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February 2018, 11(1): 119-135. doi: 10.3934/krm.2018007

## Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium

 1 Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstraße 36,8010 Graz, Austria 2 Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk, Russia

* Corresponding author: anna.zubkova@uni-graz.at

Received  November 2016 Revised  March 2017 Published  August 2017

Fund Project: The authors are supported by the Austrian Science Fund (FWF) Project P26147-N26: "Object identification problems: numerical analysis" (PION)

In this paper a mathematical model generalizing Poisson-Nernst-Planck system is considered. The generalized model presents electrokinetics of species in a two-phase medium consisted of solid particles and a pore space. The governing relations describe cross-diffusion of the charged species together with the overall electrostatic potential. At the interface between the pore and the solid phases nonlinear electro-chemical reactions are taken into account provided by jumps of field variables. The main advantage of the generalized model is that the total mass balance is kept within our setting. As the result of the variational approach, well-posedness properties of a discontinuous solution of the problem are demonstrated and supported by the energy and entropy estimates.

Citation: Victor A. Kovtunenko, Anna V. Zubkova. Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium. Kinetic & Related Models, 2018, 11 (1) : 119-135. doi: 10.3934/krm.2018007
##### References:
 [1] G. Allaire, R. Brizzi, J.-F. Dufrêche, A. Mikelić and A. Piatnitski, Ion transport in porous media: Derivation of the macroscopic equations using upscaling and properties of the effective coefficients, Comp. Geosci., 17 (2013), 479-495. doi: 10.1007/s10596-013-9342-6. [2] K. Becker-Steinberger, S. Funken, M. Landstorfer and K. Urban, A mathematical model for all solid-state lithium-ion batteries, ECS Trans., 25 (2010), 285-296. doi: 10.1149/1.3393864. [3] I. Borukhov, Charge renormalization of cylinders and spheres: Ion size effects, J. Polym. Sci. Pol. Phys., 42 (2004), 3598-3615. doi: 10.1002/polb.20204. [4] M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries, Nonlinearity, 25 (2012), 961-990. doi: 10.1088/0951-7715/25/4/961. [5] C. Chainais-Hillairet, M. Gisclon and A. Jüngel, A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors, Numer. Meth. Part. Differ. Equations, 27 (2011), 1483-1510. doi: 10.1002/num.20592. [6] S. , R. De Groot and P. Mazur, Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1962. [7] L. Desvillettes and K. Fellner, Duality and entropy methods for reversible reaction-diffusion equations with degenerate diffusion, Math. Meth. Appl. Sci., 38 (2015), 3432-3443. doi: 10.1002/mma.3407. [8] W. Dreyer, C. Guhlke and R. Müller, Overcoming the shortcomings of the Nernst-Planck model, Phys. Chem. Chem. Phys., 15 (2013), 7075-7086. doi: 10.1039/c3cp44390f. [9] W. Dreyer, C. Guhlke and R. Müller, Modeling of electrochemical double layers in thermodynamic non-equilibrium, Phys. Chem. Chem. Phys., 17 (2015), 27176-27194. doi: 10.1039/C5CP03836G. [10] M. Efendiev, Evolution equations arising in the modelling of life sciences, Internat. Ser. Numer. Math. 163 Birkhäuser/Springer, (2013), xii+217 pp. doi: 10.1007/978-3-0348-0615-2. [11] K. Fellner and V. A. Kovtunenko, A discontinuous Poisson-Boltzmann equation with interfacial transfer: homogenisation and residual error estimate, Appl. Anal., 95 (2016), 2661-2682. doi: 10.1080/00036811.2015.1105962. [12] K. Fellner and V. A. Kovtunenko, A singularly perturbed nonlinear Poisson-Boltzmann equation: uniform and super-asymptotic expansions, Math. Meth. Appl. Sci., 38 (2015), 3575-3586. doi: 10.1002/mma.3593. [13] T. R. Ferguson and M.Z. Bazant, Nonequilibrium thermodynamics of porous electrodes, J. Electrochem. Soc., 159 (2012), 1967-1985. [14] J. Fuhrmann, Comparison and numerical treatment of generalized Nernst-Planck Models, Comput. Phys. Commun., 196 (2015), 166-178. doi: 10.1016/j.cpc.2015.06.004. [15] M. Gahn, M. Neuss-Radu and P. Knabner, Homogenization of reaction-diffusion processes in a two-component porous medium with nonlinear flux conditions at the interface, SIAM J. App. Math., 76 (2016), 1819-1843. doi: 10.1137/15M1018484. [16] A. Glitzky and A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. Angew. Math. Phys., 64 (2013), 29-52. doi: 10.1007/s00033-012-0207-y. [17] M. Herz, N. Ray and P. Knabner, Existence and uniqueness of a global weak solution of a Darcy-Nernst-Planck-Poisson system, GAMM-Mitt., 35 (2012), 191-208. doi: 10.1002/gamm.201210013. [18] A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, Southampton-Boston: WIT Press, 2000. [19] V. , A. Kovtunenko, Electro-kinetic structure model with interfacial reactions, In Proc. 7th ECCOMAS Thematic Conference on Smart Structures and Materials SMART 2015, A. L. Araújo, C. A. Mota Soares et al. eds. , IDMEC, Lissabon, 2015. [20] V.A. Kovtunenko and A.V. Zubkova, On generalized Poisson-Nernst-Planck equations with inhomogeneous boundary conditions: a-priori estimates and stability, Math. Meth. Appl. Sci., 40 (2017), 2284-2299. [21] V. A. Kovtunenko and A. V. Zubkova, Solvability and Lyapunov stability of a two-component system of generalized Poisson-Nernst-Planck equations, in: Recent Trends in Operator Theory and Partial Differential Equations (The Roland Duduchava Anniversary Volume), V. Maz'ya, D. Natroshvili, E. Shargorodsky, W. L. Wendland (Eds. ), Operator Theory: Advances and Applications, 258, 173-191, Birkhaeuser, Basel, 2017. [22] O. , A. Ladyzhenskaya, The boundary value problems of mathematical physics, Appl. Math. Sci. , 49 Springer Verlag, (1985), xxx+322 pp. doi: 10.1007/978-1-4757-4317-3. [23] A. Latz, J. Zausch and O. Iliev, Modeling of species and charge transport in Li-ion batteries based on non-equilibrium thermodynamics, Lecture Notes Comput. Sci., 6046 (2011), 329-337. doi: 10.1007/978-3-642-18466-6_39. [24] P. , A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer Verlag, 1990. doi: 10.1007/978-3-7091-6961-2. [25] I. Prigogine, Étude Thermodynamique des Processes Irreversibles, Desoer, Lieg, 1947. [26] T. Roubíček, Incompressible ionized non-Newtonean fluid mixtures, SIAM J. Math. Anal., 39 (2007), 863-890. doi: 10.1137/060667335. [27] T. Roubíček, Incompressible ionized fluid mixtures: A non-Newtonian approach, IASME Trans., 2 (2005), 1190-1197. [28] S.A. Sazhenkov, E.V. Sazhenkova and A.V. Zubkova, Zubkova, Small perturbations of two-phase fluid in pores: Effective macroscopic monophasic viscoelastic behavior, Sib. Élektron. Mat. Izv., 11 (2014), 26-51. [29] M. Schmuck, Modeling and deriving porous media Stokes-Poisson-Nernst-Planck equations by a multi-scale approach, Commun. Math. Sci., 9 (2011), 685-710. doi: 10.4310/CMS.2011.v9.n3.a3.

show all references

##### References:
 [1] G. Allaire, R. Brizzi, J.-F. Dufrêche, A. Mikelić and A. Piatnitski, Ion transport in porous media: Derivation of the macroscopic equations using upscaling and properties of the effective coefficients, Comp. Geosci., 17 (2013), 479-495. doi: 10.1007/s10596-013-9342-6. [2] K. Becker-Steinberger, S. Funken, M. Landstorfer and K. Urban, A mathematical model for all solid-state lithium-ion batteries, ECS Trans., 25 (2010), 285-296. doi: 10.1149/1.3393864. [3] I. Borukhov, Charge renormalization of cylinders and spheres: Ion size effects, J. Polym. Sci. Pol. Phys., 42 (2004), 3598-3615. doi: 10.1002/polb.20204. [4] M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries, Nonlinearity, 25 (2012), 961-990. doi: 10.1088/0951-7715/25/4/961. [5] C. Chainais-Hillairet, M. Gisclon and A. Jüngel, A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors, Numer. Meth. Part. Differ. Equations, 27 (2011), 1483-1510. doi: 10.1002/num.20592. [6] S. , R. De Groot and P. Mazur, Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1962. [7] L. Desvillettes and K. Fellner, Duality and entropy methods for reversible reaction-diffusion equations with degenerate diffusion, Math. Meth. Appl. Sci., 38 (2015), 3432-3443. doi: 10.1002/mma.3407. [8] W. Dreyer, C. Guhlke and R. Müller, Overcoming the shortcomings of the Nernst-Planck model, Phys. Chem. Chem. Phys., 15 (2013), 7075-7086. doi: 10.1039/c3cp44390f. [9] W. Dreyer, C. Guhlke and R. Müller, Modeling of electrochemical double layers in thermodynamic non-equilibrium, Phys. Chem. Chem. Phys., 17 (2015), 27176-27194. doi: 10.1039/C5CP03836G. [10] M. Efendiev, Evolution equations arising in the modelling of life sciences, Internat. Ser. Numer. Math. 163 Birkhäuser/Springer, (2013), xii+217 pp. doi: 10.1007/978-3-0348-0615-2. [11] K. Fellner and V. A. Kovtunenko, A discontinuous Poisson-Boltzmann equation with interfacial transfer: homogenisation and residual error estimate, Appl. Anal., 95 (2016), 2661-2682. doi: 10.1080/00036811.2015.1105962. [12] K. Fellner and V. A. Kovtunenko, A singularly perturbed nonlinear Poisson-Boltzmann equation: uniform and super-asymptotic expansions, Math. Meth. Appl. Sci., 38 (2015), 3575-3586. doi: 10.1002/mma.3593. [13] T. R. Ferguson and M.Z. Bazant, Nonequilibrium thermodynamics of porous electrodes, J. Electrochem. Soc., 159 (2012), 1967-1985. [14] J. Fuhrmann, Comparison and numerical treatment of generalized Nernst-Planck Models, Comput. Phys. Commun., 196 (2015), 166-178. doi: 10.1016/j.cpc.2015.06.004. [15] M. Gahn, M. Neuss-Radu and P. Knabner, Homogenization of reaction-diffusion processes in a two-component porous medium with nonlinear flux conditions at the interface, SIAM J. App. Math., 76 (2016), 1819-1843. doi: 10.1137/15M1018484. [16] A. Glitzky and A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. Angew. Math. Phys., 64 (2013), 29-52. doi: 10.1007/s00033-012-0207-y. [17] M. Herz, N. Ray and P. Knabner, Existence and uniqueness of a global weak solution of a Darcy-Nernst-Planck-Poisson system, GAMM-Mitt., 35 (2012), 191-208. doi: 10.1002/gamm.201210013. [18] A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, Southampton-Boston: WIT Press, 2000. [19] V. , A. Kovtunenko, Electro-kinetic structure model with interfacial reactions, In Proc. 7th ECCOMAS Thematic Conference on Smart Structures and Materials SMART 2015, A. L. Araújo, C. A. Mota Soares et al. eds. , IDMEC, Lissabon, 2015. [20] V.A. Kovtunenko and A.V. Zubkova, On generalized Poisson-Nernst-Planck equations with inhomogeneous boundary conditions: a-priori estimates and stability, Math. Meth. Appl. Sci., 40 (2017), 2284-2299. [21] V. A. Kovtunenko and A. V. Zubkova, Solvability and Lyapunov stability of a two-component system of generalized Poisson-Nernst-Planck equations, in: Recent Trends in Operator Theory and Partial Differential Equations (The Roland Duduchava Anniversary Volume), V. Maz'ya, D. Natroshvili, E. Shargorodsky, W. L. Wendland (Eds. ), Operator Theory: Advances and Applications, 258, 173-191, Birkhaeuser, Basel, 2017. [22] O. , A. Ladyzhenskaya, The boundary value problems of mathematical physics, Appl. Math. Sci. , 49 Springer Verlag, (1985), xxx+322 pp. doi: 10.1007/978-1-4757-4317-3. [23] A. Latz, J. Zausch and O. Iliev, Modeling of species and charge transport in Li-ion batteries based on non-equilibrium thermodynamics, Lecture Notes Comput. Sci., 6046 (2011), 329-337. doi: 10.1007/978-3-642-18466-6_39. [24] P. , A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer Verlag, 1990. doi: 10.1007/978-3-7091-6961-2. [25] I. Prigogine, Étude Thermodynamique des Processes Irreversibles, Desoer, Lieg, 1947. [26] T. Roubíček, Incompressible ionized non-Newtonean fluid mixtures, SIAM J. Math. Anal., 39 (2007), 863-890. doi: 10.1137/060667335. [27] T. Roubíček, Incompressible ionized fluid mixtures: A non-Newtonian approach, IASME Trans., 2 (2005), 1190-1197. [28] S.A. Sazhenkov, E.V. Sazhenkova and A.V. Zubkova, Zubkova, Small perturbations of two-phase fluid in pores: Effective macroscopic monophasic viscoelastic behavior, Sib. Élektron. Mat. Izv., 11 (2014), 26-51. [29] M. Schmuck, Modeling and deriving porous media Stokes-Poisson-Nernst-Planck equations by a multi-scale approach, Commun. Math. Sci., 9 (2011), 685-710. doi: 10.4310/CMS.2011.v9.n3.a3.
An example domain $\Omega = Q \cup \omega \cup \partial\omega$ with two phases $Q$ and $\omega$, the boundary $\partial\Omega$, and two faces $\partial\omega^+$ and $\partial\omega^-$ of the interface $\partial\omega$ shown in zoom.
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