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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 
References:
[1] 
D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuseinterface methods in fluid mechanics, in "Annual Review of Fluid Mechanics, Vol. 30," Annu. Rev. Fluid Mech., 30, Annual Reviews, Palo Alto, CA, (1998), 139165. doi: 10.1146/annurev.fluid.30.1.139. 
[2] 
J. Barthel, H. Krienke and W. Kunz, "Physical Chemistry of Electrolyte Solutions: Modern Aspects," Springer, New York, 1998. 
[3] 
M. Z. Bazant, K. Thornton and A. Ajdari, Diffusecharge dynamics in electrochemical systems, Physical Review E, 70 (2004), 021506124. doi: 10.1103/PhysRevE.70.021506. 
[4] 
K. Besteman, M. A. G. Zevenbergen and S. G. Lemay, Charge inversion by multivalent ions: Dependence on dielectric constant and surfacecharge density, Phys. Rev. E, 72 (2005), 06150119. doi: 10.1103/PhysRevE.72.061501. 
[5] 
D. Boda, D. Henderson and D. Busath, Monte Carlo study of the selectivity of calcium channels: Improved geometrical model, Mol. Phys., 100 (2002), 23612368. 
[6] 
D. Boda, W. Nonner, D. Henderson, B. Eisenberg and D. Gillespie, Volume exclusion in calcium selective channels, Biophys. J., 94 (2008), 34863496. doi: 10.1529/biophysj.107.122796. 
[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), 034901111. doi: 10.1063/1.2212423. 
[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), 19601980. doi: 10.1529/biophysj.107.105478. 
[9] 
D. Boda, M. Valisko, D. Henderson, B. Eisenberg, D. Gillespie and W. Nonner, Ionic selectivity in Ltype calcium channels by electrostatics and hardcore repulsion, J. Gen. Physiol., 133 (2009), 497509. doi: 10.1085/jgp.200910211. 
[10] 
J. W. Cahn and S. M. Allen, A microscopic theory for domain wall motion and its experimental verification in FeAl alloy domain growth kinetics, J. Phys. Colloque, 4 (1978), C7C51. 
[11] 
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258267. doi: 10.1063/1.1744102. 
[12] 
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. III. Nucleation in a twocomponent incompressible fluid, J. Chem. Phys., 31 (1959), 688699. doi: 10.1063/1.1730447. 
[13] 
S. DurandVidal, J.P. Simonin and P. Turq, "Electrolytes at Interfaces," Kluwer, Boston, 2000. 
[14] 
S. DurandVidal, P. Turq, O. Bernard, C. Treiner and L. Blum, New perspectives in transport phenomena in electrolytes, Physica A, 231 (1996), 123143. doi: 10.1016/03784371(96)000830. 
[15] 
W. R. Fawcett, "Liquids, Solutions, and Interfaces: From Classical Macroscopic Descriptions to Modern Microscopic Details," Oxford University Press, New York, 2004. 
[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), 269357. 
[17] 
R. S. Eisenberg, Computing the field in proteins and channels, J. Membrane Biol., 150 (1996), 125. doi: 10.1007/s002329900026. 
[18] 
R. S. Eisenberg, From structure to function in open ionic channels, J. Membrane Biology, 171 (1999), 1317. 
[19] 
B. Eisenberg, Proteins, channels, and crowded ions, Biophysical Chemistry, 100 (2003), 507517. doi: 10.1016/S03014622(02)003022. 
[20] 
B. Eisenberg, Ion channels allow atomic control of macroscopic transport, Physica Status Solidi (c), 5 (2008), 708713. 
[21] 
B. Eisenberg, Crowded charges in ion channels. Advances in chemical physics,, in press, (). 
[22] 
B. Eisenberg, D. Boda, J. Giri, J. Fonseca, D. Gillespie, D. Henderson and W. Nonner, Selforganized models of selectivity in Ca and Na channels, Biophys. J., 96 (2009), 253a. doi: 10.1016/j.bpj.2008.12.1247. 
[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), 104104123. doi: 10.1063/1.3476262. 
[24] 
R. J. Flatt and P. Bowen, Electrostatic repulsion between particles in cement suspensions: Domain of validity of linearized PoissonBoltzmann equation for nonideal electrolytes, Cement and Concrete Research, 33 (2003), 781791. doi: 10.1016/S00088846(02)010591. 
[25] 
H. L. Friedman, "Ionic Solution Theory," Interscience Publishers, New York, 1962. 
[26] 
Ph. A. Martin, Sum rules in charged fluids, Reviews of Modern Physics, 60 (1988), 10751127. doi: 10.1103/RevModPhys.60.1075. 
[27] 
D. Gillespie, Energetics of divalent selectivity in a calcium channel: The ryanodine receptor case study, Biophys. J., 94 (2008), 11691184. doi: 10.1529/biophysj.107.116798. 
[28] 
D. Gillespie and D. Boda, The anomalous mole fraction effect in calcium channels: A measure of preferential selectivity, Biophys. J., 95 (2008), 26582672. doi: 10.1529/biophysj.107.127977. 
[29] 
D. Gillespie and M. Fill, Intracellular calcium release channels mediate their own countercurrent: The ryanodine receptor case study, Biophys. J., 95 (2008), 37063714. doi: 10.1529/biophysj.108.131987. 
[30] 
D. Gillespie, J. Giri and M. Fill, Reinterpreting the anomalous mole fraction effect. The ryanodine receptor case study, Biophys. J., 97 (2009), 22122221. doi: 10.1016/j.bpj.2009.08.009. 
[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), 66096626. doi: 10.1088/09538984/17/42/002. 
[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), 1559815610. 
[33] 
H. S. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 3^{rd} edition, Reinhold Publishing Corporation, New York, 1958. 
[34] 
J. R. Henderson, Statistical mechanical sum rules, in "Fundamentals of Inhomogeneous Fluids" (eds. D. Henderson), Marcel Dekker, New York, (1992), 2384. 
[35] 
B. Hille, "Ion Channels of Excitable Membranes," 3^{rd} edition, Sinauer Associates, Inc., 2001. 
[36] 
Y. Hyon, D. Y. Kwak and C. Liu, Energetic variational approach in complex fluids: Maximum dissipation principle, Discrete Continuous Dynam. Systems, 24 (2010), 12911304. 
[37] 
Y. Hyon, B. Eisenberg and C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions, Comm. Math. Sci., 9 (2011), 459475. 
[38] 
M. H. Jacobs, "Diffusion Processes," Springer Verlag, New York, 1967. 
[39] 
J. D. Jackson, "Classical Electrodynamics," 3^{rd} edition, Wiley, New York, 1998. 
[40] 
J. W. Jerome, "Analysis of Charge Transport. A Mathematical Study of Semiconductor Devices," SpringerVerlag, Berlin, 1996. 
[41] 
B. Jönsson, A. Nonat, C. Labbez, B. Cabane and H. Wennerström, Controlling the cohesion of cement paste, Langmuir, 21 (2005), 92119317. doi: 10.1021/la051048z. 
[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), 223338. 
[43] 
C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourierspectral method, Physica D, 179 (2003), 211228. doi: 10.1016/S01672789(03)000307. 
[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), 12410216. doi: 10.1063/1.2839302. 
[45] 
P. A. Markowich, "The Stationary Seminconductor Device Equations," Computational Microelectronics, SpringerVerlag, Vienna, 1986. 
[46] 
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," SpringerVerlag, Vienna, 1990. 
[47] 
W. Nonner, L. Catacuzzeno and B. Eisenberg, Binding and selectivity in Ltype Ca channels: A mean spherical approximation, Biophys. J., 79 (2000), 19761992. doi: 10.1016/S00063495(00)764460. 
[48] 
W. Nonner and B. Eisenberg, Ion permeation and glutamate residues linked by PoissonNernstPlanck theory in Ltype calcium channels, Biophys. J., 75 (1998), 12871305. doi: 10.1016/S00063495(98)740482. 
[49] 
J.H. Park and J. W. Jerome, Qualitative properties of steadystate PoissonNernstPlanck systems: Mathematical study, SIAM J. Appl. Math., 57 (1997), 609630. doi: 10.1137/S0036139995279809. 
[50] 
L. Pauling, "Nature of the Chemical Bond," 3^{rd} edition, Cornell University Press, New York, 1960. 
[51] 
J. Philibert, One and a half century of diffusion: Fick, Einstein, before and beyond, Diffusion Fundamentals, 2 (2005), 1.11.10. 
[52] 
R. F. Pierret, "Semiconductor Device Fundamentals," Addison Wesley, New York, 1996. 
[53] 
K. S. Pitzer, "Activity Coefficients in Electrolyte Solutions," CRC Press, Boca Raton, 1991. 
[54] 
K. S. Pitzer, "Thermodynamics," 3^{rd} edition, McGraw Hill, New York, 1995. 
[55] 
R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," 2^{nd} edition, Butterworths Scientific Publications, London, 1959. 
[56] 
Y. Rosenfeld, Freeenergy model for the inhomogeneous hardsphere fluid mixture and densityfunctional theory of freezing, Phys. Rev. Lett., 63 (1989), 980983. doi: 10.1103/PhysRevLett.63.980. 
[57] 
Y. Rosenfeld, Freeenergy model for the inhomogeneous hardsphere fluid in D dimensions: Structure factors for the harddisk ($D=2$) mixtures in simple explicit form, Phys. Rev. A, 42 (1990), 59785989. 
[58] 
N. Roussel, A. Lemaître, R. J. Flatt and P. Coussot, Steady state flow of cement suspensions: A micromechanical state of the art, Cement and Concrete Research, 40 (2010), 7784. doi: 10.1016/j.cemconres.2009.08.026. 
[59] 
R. Ryham, "An Energetic Variational Approach to Mathematical Modeling of Charged Fluids: Charge Phases, Simulation And Well Posedness," Ph.D thesis, Pennsylvania State University, University Park, 2006. 
[60] 
S. Selberherr, "Analysis and Simulation of Semiconductor Devices," SpringerVerlag, New York, 1984. 
[61] 
A. Singer, Z. Schuss and R. S. Eisenberg, Attenuation of the electric potential and field in disordered systems, J. Stat. Phys., 119 (2005), 13971418. doi: 10.1007/s1095500530251. 
[62] 
S. M. Sze, "Physics of Semiconductor Devices," John Wiley & Sons, New York, 1981. 
[63] 
G. M. Torrie and J. P. Valleau, Electrical double layers. I. Monte Carlo study of a uniformly charged surface, J. Chem. Phys., 73 (1980), 58075816. doi: 10.1063/1.440065. 
[64] 
Y. Tsividis, "Operation and Modeling of the MOS Transistor," Oxford, New York, 1999. 
[65] 
M. Valisko, D. Boda and D. Gillespie, Selective adsorption of ions with different diameter and valence at highly charged interfaces, J. Phys. Chem., 111 (2007), 1557515585. 
[66] 
J. Xu and L. Zikatanov, A monotone finite element scheme for convectiondiffusion equations, Math. Comp., 68 (1999), 14291446. 
[67] 
P. Yue, J. J. Feng, C. Liu and J. Shen, A diffuseinterface method of simulating twophase flows of complex fluids, J. Fluid Mech., 515 (2004), 293317. doi: 10.1017/S0022112004000370. 
show all references
References:
[1] 
D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuseinterface methods in fluid mechanics, in "Annual Review of Fluid Mechanics, Vol. 30," Annu. Rev. Fluid Mech., 30, Annual Reviews, Palo Alto, CA, (1998), 139165. doi: 10.1146/annurev.fluid.30.1.139. 
[2] 
J. Barthel, H. Krienke and W. Kunz, "Physical Chemistry of Electrolyte Solutions: Modern Aspects," Springer, New York, 1998. 
[3] 
M. Z. Bazant, K. Thornton and A. Ajdari, Diffusecharge dynamics in electrochemical systems, Physical Review E, 70 (2004), 021506124. doi: 10.1103/PhysRevE.70.021506. 
[4] 
K. Besteman, M. A. G. Zevenbergen and S. G. Lemay, Charge inversion by multivalent ions: Dependence on dielectric constant and surfacecharge density, Phys. Rev. E, 72 (2005), 06150119. doi: 10.1103/PhysRevE.72.061501. 
[5] 
D. Boda, D. Henderson and D. Busath, Monte Carlo study of the selectivity of calcium channels: Improved geometrical model, Mol. Phys., 100 (2002), 23612368. 
[6] 
D. Boda, W. Nonner, D. Henderson, B. Eisenberg and D. Gillespie, Volume exclusion in calcium selective channels, Biophys. J., 94 (2008), 34863496. doi: 10.1529/biophysj.107.122796. 
[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), 034901111. doi: 10.1063/1.2212423. 
[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), 19601980. doi: 10.1529/biophysj.107.105478. 
[9] 
D. Boda, M. Valisko, D. Henderson, B. Eisenberg, D. Gillespie and W. Nonner, Ionic selectivity in Ltype calcium channels by electrostatics and hardcore repulsion, J. Gen. Physiol., 133 (2009), 497509. doi: 10.1085/jgp.200910211. 
[10] 
J. W. Cahn and S. M. Allen, A microscopic theory for domain wall motion and its experimental verification in FeAl alloy domain growth kinetics, J. Phys. Colloque, 4 (1978), C7C51. 
[11] 
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258267. doi: 10.1063/1.1744102. 
[12] 
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. III. Nucleation in a twocomponent incompressible fluid, J. Chem. Phys., 31 (1959), 688699. doi: 10.1063/1.1730447. 
[13] 
S. DurandVidal, J.P. Simonin and P. Turq, "Electrolytes at Interfaces," Kluwer, Boston, 2000. 
[14] 
S. DurandVidal, P. Turq, O. Bernard, C. Treiner and L. Blum, New perspectives in transport phenomena in electrolytes, Physica A, 231 (1996), 123143. doi: 10.1016/03784371(96)000830. 
[15] 
W. R. Fawcett, "Liquids, Solutions, and Interfaces: From Classical Macroscopic Descriptions to Modern Microscopic Details," Oxford University Press, New York, 2004. 
[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), 269357. 
[17] 
R. S. Eisenberg, Computing the field in proteins and channels, J. Membrane Biol., 150 (1996), 125. doi: 10.1007/s002329900026. 
[18] 
R. S. Eisenberg, From structure to function in open ionic channels, J. Membrane Biology, 171 (1999), 1317. 
[19] 
B. Eisenberg, Proteins, channels, and crowded ions, Biophysical Chemistry, 100 (2003), 507517. doi: 10.1016/S03014622(02)003022. 
[20] 
B. Eisenberg, Ion channels allow atomic control of macroscopic transport, Physica Status Solidi (c), 5 (2008), 708713. 
[21] 
B. Eisenberg, Crowded charges in ion channels. Advances in chemical physics,, in press, (). 
[22] 
B. Eisenberg, D. Boda, J. Giri, J. Fonseca, D. Gillespie, D. Henderson and W. Nonner, Selforganized models of selectivity in Ca and Na channels, Biophys. J., 96 (2009), 253a. doi: 10.1016/j.bpj.2008.12.1247. 
[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), 104104123. doi: 10.1063/1.3476262. 
[24] 
R. J. Flatt and P. Bowen, Electrostatic repulsion between particles in cement suspensions: Domain of validity of linearized PoissonBoltzmann equation for nonideal electrolytes, Cement and Concrete Research, 33 (2003), 781791. doi: 10.1016/S00088846(02)010591. 
[25] 
H. L. Friedman, "Ionic Solution Theory," Interscience Publishers, New York, 1962. 
[26] 
Ph. A. Martin, Sum rules in charged fluids, Reviews of Modern Physics, 60 (1988), 10751127. doi: 10.1103/RevModPhys.60.1075. 
[27] 
D. Gillespie, Energetics of divalent selectivity in a calcium channel: The ryanodine receptor case study, Biophys. J., 94 (2008), 11691184. doi: 10.1529/biophysj.107.116798. 
[28] 
D. Gillespie and D. Boda, The anomalous mole fraction effect in calcium channels: A measure of preferential selectivity, Biophys. J., 95 (2008), 26582672. doi: 10.1529/biophysj.107.127977. 
[29] 
D. Gillespie and M. Fill, Intracellular calcium release channels mediate their own countercurrent: The ryanodine receptor case study, Biophys. J., 95 (2008), 37063714. doi: 10.1529/biophysj.108.131987. 
[30] 
D. Gillespie, J. Giri and M. Fill, Reinterpreting the anomalous mole fraction effect. The ryanodine receptor case study, Biophys. J., 97 (2009), 22122221. doi: 10.1016/j.bpj.2009.08.009. 
[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), 66096626. doi: 10.1088/09538984/17/42/002. 
[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), 1559815610. 
[33] 
H. S. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 3^{rd} edition, Reinhold Publishing Corporation, New York, 1958. 
[34] 
J. R. Henderson, Statistical mechanical sum rules, in "Fundamentals of Inhomogeneous Fluids" (eds. D. Henderson), Marcel Dekker, New York, (1992), 2384. 
[35] 
B. Hille, "Ion Channels of Excitable Membranes," 3^{rd} edition, Sinauer Associates, Inc., 2001. 
[36] 
Y. Hyon, D. Y. Kwak and C. Liu, Energetic variational approach in complex fluids: Maximum dissipation principle, Discrete Continuous Dynam. Systems, 24 (2010), 12911304. 
[37] 
Y. Hyon, B. Eisenberg and C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions, Comm. Math. Sci., 9 (2011), 459475. 
[38] 
M. H. Jacobs, "Diffusion Processes," Springer Verlag, New York, 1967. 
[39] 
J. D. Jackson, "Classical Electrodynamics," 3^{rd} edition, Wiley, New York, 1998. 
[40] 
J. W. Jerome, "Analysis of Charge Transport. A Mathematical Study of Semiconductor Devices," SpringerVerlag, Berlin, 1996. 
[41] 
B. Jönsson, A. Nonat, C. Labbez, B. Cabane and H. Wennerström, Controlling the cohesion of cement paste, Langmuir, 21 (2005), 92119317. doi: 10.1021/la051048z. 
[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), 223338. 
[43] 
C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourierspectral method, Physica D, 179 (2003), 211228. doi: 10.1016/S01672789(03)000307. 
[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), 12410216. doi: 10.1063/1.2839302. 
[45] 
P. A. Markowich, "The Stationary Seminconductor Device Equations," Computational Microelectronics, SpringerVerlag, Vienna, 1986. 
[46] 
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," SpringerVerlag, Vienna, 1990. 
[47] 
W. Nonner, L. Catacuzzeno and B. Eisenberg, Binding and selectivity in Ltype Ca channels: A mean spherical approximation, Biophys. J., 79 (2000), 19761992. doi: 10.1016/S00063495(00)764460. 
[48] 
W. Nonner and B. Eisenberg, Ion permeation and glutamate residues linked by PoissonNernstPlanck theory in Ltype calcium channels, Biophys. J., 75 (1998), 12871305. doi: 10.1016/S00063495(98)740482. 
[49] 
J.H. Park and J. W. Jerome, Qualitative properties of steadystate PoissonNernstPlanck systems: Mathematical study, SIAM J. Appl. Math., 57 (1997), 609630. doi: 10.1137/S0036139995279809. 
[50] 
L. Pauling, "Nature of the Chemical Bond," 3^{rd} edition, Cornell University Press, New York, 1960. 
[51] 
J. Philibert, One and a half century of diffusion: Fick, Einstein, before and beyond, Diffusion Fundamentals, 2 (2005), 1.11.10. 
[52] 
R. F. Pierret, "Semiconductor Device Fundamentals," Addison Wesley, New York, 1996. 
[53] 
K. S. Pitzer, "Activity Coefficients in Electrolyte Solutions," CRC Press, Boca Raton, 1991. 
[54] 
K. S. Pitzer, "Thermodynamics," 3^{rd} edition, McGraw Hill, New York, 1995. 
[55] 
R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," 2^{nd} edition, Butterworths Scientific Publications, London, 1959. 
[56] 
Y. Rosenfeld, Freeenergy model for the inhomogeneous hardsphere fluid mixture and densityfunctional theory of freezing, Phys. Rev. Lett., 63 (1989), 980983. doi: 10.1103/PhysRevLett.63.980. 
[57] 
Y. Rosenfeld, Freeenergy model for the inhomogeneous hardsphere fluid in D dimensions: Structure factors for the harddisk ($D=2$) mixtures in simple explicit form, Phys. Rev. A, 42 (1990), 59785989. 
[58] 
N. Roussel, A. Lemaître, R. J. Flatt and P. Coussot, Steady state flow of cement suspensions: A micromechanical state of the art, Cement and Concrete Research, 40 (2010), 7784. doi: 10.1016/j.cemconres.2009.08.026. 
[59] 
R. Ryham, "An Energetic Variational Approach to Mathematical Modeling of Charged Fluids: Charge Phases, Simulation And Well Posedness," Ph.D thesis, Pennsylvania State University, University Park, 2006. 
[60] 
S. Selberherr, "Analysis and Simulation of Semiconductor Devices," SpringerVerlag, New York, 1984. 
[61] 
A. Singer, Z. Schuss and R. S. Eisenberg, Attenuation of the electric potential and field in disordered systems, J. Stat. Phys., 119 (2005), 13971418. doi: 10.1007/s1095500530251. 
[62] 
S. M. Sze, "Physics of Semiconductor Devices," John Wiley & Sons, New York, 1981. 
[63] 
G. M. Torrie and J. P. Valleau, Electrical double layers. I. Monte Carlo study of a uniformly charged surface, J. Chem. Phys., 73 (1980), 58075816. doi: 10.1063/1.440065. 
[64] 
Y. Tsividis, "Operation and Modeling of the MOS Transistor," Oxford, New York, 1999. 
[65] 
M. Valisko, D. Boda and D. Gillespie, Selective adsorption of ions with different diameter and valence at highly charged interfaces, J. Phys. Chem., 111 (2007), 1557515585. 
[66] 
J. Xu and L. Zikatanov, A monotone finite element scheme for convectiondiffusion equations, Math. Comp., 68 (1999), 14291446. 
[67] 
P. Yue, J. J. Feng, C. Liu and J. Shen, A diffuseinterface method of simulating twophase flows of complex fluids, J. Fluid Mech., 515 (2004), 293317. doi: 10.1017/S0022112004000370. 
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