Kinetic description of ion transport in the system "ionic solution – porous environment"

: pp. 719–733
Received: May 09, 2022
Revised: August 04, 2022
Accepted: August 10, 2022

Mathematical Modeling and Computing, Vol. 9, No. 3, pp. 719–733 (2022)

Lviv Polytechnic National University; Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine

A kinetic approach based on a modified chain of BBGKI equations for nonequilibrium particle distribution functions was used to describe the ion transfer processes in the ionic solution – porous medium system.  A generalized kinetic equation of the revised  Enskog–Vlasov–Landau theory for the nonequilibrium ion distribution function in the model of charged solid spheres is obtained, taking into account attractive short-range interactions for the ionic solution – porous medium system.

  1. Sahimi M.  Non-linear and non-local transport processes in heterogeneous media: from long-range correlated percolation to fracture and materials breakdown.  Physics Reports.  306 (4–6), 213–395 (1998).
  2. Hatano Y., Hatano N.  Dispersive transport of ions in column experiments: An explanation of long-tailed profiles.  Water Resources Research.  34 (5), 1027–1033 (1998).
  3. Gelb L. D., Gubbins K. E., Radhakrishnan R., Sliwinska-Bartkowiak M.  Phase separation in confined systems.  Reports on Progress in Physics.  62 (12), 1573–1659 (1999).
  4. Advances in Lithium-Ion Batteries.  Eds.: W. A. van Schalkwijk, B. Scrosati.  Springer New York, NY (2002).
  5. Wagemaker M.  Structure and Dynamics of Lithium in Anatase TiO$_2$.  Delft Univer. Press, Netherland (2002).
  6. Berkowitz B., Klafter J., Metzler R., Scher H.  Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk, and fractional derivative formulations.  Water Resources Research.  38 (10), 1191 (2002).
  7. Berkowitz B., Cortis A., Dentz M., Scher H.  Modeling non-Fickian transport in geological formations as a continuous time random walk.  Reviews of Geophysics.  44 (2), RG2003 (2006).
  8. Smith J. J., Zharov I.  Ion Transport in Sulfonated Nanoporous Colloidal Films.  Langmuir.  24 (6), 2650–2654 (2008).
  9. Neuman S. P., Tartakovsky D. M.  Perspective on theories of non-Fickian transport in heterogeneous media.  Advances in Water Resources.  32 (5), 670–680 (2009).
  10. Rotenberg B., Pagonabarrag I., Frenkel D.  Coarse-grained simulations of charge, current and flow in heterogeneous media.  Faraday Discussions.  144, 223–243  (2010).
  11. Yang C., Nakayama A.  A synthesis of tortuosity and dispersion in effective thermal conductivity of porous media.  International Journal of Heat and Mass Transfer.  53 (15–16), 3222–3230 (2010).
  12. Maffeo C., Bhattacharya S., Yoo J., Wells D., Aksimentiev A.  Modeling and Simulation of Ion Channels.  Chemical Reviews.  112 (12), 6250–6284 (2012).
  13. Bijeljic B., Raeini A., Mostaghimi P., Blunt M. J.  Predictions of non-Fickian solute transport in different classes of porous media using direct simulation on pore-scale images.  Physical Review E.  87 (1), 013011 (2013).
  14. Grygorchak I. I., Kostrobij P. P., Stasiuk I. V., Tokarchuk M. V., Velychko O. V., Ivashchyshyn F. O., Markovych B. M.  Physical processes and their microscopic models in periodic inorganic / organic clathrates.  Lviv, Rastr-7 (2015), (in Ukrainian).
  15. Lithium batteries. Eds.: G.-A. Nazri, G. Pistoia. USA, Springer (2009).
  16. Tyukhova A., Dentz M., Kinzelbach W., Willmann M.  Mechanisms of Anomalous Dispersion in Flow Through Heterogeneous Porous Media.  Physical Review Fluids.  1 (7), 074002 (2016).
  17. Comolli A., Dentz M.  Anomalous dispersion in correlated porous media: a coupled continuous time random walk approach.  The European Physical Journal B.  90, 166 (2017).
  18. Waisbord N., Stoop N., Walkama D. M., Dunkel J., Guasto J. S.  Anomalous percolation flow transition of yield stress fluids in porous media.  Physical Review Fluids.  4 (6), 063303 (2019).
  19. Zhao T., Qing L., Long T., Xu X., Zhao S., Lu X.  Dynamical coupling of ion adsorption with fluid flow in nanopores.  AIChE J.  67, e17266 (2021).
  20. Bisquert J., Compte A.  Theory of the electrochemical impedance of   anomalous diffusion.  Journal of Electroanalytical Chemistry.  499 (1), 112–120 (2001).
  21. Sibatov R. T., Uchaikin V. V.  Fractional differential approach to dispersive transport in semiconductors.  Physics-Uspekhi.  52 (10), 1019–1043 (2009).
  22. Sibatov R. T.  Drobno-differencialnaja teorija anomalnoj kinetiki nositelej zarjada v neuporjadochennyh poluprovodnikovyh sistemah.  Thesis for the Degree of Doctor of Sciences in Physics and Mathematics. Uljanovsk (2012), (in Russian).
  23. Khamzin A. A., Popov I. I., Nigmatullin R. R.  Correction of the power law of ac conductivity in ion-conducting materials due to the electrode polarization effect.  Physical Review E.  89 (3), 032303 (2014).
  24. Ferguson T. R., Bazant M. Z.  Nonequilidrium Thermodynamics of Porous Electrodes.  Journal of The Electrochemical Society.  159 (12),  A1967–A1985 (2012).
  25. Xie Y., Li J., Yuan C.  Mathematical modeling of the electrochemical impedance spectroscopy in lithium ion battery cycling.  Electrochimica Acta.  127, 266–275 (2014).
  26. Sibatov R. T., Uchaikin V. V.  Fractional differential approach to the description of dispersive transfer in semiconductors.  Physics-Uspekhi.  52 (10), 1019–1043 (2009).
  27. Kostrobij P. P., Markovych B. M., Chernomorets Yu. I., Tokarchuk R. M., Tokarchuk M. V.  Statistical description of electro-diffusion processes of ions intercalation in "electrolyte–electrode" system.  Mathematical Modeling and Computing.  1 (2), 178–194 (2014).
  28. Kostrobij P., Markovych B., Viznovych O., Tokarchuk M.  Generalized electrodiffusion equation with fractality of space-time.  Mathematical Modeling and Computing.  3 (2), 163–172  (2017).
  29. Grygorchak I. I., Ivashchyshyn F. O., Tokarchuk M. V., Pokladok N. T., Viznovych O. V.  Modification of properties of GaSe $\beta$-cyclodexterin Clathrat by synthesis in superposed electric and light-wave fields. Journal of Applied Physics.  121, 185501 (2017).
  30. Kostrobij P. P., Ivashchyshyn F. O., Markovych B. M., Tokarchuk M. V.  Microscopic theory of the influence of dipole superparamagnetics (type $\langle\beta\mathrm{-CD}\langle\mathrm{FeSO}_4\rangle\rangle$) on current flow in semiconductor layered structures (type GaSe, InSe).  Mathematical Modeling and Computing.  8 (1), 89–105 (2021).
  31. Kostrobij P., Grygorchak I., Ivashchyshyn I., Markovych B., Viznovych O., Tokarchuk M.  Generalized electrodiffusion equation with fractality of space time: experiment and theory.  The Journal of Physical Chemistry A.  122 (16), 4099–4110 (2018).
  32. Jardat M., Hribar-Lee B., Vlachy V.  Self-diffusion of ions in charged nanoporous media.  Soft Matter.  8, 954–964 (2012).
  33. Jardat M., Hribar-Lee B., Dahirel V., Vlachy V.  Self-diffusion and activity coefficients of ions in charged disordered media.  The Journal of Chemical Physics.  137 (11), 114507 (2012).
  34. Omelyan I. P., Zhelem R. I., Sovyak E. M., Tokarchuk M. V.  Calculation of distribution functions and diffusion coefficients for ions in the system "initial electrolyte solution – membrane".  Condensed Matter Physics.  2 (1), 53–62 (1999).
  35. Yukhnovskii I. R., Zhelem R. I., Tokarchuk M. V.  Physical processes in the fuel containing masses interacting with aqueous solutions in the "Shelter" object.  Inhomogeneous diffusion of ions UO$_2^{2+}$, Cs$^+$ in the system "glassy nuclear magma – water".  Condensed Matter Physics.  2 (2), 351–360 (1999).
  36. Yukhnovskii I. R., Omelyan I. P., Zhelem R. I., Tokarchuk M. V.  Statistical theory for diffusion of radionuclides in ground and subterranean water.  Radiation Physics and Chemistry.  59 (4), 361–375 (2000).
  37. Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V.  Kinetic equations for dense gases and liquids.  Theoretical and Mathematical Physics.  87 (1), 412–424 (1991).
  38. Tokarchuk M. V., Omelyan I. P.  Model kinetic equations for dense gases and liquids.  Ukrainian Journal of Physics.  35 (8), 1255–1261 (1990), (in Ukrainian).
  39. Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V.  The Enskog-Landau kinetic equation for charged hard spheres.  In: Problems of atomic science and technique.  Series: Nuclear physics investigations (theory and experiment).  Kharkov, Kharkov Physico Technical Institute.  3 (24), 60–65 (1992), (in Russian).
  40. Alvarez F. X., Jou D., Sellitto A.  Pore-size dependence of the thermal conductivity of porous silicon: A phonon hydrodynamic approach.  Applied Physics Letters.  97 (3), 033103 (2010).
  41. Kobryn O. E., Omelyan I. P., Tokarchuk M. V.  The modified group expansions for constructions of solutions to the BBGKY hierarchy.  Journal of Statistical Physics.  92 (5–6), 973–994 (1998).
  42. Tokarchuk M. V., Kobryn O. E., Omelyan I. P.  Consistent description of kinetics and hydrodynamics of systems of interacting particles by means of nonequilibrium operator method.  Condensed Matter Physics.  1 (4), 687–751  (1998).
  43. Kobryn A. E., Morozov V. G., Omelyan I. P., Tokarchuk M. V.  Enskog–Landau kinetic equation. Calculation of the transport coefficient for charged hard spheres.  Physica A: Statistical Mechanics and its Applications.  230 (1–2), 189–201 (1996).
  44. Tokarchuk M. V.  To the kinetic theory of dense gases and liquids. Calculation of quasi-equilibrium particle distribution functions by the method of collective variables.  Mathematical Modeling and Computing.  9 (2), 440–458 (2022).
  45. Polewczak J., Stell G.  Transport Coefficients in Some Stochastic Models of the Revised Enskog Equation.  Journal of Statistical Physics.  109,  569–590 (2002).
  46. Polewczak J.  Hard-sphere kinetic models for inert and reactive mixtures.  Journal of Physics: Condensed Matter.  28 (41), 414022 (2016).
  47. Karkheck J., Stell G.  Maximization of entropy, kinetic equations, and irreversible thermodynamics.  Physical Review A.  25 (6),  3302–3327 (1982).
  48. Yukhnovskii I. R., Holovko M. F.  Statistical Theory of Classical Equilibrium Systems.  Naukova Dumka, Kiev (1980), (in Russian).
  49. Zubarev D. N.  Statistical thermodynammics of turbulent transport processes.  Theoretical and Mathematical Physics.  53 (1), 1004–1014 (1982).
  50. Idzyk I. M., Ighatyuk V. V., Tokarchuk M. V.  Fokker–Planck equation for nonequilibrium distribution function collective variables.  I. Calculation of statistical weight, entropy, hydrodynamic speeds.  Ukrainian Journal of Physics.  41 (5-6), 596–604 (1996), (in Ukrainian).
  51. Hlushak P., Tokarchuk M.  Chain of kinetic equations for the distribution functions of particles in simple liquid taking into account nonlinear hydrodynamic fluctuations.  Physica A: Statistical Mechanics and its Applications.  443, 231–245 (2016).
  52. Yukhnovskii I. R., Hlushak P. A., Tokarchuk M. V.  BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids.  Condensed Matter Physics.  19 (4), 43705 (2016).
  53. Ramshaw J. D.  Time-dependent direct correlashion function.  Physical Review A.  24 (3), 1567–1570 (1981).
  54. Eu B. C.  Dynamic Ornstein–Zernike Equation.  In: Transport Coefficients of Fluids. Chemical physics. Vol. 82, 221–240.  Springer, Berlin, Heidelberg (2006).
  55. Gan H. H., Eu B. C.  Theory of the nonequilibrium structure of dense simple fluids: Effects of shearing.  Physical Review A.  45 (6), 3670–3686 (1992).
  56. Brader J. M., Schmidt M.  Nonequilibrium Ornstein-Zernike relation for Brownian many-body dynamics.  The Journal of Chemical Physics.  139 (10), 104108 (2013).
  57. Holovko M. F., Kalyuzhnyi Yu. V.  On the effects of association in the statistical theory of ionic systems. Analytic solution of the PY-MSA version of the Wertheim theory.  Molecular Physics.  73 (5), 1145–1157 (1991).
  58. Kalyuzhnyi Y. V., Vlachy V., Holovko M. F., Stell G.  Multidensity integral equation theory for highly asymmetric electrolyte solutions.  The Journal of Chemical Physics.  102 (14), 5770–5780 (1995).
  59. Holovko M. F., Sovyak E. M.  On taking account of interactions in the statistical theory of electrolyte solutions.  Condensed Matter Physics.  6, 49–78 (1995).
  60. Holovko M. F., Patsahan T. M., Shmotolokha V. I.  What is liquid in random porous media: the Barker-Henderson perturbation theory.  Condensed Matter Physics.  18 (1), 13607 (2015).