Osmosis and reverse osmosis in fine-porous charged diaphragms and membranes

Abstract

The theory of osmosis and reverse osmosis in fine-porous charged membranes and diaphragms is presented in a deductive way with several well-structured levels of consideration. The analysis starts from the discontinuous version of irreversible thermodynamics where the membrane is considered as an absolutely black box without any information needed about its internal structure. As a trade-off for generality, however, the thermodynamic forces must be small. A number of qualitative conclusions is drawn proceeding from rather evident assumptions about the order of magnitude of various phenomenological coefficients. In the case of electrolyte mixtures those conclusions turn out surprisingly numerous and non-trivial including predictions of reflection coefficients larger than unity and strong negative osmosis. Both those anomalous phenomena are confirmed by experimental findings. The next step is the introduction of a ‘uniformly black’ box, namely a macroscopically homogeneous membrane with otherwise unspecified internal structure. That permits to employ the continuous version of irreversible thermodynamics thus allowing to drop the restriction of small thermodynamic forces. In the case of binary electrolytes general solutions in quadratures are obtained for the problems of apparent osmotic pressure and of reverse osmosis, and a very accurate approximate solution in quadratures is obtained for the problem of osmosis. Some of the theoretical predictions are compared with experimental data on the reverse osmosis of binary electrolytes in track-etched membranes with the pore size of 8 nm. In the case of electrolyte mixtures a semi-quantitative asymptotic analysis is performed in the limiting case of reverse osmosis at sufficiently large Péclet numbers, and in the particular case of one dominant electrolyte in the mixture an exact solution in quadratures is obtained. Those analyses make possible to predict a number of non-trivial regularities that are confronted with experimental data on the pressure-driven transport or ternary electrolyte mixtures across various charged porous membranes. A good qualitative agreement between the theory and experiments is recorded. The next step is the specification of phenomenological coefficients within the scope of a general capillary model. That makes the box grey but not transparent, yet, as the shape of capillary cross-section as well as the nature of surface forces are not specified. General expressions for phenomenological coefficients derived in terms of ion distribution and diffusion coefficients make possible a qualitative analysis of the effect of inhomogeneity of ion distribution inside pores. Finally, the capillary space charge model is introduced making the analysis completely mechanistic. Several approaches are considered to the approximate description of the structure of overlapped diffuse parts of double electric layers in fine pores. Since qualitative effects of microscopic heterogeneity appear to be not dependent on the details of the pore geometry sample numerical calculations are performed for slit-like capillaries where there is a general solution of Poisson-Boltzmann equation in quadratures. A more general model of pores in an electrically conducting gel is introduced in an attempt to quantitatively describe the results of complete characterization of cation-exchange membranes in terms of irreversible thermodynamics.