Capillary Model of a Charged Membrane with Variable Hydrophilicity and Hydrophobicity

The paper proposes a capillary model of a charged membrane, which consists of a set of plane-parallel slitlike hydrophilic and hydrophobic pores separated by an impermeable material. The zeta potential or fixed charge density and the condition of liquid no-slip can be preset on the surface of the hydrophilic pores. The hydrophobic pores differ from the hydrophilic ones in the size, zeta potential (density of the fixed charge), and the Navier slip condition. Relations are derived for the hydrodynamic and electroosmotic permeabilities and electrical conductivity of the membrane as functions of the relative hydrophilic and hydrophobic porosities, electrolyte concentration, surface charge or potential, dielectric properties of a solution, diffusion coefficients and charge numbers of ions, and the sizes of the pores of both types. In all cases, compliance with the Onsager reciprocity principle has been shown for cross coefficients L12 and L21, which are responsible for the electroosmosis velocity and the streaming current. All boundary problems for the four types of pores are solved analytically under the Debye–Hückel approximation. It has been found that, in the case of aqueous organic mixtures against the background of a weak electrolyte solution, differently directed flows of components may occur through the hydrophilic and hydrophobic pores of the membrane under the action of external pressure and electric potential gradients. The results obtained enable one to predict the transport properties of a charged membrane as depending on the ratio between the shares of the hydrophilic and hydrophobic pores.