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

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

L. Bedin 1, and  Mark Thompson 2,

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].
Keywords: Electrophoresis, Poisson equation, Navier-Stokes equation, Nernst-Planck equation..
Mathematics Subject Classification: Primary: 35J60, 92C05; Secondary: 35J50, 42B2.
Citation: L. Bedin, Mark Thompson. Existence theory for a Poisson-Nernst-Planck model of electrophoresis. Communications on Pure and 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), 2558-2568. doi: 10.1016/S0006-3495(01)75900-0.

[2]

S. A. Allison and D. Stigter, A commentary on the screened-Oseen, counterion-condensation formalism of polyion electrophoresis, Biophys. J., 78 (2000), 121-124. doi: 10.1016/S0006-3495(00)76578-7.

[3]

J. L. Anderson, Colloidal transport by interfacial forces, Ann. Rev. Fluid Mech., 21 (1989), 61-99. doi: 10.1146/annurev.fl.21.010189.000425.

[4]

L. Bedin and M. Thompson, Motion of a charged particle in ionized fluids, Math. Models & Meth. Appl. Sci., 16 (2006), 1271-1318. doi: 10.1142/S0218202506001546.

[5]

L. Bedin and M. Thompson, Weak solutions for the electrophoretic motion of charged Particles, Comp. & App. Math., 25 (2006), 1-26. doi: 10.1590/S0101-82052006000100001.

[6]

H. Brézis, "Análisis Funcional: Teoría y Aplicaciones," Alianza Editorial, Madrid, 1984.

[7]

Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175-202. doi: 10.1007/s002110050336.

[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), 130-139. doi: 10.1016/j.jcis.2009.02.051.

[9]

Y. S. Choi and S. J. Kim, Electrokinetic flow-induced currents in silica nanofluidic channels, J. Colloid Interface Sci., 333 (2009), 672-678. doi: 10.1016/j.jcis.2009.01.061.

[10]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Krieger, Malabar, 1992.

[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), 1570-1590. doi: 10.1002/(SICI)1096-987X(199710)18:13<1570::AID-JCC2>3.0.CO;2-O.

[12]

M. Daune, "Molecular Biophysics: Structures in Motion," Oxford University Press, New York, 1999.

[13]

B. Desjardins, Weak solutions of the compressible isentropic Navier-Stokes equations, App. Math. Letters, 12 (1999), 107-111. doi: 10.1016/S0893-9659(99)00109-3.

[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), 59-71. doi: 10.1007/s002050050136.

[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), 1399-1414. doi: 10.1080/03605300008821553.

[16]

R. J. Di Perna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.

[17]

L. C. Evans, "Partial Differential Equation," AMS, 1998.

[18]

M. Fixman, Charged macromolecules in external fields I. The sphere, J. Chem. Phys., 72 (1980), 5177-5186. doi: 10.1063/1.439753.

[19]

A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 2008.

[20]

L. M. Fu, R. J. Yang and G. B. Lee, Analysis of geometry effects on band spreading of microchip electrophoresis, Electrophoresis, 23 (2002), 602-612. doi: 10.1002/1522-2683(200202)23:4<602::AID-ELPS602>3.0.CO;2-N.

[21]

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

[22]

W. Hackbusch, "Integral Equations: Theory and Numerical Treatment," Birkhäuser Verlag, Basel, 1995.

[23]

J. Huang and J. Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems, J. Diff. Equations, 184 (2002), 570-586. doi: 10.1006/jdeq.2001.4154.

[24]

H. J. Keh and J. L. Anderson, Boundary effects on electrophoretic motion of colloidal spheres, J. Fluid Mech., 153 (1985), 417-439. doi: 10.1017/S002211208500132X.

[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), 318-330. doi: 10.1006/jcis.2002.8359.

[26]

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

[27]

O. A. Ladyženskaja and N. N. Ural'ceva, "Linear and Quasi-Linear Elliptic Equations," Academic Press, New York-London, 1968.

[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), 1-17. doi: 10.1063/1.2775933.

[29]

H. Nakamura, Roles of electrostatic interaction in proteins, Quart. Rev. Biophys, 29 (1996), 1-90. doi: 10.1017/S0033583500005746.

[30]

J. Necăs, "Les Méthodes Directes en Théorie des Équations Elliptiques," Masson Et Cie, Paris, 1967.

[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), 731-739. doi: 10.1016/j.jcis.2007.07.007.

[32]

A. Quarteroni and A., Valli, "Numerical Approximation of Partial Differential Equations," Springer, Berlin, 1994.

[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), 579-592. doi: 10.1016/j.jcis.2006.08.003.

[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), 3901-3912. doi: 10.1039/FT9908603901.

[35]

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

[36]

A. Sellier, A note on the electrophoresis of a uniformly charged particle, Q. J. Mech. Appl. Math., 55 (2002), 561-572. doi: 10.1093/qjmam/55.4.561.

[37]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models & Meth. Appl. Sci., 19 (2009), 993-1015. doi: 10.1142/S0218202509003693.

[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), 298-315. doi: 10.1006/jcis.1999.6143.

[39]

Y. Solomentsev and J. L. Anderson, Electrophoresis of slender particles, J. Fluid Mech., 279 (1994), 197-215. doi: 10.1017/S0022112094003885.

[40]

M. Teubner, The motion of charged particles in electrical fields, J. Chem. Phys., 76 (1982), 5564-5573. doi: 10.1063/1.442861.

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), 2558-2568. doi: 10.1016/S0006-3495(01)75900-0.

[2]

S. A. Allison and D. Stigter, A commentary on the screened-Oseen, counterion-condensation formalism of polyion electrophoresis, Biophys. J., 78 (2000), 121-124. doi: 10.1016/S0006-3495(00)76578-7.

[3]

J. L. Anderson, Colloidal transport by interfacial forces, Ann. Rev. Fluid Mech., 21 (1989), 61-99. doi: 10.1146/annurev.fl.21.010189.000425.

[4]

L. Bedin and M. Thompson, Motion of a charged particle in ionized fluids, Math. Models & Meth. Appl. Sci., 16 (2006), 1271-1318. doi: 10.1142/S0218202506001546.

[5]

L. Bedin and M. Thompson, Weak solutions for the electrophoretic motion of charged Particles, Comp. & App. Math., 25 (2006), 1-26. doi: 10.1590/S0101-82052006000100001.

[6]

H. Brézis, "Análisis Funcional: Teoría y Aplicaciones," Alianza Editorial, Madrid, 1984.

[7]

Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175-202. doi: 10.1007/s002110050336.

[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), 130-139. doi: 10.1016/j.jcis.2009.02.051.

[9]

Y. S. Choi and S. J. Kim, Electrokinetic flow-induced currents in silica nanofluidic channels, J. Colloid Interface Sci., 333 (2009), 672-678. doi: 10.1016/j.jcis.2009.01.061.

[10]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Krieger, Malabar, 1992.

[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), 1570-1590. doi: 10.1002/(SICI)1096-987X(199710)18:13<1570::AID-JCC2>3.0.CO;2-O.

[12]

M. Daune, "Molecular Biophysics: Structures in Motion," Oxford University Press, New York, 1999.

[13]

B. Desjardins, Weak solutions of the compressible isentropic Navier-Stokes equations, App. Math. Letters, 12 (1999), 107-111. doi: 10.1016/S0893-9659(99)00109-3.

[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), 59-71. doi: 10.1007/s002050050136.

[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), 1399-1414. doi: 10.1080/03605300008821553.

[16]

R. J. Di Perna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.

[17]

L. C. Evans, "Partial Differential Equation," AMS, 1998.

[18]

M. Fixman, Charged macromolecules in external fields I. The sphere, J. Chem. Phys., 72 (1980), 5177-5186. doi: 10.1063/1.439753.

[19]

A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 2008.

[20]

L. M. Fu, R. J. Yang and G. B. Lee, Analysis of geometry effects on band spreading of microchip electrophoresis, Electrophoresis, 23 (2002), 602-612. doi: 10.1002/1522-2683(200202)23:4<602::AID-ELPS602>3.0.CO;2-N.

[21]

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

[22]

W. Hackbusch, "Integral Equations: Theory and Numerical Treatment," Birkhäuser Verlag, Basel, 1995.

[23]

J. Huang and J. Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems, J. Diff. Equations, 184 (2002), 570-586. doi: 10.1006/jdeq.2001.4154.

[24]

H. J. Keh and J. L. Anderson, Boundary effects on electrophoretic motion of colloidal spheres, J. Fluid Mech., 153 (1985), 417-439. doi: 10.1017/S002211208500132X.

[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), 318-330. doi: 10.1006/jcis.2002.8359.

[26]

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

[27]

O. A. Ladyženskaja and N. N. Ural'ceva, "Linear and Quasi-Linear Elliptic Equations," Academic Press, New York-London, 1968.

[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), 1-17. doi: 10.1063/1.2775933.

[29]

H. Nakamura, Roles of electrostatic interaction in proteins, Quart. Rev. Biophys, 29 (1996), 1-90. doi: 10.1017/S0033583500005746.

[30]

J. Necăs, "Les Méthodes Directes en Théorie des Équations Elliptiques," Masson Et Cie, Paris, 1967.

[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), 731-739. doi: 10.1016/j.jcis.2007.07.007.

[32]

A. Quarteroni and A., Valli, "Numerical Approximation of Partial Differential Equations," Springer, Berlin, 1994.

[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), 579-592. doi: 10.1016/j.jcis.2006.08.003.

[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), 3901-3912. doi: 10.1039/FT9908603901.

[35]

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

[36]

A. Sellier, A note on the electrophoresis of a uniformly charged particle, Q. J. Mech. Appl. Math., 55 (2002), 561-572. doi: 10.1093/qjmam/55.4.561.

[37]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models & Meth. Appl. Sci., 19 (2009), 993-1015. doi: 10.1142/S0218202509003693.

[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), 298-315. doi: 10.1006/jcis.1999.6143.

[39]

Y. Solomentsev and J. L. Anderson, Electrophoresis of slender particles, J. Fluid Mech., 279 (1994), 197-215. doi: 10.1017/S0022112094003885.

[40]

M. Teubner, The motion of charged particles in electrical fields, J. Chem. Phys., 76 (1982), 5564-5573. doi: 10.1063/1.442861.

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