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 & Continuous Dynamical Systems - A, 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.  doi: 10.1137/S0036139995312149.  Google Scholar

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D. Bothe, A. Fischer and J. Saal, Global well-posedness and stability of electrokinetic flows,, SIAM J. Math. Anal., 46 (2014), 1263.  doi: 10.1137/120880926.  Google Scholar

[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.  doi: 10.1006/jcis.1994.1212.  Google Scholar

[6]

Y. S. Choi and R. Lui, An integro-differential equation arising from an electrochemistry model,, Quart. Appl. Math., 55 (1997), 677.   Google Scholar

[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.  doi: 10.1017/S0022112010001254.  Google Scholar

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B. Eisenberg, Ionic channels in biological membranes: Natural nanotubes,, Acc. Chem. Res., 31 (1998), 117.  doi: 10.1021/ar950051e.  Google Scholar

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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, (1953).   Google Scholar

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W. Fang and K. Ito, Existence and Uniqueness of Steady-State Solutions for an Electrochemistry Model,, P. Am. Math. Soc., 129 (2001), 1037.  doi: 10.1090/S0002-9939-00-05769-5.  Google Scholar

[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.  doi: 10.3934/dcdsb.2012.17.1939.  Google Scholar

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A. Friedman and K. Tintarev, Boundary asymptotics for solutions of the Poisson-Boltzmann equation,, J. Differential Equations, 69 (1987), 15.  doi: 10.1016/0022-0396(87)90100-8.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

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A. Glitzky and R. Hünlich, Energetic estimates and asymptotics for electro-reaction-diffusion systems,, Z. Angew. Math. Mech., 77 (1997), 823.  doi: 10.1002/zamm.19970771105.  Google Scholar

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M. J. Holst, Multilevel Methods for the Poisson-Boltzmann Equation,, Ph.D thesis, (1993).   Google Scholar

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Y. Hyon, A Mathematical Model For Electrical Activity in Cell Membrane: Energetic Variational Approach,, work in progress., ().   Google Scholar

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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.  doi: 10.1006/jcis.2002.8453.  Google Scholar

[18]

C. Koch, Biophysics of Computation,, Oxford University Press, (1999).   Google Scholar

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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.  doi: 10.1140/epje/i2008-10433-1.  Google Scholar

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L. Lanzani and Z. Shen, On the Robin boundary condition for Laplace's equation in Lipschit domains,, Commun. Partial Differ. Eq., 29 (2004), 91.  doi: 10.1081/PDE-120028845.  Google Scholar

[21]

C. C. Lee, The charge conserving Poisson-Boltzmann equations: Existence, uniqueness and maximum principle,, J. Math. Phys., 55 (2014).  doi: 10.1063/1.4878492.  Google Scholar

[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.  doi: 10.1088/0951-7715/24/2/004.  Google Scholar

[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., ().   Google Scholar

[24]

W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species,, J. Differential Equations, 246 (2009), 428.  doi: 10.1016/j.jde.2008.09.010.  Google Scholar

[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.  doi: 10.1016/j.jde.2014.10.015.  Google Scholar

[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.  doi: 10.1016/S0006-3495(98)74048-2.  Google Scholar

[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.  doi: 10.1137/S0036139995279809.  Google Scholar

[28]

I. Rubinstein, Counterion condensation as an exact limiting property of solutions of the Poisson-Boltzmann equation,, SIAM J. Appl. Math., 46 (1986), 1024.  doi: 10.1137/0146061.  Google Scholar

[29]

R. Ryham, C. Liu and Z. Q. Wang, On electro-kinetic fluids: One dimensional configurations,, Discrete Contin Dyn. Syst. B, 6 (2006), 357.  doi: 10.3934/dcdsb.2006.6.357.  Google Scholar

[30]

R. Ryham, C. Liu and L. Zikatanov, Mathematical models for the deformation of electrolyte droplets,, Discrete Contin Dyn. Syst. B, 8 (2007), 649.  doi: 10.3934/dcdsb.2007.8.649.  Google Scholar

[31]

H. Sugioka, Ion-conserving Poisson-Boltzmann theory,, Phys. Rev. E, 86 (2012).  doi: 10.1103/PhysRevE.86.016318.  Google Scholar

[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).  doi: 10.1103/PhysRevX.4.011042.  Google Scholar

[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).  doi: 10.1103/PhysRevE.81.031912.  Google Scholar

[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).  doi: 10.1103/PhysRevE.84.021901.  Google Scholar

show all references

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.  doi: 10.1137/S0036139995312149.  Google Scholar

[2]

M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems,, Phys. Rev. E, 70 (2004).  doi: 10.1103/PhysRevE.70.021506.  Google Scholar

[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.  doi: 10.1137/040609938.  Google Scholar

[4]

D. Bothe, A. Fischer and J. Saal, Global well-posedness and stability of electrokinetic flows,, SIAM J. Math. Anal., 46 (2014), 1263.  doi: 10.1137/120880926.  Google Scholar

[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.  doi: 10.1006/jcis.1994.1212.  Google Scholar

[6]

Y. S. Choi and R. Lui, An integro-differential equation arising from an electrochemistry model,, Quart. Appl. Math., 55 (1997), 677.   Google Scholar

[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.  doi: 10.1017/S0022112010001254.  Google Scholar

[8]

B. Eisenberg, Ionic channels in biological membranes: Natural nanotubes,, Acc. Chem. Res., 31 (1998), 117.  doi: 10.1021/ar950051e.  Google Scholar

[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, (1953).   Google Scholar

[10]

W. Fang and K. Ito, Existence and Uniqueness of Steady-State Solutions for an Electrochemistry Model,, P. Am. Math. Soc., 129 (2001), 1037.  doi: 10.1090/S0002-9939-00-05769-5.  Google Scholar

[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.  doi: 10.3934/dcdsb.2012.17.1939.  Google Scholar

[12]

A. Friedman and K. Tintarev, Boundary asymptotics for solutions of the Poisson-Boltzmann equation,, J. Differential Equations, 69 (1987), 15.  doi: 10.1016/0022-0396(87)90100-8.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[14]

A. Glitzky and R. Hünlich, Energetic estimates and asymptotics for electro-reaction-diffusion systems,, Z. Angew. Math. Mech., 77 (1997), 823.  doi: 10.1002/zamm.19970771105.  Google Scholar

[15]

M. J. Holst, Multilevel Methods for the Poisson-Boltzmann Equation,, Ph.D thesis, (1993).   Google Scholar

[16]

Y. Hyon, A Mathematical Model For Electrical Activity in Cell Membrane: Energetic Variational Approach,, work in progress., ().   Google Scholar

[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.  doi: 10.1006/jcis.2002.8453.  Google Scholar

[18]

C. Koch, Biophysics of Computation,, Oxford University Press, (1999).   Google Scholar

[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.  doi: 10.1140/epje/i2008-10433-1.  Google Scholar

[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.  doi: 10.1081/PDE-120028845.  Google Scholar

[21]

C. C. Lee, The charge conserving Poisson-Boltzmann equations: Existence, uniqueness and maximum principle,, J. Math. Phys., 55 (2014).  doi: 10.1063/1.4878492.  Google Scholar

[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.  doi: 10.1088/0951-7715/24/2/004.  Google Scholar

[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., ().   Google Scholar

[24]

W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species,, J. Differential Equations, 246 (2009), 428.  doi: 10.1016/j.jde.2008.09.010.  Google Scholar

[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.  doi: 10.1016/j.jde.2014.10.015.  Google Scholar

[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.  doi: 10.1016/S0006-3495(98)74048-2.  Google Scholar

[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.  doi: 10.1137/S0036139995279809.  Google Scholar

[28]

I. Rubinstein, Counterion condensation as an exact limiting property of solutions of the Poisson-Boltzmann equation,, SIAM J. Appl. Math., 46 (1986), 1024.  doi: 10.1137/0146061.  Google Scholar

[29]

R. Ryham, C. Liu and Z. Q. Wang, On electro-kinetic fluids: One dimensional configurations,, Discrete Contin Dyn. Syst. B, 6 (2006), 357.  doi: 10.3934/dcdsb.2006.6.357.  Google Scholar

[30]

R. Ryham, C. Liu and L. Zikatanov, Mathematical models for the deformation of electrolyte droplets,, Discrete Contin Dyn. Syst. B, 8 (2007), 649.  doi: 10.3934/dcdsb.2007.8.649.  Google Scholar

[31]

H. Sugioka, Ion-conserving Poisson-Boltzmann theory,, Phys. Rev. E, 86 (2012).  doi: 10.1103/PhysRevE.86.016318.  Google Scholar

[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).  doi: 10.1103/PhysRevX.4.011042.  Google Scholar

[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).  doi: 10.1103/PhysRevE.81.031912.  Google Scholar

[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).  doi: 10.1103/PhysRevE.84.021901.  Google Scholar

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