The theory of semipermeable vesicles and membranes: An integral-equation approach. I. General formalism and application to a hard-sphere mixture

Abstract

An exact integral-equation formalism for a system of a binary hard-sphere mixture interacting with a spherical semipermeable vesicle (SPV) and plane semipermeable membrane (SPM) is derived by using the Ornstein–Zernike (OZ) equation with appropriate closures. The Percus–Yevick (PY) closure or the hypernetted chain (HNC) closure, in which the bulk correlation is obtained by the PY approximation, are considered as examples. We refer to these as the PY/PY and HNC/PY approximations, respectively. The mixture contains solvent particles, which are permeable to the membrane, and solute particles (‘‘protein’’ or ‘‘polymer’’ particles), which can not pass through the membrane. We develop an exact general formalism for this problem and as an illustration of its use give quantative results for solvent and solute particles modeled as hard spheres of different diameters. An analytical expression for the density ratio in the PY/PY and HNC/PY approximations between two sides of a plane SPM is obtained. Results obtained from these expressions agree very well with results obtained by equating chemical potentials in the region of interest. It turns out that the protein–membrane direct correlation function can be given by a simple analytic expression for the limit of a point solvent in the PY/PY approximation. The osmotic pressure and density profiles for the system containing an ideal spherical SPV or plane SPM in the PY/PY approximation are evaluated. Extension to the nonlocal density-functional closures previously introduced by Blum and Stell is discussed. Finally, we note that certain impenetrable-wall problems previously considered elsewhere can be regarded as semipermeable membrane problems treated via McMillan–Mayer formalism in the continuum–solvent approximation.