Electrostatic Interaction of Dielectric Particles in an Electrolytic Solution

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Abstract

On the basis of the Poisson-Boltzmann equation the electrostatic interaction between two charged dielectric spherical particles in a symmetric electrolyte solution is considered. The interaction forces between particles of the same radius under the condition of uniform charge distribution on their surfaces in the absence of an external field have been calculated by the finite element method. The dependence of the electrostatic repulsion forces between the particles on the magnitude of the particle charges and the dielectric permittivities of the particle materials and the surrounding medium has been analyzed.

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About the authors

S. I. Grashchenkov

Псковский государственный университет

Author for correspondence.
Email: grasi@mail.ru
Russian Federation, Псков

References

  1. Israelachvili J. N. Intermolecular and surface forces. San Diego: Academic press, 2011
  2. Ledbetter J.E., Croxton T.L., McQuarrie D.A. The interaction of two charged spheres in the Poisson–Boltzmann equation // Can. J. Chem. 1981. V. 59. № 13. P. 1860–1864. https://doi.org/10.1139/v81-277
  3. Gouy M. Sur la constitution de la charge électrique à la surface d’un électrolyte // J. Phys. Theor. Appl. 1910. V. 9. №. 1. P. 457–468. https://hal.science/jpa-00241565/document (accessed on March 06, 2024)
  4. Huckel E., Debye P. Zur theorie der elektrolyte.I. Gefrierpunktserniedrigung und verwandte erscheinungen // Phys. Z. 1923. V. 24. S. 185–206.
  5. Lamm G. The Poisson–Boltzmann equation // Reviews in computational chemistry. 2003. V. 19. P. 147–365. https://doi.org/10.1002/0471466638.ch4
  6. Derjaguin B., Landau L. Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes // Progress in Surface Science. 1993. V. 43. P. 30−59. https://doi.org/10.1016/0079-6816(93)90013-L
  7. Ether D. S. et al. Double-layer force suppression between charged microspheres // Physical Review E. 2018. V. 97. №. 2 P. 022611. https://doi.org/10.1103/PhysRevE.97.022611
  8. Schnitzer O., Morozov M. A generalized Derjaguin approximation for electrical-double-layer interactions at arbitrary separations // The Journal of Chemical Physics. 2015. V. 142. №. 24. P. 244102. http://dx.doi.org/10.1063/1.4922546
  9. Derjaguin B. On the repulsive forces between charged colloid particles and on the theory of slow coagulation and stability of lyophobe sols // Transactions of the Faraday Society. 1940. V. 35. P. 203–215. https://doi.org/10.1016/0079-6816(93)90011-J
  10. Derbenev I. N. et al. Electrostatic interactions between charged dielectric particles in an electrolyte solution // The Journal of chemical physics. 2016. V. 145. №. 8. P. 084103. http://dx.doi.org/10.1063/1.4961091
  11. Filippov A. V., Starov V. Interaction of nanoparticles in electrolyte solutions // The Journal of Physical Chemistry B. 2023. V. 127. № 29. P. 6562–6572. https://doi.org/10.1021/acs.jpcb.3c01220
  12. Гращенков С.И. О силе электростатического взаимодействия двух сфероидальных макрочастиц в модели Пуассона-Больцмана // Журнал технической физики. 2022. Т. 92. № 12. С. 1770–1775. http://dx.doi.org/10.21883/JTF.2022.12.53742.145-22
  13. Ledbetter J.E., Croxton T.L., McQuarrie D.A. The interaction of two charged spheres in the Poisson–Boltzmann equation // Canadian Journal of Chemistry. 1981. V. 59. №. 13. P. 1860–1864. https://doi.org/10.1139/v81-277
  14. Carnie S. L., Chan D. Y. C., Stankovich J. Computation of forces between spherical colloidal particles: nonlinear Poisson-Boltzmann theory // Journal of colloid and interface science. 1994. V. 165. № 1. P. 116–128. https://doi.org/10.1006/jcis.1994.1212
  15. Lima E.R.A., Tavares F.W., Biscaia Jr.E.C. Finite volume solution of the modified Poisson–Boltzmann equation for two colloidal particles // Physical Chemistry Chemical Physics. 2007. V. 9. № 24. P. 3174–3180. https://doi.org/10.1039/b701170a
  16. Brenner S.L., Roberts R.E. Variational solution of the Poisson-Boltzmann Equation for a spherical colloidal particle // The Journal of Physical Chemistry. 1973. V. 77. № 20. P. 23672370. https://doi.org/10.1021/j100639a001
  17. Chan D.Y.C., BKC C. Electrical double-layer interaction between spherical colloidal particles: An exact solution Journal of Colloid and Interface Science. 1983. V. 92. № 1. P 281283. https://doi.org/10.1016/0021-9797(83)90143-1
  18. Dyshlovenko P. Adaptive mesh enrichment for the Poisson–Boltzmann equation // Journal of Computational Physics. 2001. V. 172. № 1. P. 198–208. https://doi.org/10.1006/jcph.2001.6820
  19. Qiao Z., Li Z., Tang T. A finite difference scheme for solving the nonlinear Poisson-Boltzmann equation modeling charged spheres // Journal of Computational Mathematics. 2006. № 3. P. 252–264.
  20. Merrill J.W., Sainis S.K., Dufresne E.R. Many-body electrost atic forces between colloidal particles at vanishing ionic strength // Physical review letters. 2009. V. 103. № 13. P. 138301. https://doi.org/10.1103/PhysRevLett.103.138301
  21. Russ C. et al. Three-body forces between charged colloidal particles // Physical Review E. 2002. V. 66. № 1. P. 011402 https://doi.org/10.1103/PhysRevLett.103.138301
  22. Warszyński P., Adamczyk Z. Calculations of double-layer electrostatic interactions for the sphere/plane geometry // Journal of colloid and interface science. 1997. V. 187. № 2. P. 283–295. https://doi.org/10.1006/jcis.1996.4671
  23. Lu B. Finite element modeling of biomolecular systems in ionic solution // Image-Based Geometric Modeling and Mesh Generation. Dordrecht: Springer Netherlands. 2013. P. 271–301. https://doi.org/10.1007/978-94-007-4255-0_14
  24. Фортов В.Е., Храпак А.Г., Храпак С.А., Молотков В.И., Петров О.Ф. Пылевая плазма. // Успехи физических наук. 2004, V. 174. № 5. P. 495–544. https://doi.org/10.3367/UFNr.0174.200405b.0495
  25. Gangl P. et al. Fully and semi-automated shape differentiation in NGSolve // Structural and multidisciplinary optimization. 2021. V. 63. №. 3. P. 1579–1607. https://doi.org/10.1007/s00158-020-02742-w
  26. Carnie S.L., Chan D.Y.C. Interaction free energy between identical spherical colloidal particles: The linearized Poisson-Boltzmann theory // Journal of colloid and interface science. 1993. V. 155. №. 2. P. 297–312. https://doi.org/10.1006/jcis.1993.1039
  27. Bell G.M., Levine S., McCartney L.N. Approximate methods of determining the double-layer free energy of interaction between two charged colloidal spheres // Journal of Colloid and Interface Science. 1970. V. 33. №. 3. P. 335–359. https://doi.org/10.1016/0021-9797(70)90228-6
  28. Sainis S.K., Merrill J.W., Dufresne E.R. Electrostatic interactions of colloidal particles at vanishing ionic strength // Langmuir. 2008. V. 24. № 23. P. 13334–13337.
  29. Choi K.H. et al. Direct measurement of electrostatic interactions between poly (methyl methacrylate) microspheres with optical laser tweezers // Soft matter. 2019. V. 15. № 40. P. 8051–8058. https://doi.org/10.1039/c9sm01374a

Supplementary files

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2. Fig. 1. Structure of the initial computational domain

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3. Fig. 2. Dependence of the normalised force of electrostatic interaction of particles on the normalised distance between particle surfaces at k = 0.1: 1 - f = 1, ε = 0.1; 2 - f = 1, ε = 1; 3 - f = 1, ε = 10; 4 - f = 10, ε = 0.1; 5 - f = 10, ε = 1; 6 - f = 10, ε = 10

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4. Fig. 3. Dependence of the normalised force of electrostatic interaction of particles on the normalised distance between the particle surfaces at k = 1: 1 - f = 1, ε = 0.1; 2 - f = 1, ε = 1; 3 - f = 1, ε = 10; 4 - f = 10, ε = 0.1; 5 - f = 10, ε = 1; 6 - f = 10, ε = 10; dotted line - LS approximation

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5. Fig. 4. Dependence of the normalised force of electrostatic interaction of particles on the normalised distance between the surfaces of particles at k = 10: 1 - f = 1, ε = 0.1; 2 - f = 1, ε = 1; 3 - f = 1, ε = 10; 4 - f = 10, ε = 0.1; 5 - f = 10, ε = 1; 6 - f = 10, ε = 10

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6. Fig. 5. Dependence of the normalised force of electrostatic interaction of particles on f at h/a = 0.1: 1 - k = 0.1, ε = 0.1; 2 - k = 0.1, ε = 1; 3 - k = 0.1, ε = 10; 4 - k = 1, ε = 0.1; 5 - k = 1, ε = 1; 6 - k = 1, ε = 10

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7. Fig. 6. Dependence of the normalised force of electrostatic interaction of particles on f at h/a = 0.1 and k = 10: 1 - ε = 0.1; 2 - ε = 1; 3 - ε = 10

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8. Fig. 7. Dependence of the normalised force of electrostatic interaction of particles on f at h/a = 2: 1 - k = 0.1, ε = 0.1; 2 - k = 0.1, ε = 1; 3 - k = 0.1, ε = 10; 4 - k = 1, ε = 0.1; 5 - k = 1, ε = 10

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9. Fig. 8. Dependence of the force of electrostatic interaction of particles on the distance between the particle centres at k = 0.59, f = 7.8, ε = 1.26. The rhombuses represent experimental data [20]

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