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Charged Membranes: Poisson-Boltzmann theory, DLVO paradigm and beyond
In this chapter we review the electrostatic properties of charged membranes
in aqueous solutions, with or without added salt, employing simple physical
models. The equilibrium ionic profiles close to the membrane are governed by
the well-known Poisson-Boltzmann (PB) equation. We analyze the effect of
different boundary conditions, imposed by the membrane, on the ionic profiles
and the corresponding osmotic pressure. The discussion is separated into the
single membrane case and that of two interacting membranes. For one membrane
setup, we show the different solutions of the PB equation and discuss the
interplay between constant-charge and constant-potential boundary conditions. A
modification of the PB theory is presented to treat the extremely high
counter-ion concentration in the vicinity of a charge membrane. For two
equally-charged membranes, we analyze the different pressure regimes for the
constant-charge boundary condition, and discuss the difference in the osmotic
pressure for various boundary conditions. The non-equal charged membranes is
reviewed as well, and the crossover from repulsion to attraction is calculated
analytically. We then examine the charge-regulation boundary condition and
discuss its effects on the ionic profiles and the osmotic pressure for two
equally-charged membranes. In the last section, we briefly review the van der
Waals (vdW) interactions and their effect on the free energy between two planar
membranes. We explain the simple Hamaker pair-wise summation procedure, and
introduce the more rigorous Lifshitz theory. The latter is a key ingredient in
the DLVO theory, which combines repulsive electrostatic with attractive vdW
interactions, and offers a simple explanation for colloidal or membrane
stability. Finally, the chapter ends by a short account of the limitations of
the approximations inherent in the PB theory.Comment: 57 pages, 19 figures, From the forthcoming Handbook of Lipid
Membranes: Molecular, Functional, and Materials Aspects. Edited by Cyrus
Safinya and Joachim Radler, Taylor & Francis/CRC Press, 201
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