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Effects of electrostatic correlations and ion–solvent interactions of finite-sized ions on the electrophoresis of a soft particle

Published online by Cambridge University Press:  06 March 2025

Bapan Mondal Open the ORCID record for Bapan Mondal [Opens in a new window]  and
Somnath Bhattacharyya Open the ORCID record for Somnath Bhattacharyya [Opens in a new window]
Bapan Mondal
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, West Bengal, India
Somnath Bhattacharyya*
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, West Bengal, India
*
Corresponding author: Somnath Bhattacharyya, somnath@maths.iitkgp.ac.in
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Abstract

A numerical study supplemented with theoretical analysis is made, to analyse the electrophoresis of highly charged soft particles in electrolytes with trivalent counterions. The electrokinetic model is devised under the continuum hypothesis, which incorporates the ion–ion electrostatic correlations, hydrodynamic steric interactions of finite sized ions and ion–solvent interactions. The governing equations for ion transport and electric field are derived from the volumetric free energy of the system, which includes the first-order correction for the non-local electrostatic correlations of interacting ions, excess electrochemical potential due to finite ion size as well as the Born energy difference of ions due to dielectric permittivity variation. The electrolyte viscosity is considered to be a function of the local volume fraction of finite-sized ions, which causes the diffusivity of ions to vary locally. The occurrence of mobility reversal of a soft particle having the same polarity of its core and soft shell charge and formation of a coion-dominated zone in the soft layer is elaborated through this study. This can explain the mechanisms for the attraction between like-charged soft particles, as seen in the condensation of DNAs. The impact of ion–ion correlations and ion–solvent interactions of finite-sized ions are analysed by comparing them with the results based on the standard model. At a higher range of the core charge density, the ion–ion correlations induce a condensed layer of counterions on the outer surface of the core, which draws coions in the electric double layer, leading to an inversion in polarity of the charge density and mobility reversal. The dielectric decrement and ion steric interactions create a saturation in ion distribution and hence, modify the condensed layer of counterions. The enhanced fixed charge density of the polyelectrolyte layer diminishes the ion correlations due to the stronger screening effects and prevents the formation of a coion dominated zone in the Debye layer. The impact of the counterion size and the mixture of monovalent and trivalent counterions on mobility is analysed.

JFM classification

Micro-/Nano-fluid dynamics: Microfluidics MHD and Electrohydrodynamics: Electrokinetic flows Complex Fluids: Colloids

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JFM Papers
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Journal of Fluid Mechanics , Volume 1006 , 10 March 2025 , A9
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© The Author(s), 2025. Published by Cambridge University Press

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References

Agrawal, N. & Wang, R. 2022 Electrostatic correlation induced ion condensation and charge inversion in multivalent electrolytes. J. Chem. Theor. Comput. 18 (10), 6271–6280.CrossRefGoogle ScholarPubMed
Anderson, J. L. 1989 Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21 (1), 61–99.CrossRefGoogle Scholar
Andrade, E. N. D. C. & Dodd, C. 1951 The effect of an electric field on the viscosity of liquids. ii. Proc. R. Soc. Lond. A. Math. Phys. Sci. 204, 449–464.Google Scholar
Angelini, T. E., Liang, H., Wriggers, W. & Wong, G. C. 2003 Like-charge attraction between polyelectrolytes induced by counterion charge density waves. Proc. Natl. Acad. Sci. USA 100 (15), 8634–8637.CrossRefGoogle ScholarPubMed
Batchelor, G. & Green, J. 1972 The determination of the bulk stress in a suspension of spherical particles to order c2. J. Fluid Mech. 56 (3), 401–427.CrossRefGoogle Scholar
Bazant, M. Z., Kilic, M. S., Storey, B. D. & Ajdari, A. 2009 Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions. Adv. Colloid Interface Sci. 152 (1–2), 48–88.CrossRefGoogle Scholar
Bazant, M. Z., Storey, B. D. & Kornyshev, A. A. 2011 Double layer in ionic liquids: overscreening versus crowding. Phys. Rev. Lett. 106 (4), 046102.CrossRefGoogle ScholarPubMed
Cardenas-Benitez, B., Jind, B., Gallo-Villanueva, R. C., Martinez-Chapa, S. O., Lapizco-Encinas, B. H. & Perez-Gonzalez, V. H. 2020 Direct current electrokinetic particle trapping in insulator-based microfluidics: theory and experiments. Anal. Chem. 92 (19), 12871–12879.CrossRefGoogle Scholar
Carrique, F., Ruiz-Reina, E., Arroyo, F., López-García, J. & Delgado, A. 2024 Effects of finite counterion size and nonhomogeneous permittivity and viscosity of the solution on the electrokinetics of a concentrated salt-free colloid. Phys. Rev. E 110 (1), 014601.CrossRefGoogle ScholarPubMed
Celebi, A. T., Cetin, B. & Beskok, A. 2019 Molecular and continuum perspectives on intermediate and flow reversal regimes in electroosmotic transport. J. Phys. Chem. C 123 (22), 14024–14035.CrossRefGoogle Scholar
Chen, Y., Ni, Z., Wang, G., Xu, D. & Li, D. 2008 Electroosmotic flow in nanotubes with high surface charge densities. Nano Lett. 8 (1), 42–48.CrossRefGoogle ScholarPubMed
Cobos, R. & Khair, A. S. 2023 Nonlinear electrophoretic velocity of a spherical colloidal particle. J. Fluid Mech. 968, A14.CrossRefGoogle Scholar
de Souza, J. & Bazant, M. Z. 2020 Continuum theory of electrostatic correlations at charged surfaces. J. Phys. Chem. C 124 (21), 11414–11421.CrossRefGoogle Scholar
Diehl, A. & Levin, Y. 2006 Smoluchowski equation and the colloidal charge reversal. J. Chem. Phys. 125 (5), 054902.CrossRefGoogle ScholarPubMed
Duval, J. F. & Gaboriaud, F. 2010 Progress in electrohydrodynamics of soft microbial particle interphases. Curr. Opin. Colloid Interface Sci. 15 (3), 184–195.CrossRefGoogle Scholar
Duval, J. F. & Ohshima, H. 2006 Electrophoresis of diffuse soft particles. Langmuir 22 (8), 3533–3546.CrossRefGoogle ScholarPubMed
Duval, J. F., Werner, C. & Zimmermann, R. 2016 Electrokinetics of soft polymeric interphases with layered distribution of anionic and cationic charges. Curr. Opin. Colloid Interface Sci. 24, 1–12.CrossRefGoogle Scholar
Fletcher, C. A. 2012 Computational Techniques for Fluid Dynamics: Specific Techniques for Different Flow Categories. Springer Science & Business Media.Google Scholar
Ganguly, A., Roychowdhury, S. & Gupta, A. 2024 Unified mobility expressions for externally driven and self-phoretic propulsion of particles. J. Fluid Mech. 994, A2.CrossRefGoogle Scholar
Gavryushov, S., & Linse, P. 2003 Polarization deficiency and excess free energy of ion hydration in electric fields. J. Phys. Chem. B 107 (29), 7135–7142.CrossRefGoogle Scholar
Gupta, A., Rajan, G., Ananth, C., Emily, A. & Stone, H.A 2020 b Thermodynamics of electrical double layers with electrostatic correlations. J. Phys. Chem. C 124 (49), 26830–26842.CrossRefGoogle Scholar
Gupta, A., Rajan, G., Ananth, C., Emily, A. & Stone, H. A. 2020 a A Ionic layering and overcharging in electrical double layers in a Poisson–Boltzmann model. Phys. Rev. Lett. 125 (18), 188004.CrossRefGoogle Scholar
Gupta, A. & Stone, H. A. 2018 Electrical double layers: effects of asymmetry in electrolyte valence on steric effects, dielectric decrement, and ion–ion correlations. Langmuir 34 (40), 11971–11985.CrossRefGoogle ScholarPubMed
Hasted, J., Ritson, D. & Collie, C. 1948 Dielectric properties of aqueous ionic solutions. Parts I and II. J. Chem. Phys. 16 (1), 1–21.CrossRefGoogle Scholar
Hatlo, M. M., Van Roij, R. & Lue, L. 2012 The electric double layer at high surface potentials: the influence of excess ion polarizability. Europhys. Lett. 97 (2), 28010.CrossRefGoogle Scholar
Hennequin, T., Manghi, M. & Palmeri, J. 2021 Competition between born solvation, dielectric exclusion, and coulomb attraction in spherical nanopores. Phys. Rev. E 104 (4), 044601.CrossRefGoogle ScholarPubMed
Hsiao, P.-Y. 2008 Overcharging, charge inversion, and reentrant condensation: using highly charged polyelectrolytes in tetravalent salt solutions as an example of study. J. Phys. Chem. B 112 (25), 7347–7350.CrossRefGoogle Scholar
Hsu, W.-L., Daiguji, H., Dunstan, D. E., Davidson, M. R. & Harvie, D. J. 2016 Electrokinetics of the silica and aqueous electrolyte solution interface: viscoelectric effects. Adv. Colloid Interface Sci. 234, 108–131.CrossRefGoogle ScholarPubMed
Hunter, R. J. 2013 Zeta Potential in Colloid Science: Principles and Applications, vol. 2. Academic Press.Google Scholar
Kavokine, N., Netz, R. R. & Bocquet, L. 2021 Fluids at the nanoscale: from continuum to subcontinuum transport. Annu. Rev. Fluid Mech. 53 (1), 377–410.CrossRefGoogle Scholar
Khair, A. S. & Squires, T. M. 2009 Ion steric effects on electrophoresis of a colloidal particle. J. Fluid Mech. 640, 343–356.CrossRefGoogle Scholar
Kubíčková, A., Křížek, T., Coufal, P., Vazdar, M., Wernersson, E., Heyda, J. & Jungwirth, P. 2012 Overcharging in biological systems: reversal of electrophoretic mobility of aqueous polyaspartate by multivalent cations. Phys. Rev. Lett. 108 (18), 186101.CrossRefGoogle ScholarPubMed
Landau, L. D. & Lifshitz, E. M. 2013 Fluid Mechanics: Landau and Lifshitz: Course of Theoretical Physics, vol. 6. Elsevier.Google Scholar
Lau, A. W., Lukatsky, D. B., Pincus, P. & Safran, S.A. 2002 Charge fluctuations and counterion condensation. Phys. Rev. E 65 (5), 051502.CrossRefGoogle ScholarPubMed
Lee, Y.-F., Chang, W.-C., Wu, Y., Fan, L. & Lee, E. 2021 Diffusiophoresis of a highly charged soft particle in electrolyte solutions. Langmuir 37 (4), 1480–1492.CrossRefGoogle ScholarPubMed
Lenz, O. & Holm, C. 2008 Simulation of charge reversal in salty environments: giant overcharging? Eur. Phys. J. E 26 (1-2), 191–195.CrossRefGoogle ScholarPubMed
Lesniewska, N., Beaussart, A. & Duval, J. F. 2023 Electrostatics of soft (bio) interfaces: corrections of mean-field Poisson–Boltzmann theory for ion size, dielectric decrement and ion–ion correlations. J. Colloid Interface Sci. 642, 154–168.CrossRefGoogle ScholarPubMed
Li, F., Allison, S. A. & Hill, R. J. 2014 Nanoparticle gel electrophoresis: Soft spheres in polyelectrolyte hydrogels under the Debye–Hückel approximation. J. Colloid Interface Sci. 423, 129–142.CrossRefGoogle ScholarPubMed
Li, G. & Koch, D. L. 2020 Electrophoresis in dilute polymer solutions. J. Fluid Mech. 884, A9.CrossRefGoogle Scholar
Liu, X. & Lu, B. 2017 Incorporating Born solvation energy into the three-dimensional Poisson–Nernst–Planck model to study ion selectivity in KcsA K + channels. Phys. Rev. E 96 (6), 062416.CrossRefGoogle ScholarPubMed
López-Garcıa, J., Grosse, C. & Horno, J. 2003 Numerical study of colloidal suspensions of soft spherical particles using the network method: 1. DC electrophoretic mobility. J. Colloid Interface Sci. 265 (2), 327–340.CrossRefGoogle Scholar
López-García, J., Horno, J. & Grosse, C. 2016 Ion size effects on the dielectric and electrokinetic properties in aqueous colloidal suspensions. Curr. Opin. Colloid Interface Sci. 24, 23–31.CrossRefGoogle Scholar
López-García, J., Horno, J. & Grosse, C. 2019 Ionic size, permittivity, and viscosity-related effects on the electrophoretic mobility: a modified electrokinetic model. Phys. Rev. Fluids 4 (10), 103702.CrossRefGoogle Scholar
López-Viota, J., Mandal, S., Delgado, A. V., Toca-Herrera, J. L., Möller, M., Zanuttin, F., Balestrino, M. & Krol, S. 2009 Electrophoretic characterization of gold nanoparticles functionalized with human serum albumin (HSA) and creatine. J. Colloid Interface Sci. 332 (1), 215–223.CrossRefGoogle ScholarPubMed
Luan, B. & Aksimentiev, A. 2010 Electric and electrophoretic inversion of the DNA charge in multivalent electrolytes. Soft Matt. 6 (2), 243–246.CrossRefGoogle ScholarPubMed
Martın-Molina, A., Quesada-Pérez, M., Galisteo-González, F., & Hidalgo-Á lvarez, R. 2003 Primitive models and electrophoresis: an experimental study. Colloids Surf. A 222 (1–3), 155–164.CrossRefGoogle Scholar
Mondal, B. & Bhattacharyya, S. 2024 Diffusiophoresis of charge-regulated nanoparticles comprising finite ion size and electrostatic correlation effects. Phys. Fluids 36 (2), 022022.CrossRefGoogle Scholar
Mondal, B., Bhattacharyya, S., Majhi, S. & Ohshima, H. 2023 Diffusiophoresis of a soft particle incorporating ion partitioning and hydrophobic core. Phys. Fluids 35 (6), 062017.CrossRefGoogle Scholar
Morales, M. C., Lin, H. & Zahn, J. D. 2012 Continuous microfluidic DNA and protein trapping and concentration by balancing transverse electrokinetic forces. Lab Chip 12 (1), 99–108.CrossRefGoogle ScholarPubMed
Moussa, M., Caillet, C., Town, R. M. & Duval, J. F. 2015 Remarkable electrokinetic features of charge-stratified soft nanoparticles: mobility reversal in monovalent aqueous electrolyte. Langmuir 31 (20), 5656–5666.CrossRefGoogle ScholarPubMed
Mukhina, T., Hemmerle, A., Rondelli, V., Gerelli, Y., Fragneto, G., Daillant, J. & Charitat, T. 2019 Attractive interaction between fully charged lipid bilayers in a strongly confined geometry. J. Phys. Chem. Lett. 10 (22), 7195–7199.CrossRefGoogle Scholar
Nakayama, Y. 2023 Nonlinear dielectric decrement of electrolyte solutions: an effective medium approach. J. Colloid Interface Sci. 646, 354–360.CrossRefGoogle ScholarPubMed
Nakayama, Y. & Andelman, D. 2015 Differential capacitance of the electric double layer: the interplay between ion finite size and dielectric decrement. J. Chem. Phys. 142 (4), 044706.CrossRefGoogle ScholarPubMed
Netz, R. R. 2001 Electrostatistics of counter-ions at and between planar charged walls: from Poisson-Boltzmann to the strong-coupling theory. Eur. Phys. J. E 5 (5), 557–574.CrossRefGoogle Scholar
Nguyen, T. H., Easter, N., Gutierrez, L., Huyett, L., Defnet, E., Mylon, S. E., Ferri, J. K. & Viet, N. A. 2011 The RNA core weakly influences the interactions of the bacteriophage MS2 at key environmental interfaces. Soft Matt. 7 (21), 10449–10456.CrossRefGoogle Scholar
Nishiya, M., Sugimoto, T. & Kobayashi, M. 2016 Electrophoretic mobility of carboxyl latex particles in the mixed solution of 1: 1 and 2: 1 electrolytes or 1: 1 and 3: 1 electrolytes: experiments and modeling. Colloids Surf. A 504, 219–227.CrossRefGoogle Scholar
O’Brien, R. W. & White, L. R. 1978 Electrophoretic mobility of a spherical colloidal particle. J. Chem. Soc. Faraday Trans. 2: Mol. Chem. Phys. 74, 1607–1626.CrossRefGoogle Scholar
Ohshima, H. 2000 On the general expression for the electrophoretic mobility of a soft particle. J. Colloid Interface. Sci. 228 (1), 190–193.CrossRefGoogle ScholarPubMed
Ohshima, H. 2013 Electrokinetic phenomena of soft particles. Curr. Opin. Colloid Interface Sci. 18 (2), 73–82.CrossRefGoogle Scholar
Pianegonda, S., Barbosa, M. C. & Levin, Y. 2005 Charge reversal of colloidal particles. Europhys. Lett. 71 (5), 831–837.CrossRefGoogle Scholar
Quesada-Pérez, M., González-Tovar, E., Martín-Molina, A.,Lozada-Cassou, M., & Hidalgo-A Ivarez, R. 2005 Ion size correlations and charge reversal in real colloids. Colloids Surf. A 267 (1–3), 24–30.CrossRefGoogle Scholar
Raafatnia, S., Hickey, O. A. & Holm, C. 2014 Mobility reversal of polyelectrolyte-grafted colloids in monovalent salt solutions. Phys. Rev. Lett. 113 (23), 238301.CrossRefGoogle ScholarPubMed
Raafatnia, S., Hickey, O. A. & Holm, C. 2015 Electrophoresis of a spherical polyelectrolyte-grafted colloid in monovalent salt solutions: comparison of molecular dynamics simulations with theory and numerical calculations. Macromolecules 48 (3), 775–787.CrossRefGoogle Scholar
Saville, D. 2000 Electrokinetic properties of fuzzy colloidal particles. J. Colloid Interface. Sci. 222 (1), 137–145.CrossRefGoogle ScholarPubMed
Saville, D. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29 (1), 27–64.CrossRefGoogle Scholar
Sawatzky, R. & Babchin, A. 1993 Hydrodynamics of electrophoretic motion in an alternating electric field. J. Fluid Mech. 246, 321–334.CrossRefGoogle Scholar
Schnitzer, O., Frankel, I. & Yariv, E. 2014 Electrophoresis of bubbles. J. Fluid Mech. 753, 49–79.CrossRefGoogle Scholar
Schnitzer, O., Yariv, E. 2015 the Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description. J. Fluid Mech. 773, 1–33.CrossRefGoogle Scholar
Schnitzer, O. & Yariv, E. 2012 Strong-field electrophoresis. J. Fluid Mech. 701, 333–351.CrossRefGoogle Scholar
Semenov, I., Raafatnia, S., Sega, M., Lobaskin, V., Holm, C. & Kremer, F. 2013 Electrophoretic mobility and charge inversion of a colloidal particle studied by single-colloid electrophoresis and molecular dynamics simulations. Phys. Rev. E 87 (2), 022302.CrossRefGoogle ScholarPubMed
Sherwood, J. 2011 Non-Newtonian stress in an electrolyte. J. Phys. Chem. B 115 (5), 1084–1088.CrossRefGoogle Scholar
Shin, S., Ault, J. T., Toda-Peters, K. & Shen, A. Q. 2020 Particle trapping in merging flow junctions by fluid-solute-colloid-boundary interactions. Phys. Rev. Fluids 5 (2), 024304.CrossRefGoogle Scholar
Šiber, A., Božič, A. L. & Podgornik, R. 2012 Energies and pressures in viruses: contribution of nonspecific electrostatic interactions. Phys. Chem. Chem. Phys. 14 (11), 3746–3765.CrossRefGoogle ScholarPubMed
Storey, B. D. & Bazant, M. Z. 2012 Effects of electrostatic correlations on electrokinetic phenomena. Phys. Rev. E 86 (5), 056303.CrossRefGoogle ScholarPubMed
Stout, R. F. & Khair, A. S. 2014 A continuum approach to predicting electrophoretic mobility reversals. J. Fluid Mech. 752, R1.CrossRefGoogle Scholar
Stout, R. F. & Khair, A. S. 2017 Influence of ion sterics on diffusiophoresis and electrophoresis in concentrated electrolytes. Phys. Rev. Fluids 2 (1), 014201.CrossRefGoogle Scholar
Tottori, S., Misiunas, K., Keyser, U. F. & Bonthuis, D. J. 2019 Nonlinear electrophoresis of highly charged nonpolarizable particles. Phys. Rev. Lett. 123 (1), 014502.CrossRefGoogle ScholarPubMed
Versteeg, H. K. 2007 An Introduction to Computational Fluid Dynamics the Finite Volume Method, 2/E. Pearson Education India.Google Scholar
Yang, X., Buyukdagli, S., Scacchi, A., Sammalkorpi, M. & Ala-Nissila, T. 2023 Theoretical and computational analysis of the electrophoretic polymer mobility inversion induced by charge correlations. Phys. Rev. E 107 (3), 034503.CrossRefGoogle ScholarPubMed
Yeh, L.-H., Fang, K.-Y., Hsu, J.-P. & Tseng, S. 2011 Influence of boundary on the effect of double-layer polarization and the electrophoretic behavior of soft biocolloids. Colloids Surf. B 88 (2), 559–567.CrossRefGoogle ScholarPubMed
Zhang, S. & Chu, H. C. 2024 Diffusioosmotic flow reversals due to ion–ion electrostatic correlations. Nanoscale 16 (19), 9367–9381.CrossRefGoogle ScholarPubMed
Zhao, H. & Zhai, S. 2013 The influence of dielectric decrement on electrokinetics. J. Fluid Mech. 724, 69–94.CrossRefGoogle ScholarPubMed
Zimmermann, R., Gunkel-Grabole, G., Bünsow, J., Werner, C., Huck, W. T. & Duval, J. F. 2017 Evidence of ion-pairing in cationic brushes from evaluation of brush charging and structure by electrokinetic and surface conductivity analysis. J. Phys. Chem. C 121 (5), 2915–2922.CrossRefGoogle Scholar