# American Institute of Mathematical Sciences

June  2016, 36(6): 3251-3276. doi: 10.3934/dcds.2016.36.3251

## Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients

 1 Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu City 30014, Taiwan

Received  September 2015 Revised  October 2015 Published  December 2015

Concerning the influence of the dielectric constant on the electrostatic potential in the bulk of electrolyte solutions, we investigate a charge conserving Poisson-Boltzmann (CCPB) equation [31,32] with a variable dielectric coefficient and a small parameter $\epsilon$ (related to the Debye screening length) in a bounded connected domain with smooth boundary. Under the Robin boundary condition with a given applied potential, the limiting behavior (as $\epsilon\downarrow0$) of the solution (the electrostatic potential) has been rigorously studied. In particular, under the charge neutrality constraint, our result exactly shows the effects of the dielectric coefficient and the applied potential on the limiting value of the solution in the interior domain. The main approach is the Pohozaev's identity of this model. On the other hand, under the charge non-neutrality constraint, we show that the maximum difference between the boundary and interior values of the solution has a lower bound $\log\frac{1}{\epsilon}$ as $\epsilon$ goes to zero. Such an asymptotic blow-up behavior describes an unstable phenomenon which is totally different from the behavior of the solution under the charge neutrality constraint.
Citation: Chiun-Chang Lee. Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3251-3276. doi: 10.3934/dcds.2016.36.3251
##### References:
 [1] V. Barcilon, D.-P. Chen, R. S. Eisenberg and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study, SIAM J. Appl. Math., 57 (1997), 631-648. doi: 10.1137/S0036139995312149. [2] M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E, 70 (2004), 021506. doi: 10.1103/PhysRevE.70.021506. [3] M. Z. Bazant, K. T. Chu and B. J. Bayly, Current-Voltage relations for electrochemical thin films, SIAM J. Appl. Math., 65 (2005), 1463-1484. doi: 10.1137/040609938. [4] D. Bothe, A. Fischer and J. Saal, Global well-posedness and stability of electrokinetic flows, SIAM J. Math. Anal., 46 (2014), 1263-1316. doi: 10.1137/120880926. [5] S. L. Carnie, D. Y. C. Chan and J. Stankovich, Computation of forces between spherical colloidal particles: Nonlinear Poisson-Boltzmann theory, J. Colloid Interface Sci., 165 (1994), 116-128. doi: 10.1006/jcis.1994.1212. [6] Y. S. Choi and R. Lui, An integro-differential equation arising from an electrochemistry model, Quart. Appl. Math., 55 (1997), 677-686. [7] D. T. Conroy, R. V. Craster, O. K. Matar and D. T. Papageorgiou, Dynamics and stability of an annular electrolyte film, J. Fluid Mech., 656 (2010), 481-506. doi: 10.1017/S0022112010001254. [8] B. Eisenberg, Ionic channels in biological membranes: Natural nanotubes, Acc. Chem. Res., 31 (1998), 117-123. doi: 10.1021/ar950051e. [9] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions II, Based, in part, on notes left by Harry Bateman, McGraw-Hill, New York, 1953. Available from: http://authors.library.caltech.edu/43491/. [10] W. Fang and K. Ito, Existence and Uniqueness of Steady-State Solutions for an Electrochemistry Model, P. Am. Math. Soc., 129 (2001), 1037-1040. doi: 10.1090/S0002-9939-00-05769-5. [11] M. A. Fontelos and L. B. Gamboa, On the structure of double layers in Poisson-Boltzmann equation, Discrete Contin Dyn. Syst. B, 17 (2012), 1939-1967. doi: 10.3934/dcdsb.2012.17.1939. [12] A. Friedman and K. Tintarev, Boundary asymptotics for solutions of the Poisson-Boltzmann equation, J. Differential Equations, 69 (1987), 15-38. doi: 10.1016/0022-0396(87)90100-8. [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, 2001. doi: 10.1007/978-3-642-61798-0. [14] A. Glitzky and R. Hünlich, Energetic estimates and asymptotics for electro-reaction-diffusion systems, Z. Angew. Math. Mech., 77 (1997), 823-832. doi: 10.1002/zamm.19970771105. [15] M. J. Holst, Multilevel Methods for the Poisson-Boltzmann Equation, Ph.D thesis, Numerical Computing Group, University of Illinois at Urbana-Champaign, 1993. Available from: http://ccom.ucsd.edu/~mholst/pubs/dist/Hols94d.pdf. [16] Y. Hyon, A Mathematical Model For Electrical Activity in Cell Membrane: Energetic Variational Approach, work in progress. [17] Y. J. Kang, C. Yang and X. Y. Huang, Electroosmotic flow in a capillary annulus with high zeta potentials, J. Colloid Interface Sci., 253 (2002), 285-294. doi: 10.1006/jcis.2002.8453. [18] C. Koch, Biophysics of Computation, Oxford University Press, Canada, 1999. Available from: http://www.oupcanada.com/catalog/9780195181999.html. [19] D. Lacoste, G. I. Menon, M. Z. Bazant and J. F. Joanny, Electrostatic and electrokinetic contributions to the elastic moduli of a driven membrane, Eur. Phys. J. E, 28 (2009), 243-264. doi: 10.1140/epje/i2008-10433-1. [20] L. Lanzani and Z. Shen, On the Robin boundary condition for Laplace's equation in Lipschit domains, Commun. Partial Differ. Eq., 29 (2004), 91-109. doi: 10.1081/PDE-120028845. [21] C. C. Lee, The charge conserving Poisson-Boltzmann equations: Existence, uniqueness and maximum principle, J. Math. Phys., 55 (2014), 051503, 16pp. doi: 10.1063/1.4878492. [22] C. C. Lee, H. Lee, Y. Hyon, T. C. Lin and C. Liu, New Poisson-Boltzmann type equations: One-dimensional solutions, Nonlinearity, 24 (2011), 431-458. doi: 10.1088/0951-7715/24/2/004. [23] C. C. Lee, H. Lee, Y. Hyon, T. C. Lin and C. Liu, Boundary layer solutions of Charge Conserving Poisson-Boltzmann equations: One-dimensional case, to appear in Commun. Math. Sci., arXiv:1504.06951v1. [24] W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differential Equations, 246 (2009), 428-451. doi: 10.1016/j.jde.2008.09.010. [25] W. Liu and H. Xu, A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow, J. Differential Equations, 258 (2015), 1192-1228. doi: 10.1016/j.jde.2014.10.015. [26] W. Nonner and B. Eisenberg, Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type calcium channels, Biophys. J., 75 (1998), 1287-1305. doi: 10.1016/S0006-3495(98)74048-2. [27] J. H. Park and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study, SIAM J. Appl. Math., 57 (1997), 609-630. doi: 10.1137/S0036139995279809. [28] I. Rubinstein, Counterion condensation as an exact limiting property of solutions of the Poisson-Boltzmann equation, SIAM J. Appl. Math., 46 (1986), 1024-1038. doi: 10.1137/0146061. [29] R. Ryham, C. Liu and Z. Q. Wang, On electro-kinetic fluids: One dimensional configurations, Discrete Contin Dyn. Syst. B, 6 (2006), 357-371. doi: 10.3934/dcdsb.2006.6.357. [30] R. Ryham, C. Liu and L. Zikatanov, Mathematical models for the deformation of electrolyte droplets, Discrete Contin Dyn. Syst. B, 8 (2007), 649-661. doi: 10.3934/dcdsb.2007.8.649. [31] H. Sugioka, Ion-conserving Poisson-Boltzmann theory, Phys. Rev. E, 86 (2012), 016318. doi: 10.1103/PhysRevE.86.016318. [32] L. Wan, S. Xu, M. Liao, C. Liu and P. Sheng, Self-consistent approach to global charge neutrality in electrokinetics: A surface potential trap model, Phys. Rev. X, 4 (2014), 011042. doi: 10.1103/PhysRevX.4.011042. [33] F. Ziebert, M. Z. Bazant and D. Lacoste, Effective zero-thickness model for a conductive membrane driven by an electric field, Phys. Rev. E, 81 (2010), 031912. doi: 10.1103/PhysRevE.81.031912. [34] S. Zhou, Z. Wang and B. Li, Mean-field description of ionic size effects with non-uniform ionic sizes: A numerical approach, Phys. Rev. E, 84 (2011), 021901. doi: 10.1103/PhysRevE.84.021901.

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##### References:
 [1] V. Barcilon, D.-P. Chen, R. S. Eisenberg and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study, SIAM J. Appl. Math., 57 (1997), 631-648. doi: 10.1137/S0036139995312149. [2] M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E, 70 (2004), 021506. doi: 10.1103/PhysRevE.70.021506. [3] M. Z. Bazant, K. T. Chu and B. J. Bayly, Current-Voltage relations for electrochemical thin films, SIAM J. Appl. Math., 65 (2005), 1463-1484. doi: 10.1137/040609938. [4] D. Bothe, A. Fischer and J. Saal, Global well-posedness and stability of electrokinetic flows, SIAM J. Math. Anal., 46 (2014), 1263-1316. doi: 10.1137/120880926. [5] S. L. Carnie, D. Y. C. Chan and J. Stankovich, Computation of forces between spherical colloidal particles: Nonlinear Poisson-Boltzmann theory, J. Colloid Interface Sci., 165 (1994), 116-128. doi: 10.1006/jcis.1994.1212. [6] Y. S. Choi and R. Lui, An integro-differential equation arising from an electrochemistry model, Quart. Appl. Math., 55 (1997), 677-686. [7] D. T. Conroy, R. V. Craster, O. K. Matar and D. T. Papageorgiou, Dynamics and stability of an annular electrolyte film, J. Fluid Mech., 656 (2010), 481-506. doi: 10.1017/S0022112010001254. [8] B. Eisenberg, Ionic channels in biological membranes: Natural nanotubes, Acc. Chem. Res., 31 (1998), 117-123. doi: 10.1021/ar950051e. [9] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions II, Based, in part, on notes left by Harry Bateman, McGraw-Hill, New York, 1953. Available from: http://authors.library.caltech.edu/43491/. [10] W. Fang and K. Ito, Existence and Uniqueness of Steady-State Solutions for an Electrochemistry Model, P. Am. Math. Soc., 129 (2001), 1037-1040. doi: 10.1090/S0002-9939-00-05769-5. [11] M. A. Fontelos and L. B. Gamboa, On the structure of double layers in Poisson-Boltzmann equation, Discrete Contin Dyn. Syst. B, 17 (2012), 1939-1967. doi: 10.3934/dcdsb.2012.17.1939. [12] A. Friedman and K. Tintarev, Boundary asymptotics for solutions of the Poisson-Boltzmann equation, J. Differential Equations, 69 (1987), 15-38. doi: 10.1016/0022-0396(87)90100-8. [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, 2001. doi: 10.1007/978-3-642-61798-0. [14] A. Glitzky and R. Hünlich, Energetic estimates and asymptotics for electro-reaction-diffusion systems, Z. Angew. Math. Mech., 77 (1997), 823-832. doi: 10.1002/zamm.19970771105. [15] M. J. Holst, Multilevel Methods for the Poisson-Boltzmann Equation, Ph.D thesis, Numerical Computing Group, University of Illinois at Urbana-Champaign, 1993. Available from: http://ccom.ucsd.edu/~mholst/pubs/dist/Hols94d.pdf. [16] Y. Hyon, A Mathematical Model For Electrical Activity in Cell Membrane: Energetic Variational Approach, work in progress. [17] Y. J. Kang, C. Yang and X. Y. Huang, Electroosmotic flow in a capillary annulus with high zeta potentials, J. Colloid Interface Sci., 253 (2002), 285-294. doi: 10.1006/jcis.2002.8453. [18] C. Koch, Biophysics of Computation, Oxford University Press, Canada, 1999. Available from: http://www.oupcanada.com/catalog/9780195181999.html. [19] D. Lacoste, G. I. Menon, M. Z. Bazant and J. F. Joanny, Electrostatic and electrokinetic contributions to the elastic moduli of a driven membrane, Eur. Phys. J. E, 28 (2009), 243-264. doi: 10.1140/epje/i2008-10433-1. [20] L. Lanzani and Z. Shen, On the Robin boundary condition for Laplace's equation in Lipschit domains, Commun. Partial Differ. Eq., 29 (2004), 91-109. doi: 10.1081/PDE-120028845. [21] C. C. Lee, The charge conserving Poisson-Boltzmann equations: Existence, uniqueness and maximum principle, J. Math. Phys., 55 (2014), 051503, 16pp. doi: 10.1063/1.4878492. [22] C. C. Lee, H. Lee, Y. Hyon, T. C. Lin and C. Liu, New Poisson-Boltzmann type equations: One-dimensional solutions, Nonlinearity, 24 (2011), 431-458. doi: 10.1088/0951-7715/24/2/004. [23] C. C. Lee, H. Lee, Y. Hyon, T. C. Lin and C. Liu, Boundary layer solutions of Charge Conserving Poisson-Boltzmann equations: One-dimensional case, to appear in Commun. Math. Sci., arXiv:1504.06951v1. [24] W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differential Equations, 246 (2009), 428-451. doi: 10.1016/j.jde.2008.09.010. [25] W. Liu and H. Xu, A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow, J. Differential Equations, 258 (2015), 1192-1228. doi: 10.1016/j.jde.2014.10.015. [26] W. Nonner and B. Eisenberg, Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type calcium channels, Biophys. J., 75 (1998), 1287-1305. doi: 10.1016/S0006-3495(98)74048-2. [27] J. H. Park and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study, SIAM J. Appl. Math., 57 (1997), 609-630. doi: 10.1137/S0036139995279809. [28] I. Rubinstein, Counterion condensation as an exact limiting property of solutions of the Poisson-Boltzmann equation, SIAM J. Appl. Math., 46 (1986), 1024-1038. doi: 10.1137/0146061. [29] R. Ryham, C. Liu and Z. Q. Wang, On electro-kinetic fluids: One dimensional configurations, Discrete Contin Dyn. Syst. B, 6 (2006), 357-371. doi: 10.3934/dcdsb.2006.6.357. [30] R. Ryham, C. Liu and L. Zikatanov, Mathematical models for the deformation of electrolyte droplets, Discrete Contin Dyn. Syst. B, 8 (2007), 649-661. doi: 10.3934/dcdsb.2007.8.649. [31] H. Sugioka, Ion-conserving Poisson-Boltzmann theory, Phys. Rev. E, 86 (2012), 016318. doi: 10.1103/PhysRevE.86.016318. [32] L. Wan, S. Xu, M. Liao, C. Liu and P. Sheng, Self-consistent approach to global charge neutrality in electrokinetics: A surface potential trap model, Phys. Rev. X, 4 (2014), 011042. doi: 10.1103/PhysRevX.4.011042. [33] F. Ziebert, M. Z. Bazant and D. Lacoste, Effective zero-thickness model for a conductive membrane driven by an electric field, Phys. Rev. E, 81 (2010), 031912. doi: 10.1103/PhysRevE.81.031912. [34] S. Zhou, Z. Wang and B. Li, Mean-field description of ionic size effects with non-uniform ionic sizes: A numerical approach, Phys. Rev. E, 84 (2011), 021901. doi: 10.1103/PhysRevE.84.021901.
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