FEATURES OF THE DOUBLE ELECTRIC LAYER AROUND SPHERICAL PARTICLES. THE POISSON-HELMHOLTZ-BOLTZMANN MODEL

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Abstract

The Poisson- Helmholtz- Boltzmann model is used to study the properties of a double electric layer formed near a single weakly charged spherical particle surrounded by a 1:1 electrolyte solution. Dividing into Coulomb and non- Coulomb (defined by the Yukawa potential) interactions between ions in solution, as well as between ions and a particle, we obtain mathematical expressions for the profiles of the corresponding potentials near the particle as a function of the main parameters of the model. When varying the values of key parameters, we find both monotonic and nonmonotonic profiles of the electrostatic potential, and we observe a change in the sign of the potential, resulting in the phenomena of inversion and reversal of charge. The conditions under which the inversion and reversal of the sign of the particle potential occur are determined. The dependence of the zero charge potential on the particle size, the concentration of a monovalent electrolyte solution, and the surface density of a non- Coulomb force source is considered.

About the authors

A. I Dolinyi

Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences

Email: dolinyi@mail.ru
Moscow, Russia

References

  1. Verwey E.J.W. Overbeek J.Th.G. Theory of the Stability of Lyophobic Colloids. NY: Elsevier, 1948.
  2. Делахей П. Двойной слой и кинетика электродных процессов. М.: Мир, 1967.
  3. Дерягин Б.В., Чураев Н.В., Муллер В.М. Поверхностные силы. М.: Наука, 1985.
  4. Lyklema J. Fundamentals of Interface and Colloid Science. San Diego: Elsevier Academic Press, 2005. Vol. 2. Chapter 3.
  5. Markovich T., Andelman D., Podgornik R. Charged membranes: Poisson-Boltzmann theory. The DLVO paradigm, and beyond. In: Handbook of Lipid Membranes. Molecular, Functional, and Materials Aspects. Eds. by Safinya C.R., Radler J.O. CRC Press. 2021.
  6. Israelashvili J.N. Intermolecular and Surface Forces. Third edition. Amsterdam: Elsevier, 2011.
  7. Wu J. Understanding the electric double-layer structure, capacitance, and charging dynamics // Chem. Rev. 2022. V. 122. N: 12. P. 10821-10859. https://doi.org/10.1021/acs.chemrev.2c00097
  8. Fleischmann S., Mitchell J.B., Wang R., Zhan Ch., Jiang D., Presser V., Augustyn V. Pseudocapacitance: From fundamental understanding to high power energy storage materials // Chem. Rev. 2020. V. 120. N: 14. P. 6738-6782. https://doi.org/10.1021/acs.chemrev.0c00170
  9. Henrique F., Zuk P.J., Gupta A. Charging dynamics of electrical double layers inside a cylindrical pore: predicting the effects of arbitrary pore size // Soft Matter. 2022. V. 18. N: 1. P. 198-213. https://doi.org/10.1039/D1SM01239H
  10. Rajan A.G., Martinez J.M.P., Carter E.A. Why do we use the materials and operating conditions we use for heterogeneous (photo)electrochemical water splitting? // ACS Catalysis. 2020. V. 10. N: 19. P. 11177-11234. https://doi.org/10.1021/acscatal.0c01862
  11. Biesheuvel P., Bazant M. Nonlinear dynamics of capacitive charging and desalination by porous electrodes // Phys. Rev. E. 2010. V. 81. N: 3. P. 031502. https://doi.org/10.1103/PhysRevE.81.031502
  12. Levin Y. Electrostatic correlations: from plasma to biology // Rep. Prog. Phys. 2002. V. 65. N: 11. P. 1577-1632. https://doi.org/10.1088/0034-4885/65/11/201
  13. Grosberg A.Y., Nguyen T.T., Shklovskii B I. Colloquium: The physics of charge inversion in chemical and biological systems // Rev. Modern Phys. 2002. V. 74. N: 2. P. 329-345. https://doi.org/10.1103/RevModPhys.74.329
  14. Quezada-Perez M., Gonzalez-Tovar E., Martin-Molina A., Lozada-Cassou M., Hidalgo-Alvarez R. Overcharging in colloids: beyond the Poisson-Boltzmann approach // ChemPhysChem. 2003. V. 4. N: 3. P. 234-248. https://doi.org/10.1002/cphc.200390040
  15. Lyklema J. Overcharging, charge reversal: Chemistry or physics? // Colloids Surf. A 2006. V. 291. N: 1-3. P. 3-12. https://doi.org/10.1016/j.colsurfa.2006.06.043
  16. Strauss U.P., Gershfeld N.L., Spiera H. Charge reversal of cationic poly-4-vinylpyridine derivatives in KBr solutions // J. Am. Chem. Soc. 1954. V. 76. N: 23. P. 5909-5911. https://doi.org/10.1021/ja01652a004
  17. Elimelech M., O'Melia C.R. Kinetics of deposition of colloidal particles in porous media // Environ. Sci. Technol. 1990. V. 24. N: 10. P. 1528-1536. https://doi.org/10.1021/es00080a012
  18. Martin-Molina A., Quezada-Perez M., Galisteo-Gonzalez F., Hidalgo-Alvarez R. Looking into overcharging in model colloids through electrophoresis: Asymmetric electrolytes // J. Chem. Phys. 2003. V. 118. N: 9. P. 4183-4189. https://doi.org/10.1063/1.1540631
  19. Quesada-Perez M., Gonzalez-Tovar E., Martin-Molina A., Lozada-Cassou M., Hidalgo-Alvarez R. Ion size correlations and charge reversal in real colloids // Colloids Surf. A 2005. V. 267. N: 1-3. P. 24-30. https://doi.org/10.1016/j.colsurfa.2005.06.034
  20. Besteman K., Zevenbergen M.A.G., Heering H.A., Lemay S.G. Direct observation of charge inversion by multivalent ions as a universal electrostatic phenomenon // Phys. Rev. Lett. 2004. V. 93. N: 17. P. 170802. https://doi.org/10.1103/PhysRevLett.93.170802
  21. Besteman K., Zevenbergen M.A.G., Lemay S. Charge inversion by multivalent ions: Dependence on dielectric constant and surface-charge density // Phys. Rev. E. 2005. V. 72. N6. P. 061501. https://doi.org/10.1103/PhysRevE.72.061501
  22. Besteman K., van Eijk K., Lemay S.G. Charge inversion accompanies DNA condensation by multivalent ions // Nat. Phys. 2007. V. 3. P. 641-644. https://doi.org/10.1038/nphys697
  23. Vaknin D., Krüger P., Lösche M. Anomalous X-ray reflectivity characterization of ion distribution at biomimetic membranes // Phys. Rev. Lett. 2003. V. 90. N6. P. 178102. https://doi.org/10.1103/PhysRevLett.90.178102
  24. Pittler J., Bu W., Vaknin D., Travesset A., McGillivray D.J., Lösche M. Charge inversion at minute electrolyte concentrations // Phys. Rev. Lett. 2006. V. 97. N6. P. 046102. https://doi.org/10.1103/PhysRevLett.97.046102
  25. Radeva T. Physical Chemistry of Polyelectrolytes. NY: Marcel Dekker, 2001.
  26. Ladam G., Schaad P., Voegel J.C., Schaaf P., Decher G., Cuisinier F. In situ determination of the structural properties of initially deposited polyelectrolyte multilayers // Langmuir 2000. V. 16. N6. P. 1249-1255. https://doi.org/10.1021/la990650k
  27. Paton-Morales P., Talens-Alesson F.I. Effect of competitive adsorption of Zn2+ on the flocculation of Lauryl Sulfate micelles by Al3+ // Langmuir 2002. V. 18. N6. P. 8295-8301. https://doi.org/10.1021/la0200820
  28. Gelbart W.M., Bruinsma R.F., Pincus P.A., Parsegian V.A. DNA-inspired electrostatics // Phys. Today 2000. V. 53. N6. P. 38-44. https://doi.org/10.1063/1.1325230
  29. Attard P. Ion condensation in the electric double layer and the corresponding Poisson-Boltzmann effective surface charge // J. Phys. Chem. 1995. V. 99. N6. P. 14174-14181. https://doi.org/10.1021/j100038a060
  30. Tanaka V., Grosberg A.Y. Giant charge inversion of a macroion due to multivalent counterions and monovalent coions: Molecular dynamics study // J. Chem. Phys. 2001. V. 115. N6. P. 567-574. https://doi.org/10.1063/1.1377033
  31. Terao T., Nakayama T. Charge inversion of colloidal particles in an aqueous solution: Screening by multivalent ions // Phys. Rev. E. 2001. V. 63. N6. P. 041401. https://doi.org/10.1103/PhysRevE.63.041401
  32. Jimenez-Angeles F., Lozada-Cassou M. A Model macroion solution next to a charged wall: overcharging, charge reversal, and charge inversion by macroions // J. Phys. Chem. B 2004. V. 108. N6. P. 7286-7296. https://doi.org/10.1021/jp036464b
  33. Semenov I., Raafatnia S., Sega M., Lobaskin V., Holm C., Kremer F. Electrophoretic mobility and charge inversion of a colloidal particle studied by single-colloid electrophoresis and molecular dynamics simulations // Phys. Rev. E. 2013. V. 87. N6. P. 022302. https://doi.org/10.1103/PhysRevE.87.022302.
  34. Nguyen T. T., Grosberg A. Y., Shklovskii B. I. Macroions in salty water with multivalent ions: Giant inversion of charge // Phys. Rev. Lett. 2000. V. 85. N6. P. 1568-1571. https://doi.org/10.1103/PhysRevLett.85.1568
  35. Kjellander R. Ion-ion correlations and effective charges in electrolyte and macroion systems // Ber. Bunsenges. Phys. Chem. 1996. V. 100. N6. P. 894-904. https://doi.org/10.1002/bbpc.19961000635
  36. Lozada-Cassou M., Saavedra-Barrera R., Henderson D. The application of the hypernetted chain approximation to the electrical double layer: Comparison with Monte Carlo results for symmetric salts // J. Chem. Phys. 1982. V. 77. N6. P. 5150-5156. https://doi.org/10.1063/1.443691
  37. González-Tovar E., Lozada-Cassou M. The spherical double layer: A hypernetted chain mean spherical approximation calculation for a model spherical colloid particle // J. Phys. Chem. 1989. V. 93. N6. P. 3761-3768. https://doi.org/10.1021/j100346a076
  38. Torrie G.M., Valleau J.P. Electrical double layers. 4. Limitations of the Gouy-Chapman theory // J. Phys. Chem. 1982. V. 86. N6. P. 3251-3257. https://doi.org/10.1021/j100213a035
  39. Das T., Bratko D., Bhuiyan L.B., Outhwaite C.W. Polyelectrolyte solutions containing mixed valency ions in the cell model: A simulation and modified Poisson-Boltzmann study // J. Chem. Phys. 1997. V. 107. N. 21. P. 9197-9207. https://doi.org/10.1063/1.475211
  40. Vlachy V. Ionic effects beyond Poisson-Boltzmann theory // Annu. Rev. Phys. Chem. 1999. V. 50. P. 145-165. https://doi.org/10.1146/annurev.physchem.50.1.145
  41. Li Z., Wu J. Density-functional theory for the structures and thermodynamic properties of highly asymmetric electrolyte and neutral component mixtures // Phys. Rev. E. 2004. V. 70. N. 3. P. 031109. https://doi.org/10.1103/PhysRevE.70.031109
  42. Henderson D., Lamperski S., Jin Z., Wu J. Density functional study of the electric double layer formed by a high density electrolyte // J. Phys. Chem. B 2011. V. 115. N. 44. P. 12911-12914. https://doi.org/10.1021/jp2078105
  43. Lee J.W., Nilson R.H., Templeton J.A., Griffiths S.K., Kung A., Wong B.M. Comparison of molecular dynamics with classical density functional and Poisson-Boltzmann theories of the electric double layer in nanochannels // J. Chem. Theory Comput. 2012. V. 8. N. 6. P. 2012-2022. https://doi.org/10.1021/ct3001156
  44. Jimenez-Angeles F., Lozada-Cassou M. On the regimes of charge reversal // J. Chem. Phys. 2008. V. 128. N. 17. P. 174701. https://doi.org/10.1063/1.2911923
  45. Torrie G.M., Valleau J.P. Electrical double layers. I. Monte Carlo study of a uniformly charged surface // J. Chem. Phys. 1980. V. 73. N. 11. P. 5807-5816. https://doi.org/10.1063/1.440065
  46. Valleau J.P., Torrie G.M. The electrical double layer. V. Asymmetric ion-wall interactions // J. Chem. Phys. 1984. V. 81. N. 12. P. 6291-6296. https://doi.org/10.1063/1.447535
  47. Henderson D., Boda D. Insights from theory and simulation on the electrical double layer // Phys. Chem. Chem. Phys. 2009. V. 11. N. 20. P. 3822-3830. https://doi.org/10.1039/B815946G
  48. Boda D., Fawcett W. R., Henderson D., Sokolowski S. Monte Carlo, density functional theory, and Poisson-Boltzmann theory study of the structure of an electrolyte near an electrode // J. Chem. Phys. 2002. V. 116. N. 16. P. 7170-7176. https://doi.org/10.1063/1.1464826
  49. Martin-Molina A., Maroto-Centeno J.A., Hidalgo-Alvarez R., Quesada-Perez M. Charge reversal in real colloids: Experiments, theory and simulations // Colloids Surf. A 2008. V. 319. N. 1-3. P. 103-108. https://doi.org/10.1016/j.colsurfa.2007.09.041
  50. Crozier P.S., Rowley R.L., Henderson D. Molecular dynamics simulations of ion size effects on the fluid structure of aqueous electrolyte systems between charged model electrodes // J. Chem. Phys. 2001. V. 114. N. 17. P. 7513-7517. https://doi.org/10.1063/1.1362290
  51. Messina R., Gonzalez-Tovar E., Lozada-Cassou M., Holm C. Overcharging: The crucial role of excluded volume // Europhys. Lett. 2002. V. 60. N. 3. P. 383-389. https://doi.org/10.1209/epl/i2002-00275-y
  52. Kubičková A., Křížek T., Coufal P., Vazdar M., Wernersson E., Heyda J., Jungwirth P. Overcharging in biological systems: Reversal of electrophoretic mobility of aqueous polysaptrate by multivalent cations // Phys. Rev. Lett. 2012. V. 108. N. 18. P. 186101. https://doi.org/10.1103/PhysRevLett.108.186101
  53. Bazant M.Z., Storey B.D., Kornyshev A.A. Double layer in ionic liquids: Overscreening versus crowding // Phys. Rev. Lett. 2011. V. 106. N. 4. P. 046102. https://doi.org/10.1103/PhysRevLett.106.046102
  54. Storey B.D., Bazant M.Z. Effects of electrostatic correlations on electrokinetic phenomena // Phys. Rev. E. 2012. V. 86. N. 5. P. 056303. https://doi.org/10.1103/PhysRevE.86.056303
  55. de Souza J.P., Bazant M.Z. Continuum theory of electrostatic correlations at changed surfaces // J. Phys. Chem. C 2020. V. 124. N. 21. P. 11414-11421. https://doi.org/10.1021/acs.jpcc.0c01261
  56. Parsons D.F., Ninham B.W. Surface charge reversal and hydration forces explained by ionic dispersion forces and surface hydration // Colloids Surf. A. 2011. V. 383. № 1-3. P. 2-9. https://doi.org/10.1016/j.colsurfa.2010.12.025
  57. Parsons D.F., Ninham B.W. Charge reversal of surfaces in divalent electrolytes: The role of ionic dispersion interactions // Langmuir 2010. V. 26. № 9. P. 6430-6436. https://doi.org/10.1021/la9041265
  58. Tavares F.W., Bostrom M., Lima E.R.A., Biscaia E.C.Jr. Ion-specific thermodynamic properties of colloids and proteins // Fluid Phase Equilibria 2010. V. 296. № 2. P. 99-105. https://doi.org/10.1016/j.fluid.2010.02.031
  59. Bohinc K., Shrestha A., May S. The Poisson-Helmholtz-Boltzmann model // Eur. Phys. J. E 2011. V. 34. № 10. P. 108. https://doi.org/10.1140/epjc/i2011-11108-6
  60. Bohinc K., Shrestha A., Brumen M., May S. Poisson-Helmholtz-Boltzmann model of the electric double layer: Analysis of monovalent ionic mixtures // Phys. Rev. E. 2012. V. 85. № 3. P. 031130. https://doi.org/10.1103/PhysRevE.85.031130
  61. Zhao H. Influence of nonelectrostatic ion-ion interactions on double-layer capacitance // Phys. Rev. E. 2012. V. 86. № 5. P. 051502. https://doi.org/10.1103/PhysRevE.86.051502
  62. Brown M.A., Bossa G.V., May S. Emergence of a Stern layer from the incorporation of hydration interactions into the Gouy-Chapman model of the electrical double layer // Langmuir 2015. V. 31. № 42. P. 11477-11483. https://doi.org/10.1021/acs.langmuir.5b02389
  63. Goswami P., Dhar J., Ghosh U., Chakraborty S. Solvent-mediated non-electrostatic ion-ion interactions predicting anomalies in electrophoresis // Electrophoresis 2014. V. 38. № 5. P. 712-719. https://doi.org/10.1002/elps.201600394
  64. Berntson B.K., Downing R., Bossa G.V., May S. Debye-Hückel theory of weakly curved macroions: Implementing ion specificity through a composite Coulomb-Yukawa interaction potential // Phys. Rev. E. 2018. V. 98. № 2. P. 022609. https://doi.org/10.1103/PhysRevE.98.022609
  65. Brown M.A., Abbas Z., Kleibert A., Green R.G., Goel A., May S., Squires T.M. Determination of surface potential and electrical double-layer structure at the aqueous electrolyte-nanoparticle interface // Phys. Rev. X. 2016. V. 6. № 1. P. 011007. https://doi.org/10.1103/PhysRevX.6.011007
  66. Spaight J., Downing R., May S., de Carvalho S.J., Bossa G.V. Modeling hydration-mediated ion-ion interactions in electrolytes through oscillating Yukawa potentials // Phys. Rev. E. 2020. V. 101. № 5. P. 052603. https://doi.org/10.1103/PhysRevE.101.052603
  67. Caetano D.L.Z., Bossa G.V., de Oliveira V.M., Brown M.A., de Carvalho S.J., May S. Differential capacitance of an electric double layer with asymmetric solvent-mediated interactions: mean-field theory and Monte Carlo simulations // Phys. Chem. Chem. Phys. 2017. V. 19. № 35. P. 23971-23981. https://doi.org/10.1039/C7CP04672C
  68. Valleau J.P., Torrie G.M. The electrical double layer. III. Modified Gouy-Chapman theory with unequal ion sizes // J. Chem. Phys. 1982. V. 76. № 9. P. 4623-4630. https://doi.org/10.1063/1.443542
  69. Bhuiyan L.B., Blum L., Henderson D. The application of the modified Gouy-Chapman theory to an electrical double layer containing asymmetric ions // J. Chem. Phys. 1983. V. 78. № 1. P. 442-445. https://doi.org/10.1063/1.444523
  70. Valiskó M., Henderson D., Boda D. Competition between the effects of asymmetries in ion diameters and charges in an electrical double layer studied by Monte Carlo simulations and theories // J. Phys. Chem. B 2004. V. 108. № 42. P. 16548-16555. https://doi.org/10.1021/jp0473873

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