January  2013, 12(1): 157-206. doi: 10.3934/cpaa.2013.12.157

Existence theory for a Poisson-Nernst-Planck model of electrophoresis

1. 

Departamento de Matemática, Universidade Federal de Santa Catarina, Campus Trindade, Florianópolis-SC, Brazil, CEP 88040-900, Brazil

2. 

Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonáalves 9500, Porto Alegre-RS, Brazil, CEP 91509-900, Brazil

Received  February 2011 Revised  January 2012 Published  September 2012

A system modeling the electrophoretic motion of a charged rigid macromolecule immersed in a incompressible ionized fluid is considered. The ionic concentration is governing by the Nernst-Planck equation coupled with the Poisson equation for the electrostatic potential, Navier-Stokes and Newtonian equations for the fluid and the macromolecule dynamics, respectively. A local in time existence result for suitable weak solutions is established, following the approach of [15].
Citation: L. Bedin, Mark Thompson. Existence theory for a Poisson-Nernst-Planck model of electrophoresis. Communications on Pure & Applied Analysis, 2013, 12 (1) : 157-206. doi: 10.3934/cpaa.2013.12.157
References:
[1]

S. A. Allison, C. Chen and D. Stigter, The length dependence of translational diffusion, free solution electrophoretic mobility, and electrophoretic tether force of rigid rod-like model duplex DNA,, Biophys. J., 81 (2001), 2558. doi: 10.1016/S0006-3495(01)75900-0. Google Scholar

[2]

S. A. Allison and D. Stigter, A commentary on the screened-Oseen, counterion-condensation formalism of polyion electrophoresis,, Biophys. J., 78 (2000), 121. doi: 10.1016/S0006-3495(00)76578-7. Google Scholar

[3]

J. L. Anderson, Colloidal transport by interfacial forces,, Ann. Rev. Fluid Mech., 21 (1989), 61. doi: 10.1146/annurev.fl.21.010189.000425. Google Scholar

[4]

L. Bedin and M. Thompson, Motion of a charged particle in ionized fluids,, Math. Models $&$ Meth. Appl. Sci., 16 (2006), 1271. doi: 10.1142/S0218202506001546. Google Scholar

[5]

L. Bedin and M. Thompson, Weak solutions for the electrophoretic motion of charged Particles,, Comp. $&$ App. Math., 25 (2006), 1. doi: 10.1590/S0101-82052006000100001. Google Scholar

[6]

H. Brézis, "Análisis Funcional: Teoría y Aplicaciones,", Alianza Editorial, (1984). Google Scholar

[7]

Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems,, Numer. Math., 79 (1998), 175. doi: 10.1007/s002110050336. Google Scholar

[8]

W. L. Cheng, Y. Y. He and E. Lee, Electrophoresis of a soft particle normal to a plane,, J. Colloid Interface Sci., 335 (2009), 130. doi: 10.1016/j.jcis.2009.02.051. Google Scholar

[9]

Y. S. Choi and S. J. Kim, Electrokinetic flow-induced currents in silica nanofluidic channels,, J. Colloid Interface Sci., 333 (2009), 672. doi: 10.1016/j.jcis.2009.01.061. Google Scholar

[10]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", Krieger, (1992). Google Scholar

[11]

C. M. Cortis and R. A. Friesner, An automatic three-dimensional finit element mesh generation system for the Poisson-Boltzmann equation,, J. Comp. Chem., 18 (1997), 1570. doi: 10.1002/(SICI)1096-987X(199710)18:13<1570::AID-JCC2>3.0.CO;2-O. Google Scholar

[12]

M. Daune, "Molecular Biophysics: Structures in Motion,", Oxford University Press, (1999). Google Scholar

[13]

B. Desjardins, Weak solutions of the compressible isentropic Navier-Stokes equations,, App. Math. Letters, 12 (1999), 107. doi: 10.1016/S0893-9659(99)00109-3. Google Scholar

[14]

B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid,, Arch. Rational Mech. Anal., 146 (1999), 59. doi: 10.1007/s002050050136. Google Scholar

[15]

B. Desjardins and M. J. Esteban, On weak solutions for fluid-rigid structure interaction: compressible and incompressible models,, Comm. Partial Diff. Eq., 25 (2000), 1399. doi: 10.1080/03605300008821553. Google Scholar

[16]

R. J. Di Perna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511. doi: 10.1007/BF01393835. Google Scholar

[17]

L. C. Evans, "Partial Differential Equation,", AMS, (1998). Google Scholar

[18]

M. Fixman, Charged macromolecules in external fields I. The sphere,, J. Chem. Phys., 72 (1980), 5177. doi: 10.1063/1.439753. Google Scholar

[19]

A. Friedman, "Partial Differential Equations of Parabolic Type,", Dover, (2008). Google Scholar

[20]

L. M. Fu, R. J. Yang and G. B. Lee, Analysis of geometry effects on band spreading of microchip electrophoresis,, Electrophoresis, 23 (2002), 602. doi: 10.1002/1522-2683(200202)23:4<602::AID-ELPS602>3.0.CO;2-N. Google Scholar

[21]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (2001). Google Scholar

[22]

W. Hackbusch, "Integral Equations: Theory and Numerical Treatment,", Birkh\, (1995). Google Scholar

[23]

J. Huang and J. Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems,, J. Diff. Equations, 184 (2002), 570. doi: 10.1006/jdeq.2001.4154. Google Scholar

[24]

H. J. Keh and J. L. Anderson, Boundary effects on electrophoretic motion of colloidal spheres,, J. Fluid Mech., 153 (1985), 417. doi: 10.1017/S002211208500132X. Google Scholar

[25]

J. Y. Kim and B. J. Yoon, Electrophoretic motion of a slightly deformed sphere with a nonuniform zeta potential distribution,, J. Colloid Interface Sci., 251 (2002), 318. doi: 10.1006/jcis.2002.8359. Google Scholar

[26]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,", AMS, (1988). Google Scholar

[27]

O. A. Ladyženskaja and N. N. Ural'ceva, "Linear and Quasi-Linear Elliptic Equations,", Academic Press, (1968). Google Scholar

[28]

B. Lu, Y. C. Zhou, G. A. Huber, S. D. Bond, M. J. Holst and J. A. McCannon, Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution,, J. Chem. Phys., 127 (2007), 1. doi: 10.1063/1.2775933. Google Scholar

[29]

H. Nakamura, Roles of electrostatic interaction in proteins,, Quart. Rev. Biophys, 29 (1996), 1. doi: 10.1017/S0033583500005746. Google Scholar

[30]

J. Necăs, "Les Méthodes Directes en Théorie des Équations Elliptiques,", Masson Et Cie, (1967). Google Scholar

[31]

H. M. Park, J. S. Lee and T. W. Kim, Comparison of the Nernst-Planck model and the Poisson-Boltzmann model for electroosmotic flows in microchannels,, J. Colloid Interface Sci., 315 (2007), 731. doi: 10.1016/j.jcis.2007.07.007. Google Scholar

[32]

A. Quarteroni and A., Valli, "Numerical Approximation of Partial Differential Equations,", Springer, (1994). Google Scholar

[33]

S. Qian, A. Wang and J. K. Afonien, Electrophoretic motion of a spherical particle in a converging-diverging nanotube,, J. Colloid Interface Sci., 303 (2006), 579. doi: 10.1016/j.jcis.2006.08.003. Google Scholar

[34]

S. E. Reiner and C. J. Radke, Variational Approach to the Electrostatic Free Energy in Charged Colloidal Suspensions: General Theory for Open Systems,, J. Chem. Faraday Trans., 86 (1990), 3901. doi: 10.1039/FT9908603901. Google Scholar

[35]

W. B. Russel, D. A. Saville and W. R. Schowalter, "Colloidal Dispersions,", Cambridge University Press, (1995). Google Scholar

[36]

A. Sellier, A note on the electrophoresis of a uniformly charged particle,, Q. J. Mech. Appl. Math., 55 (2002), 561. doi: 10.1093/qjmam/55.4.561. Google Scholar

[37]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system,, Math. Models $&$ Meth. Appl. Sci., 19 (2009), 993. doi: 10.1142/S0218202509003693. Google Scholar

[38]

A. A. Shugai and S. L. Carnie, Electrophoretic motion of a spherical particle with a thick double layer in bounded flows,, J. Colloid Interface Sci., 213 (1999), 298. doi: 10.1006/jcis.1999.6143. Google Scholar

[39]

Y. Solomentsev and J. L. Anderson, Electrophoresis of slender particles,, J. Fluid Mech., 279 (1994), 197. doi: 10.1017/S0022112094003885. Google Scholar

[40]

M. Teubner, The motion of charged particles in electrical fields,, J. Chem. Phys., 76 (1982), 5564. doi: 10.1063/1.442861. Google Scholar

show all references

References:
[1]

S. A. Allison, C. Chen and D. Stigter, The length dependence of translational diffusion, free solution electrophoretic mobility, and electrophoretic tether force of rigid rod-like model duplex DNA,, Biophys. J., 81 (2001), 2558. doi: 10.1016/S0006-3495(01)75900-0. Google Scholar

[2]

S. A. Allison and D. Stigter, A commentary on the screened-Oseen, counterion-condensation formalism of polyion electrophoresis,, Biophys. J., 78 (2000), 121. doi: 10.1016/S0006-3495(00)76578-7. Google Scholar

[3]

J. L. Anderson, Colloidal transport by interfacial forces,, Ann. Rev. Fluid Mech., 21 (1989), 61. doi: 10.1146/annurev.fl.21.010189.000425. Google Scholar

[4]

L. Bedin and M. Thompson, Motion of a charged particle in ionized fluids,, Math. Models $&$ Meth. Appl. Sci., 16 (2006), 1271. doi: 10.1142/S0218202506001546. Google Scholar

[5]

L. Bedin and M. Thompson, Weak solutions for the electrophoretic motion of charged Particles,, Comp. $&$ App. Math., 25 (2006), 1. doi: 10.1590/S0101-82052006000100001. Google Scholar

[6]

H. Brézis, "Análisis Funcional: Teoría y Aplicaciones,", Alianza Editorial, (1984). Google Scholar

[7]

Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems,, Numer. Math., 79 (1998), 175. doi: 10.1007/s002110050336. Google Scholar

[8]

W. L. Cheng, Y. Y. He and E. Lee, Electrophoresis of a soft particle normal to a plane,, J. Colloid Interface Sci., 335 (2009), 130. doi: 10.1016/j.jcis.2009.02.051. Google Scholar

[9]

Y. S. Choi and S. J. Kim, Electrokinetic flow-induced currents in silica nanofluidic channels,, J. Colloid Interface Sci., 333 (2009), 672. doi: 10.1016/j.jcis.2009.01.061. Google Scholar

[10]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", Krieger, (1992). Google Scholar

[11]

C. M. Cortis and R. A. Friesner, An automatic three-dimensional finit element mesh generation system for the Poisson-Boltzmann equation,, J. Comp. Chem., 18 (1997), 1570. doi: 10.1002/(SICI)1096-987X(199710)18:13<1570::AID-JCC2>3.0.CO;2-O. Google Scholar

[12]

M. Daune, "Molecular Biophysics: Structures in Motion,", Oxford University Press, (1999). Google Scholar

[13]

B. Desjardins, Weak solutions of the compressible isentropic Navier-Stokes equations,, App. Math. Letters, 12 (1999), 107. doi: 10.1016/S0893-9659(99)00109-3. Google Scholar

[14]

B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid,, Arch. Rational Mech. Anal., 146 (1999), 59. doi: 10.1007/s002050050136. Google Scholar

[15]

B. Desjardins and M. J. Esteban, On weak solutions for fluid-rigid structure interaction: compressible and incompressible models,, Comm. Partial Diff. Eq., 25 (2000), 1399. doi: 10.1080/03605300008821553. Google Scholar

[16]

R. J. Di Perna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511. doi: 10.1007/BF01393835. Google Scholar

[17]

L. C. Evans, "Partial Differential Equation,", AMS, (1998). Google Scholar

[18]

M. Fixman, Charged macromolecules in external fields I. The sphere,, J. Chem. Phys., 72 (1980), 5177. doi: 10.1063/1.439753. Google Scholar

[19]

A. Friedman, "Partial Differential Equations of Parabolic Type,", Dover, (2008). Google Scholar

[20]

L. M. Fu, R. J. Yang and G. B. Lee, Analysis of geometry effects on band spreading of microchip electrophoresis,, Electrophoresis, 23 (2002), 602. doi: 10.1002/1522-2683(200202)23:4<602::AID-ELPS602>3.0.CO;2-N. Google Scholar

[21]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (2001). Google Scholar

[22]

W. Hackbusch, "Integral Equations: Theory and Numerical Treatment,", Birkh\, (1995). Google Scholar

[23]

J. Huang and J. Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems,, J. Diff. Equations, 184 (2002), 570. doi: 10.1006/jdeq.2001.4154. Google Scholar

[24]

H. J. Keh and J. L. Anderson, Boundary effects on electrophoretic motion of colloidal spheres,, J. Fluid Mech., 153 (1985), 417. doi: 10.1017/S002211208500132X. Google Scholar

[25]

J. Y. Kim and B. J. Yoon, Electrophoretic motion of a slightly deformed sphere with a nonuniform zeta potential distribution,, J. Colloid Interface Sci., 251 (2002), 318. doi: 10.1006/jcis.2002.8359. Google Scholar

[26]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,", AMS, (1988). Google Scholar

[27]

O. A. Ladyženskaja and N. N. Ural'ceva, "Linear and Quasi-Linear Elliptic Equations,", Academic Press, (1968). Google Scholar

[28]

B. Lu, Y. C. Zhou, G. A. Huber, S. D. Bond, M. J. Holst and J. A. McCannon, Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution,, J. Chem. Phys., 127 (2007), 1. doi: 10.1063/1.2775933. Google Scholar

[29]

H. Nakamura, Roles of electrostatic interaction in proteins,, Quart. Rev. Biophys, 29 (1996), 1. doi: 10.1017/S0033583500005746. Google Scholar

[30]

J. Necăs, "Les Méthodes Directes en Théorie des Équations Elliptiques,", Masson Et Cie, (1967). Google Scholar

[31]

H. M. Park, J. S. Lee and T. W. Kim, Comparison of the Nernst-Planck model and the Poisson-Boltzmann model for electroosmotic flows in microchannels,, J. Colloid Interface Sci., 315 (2007), 731. doi: 10.1016/j.jcis.2007.07.007. Google Scholar

[32]

A. Quarteroni and A., Valli, "Numerical Approximation of Partial Differential Equations,", Springer, (1994). Google Scholar

[33]

S. Qian, A. Wang and J. K. Afonien, Electrophoretic motion of a spherical particle in a converging-diverging nanotube,, J. Colloid Interface Sci., 303 (2006), 579. doi: 10.1016/j.jcis.2006.08.003. Google Scholar

[34]

S. E. Reiner and C. J. Radke, Variational Approach to the Electrostatic Free Energy in Charged Colloidal Suspensions: General Theory for Open Systems,, J. Chem. Faraday Trans., 86 (1990), 3901. doi: 10.1039/FT9908603901. Google Scholar

[35]

W. B. Russel, D. A. Saville and W. R. Schowalter, "Colloidal Dispersions,", Cambridge University Press, (1995). Google Scholar

[36]

A. Sellier, A note on the electrophoresis of a uniformly charged particle,, Q. J. Mech. Appl. Math., 55 (2002), 561. doi: 10.1093/qjmam/55.4.561. Google Scholar

[37]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system,, Math. Models $&$ Meth. Appl. Sci., 19 (2009), 993. doi: 10.1142/S0218202509003693. Google Scholar

[38]

A. A. Shugai and S. L. Carnie, Electrophoretic motion of a spherical particle with a thick double layer in bounded flows,, J. Colloid Interface Sci., 213 (1999), 298. doi: 10.1006/jcis.1999.6143. Google Scholar

[39]

Y. Solomentsev and J. L. Anderson, Electrophoresis of slender particles,, J. Fluid Mech., 279 (1994), 197. doi: 10.1017/S0022112094003885. Google Scholar

[40]

M. Teubner, The motion of charged particles in electrical fields,, J. Chem. Phys., 76 (1982), 5564. doi: 10.1063/1.442861. Google Scholar

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