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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

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