Normal stresses in colloidal dispersions

Abstract

Normal stresses in colloidal dispersions at low shear rates are determined theoretically for both dilute and concentrated suspensions of Brownian hard spheres. An evolution equation for the pair‐distribution function is developed and the perturbation to the microstructure in a general linear flow is shown to be regular to O(Pe2), where Pe=γa2/D0. Here, γ is the characteristic shear rate, a is the particle size, and D0 is the bare‐particle diffusivity. The next term in the perturbation of the microstructure is shown to be O(Pe5/2). The bulk stress (nondimensionalized by ηγ, where η is the viscosity of the suspending fluid) for a dilute suspension in a general linear flow is determined to O(φ2Pe). For simple shear flow the theory predicts normal stress differences of N1/ηγ=0.899φ2Pe and N2/ηγ=−0.788φ2Pe; there is no correction to the shear viscosity at O(Pe), however. A scaling theory is also presented for concentrated suspensions using the corrected time scale a2/Ds0(φ), where Ds0(φ) is the short‐time self‐diffusivity at the volume fraction φ.The appropriate Peclet number is now Pe=γa2/Ds0(φ). The scaling theory predicts that the dominant contribution to the stress comes from Brownian motion and scales as Peg(2;φ)/Ds0(φ), where g(2;φ) is the equilibrium radial‐distribution function at contact and Ds0(φ)=Ds0(φ)/D0. As maximum packing is approached, φm, the normal stress differences are predicted to diverge as (1−φ/φm)−2Pe, Pe≪1. In the presence of interparticle forces there is an additional contribution to the stress analogous to the Brownian contribution. When the length scale of the interparticle force is comparable to the particle size, there is no qualitative change for the divergence of the normal stress differences near maximum packing. For a strongly repulsive interparticle force characterized by a length scale b(≫a), the theory predicts that the appropriate Peclet number is now Peb=γb2/D0 and that near maximum packing based on the thermodynamic volume fraction φb=4πnb3/3, the normal stress differences diverge as (1−φb/φbm)−1Peb, Peb≪1.