The sedimentation behaviour of gels — the generalised Lamm’s differential equation

Abstract

Lamm’s differential equation for polymer solutions is a well-known tool to describe the time-dependent change of the polymer concentration as a function of diffusion and sedimentation of the polymer component in a centrifugal field. Based on the phenomenological equations, which describe the flux of the polymer component as the sum of the products of the phenomenological coefficients and the generalised specific forces, this was derived. The phenomenological definition of the flux is valid for polymer solutions as well as for gels. It is shown that the phenomenological equation in the case of gels leads to a “generalised Lamm differential equation”, which describes the change of the concentration with respect to the time as a function of the diffusion, the sedimentation and a so-called “elastically active coefficient”. All changes between the sedimentation behaviour of a polymer in solution and a polymer in a swollen elastic network can be attributed to this coefficient. Ultracentrifugal measurements of gelatin gels (physical networks) yield the ratio of the diffusion coefficient and the sedimentation coefficient of the polymer at different overall concentrations of the gels. From the literature values of non-cross-linked and cross-linked polystyrene in chlorobenzene the ratio of the mobilities and sedimentation coefficients are calculated and discussed.