Markus Schmuck, Martin Z. Bazant
Effective Poisson-Nernst-Planck (PNP) equations are derived for macroscopic
ion transport in charged porous media. Homogenization analysis is performed for
a two-component pe- riodic composite consisting of a dilute electrolyte
continuum (described by standard PNP equations) and a continuous dielectric
matrix, which is impermeable to the ions and carries a given surface charge.
Three new features arise in the upscaled equations: (i) the effective ionic
diffusivities and mobilities become tensors, related to the microstructure;
(ii) the effective permittivity is also a tensor, depending on the
electrolyte/matrix permittivity ratio and the ratio of the Debye screening
length to mean pore size; and (iii) the surface charge per volume appears as a
continuous "background charge density". The coeffcient tensors in the
macroscopic PNP equations can be calculated from periodic reference cell
problem, and several examples are considered. For an insulating solid matrix,
all gradients are corrected by a single tortuosity tensor, and the Einstein
relation holds at the macroscopic scale, which is not generally the case for a
polarizable matrix. In the limit of thin double layers, Poisson's equation is
replaced by macroscopic electroneutrality (balancing ionic and surface
charges). The general form of the macroscopic PNP equations may also hold for
concentrated solution theories, based on the local-density and mean-field
approximations. These results have broad applicability to ion transport in
porous electrodes, separators, membranes, ion-exchange resins, soils, porous
rocks, and biological tissues.
View original:
http://arxiv.org/abs/1202.1916
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