A Cell Model of the Ion-Exchange Membrane. Electrical Conductivity and Electroosmotic Permeability
Electroosmotic permeability and specific electrical conductivity of an ion-exchange membrane have been calculated in terms of the thermodynamics of nonequilibrium processes on the basis of a cell model that was previously proposed for a charged membrane. The calculated parameters have been considered as kinetic coefficients of the Onsager matrix. The membrane has been considered to be an ordered set of porous spherical charged particles placed into spherical shells filled with a binary electrolyte solution. The boundary value problems have been analytically solved to determine the electroosmotic permeability and electrical conductivity of the membrane for the case of the Kuwabara boundary condition imposed on the cell surface. The consideration has been carried out within the framework of a small deviation of system parameters from their equilibrium values upon imposition of external fields. Different particular cases of the derived exact analytical equations, including those for a binary symmetric electrolyte and an ideally selective membrane, have been analyzed. It has been shown that, as electrolyte concentration increases, the specific electrical conductivity (direct kinetic coefficient) of a cation-exchange membrane may monotonically grow in different manners, i.e., with an inflection point in a plot (similarly to a current–voltage curve) or without it. The behavior of the electroosmotic permeability upon increasing electrolyte concentration depends on the deviation of the distribution coefficient of electrolyte molecules from unity and the difference between the diffusion coefficient ratios of different ions in a dilute solution and in the membrane: the permeability may monotonically grow, increase reaching a plateau, or pass through a maximum.
This work was supported by the Russian Foundation for Basic Research (project no. 17-08-01287) (theoretical part) and the Ministry of Education and Science of the Russian Federation (project no. 14.Z50.31.0035) (experimental data processing).
- 1.Happel, D. and Brenner, G., Low Reynolds Number Hydrodynamics, Leyden: Noordhoff, 1965; Moscow: Mir, 1976.Google Scholar
- 2.Filippov, A.N., Colloid J., 2018, vol. 80, p. 716.Google Scholar
- 3.Shilov, V.N., Zharkikh, N.I., and Borkovskaya, Yu.B., Kolloidn. Zh., 1981, vol. 43, p. 540.Google Scholar
- 4.Zharkikh, N.I. and Borkovskaya, Yu.B., Kolloidn. Zh., 1981, vol. 43, p. 652.Google Scholar
- 5.Filippov, A.N. and Shkirskaya, S.A., Membr. Membr. Tekhnol., 2018, vol. 8, p. 254.Google Scholar
- 6.Brinkman, H.C., Appl. Sci. Res. A1, 1947, p. 27.Google Scholar
- 14.Pismenskaya, N., Nikonenko, V., Sarapulova, V., Shkorkina, I., Titorova, V., Butylskii, D., and Tongwen Xu, Abstracts of Papers, Conf. on Ion Transport in Organic and Inorganic Membranes, Sochi, 2017, p. 21.Google Scholar
- 16.Moelwyn-Hughes, E.A., Physical Chemistry, London: Pergamon, 1961, vol. 2.Google Scholar
- 17.Borkovskaya, Yu.B., Zharkikh, N.I., and Dudkina, L.M., Kolloidn. Zh., 1982, vol. 44, p. 645.Google Scholar
- 18.Zharkikh, N.I. and Shilov, V.N., Kolloidn. Zh., 1981, vol. 43, p. 1061.Google Scholar
- 19.Nikonenko, V.V., Mareev S.A., Pis'menskaya, N.D., Uzdenova, A.M., Kovalenko, A.V., Urtenov, M.Kh., and Pourcelly, G., Russian J. of Electrochemistry, 2017, vol. 53, p. 1122.Google Scholar