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 (∂ρi/∂μj)μ; i,j=1,...,p, where ρi is the particle number density of i and μj the chemical potential of j. This partial derivative measures the susceptibility of δρi to variations in a weak external (centrifugal) field working on species j and is expressed as an integral over the total correlation function hij(r) via a relation of Yvon. Second, the sedimentation equation is expressed in terms of matrices containing (∂μi/∂ρi)ρ, each connected with an integral over the direct correlation function cij(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 Qij, which have a simpler mathematical structure than the derivatives ∂μi/∂ρj 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 (∂ρi/∂μk)μ in the Percus–Yevick approximation (compressibility version).