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Understanding Depletion Induced Like-Charge Attraction from Self-Consistent Field Model

  • Pei Liu (a1), Manman Ma (a1) and Zhenli Xu (a1) (a2)

The interaction force between likely charged particles/surfaces is usually repulsive due to the Coulomb interaction. However, the counterintuitive like-charge attraction in electrolytes has been frequently observed in experiments, which has been theoretically debated for a long time. It is widely known that the mean field Poisson-Boltzmann theory cannot explain and predict this anomalous feature since it ignores many-body properties. In this paper, we develop efficient algorithm and perform the force calculation between two interfaces using a set of self-consistent equations which properly takes into account the electrostatic correlation and the dielectric-boundary effects. By solving the equations and calculating the pressure with the Debye-charging process, we show that the self-consistent equations could be used to study the attraction between like-charge surfaces from weak-coupling to mediate-coupling regimes, and that the attraction is due to the electrostatics-driven entropic force which is significantly enhanced by the dielectric depletion of mobile ions. A systematic investigation shows that the interaction forces can be tuned by material permittivity, ionic size and valence, and salt concentration, and that the like-charge attraction exists only for specific regime of these parameters.

Corresponding author
*Corresponding author. Email addresses: (P. Liu), (M. Ma), (Z. Xu)
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[1] Angelini, T. E., Liang, H., Wriggers, W., and Wong, G. C. L.. Like-charge attraction between polyelectrolytes induced by counterion charge density waves. Proc. Nat. Acad. Sci. USA, 100:86348637, 2003.
[2] Asakura, S. and Oosawa, F.. On interaction between two bodies immersed in a solution of macromolecules. J. Chem. Phys., 22(7):12551256, 1954.
[3] Avdeev, S. M. and Martynov, G. A.. Influence of image forces on the electrostatic component of the disjoining pressure. Colloid J. USSR, 48:535542, 1986.
[4] Azuara, C., Orland, H., Bon, M., Koehl, P., and Delarue, M.. Incorporating dipolar solvents with variable density in poisson-boltzmann electrostatics. Biophys. J., 95(12):55875605, 2008.
[5] Barros, K. and Luijten, E.. Dielectric effects in the self-assembly of binary colloidal aggregates. Phys. Rev. Lett., 113(1):017801, 2014.
[6] Bell, G. M. and Levine, S.. Electrical forces between uncharged plates in ionic solutions. J. Chem. Phys., 49(10):45844599, 1968.
[7] Booth, F.. The dielectric constant of water and the saturation effect. J. Chem. Phys., 19(4):391394, 1951.
[8] Booth, F.. Dielectric constant of polar liquids at high field strengths. J. Chem. Phys., 23(3):453457, 1955.
[9] Born, M.. Volumes and heats of hydration of ions. Z. Phys., 1:4548, 1920.
[10] Boroudjerdi, H., Kim, Y.-W., Naji, A., Netz, R. R., Schlagberger, X., and Serr, A.. Statics and dynamics of strongly charged soft matter. Phys. Rep., 416:129199, 2005.
[11] Bratko, D. and Henderson, D.. Osmotic interactions between neutral surfaces in an electrolyte solution. Phys. Rev. E, 49:41404144, May 1994.
[12] Buyukdagli, S., Achim, C. V., and Ala-Nissila, T.. Electrostatic correlations in inhomogeneous charged fluids beyond loop expansion. J. Chem. Phys., 137:104902, 2012.
[13] Buyukdagli, S., Manghi, M., and Palmeri, J.. Variational approach for electrolyte solutions: From dielectric interfaces to charged nanopores. Phys. Rev. E, 81:041601, 2010.
[14] Casimir, H. B. G.. On the attraction between two perfectly conducting plates. In Proc. K. Ned. Akad. Wet., volume 51(7), pages 793795, 1948.
[15] Curtis, R. A. and Lue, L.. Depletion forces due to image charges near dielectric discontinuities. Curr. Opin. Colloid Interf. Sci., 20:1923, 2015.
[16] Derbenev, I. N., Filippov, A. V., Stace, A. J., and Besley, E.. Electrostatic interactions between charged dielectric particles in an electrolyte solution. J. Chem. Phys., 145(8), 2016.
[17] Diehl, A., dos Santos, A. P., and Levin, Y.. Surface tension of an electrolyte-air interface: a Monte Carlo study. J. Phys.: Condens. Matter, 24:284115, 2012.
[18] Diehl, A., Tamashiro, M. N., Barbosa, M. C., and Levin, Y.. Density-functional theory for attraction between like-charged plates. Physica A, 274(3):433445, 1999.
[19] Dishon, M., Zohar, O., and Sivan, U.. From repulsion to attraction and back to repulsion: The effect of nacl, kcl, and cscl on the force between silica surfaces in aqueous solution. Langmuir, 25(5):28312836, 2009.
[20] Emelyanenko, K. A., Emelyanenko, A. M., and Boinovich, L. B.. Image charge effects in the wetting behavior of alkanes on water with accounting for water solubility. Materials, 9(3):177, 2016.
[21] French, R. H., Parsegian, V. A., Podgornik, R., Rajter, R. F., Jagota, A., Luo, J., Asthagiri, D., Chaudhury, M. K., Chiang, Y.-M., Granick, S., Kalinin, S., Kardar, M., Kjellander, R., Langreth, D. C., Lewis, J., Lustig, S., Wesolowski, D., Wettlaufer, J. S., Ching, W.-Y., Finnis, M., Houlihan, F., von Lilienfeld, O. A., van Oss, C. J., and Zemb, T.. Long range interactions in nanoscale science. Rev. Mod. Phys., 82(2):18871944, 2010.
[22] Frydel, D. and Ma, M.. Density functional formulation of the random phase approximation for inhomogeneous fluids: application to the Gaussian core and Coulomb particles. Phys. Rev. E, 93:062112, 2016.
[23] Frydel, D. and Oettel, M.. Charged particles at fluid interfaces as a probe into structural details of a double layer. Phys. Chem. Chem. Phys., 13(9):41094118, 2011.
[24] Gan, Z., Wu, H., Barros, K., Xu, Z., and Luijten, E.. Comparison of efficient techniques for the simulation of dielectric objects in electrolytes. J. Comput. Phys., 291:317333, 2015.
[25] Gelbart, W.M., Bruinsma, R. F., Pincus, P. A., and Parsegian, V. A.. DNA-inspired electrostatics. Phys. Today, 53:3844, 2000.
[26] Grosberg, A. Y., Nguyen, T. T., and Shklovskii, B. I.. The physics of charge inversion in chemical and biological systems. Rev. Mod. Phys., 74:329345, 2002.
[27] Hansen, J. P. and McDonald, I. R.. Theory of simple liquids. Academic Press, Amsterdam, 2006.
[28] Hatlo, M. M., Curtis, R., and Lue, L.. Electrostatic depletion forces between planar surfaces. J. Chem. Phys., 128:164717, 2008.
[29] Hatlo, M. M. and Lue, L.. The role of image charges in the interactions between colloidal particles. Soft Matter, 4:15821596, 2008.
[30] Jadhao, V., Solis, F. J., and de la Cruz, M. O.. A variational formulation of electrostatics in a medium with spatially varying dielectric permittivity. J. Chem. Phys., 138(5):054119, 2013.
[31] Jancovici, B. and Šamaj, L.. Screening of classical casimir forces by electrolytes in semi-infinite geometries. J. Stat. Mech, 2004(08):P08006, 2004.
[32] Jho, Y. S., Kanduč, M., Naji, A., Podgornik, R., Kim, M. W., and Pincus, P. A.. Strong-coupling electrostatics in the presence of dielectric inhomogeneities. Phys. Rev. Lett., 101:188101, 2008.
[33] Ji, H., Zhao, X., Qiao, Z., Jung, J., Zhu, Y., Lu, Y., Zhang, L. L., MacDonald, A. H., and Ruoff, R. S.. Capacitance of carbon-based electric double-layer capacitors. Nature Commun., 5:3317, 2014.
[34] Krishnan, M., Petrášek, Z., Mönch, I., and Schwille, P.. Electrostatic self-assembly of charged colloids and macromolecules in a fluidic nanoslit. Small, 4(11):19001906, 2008.
[35] Larsen, A. E. and Grier, D. G.. Like-charge attractions in metastable colloidal crystallites. Nature, 385:230233, 1997.
[36] Levin, Y.. Electrostatic corrections: from plasma to biology. Rep. Prog. Phys., 65:15771632, 2002.
[37] Lin, L., Yang, C., Meza, J. C., Lu, J., Ying, L., and W. E., SelInv—An algorithm for selected inversion of a sparse symmetric matrix. ACM Trans. Math. Softw., 37:40:1–40:19, 2011.
[38] Linse, P.. Structure, phase stability, and thermodynamics in charged colloidal solutions. J. Chem. Phys., 113(10):43594373, 2000.
[39] Linse, P. and Lobaskin, V.. Electrostatic attraction and phase separation in solutions of like-charged colloidal particles. Phys. Rev. Lett., 83(20):4208, 1999.
[40] Lu, B.-S. and Xing, X.. Correlation potential of a test ion near a strongly charged plate. Phys. Rev. E, 89:032305, Mar 2014.
[41] Ma, M. and Xu, Z.. Self-consistent field model for strong electrostatic correlations and inhomogeneous dielectric media. J. Chem. Phys., 141(24):244903, 2014.
[42] Ma, M., Zhao, S., and Xu, Z.. Investigation of dielectric decrement and correlation effects on electric double-layer capacitance by self-consistent field model. Comm. Comp. Phys., 20:441458, 2016.
[43] Messina, R.. Electrostatics in soft matter. J. Phys. Condens. Matter, 21:113102, 2009.
[44] Messina, R., Holm, C., and Kremer, K.. Strong attraction between charged spheres due to metastable ionized states. Phys. Rev. Lett., 85(4):872, 2000.
[45] Naji, A., Kanduč, M., Forsman, J., and Podgornik, R.. Perspective: Coulomb fluids–weak coupling, strong coupling, in between and beyond. J. Chem. Phys., 139(15):150901, 2013.
[46] Naji, A. and Netz, R. R.. Attraction of like-charged macroions in the strong-coupling limit. Eur. Phys. J. E, 13(1):4359, 2004.
[47] Netz, R. R. and Orland, H.. Variational charge renormalization in charged systems. Eur. Phys. J. E, 11:301311, 2003.
[48] Neu, J. C.. Wall-mediated forces between like-charged bodies in an electrolyte. Phys. Rev. Lett., 82:10721074, 1999.
[49] Olivares, W. and McQuarrie, D.. A variational approach to the theory of ionic solutions. J. Chem. Phys., 65(9):36043610, 1976.
[50] Podgornik, R.. Electrostatic correlation forces between surfaces with surface specific ionic interactions. J. Chem. Phys., 91(9):58405849, 1989.
[51] Rosenfeld, Y.. Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing. Phys. Rev. Lett., 63(9):980983, 1989.
[52] Roth, R., Evans, R., Lang, A., and Kahl, G.. Fundamental measure theory for hard-sphere mixtures revisited: the white bear version. J. Phys.: Condens. Matter, 14(46):12063, 2002.
[53] Roux, B.. Influence of the membrane potential on the free energy of an intrinsic protein. Biophys. J., 73(6):2980, 1997.
[54] Sarabadani, J., Dogahe, B. O., and Podgornik, R.. Repulsive casimir interaction: Boyer oscillators at nanoscale. EPL, 112(4):41001, 2015.
[55] Tamashiro, M. N. and Pincus, P.. Electrolytic depletion interactions. Phys. Rev. E, 60(6):6549, 1999.
[56] Trizac, E.. Effective interactions between like-charged macromolecules. Phys. Rev. E, 62(2):R1465, 2000.
[57] Wang, R. and Wang, Z.-G.. Effects of image charges on double layer structure and forces. J. Chem. Phys., 139:124702, 2013.
[58] Wang, Z. G.. Fluctuation in electrolyte solutions: The self energy. Phys. Rev. E, 81:021501, 2010.
[59] Wang, Z. Y. and Ma, Y. Q.. Impact of head group charges, ionic sizes, and dielectric images on charge inversion: A Monte Carlo simulation study. J. Phys. Chem. B, 114:1338613392, 2010.
[60] Xu, Z., Ma, M., and Liu, P.. Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches. Phys. Rev. E, 90(1):013307, 2014.
[61] Xu, Z. and Maggs, A.. Solving fluctuation-enhanced Poisson-Boltzmann equations. J. Comput. Phys., 275:310322, 2014.
[62] Yu, Y.-X. and Wu, J.. Structures of hard-sphere fluids from a modified fundamental-measure theory. J. Chem. Phys., 117(22):1015610164, 2002.
[63] Zwanikken, J. W. and de la Cruz, M. O.. Tunable soft structure in charged fluids confined by dielectric interfaces. Proc. Nat. Acad. Sci. USA, 110:53015308, 2013.
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