b, with a weaker exponent of −1. The viscoelastic response now occurs on the time scale b2/D0s(φ), but is of the same form as for Brownian dispersions.