American Institute of Mathematical Sciences

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), 2558-2568. 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), 121-124. 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), 61-99. 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), 1271-1318. 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), 1-26. doi: 10.1590/S0101-82052006000100001.  Google Scholar [6] H. Brézis, "Análisis Funcional: Teoría y Aplicaciones," Alianza Editorial, Madrid, 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), 175-202. 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), 130-139. 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), 672-678. doi: 10.1016/j.jcis.2009.01.061.  Google Scholar [10] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Krieger, Malabar, 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), 1570-1590. 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, New York, 1999. Google Scholar [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.  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), 59-71. 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), 1399-1414. 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), 511-547. 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), 5177-5186. doi: 10.1063/1.439753.  Google Scholar [19] A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 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), 602-612. 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, Berlin, 2001.  Google Scholar [22] W. Hackbusch, "Integral Equations: Theory and Numerical Treatment," Birkhäuser Verlag, Basel, 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), 570-586. 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), 417-439. 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), 318-330. 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, Providence, 1988.  Google Scholar [27] O. A. Ladyženskaja and N. N. Ural'ceva, "Linear and Quasi-Linear Elliptic Equations," Academic Press, New York-London, 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), 1-17. doi: 10.1063/1.2775933.  Google Scholar [29] H. Nakamura, Roles of electrostatic interaction in proteins, Quart. Rev. Biophys, 29 (1996), 1-90. doi: 10.1017/S0033583500005746.  Google Scholar [30] J. Necăs, "Les Méthodes Directes en Théorie des Équations Elliptiques," Masson Et Cie, Paris, 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), 731-739. doi: 10.1016/j.jcis.2007.07.007.  Google Scholar [32] A. Quarteroni and A., Valli, "Numerical Approximation of Partial Differential Equations," Springer, Berlin, 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), 579-592. 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), 3901-3912. 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), 561-572. 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), 993-1015. 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), 298-315. doi: 10.1006/jcis.1999.6143.  Google Scholar [39] Y. Solomentsev and J. L. Anderson, Electrophoresis of slender particles, J. Fluid Mech., 279 (1994), 197-215. doi: 10.1017/S0022112094003885.  Google Scholar [40] M. Teubner, The motion of charged particles in electrical fields, J. Chem. Phys., 76 (1982), 5564-5573. 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), 2558-2568. 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), 121-124. 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), 61-99. 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), 1271-1318. 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), 1-26. doi: 10.1590/S0101-82052006000100001.  Google Scholar [6] H. Brézis, "Análisis Funcional: Teoría y Aplicaciones," Alianza Editorial, Madrid, 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), 175-202. 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), 130-139. 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), 672-678. doi: 10.1016/j.jcis.2009.01.061.  Google Scholar [10] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Krieger, Malabar, 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), 1570-1590. 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, New York, 1999. Google Scholar [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.  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), 59-71. 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), 1399-1414. 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), 511-547. 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), 5177-5186. doi: 10.1063/1.439753.  Google Scholar [19] A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 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), 602-612. 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, Berlin, 2001.  Google Scholar [22] W. Hackbusch, "Integral Equations: Theory and Numerical Treatment," Birkhäuser Verlag, Basel, 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), 570-586. 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), 417-439. 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), 318-330. 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, Providence, 1988.  Google Scholar [27] O. A. Ladyženskaja and N. N. Ural'ceva, "Linear and Quasi-Linear Elliptic Equations," Academic Press, New York-London, 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), 1-17. doi: 10.1063/1.2775933.  Google Scholar [29] H. Nakamura, Roles of electrostatic interaction in proteins, Quart. Rev. Biophys, 29 (1996), 1-90. doi: 10.1017/S0033583500005746.  Google Scholar [30] J. Necăs, "Les Méthodes Directes en Théorie des Équations Elliptiques," Masson Et Cie, Paris, 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), 731-739. doi: 10.1016/j.jcis.2007.07.007.  Google Scholar [32] A. Quarteroni and A., Valli, "Numerical Approximation of Partial Differential Equations," Springer, Berlin, 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), 579-592. 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), 3901-3912. 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), 561-572. 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), 993-1015. 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), 298-315. doi: 10.1006/jcis.1999.6143.  Google Scholar [39] Y. Solomentsev and J. L. Anderson, Electrophoresis of slender particles, J. Fluid Mech., 279 (1994), 197-215. doi: 10.1017/S0022112094003885.  Google Scholar [40] M. Teubner, The motion of charged particles in electrical fields, J. Chem. Phys., 76 (1982), 5564-5573. doi: 10.1063/1.442861.  Google Scholar
 [1] Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078 [2] Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237 [3] Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073 [4] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [5] I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191 [6] Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67 [7] Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159 [8] Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681 [9] Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 [10] Fuzhi Li, Dongmei Xu, Jiali Yu. Regular measurable backward compact random attractor for $g$-Navier-Stokes equation. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3137-3157. doi: 10.3934/cpaa.2020136 [11] Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107 [12] Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations & Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1 [13] Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465 [14] Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215 [15] Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5421-5448. doi: 10.3934/dcdsb.2020352 [16] Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 [17] Hongyong Cui, Mirelson M. Freitas, José A. Langa. Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1297-1324. doi: 10.3934/dcdsb.2018152 [18] Chia-Yu Hsieh. Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2657-2681. doi: 10.3934/dcdsb.2018269 [19] Mohamad Rachid. Incompressible Navier-Stokes-Fourier limit from the Landau equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021017 [20] Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185

2019 Impact Factor: 1.105