The theory of semipermeable vesicles and membranes: An integral-equation approach. II. Donnan equilibrium.

Integral equations that yield the charge and density profiles are derived for a Donnan system, in which an ionic solution is separated into two regions by a semipermeable membrane (SPM) or a spherical semipermeable vesicle (SPV). These equations are obtained from the Ornstein–Zernike (OZ) equation. We show how quantitative results can be obtained from either the mean spherical approximation (MSA) closure or the hypernetted-chain (HNC) closure for profiles. Use is made of bulk-correlation input obtained by means of the Debye–Hückel approximation, the MSA approximation, or the HNC approximation. The resulting approximations will be referred as MSA/DH, HNC/DH, MSA/MSA, etc. The system on which we focus contains three charged hard-sphere species: cation, anion, and a large ion (a protein or polymer ion) separated by a plane SPM, through which the large ion cannot pass, and to one side of which all large ions are confined, or a spherical SPV, outside of which the large ions are confined. Analytical expressions for the bulk density ratio between the two sides of a plane membrane as well as the membrane potential in various approximations are obtained. Results obtained from these expresssions are compared with the results obtained by equating electrochemical potentials. A new contact-value theorem is provided for the plane SPM system. Analytical solutions for the charge profile and the potential profile in the MSA/DH approximation are obtained. It turns out that results obtained in the HNC/DH approximation are exactly the same as those obtained by using 1D nonlinear Poisson–Boltzmann equations if the repulsive cores of the macroions are neglected. [ABSTRACT FROM AUTHOR]