# American Institute of Mathematical Sciences

November  2012, 17(8): 2725-2743. doi: 10.3934/dcdsb.2012.17.2725

## Energy variational approach to study charge inversion (layering) near charged walls

 1 Department of Mechanical Engineering, University of Nevada, Reno, Reno, NV 89557, United States 2 Department of Molecular Biophysics & Physiology Rush Medical Center, 1653 West Congress, Parkway, Chicago, IL 60612, United States, United States 3 Department of Mathematics and Center for Materials Physics, Penn State University, University Park, PA 16802

Received  April 2011 Revised  September 2011 Published  July 2012

We introduce a mathematical model, which describes the charge inversion phenomena in systems with a charged wall or boundary. This model may prove helpful in understanding semiconductor devices, ion channels, and electrochemical systems like batteries that depend on complex distributions of charge for their function. The mathematical model is derived using the energy variational approach that takes into account ion diffusion, electrostatics, finite size effects, and specific boundary behavior. In ion dynamic theory, a well-known system of equations is the Poisson-Nernst-Planck (PNP) equation that includes entropic and electrostatic energy. The PNP type of equation can also be derived by the energy variational approach. However, the PNP equations have not produced the charge inversion/layering in charged wall situations presumably because the conventional PNP does not include the finite size of ions and other physical features needed to create the charge inversion. In this paper, we investigate the key features needed to produce the charge inversion phenomena using a mathematical model, the energy variational approach. One of the key features is a finite size (finite volume) effect, which is an unavoidable property of ions important for their dynamics on small scales. The other is an interfacial constraint to capture the spatial variation of electroneutrality in systems with charged walls. The interfacial constraint is established by the diffusive interface approach that approximately describes the boundary effect produced by the charged wall. The energy variational approach gives us a mathematically self-consistent way to introduce the interfacial constraint. We mainly discuss those two key features in this paper. Employing the energy variational approach, we derive a non-local partial differential equation with a total energy consisting of the entropic energy, electrostatic energy, repulsion energy representing the excluded volume effect, and the contribution of an interfacial constraint related to overall electroneutrality between bulk/bath and wall. The resulting mathematical model produces the charge inversion phenomena near charged walls. We compare the computational results of the mathematical model to those of Monte-Carlo computations.
Citation: YunKyong Hyon, James E. Fonseca, Bob Eisenberg, Chun Liu. Energy variational approach to study charge inversion (layering) near charged walls. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2725-2743. doi: 10.3934/dcdsb.2012.17.2725
##### References:
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Fawcett, "Liquids, Solutions, and Interfaces: From Classical Macroscopic Descriptions to Modern Microscopic Details," Oxford University Press, New York, 2004. Google Scholar [16] R. S. Eisenberg, Atomic biology, electrostatics, and ionic channels, in "Recent Developments in Theoretical Studies of Proteins, Vol. 7" (eds. R. Elber), World Scientific, Philadelphia, (1996), 269-357. Google Scholar [17] R. S. Eisenberg, Computing the field in proteins and channels, J. Membrane Biol., 150 (1996), 1-25. doi: 10.1007/s002329900026.  Google Scholar [18] R. S. Eisenberg, From structure to function in open ionic channels, J. Membrane Biology, 171 (1999), 1-317. Google Scholar [19] B. Eisenberg, Proteins, channels, and crowded ions, Biophysical Chemistry, 100 (2003), 507-517. doi: 10.1016/S0301-4622(02)00302-2.  Google Scholar [20] B. Eisenberg, Ion channels allow atomic control of macroscopic transport, Physica Status Solidi (c), 5 (2008), 708-713. Google Scholar [21] B. 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##### References:
 [1] D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, in "Annual Review of Fluid Mechanics, Vol. 30," Annu. Rev. Fluid Mech., 30, Annual Reviews, Palo Alto, CA, (1998), 139-165. doi: 10.1146/annurev.fluid.30.1.139.  Google Scholar [2] J. Barthel, H. Krienke and W. Kunz, "Physical Chemistry of Electrolyte Solutions: Modern Aspects," Springer, New York, 1998. Google Scholar [3] M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Physical Review E, 70 (2004), 021506-1-24. doi: 10.1103/PhysRevE.70.021506.  Google Scholar [4] K. Besteman, M. A. G. Zevenbergen and S. G. Lemay, Charge inversion by multivalent ions: Dependence on dielectric constant and surface-charge density, Phys. Rev. E, 72 (2005), 061501-1-9. doi: 10.1103/PhysRevE.72.061501.  Google Scholar [5] D. Boda, D. Henderson and D. Busath, Monte Carlo study of the selectivity of calcium channels: Improved geometrical model, Mol. Phys., 100 (2002), 2361-2368. Google Scholar [6] D. Boda, W. Nonner, D. Henderson, B. Eisenberg and D. Gillespie, Volume exclusion in calcium selective channels, Biophys. J., 94 (2008), 3486-3496. doi: 10.1529/biophysj.107.122796.  Google Scholar [7] D. Boda, M. Valisko, B. Eisenberg, W. Nonner, D. Henderson and D. Gillespie, The effect of protein dielectric coefficient on the ionic selectivity of a calcium channel, J. Chem. Phys., 125 (2006), 034901-1-11. doi: 10.1063/1.2212423.  Google Scholar [8] D. Boda, W. Nonner, M. Valisko, D. Henderson, B. Eisenberg and D. Gillespie, Steric selectivity in Na channels arising from protein polarization and mobile side chains, Biophys. J., 93 (2007), 1960-1980. doi: 10.1529/biophysj.107.105478.  Google Scholar [9] D. Boda, M. Valisko, D. Henderson, B. Eisenberg, D. Gillespie and W. Nonner, Ionic selectivity in L-type calcium channels by electrostatics and hard-core repulsion, J. Gen. Physiol., 133 (2009), 497-509. doi: 10.1085/jgp.200910211.  Google Scholar [10] J. W. Cahn and S. M. Allen, A microscopic theory for domain wall motion and its experimental verification in Fe-Al alloy domain growth kinetics, J. Phys. Colloque, 4 (1978), C7-C51. Google Scholar [11] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.  Google Scholar [12] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699. doi: 10.1063/1.1730447.  Google Scholar [13] S. Durand-Vidal, J.-P. Simonin and P. Turq, "Electrolytes at Interfaces," Kluwer, Boston, 2000. Google Scholar [14] S. Durand-Vidal, P. Turq, O. Bernard, C. Treiner and L. Blum, New perspectives in transport phenomena in electrolytes, Physica A, 231 (1996), 123-143. doi: 10.1016/0378-4371(96)00083-0.  Google Scholar [15] W. R. Fawcett, "Liquids, Solutions, and Interfaces: From Classical Macroscopic Descriptions to Modern Microscopic Details," Oxford University Press, New York, 2004. Google Scholar [16] R. S. Eisenberg, Atomic biology, electrostatics, and ionic channels, in "Recent Developments in Theoretical Studies of Proteins, Vol. 7" (eds. R. Elber), World Scientific, Philadelphia, (1996), 269-357. Google Scholar [17] R. S. Eisenberg, Computing the field in proteins and channels, J. Membrane Biol., 150 (1996), 1-25. doi: 10.1007/s002329900026.  Google Scholar [18] R. S. Eisenberg, From structure to function in open ionic channels, J. Membrane Biology, 171 (1999), 1-317. Google Scholar [19] B. Eisenberg, Proteins, channels, and crowded ions, Biophysical Chemistry, 100 (2003), 507-517. doi: 10.1016/S0301-4622(02)00302-2.  Google Scholar [20] B. Eisenberg, Ion channels allow atomic control of macroscopic transport, Physica Status Solidi (c), 5 (2008), 708-713. Google Scholar [21] B. Eisenberg, Crowded charges in ion channels. Advances in chemical physics,, in press, ().   Google Scholar [22] B. Eisenberg, D. Boda, J. Giri, J. Fonseca, D. Gillespie, D. Henderson and W. Nonner, Self-organized models of selectivity in Ca and Na channels, Biophys. J., 96 (2009), 253a. doi: 10.1016/j.bpj.2008.12.1247.  Google Scholar [23] B. Eisenberg, Y. Hyon and C. Liu, Energy variational analysis EnVarA of ions in water and channels: Field theory for primitive models of complex ionic fluids, Journal of Chemical Physics, 133 (2010), 104104-1-23. doi: 10.1063/1.3476262.  Google Scholar [24] R. J. Flatt and P. Bowen, Electrostatic repulsion between particles in cement suspensions: Domain of validity of linearized Poisson-Boltzmann equation for nonideal electrolytes, Cement and Concrete Research, 33 (2003), 781-791. doi: 10.1016/S0008-8846(02)01059-1.  Google Scholar [25] H. L. Friedman, "Ionic Solution Theory," Interscience Publishers, New York, 1962. Google Scholar [26] Ph. A. Martin, Sum rules in charged fluids, Reviews of Modern Physics, 60 (1988), 1075-1127. doi: 10.1103/RevModPhys.60.1075.  Google Scholar [27] D. Gillespie, Energetics of divalent selectivity in a calcium channel: The ryanodine receptor case study, Biophys. J., 94 (2008), 1169-1184. doi: 10.1529/biophysj.107.116798.  Google Scholar [28] D. Gillespie and D. Boda, The anomalous mole fraction effect in calcium channels: A measure of preferential selectivity, Biophys. J., 95 (2008), 2658-2672. doi: 10.1529/biophysj.107.127977.  Google Scholar [29] D. Gillespie and M. Fill, Intracellular calcium release channels mediate their own countercurrent: The ryanodine receptor case study, Biophys. J., 95 (2008), 3706-3714. doi: 10.1529/biophysj.108.131987.  Google Scholar [30] D. Gillespie, J. Giri and M. Fill, Reinterpreting the anomalous mole fraction effect. The ryanodine receptor case study, Biophys. J., 97 (2009), 2212-2221. doi: 10.1016/j.bpj.2009.08.009.  Google Scholar [31] D. Gillespie, M. Valisk\'o and D. Boda, Density functional theory of the electrical double layer: The RFD functional, J. Phys.: Condens. Matter, 17 (2005), 6609-6626. doi: 10.1088/0953-8984/17/42/002.  Google Scholar [32] D. Gillespie, L. Xu, Y. Wang and G. Meissner, (De)construcing the ryanodine receptor: Modeling ion permeation and selectivity of the calcium release channel, Journal of Physical Chemistry, 109 (2005), 15598-15610. Google Scholar [33] H. S. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 3rd edition, Reinhold Publishing Corporation, New York, 1958. Google Scholar [34] J. R. Henderson, Statistical mechanical sum rules, in "Fundamentals of Inhomogeneous Fluids" (eds. D. Henderson), Marcel Dekker, New York, (1992), 23-84. Google Scholar [35] B. Hille, "Ion Channels of Excitable Membranes," 3rd edition, Sinauer Associates, Inc., 2001. Google Scholar [36] Y. Hyon, D. Y. Kwak and C. Liu, Energetic variational approach in complex fluids: Maximum dissipation principle, Discrete Continuous Dynam. Systems, 24 (2010), 1291-1304.  Google Scholar [37] Y. Hyon, B. Eisenberg and C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions, Comm. Math. Sci., 9 (2011), 459-475.  Google Scholar [38] M. H. Jacobs, "Diffusion Processes," Springer Verlag, New York, 1967. Google Scholar [39] J. D. Jackson, "Classical Electrodynamics," 3rd edition, Wiley, New York, 1998. Google Scholar [40] J. W. Jerome, "Analysis of Charge Transport. A Mathematical Study of Semiconductor Devices," Springer-Verlag, Berlin, 1996.  Google Scholar [41] B. Jönsson, A. Nonat, C. Labbez, B. Cabane and H. Wennerström, Controlling the cohesion of cement paste, Langmuir, 21 (2005), 9211-9317. doi: 10.1021/la051048z.  Google Scholar [42] J. C. Justice, Conductance of electrolyte solutions, in "Thermondynbamic and Transport Properties of Aqueous and Molten Electrolytes, Vol. 7" (eds. B. E. Conway, J. O. M. Bockris and E. Yaeger), Plenum, New York, (1983), 223-338. Google Scholar [43] C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228. doi: 10.1016/S0167-2789(03)00030-7.  Google Scholar [44] A. Malasics, D. Gillespie and D. Boda, Simulating prescribed particle densities in the grand canonical ensemble using iterative algorithms, J. Chem. Phys., 128 (2008), 124102-1-6. doi: 10.1063/1.2839302.  Google Scholar [45] P. A. Markowich, "The Stationary Seminconductor Device Equations," Computational Microelectronics, Springer-Verlag, Vienna, 1986.  Google Scholar [46] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990.  Google Scholar [47] W. Nonner, L. Catacuzzeno and B. Eisenberg, Binding and selectivity in L-type Ca channels: A mean spherical approximation, Biophys. J., 79 (2000), 1976-1992. doi: 10.1016/S0006-3495(00)76446-0.  Google Scholar [48] W. Nonner and B. Eisenberg, Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type calcium channels, Biophys. J., 75 (1998), 1287-1305. doi: 10.1016/S0006-3495(98)74048-2.  Google Scholar [49] J.-H. Park and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study, SIAM J. Appl. Math., 57 (1997), 609-630. doi: 10.1137/S0036139995279809.  Google Scholar [50] L. Pauling, "Nature of the Chemical Bond," 3rd edition, Cornell University Press, New York, 1960. Google Scholar [51] J. Philibert, One and a half century of diffusion: Fick, Einstein, before and beyond, Diffusion Fundamentals, 2 (2005), 1.1-1.10. Google Scholar [52] R. F. Pierret, "Semiconductor Device Fundamentals," Addison Wesley, New York, 1996. Google Scholar [53] K. S. 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