Sedimentation equilibrium in concentrated, multicomponent particle dispersions. Hard spheres in the Percus–Yevick approximation.

Abstract

An equation is derived for the sedimentation equilibrium of (colloidal or macromolecular) particles with hard‐sphere interparticle potential in a low‐molecular weight solvent, in the Percus–Yevick and Carnahan–Starling approximation. The concentration–distance profile has a sigmoidal shape. It is shown that the turbidity of the mixture shows a maximum at the inflection point of this profile, a phenomenon that is sometimes observed in practice. The treatment is subsequently extended to a p‐component mixture of hard spheres (HS) with different HS diameters. First, the explicit role of the low‐molecular weight solvent is removed and the sedimentation differential equation is expressed in terms of partial derivatives (∂ρ_i/∂μ_j)_μ; i,j=1,...,p, where ρ_i is the particle number density of i and μ_j the chemical potential of j. This partial derivative measures the susceptibility of δρ_i to variations in a weak external (centrifugal) field working on species j and is expressed as an integral over the total correlation function h_ij(r) via a relation of Yvon. Second, the sedimentation equation is expressed in terms of matrices containing (∂μ_i/∂ρ_i)_ρ, each connected with an integral over the direct correlation function c_ij(r). Third, Baxter’s factorization of the direct correlation matrix is used to derive a new sedimentation equation, also in factorized form, containing matrices with elements Q_ij, which have a simpler mathematical structure than the derivatives ∂μ_i/∂ρ_j in some theories. It turns out that for hard spheres treated in the Percus–Yevick approximation these Q matrices reduce to simple, closed expressions even for a p‐component HS mixture. This leads to p differential equations for the concentration profiles that must be solved numerically. Finally a closed expression is given for (∂ρ_i/∂μ_k)_μ in the Percus–Yevick approximation (compressibility version).