American Institute of Mathematical Sciences

Advanced
September  2012, 17(6): 1939-1967. doi: 10.3934/dcdsb.2012.17.1939

On the structure of double layers in Poisson-Boltzmann equation

Marco A. Fontelos 1, and  Lucía B. Gamboa 2,

1. 

ICMAT (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 15, Madrid, 28049
2. 

Instituto de Ciencias Matemáticas (CSIC - UAM - UC3M - UCM), C/Nicolás Cabrera, 13-15, Campus de Cantoblanco, 28049 Madrid, Spain

Received  May 2011 Revised  September 2011 Published  May 2012

We study the solutions to Poisson-Boltzmann equation for electrolytic solutions in a domain $\Omega$, surrounded by an uncharged dielectric medium. We establish existence, uniqueness and regularity of solutions and study in detail their asymptotic behaviour close to $\partial\Omega$ when a characteristic length, called the Debye length, is sufficiently small. This is a double layer with a thickness that changes from point to point along $\partial\Omega$ depending on the normal derivative of a harmonic function outside $\Omega$ and the mean curvature of $\partial\Omega$. We also provide numerical evidence of our results based on a finite elements approximation of the problem.
Keywords: electrokinetic equations., finite element method, Poisson-Boltzmann equation, non-local elliptic equation, Debye boundary layer.
Mathematics Subject Classification: Primary: 35Q35, 35J15; Secondary: 35B4.
Citation: Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939
References:
[1]

S. I. Betelú, M. A. Fontelos, U. Kindelán and O. Vantzos, Sigularities on charged drops,, Phys. Fluids, 18 (2006).   Google Scholar

[2]

D. Duft, T. Achtzehn, R. Müller, B. A. Huber and T. Leisner, Rayleigh jets from levitated microdroplets,, Nature, 421 (2003).  doi: 10.1038/421128a.  Google Scholar

[3]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998).   Google Scholar

[4]

J. Fernández de la Mora, The fluid dynamics of Taylor cones,, Annual Review of Fluid Mechanics, 39 (2007), 217.  doi: 10.1146/annurev.fluid.39.050905.110159.  Google Scholar

[5]

M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of charged drops,, Arch. Ration. Mech. Anal., 172 (2004), 267.  doi: 10.1007/s00205-003-0298-x.  Google Scholar

[6]

M. A. Fontelos, U. Kindelán and O. Vantzos, Evolution of neutral and charged droplets in an electric field,, Phys. Fluids, 20 (2008).  doi: 10.1063/1.2980030.  Google Scholar

[7]

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

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar

[9]

J. H. Masliyah and S. Bhattacharjee, "Electrokinetic and Colloid Transport Phenomena,", Wiley, (2006).   Google Scholar

[10]

F. Mugele and J. C. Baret, Electrowetting: From basics to applications,, J. Phys. Condens. Matter, 17 (2005).  doi: 10.1088/0953-8984/17/28/R01.  Google Scholar

[11]

C. Quillet and B. Berge, Electrowetting: A recent outbreak,, Current Opinion in Colloid & Interface Science, 6 (2001), 34.   Google Scholar

[12]

I. Rubinstein, "Electro-Diffusion of Ions,", SIAM Studies in Applied Mathematics, 11 (1990).   Google Scholar

[13]

R. J. Ryham, Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electro-hydrodinamics,, preprint., ().   Google Scholar

[14]

D. A. Saville, Electrohydrodynamics: The Taylor-Melcher Leaky dielectric model,, Annual Review of Fluid Mechanics, 29 (1997), 27.  doi: 10.1146/annurev.fluid.29.1.27.  Google Scholar

[15]

H. A. Stone, A. D. Stroock and A. Ajdari, Engineering flows in small devices: Microfluidics toward a lab-on-a-chip,, Annu. Rev. Fluid Mech., 36 (2004), 381.   Google Scholar

show all references

References:
[1]

S. I. Betelú, M. A. Fontelos, U. Kindelán and O. Vantzos, Sigularities on charged drops,, Phys. Fluids, 18 (2006).   Google Scholar

[2]

D. Duft, T. Achtzehn, R. Müller, B. A. Huber and T. Leisner, Rayleigh jets from levitated microdroplets,, Nature, 421 (2003).  doi: 10.1038/421128a.  Google Scholar

[3]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998).   Google Scholar

[4]

J. Fernández de la Mora, The fluid dynamics of Taylor cones,, Annual Review of Fluid Mechanics, 39 (2007), 217.  doi: 10.1146/annurev.fluid.39.050905.110159.  Google Scholar

[5]

M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of charged drops,, Arch. Ration. Mech. Anal., 172 (2004), 267.  doi: 10.1007/s00205-003-0298-x.  Google Scholar

[6]

M. A. Fontelos, U. Kindelán and O. Vantzos, Evolution of neutral and charged droplets in an electric field,, Phys. Fluids, 20 (2008).  doi: 10.1063/1.2980030.  Google Scholar

[7]

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

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar

[9]

J. H. Masliyah and S. Bhattacharjee, "Electrokinetic and Colloid Transport Phenomena,", Wiley, (2006).   Google Scholar

[10]

F. Mugele and J. C. Baret, Electrowetting: From basics to applications,, J. Phys. Condens. Matter, 17 (2005).  doi: 10.1088/0953-8984/17/28/R01.  Google Scholar

[11]

C. Quillet and B. Berge, Electrowetting: A recent outbreak,, Current Opinion in Colloid & Interface Science, 6 (2001), 34.   Google Scholar

[12]

I. Rubinstein, "Electro-Diffusion of Ions,", SIAM Studies in Applied Mathematics, 11 (1990).   Google Scholar

[13]

R. J. Ryham, Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electro-hydrodinamics,, preprint., ().   Google Scholar

[14]

D. A. Saville, Electrohydrodynamics: The Taylor-Melcher Leaky dielectric model,, Annual Review of Fluid Mechanics, 29 (1997), 27.  doi: 10.1146/annurev.fluid.29.1.27.  Google Scholar

[15]

H. A. Stone, A. D. Stroock and A. Ajdari, Engineering flows in small devices: Microfluidics toward a lab-on-a-chip,, Annu. Rev. Fluid Mech., 36 (2004), 381.   Google Scholar

[1]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052
[2]

Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179
[3]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095
[4]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[5]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221
[6]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120
[7]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097
[8]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052
[9]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004
[10]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126
[11]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495
[12]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061
[13]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448
[14]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089
[15]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127
[16]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351
[17]

Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349
[18]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003
[19]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[20]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences