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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. AllaireR. BrizziJ.-F. DufrêcheA. 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.  Google Scholar

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M. BurgerB. 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.  Google Scholar

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C. Chainais-HillairetM. 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.  Google Scholar

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W. DreyerC. Guhlke and R. Müller, Overcoming the shortcomings of the Nernst-Planck model, Phys. Chem. Chem. Phys., 15 (2013), 7075-7086.  doi: 10.1039/c3cp44390f.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. GahnM. 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.  Google Scholar

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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.  Google Scholar

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M. HerzN. 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.  Google Scholar

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A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, Southampton-Boston: WIT Press, 2000. Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[23]

A. LatzJ. 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.  Google Scholar

[24]

P. , A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer Verlag, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[25]

I. Prigogine, Étude Thermodynamique des Processes Irreversibles, Desoer, Lieg, 1947. Google Scholar

[26]

T. Roubíček, Incompressible ionized non-Newtonean fluid mixtures, SIAM J. Math. Anal., 39 (2007), 863-890.  doi: 10.1137/060667335.  Google Scholar

[27]

T. Roubíček, Incompressible ionized fluid mixtures: A non-Newtonian approach, IASME Trans., 2 (2005), 1190-1197.   Google Scholar

[28]

S.A. SazhenkovE.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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

G. AllaireR. BrizziJ.-F. DufrêcheA. 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.  Google Scholar

[2]

K. Becker-SteinbergerS. FunkenM. 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.  Google Scholar

[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.  Google Scholar

[4]

M. BurgerB. 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.  Google Scholar

[5]

C. Chainais-HillairetM. 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.  Google Scholar

[6]

S. , R. De Groot and P. Mazur, Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1962.  Google Scholar

[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.  Google Scholar

[8]

W. DreyerC. Guhlke and R. Müller, Overcoming the shortcomings of the Nernst-Planck model, Phys. Chem. Chem. Phys., 15 (2013), 7075-7086.  doi: 10.1039/c3cp44390f.  Google Scholar

[9]

W. DreyerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

T. R. Ferguson and M.Z. Bazant, Nonequilibrium thermodynamics of porous electrodes, J. Electrochem. Soc., 159 (2012), 1967-1985.   Google Scholar

[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.  Google Scholar

[15]

M. GahnM. 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.  Google Scholar

[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.  Google Scholar

[17]

M. HerzN. 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.  Google Scholar

[18]

A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, Southampton-Boston: WIT Press, 2000. Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[23]

A. LatzJ. 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.  Google Scholar

[24]

P. , A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer Verlag, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[25]

I. Prigogine, Étude Thermodynamique des Processes Irreversibles, Desoer, Lieg, 1947. Google Scholar

[26]

T. Roubíček, Incompressible ionized non-Newtonean fluid mixtures, SIAM J. Math. Anal., 39 (2007), 863-890.  doi: 10.1137/060667335.  Google Scholar

[27]

T. Roubíček, Incompressible ionized fluid mixtures: A non-Newtonian approach, IASME Trans., 2 (2005), 1190-1197.   Google Scholar

[28]

S.A. SazhenkovE.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.   Google Scholar

[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.  Google Scholar

Figure 1.  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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