Dynamic ultrafiltration model for charged colloidal dispersions: A Wigner-Seitz cell approach

Abstract

A rigorous, dynamic mathematical model for predicting the rate of ultrafiltration of charged colloidal dispersions is developed. The model is based on sophisticated descriptions of the particle-particle interactions within filter cakes which are responsible for controlling permeation rates. Electrostatic (double layer) interactions are accounted for by means of a Wigner-Seitz cell approach, including a numerical solution of the non-linear Poisson-Boltzmann equation, which is known to give an excellent description of the configurational electrostatic interaction energy of particle assemblages. London-van der Waals forces are calculated using a computationally efficient means of approximating screened, retarded Lifshitz-Hamaker constants. Hydration forces are included by utilising mathematical expressions derived from the latest results obtained with surface-forces apparatus. Configurational entropy effects are calculated using an equation of state giving excellent agreement with molecular dynamic data. Electroviscous effects are also accounted for. These descriptions of particle-particle interactions in assemblages are used to develop an a priori model, with no adjustable parameters, that allows quantitative prediction of the rate of filtration of charged colloidal dispersions as a function of zeta-potential, particle composition (through the Hamaker constant), ionic strength, applied pressure, particle radius and membrane resistance. This is a dynamic model which takes into account the variation of local specific cake resistance as a function of both position in the cake and time. The predictions of the model are systematically investigated, showing the great importance of taking particle-particle interactions into account for ultrafiltration. A comparison with experimental filtration data for colloidal silica shows that the model is in excellent agreement with such data.