Colloid Journal

, Volume 80, Issue 6, pp 728–738 | Cite as

A Cell Model of the Ion-Exchange Membrane. Electrical Conductivity and Electroosmotic Permeability

  • A. N. FilippovEmail author


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. 1.
    Happel, D. and Brenner, G., Low Reynolds Number Hydrodynamics, Leyden: Noordhoff, 1965; Moscow: Mir, 1976.Google Scholar
  2. 2.
    Filippov, A.N., Colloid J., 2018, vol. 80, p. 716.Google Scholar
  3. 3.
    Shilov, V.N., Zharkikh, N.I., and Borkovskaya, Yu.B., Kolloidn. Zh., 1981, vol. 43, p. 540.Google Scholar
  4. 4.
    Zharkikh, N.I. and Borkovskaya, Yu.B., Kolloidn. Zh., 1981, vol. 43, p. 652.Google Scholar
  5. 5.
    Filippov, A.N. and Shkirskaya, S.A., Membr. Membr. Tekhnol., 2018, vol. 8, p. 254.Google Scholar
  6. 6.
    Brinkman, H.C., Appl. Sci. Res. A1, 1947, p. 27.Google Scholar
  7. 7.
    Saffman, P.G., Stud. Appl. Math., 1971, vol. 50, p. 93.CrossRefGoogle Scholar
  8. 8.
    Starov, V.M. and Churaev, N.V., Adv. Colloid Interface Sci., 1993, vol. 43, p. 145.CrossRefGoogle Scholar
  9. 9.
    Vasin, S.I. and Filippov, A.N., Colloid J., 2009, vol. 71, p. 31.CrossRefGoogle Scholar
  10. 10.
    Vasin, S.I., Filippov, A.N., and Starov, V.M., Adv. Colloid Interface Sci., 2008, vol. 139, p. 83.CrossRefGoogle Scholar
  11. 11.
    Filippov, A., Afonin, D., Kononenko, N., Lvov, Y., and Vinokurov, V., Colloids Surf. A, 2017, vol. 521, p. 251.CrossRefGoogle Scholar
  12. 12.
    Filippov, A., Petrova, D., Falina, I., Kononenko, N., Ivanov, E., Lvov, Y., and Vinokurov, V., Polymers, 2018, vol. 10, Article no. 366.CrossRefGoogle Scholar
  13. 13.
    Sidorova, M.P., Ermakova, L.E., Savina, I.A., and Fridrikhsberg, D.A., J. Membr. Sci., 1993, vol. 79, p. 159.CrossRefGoogle Scholar
  14. 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
  15. 15.
    Zholkovskiy, E.K., Shilov, V.N., Masliyah, J.H., and Bondarenko, M.P., Can. J. Chem. Eng., 2007, vol. 85, p. 701.CrossRefGoogle Scholar
  16. 16.
    Moelwyn-Hughes, E.A., Physical Chemistry, London: Pergamon, 1961, vol. 2.Google Scholar
  17. 17.
    Borkovskaya, Yu.B., Zharkikh, N.I., and Dudkina, L.M., Kolloidn. Zh., 1982, vol. 44, p. 645.Google Scholar
  18. 18.
    Zharkikh, N.I. and Shilov, V.N., Kolloidn. Zh., 1981, vol. 43, p. 1061.Google Scholar
  19. 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

Copyright information

© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  1. 1.National University of Oil and Gas “Gubkin University”MoscowRussia

Personalised recommendations