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Molecular models for permeation through thin membranes: the effect of hydrodynamic interaction on permeability

Published online by Cambridge University Press:  29 March 2006

George H. Malone ,
Thomas E. Hutchinson  and
Stephen Prager
George H. Malone
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis
Thomas E. Hutchinson
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis
Stephen Prager
Affiliation:
Department of Chemistry, University of Minnesota, Minneapolis
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Abstract

A variational method involving minimization of the energy dissipation rate which was previously developed for transport in polymer systems is applied here to flow of a continuum solvent through a thin membrane. The membrane is represented by an array of spherical particles undergoing Brownian motion, subject to various interactions with one another and with the motion of the solvent. General upper bounds on the solvent permeability of the membrane are obtained in terms of equilibrium distribution functions, and applications of the method are illustrated for the case where membrane elements are confined to a plane. Calculations which treat all beads equivalently give permeability estimates whose dependence on the number n of beads per unit area of membrane has the form, at low n, \[ \kappa = (6\pi\eta n)^{-1}(1-\alpha n+\ldots), \] where η is the solvent viscosity and α is a constant. More-elaborate trials which allow the drag on a bead to be influenced by the distribution of other beads in the vicinity give the stronger bounding estimate \[ \kappa = (6\pi\eta n)^{-1}(1+\alpha^{\prime}n\ln n+\ldots). \] Comparison with a self-consistent field approach suggests that this logarithmic behaviour is the true first-order correction.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 65 , Issue 4 , 2 October 1974 , pp. 753 - 767
Copyright
© 1974 Cambridge University Press

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