Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-06-09T01:28:58.972Z Has data issue: false hasContentIssue false
  • English
  • Français

New insights on the interfacial tension of electrochemical interfaces and the Lippmann equation

Published online by Cambridge University Press:  13 December 2017

W. DREYER ,
C. GUHLKE ,
M. LANDSTORFER  and
R. MÜLLER
W. DREYER
Affiliation:
Weierstrass-Institute, Mohrenstr. 39, 10117 Berlin, Germany emails: Wolfgang.Dreyer@wias-berlin.de, Clemens.Guhlke@wias-berlin.de, Manuel.Landstrofer@wias-berlin.de, Ruediger.Mueller@wias-berlin.de
C. GUHLKE
Affiliation:
Weierstrass-Institute, Mohrenstr. 39, 10117 Berlin, Germany emails: Wolfgang.Dreyer@wias-berlin.de, Clemens.Guhlke@wias-berlin.de, Manuel.Landstrofer@wias-berlin.de, Ruediger.Mueller@wias-berlin.de
M. LANDSTORFER
Affiliation:
Weierstrass-Institute, Mohrenstr. 39, 10117 Berlin, Germany emails: Wolfgang.Dreyer@wias-berlin.de, Clemens.Guhlke@wias-berlin.de, Manuel.Landstrofer@wias-berlin.de, Ruediger.Mueller@wias-berlin.de
R. MÜLLER
Affiliation:
Weierstrass-Institute, Mohrenstr. 39, 10117 Berlin, Germany emails: Wolfgang.Dreyer@wias-berlin.de, Clemens.Guhlke@wias-berlin.de, Manuel.Landstrofer@wias-berlin.de, Ruediger.Mueller@wias-berlin.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Lippmann equation is considered as universal relationship between interfacial tension, double layer charge, and cell potential. Based on the framework of continuum thermo-electrodynamics, we provide some crucial new insights to this relation. For general interfaces such that the local curvature radius is large compared to the Debye length, we apply asymptotic analysis methods to obtain the Lippmann equation. We give precise definitions of the involved quantities and show that the interfacial tension of the Lippmann equation is composed of the surface tension of our general model, and contributions arising from the adjacent space charge layers that can only lower the interfacial tension. Moreover, it turns out that surface reactions can be consistently incorporated into the Lippmann equation, provided that there is no charge transfer from one side of the interface to the other. We apply the model to curved liquid metal electrodes and compare our model to experimental data of several mercury–electrolyte interfaces. We obtain qualitative and quantitative agreement in the 2 V potential range for various salt concentrations.

Keywords

Lippmann equationelectrochemistryliquid-liquid interfaceasymptotic analysis
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 4 , August 2018 , pp. 708 - 753
Copyright
Copyright © Cambridge University Press 2017 

References

[1] Adamson, A. W. & Gast, A. P. (1997) Physical Chemistry of Surfaces, 6st ed., John Wiley & Sons, New York.Google Scholar
[2] Albano, A. M. & Bedeaux, D. (1987) Non-equilibrium electro-thermodynamics of polarizable multicomponent fluids with an interface. Physica A 147 (1–2), 407–435.CrossRefGoogle Scholar
[3] Bard, A. J. & Faulkner, L. R. (2001) Electrochemical Methods: Fundamentals and Applications, 2nd ed., Wiley, New York.Google Scholar
[4] Bazant, M. Z., Thornton, K. & Ajdari, A. (2004) Diffuse-charge dynamics in electrochemical systems. Phys. Rev. E 70, 021506.CrossRefGoogle ScholarPubMed
[5] Bazant, M. Z., Chu, K. T. & Bayly, B. J. (2005) Current-voltage relations for electrochemical thin films. SIAM J. Appl. Math. 65 (5), 1463–1484.Google Scholar
[6] Bedeaux, D. (1986) Nonequilibrium thermodynamics and statistical physics of surfaces. In: Ilya, Prigogine & Rice, S. A. (editors), Advances in Chemical Physics, Vol. 64, John Wiley & Sons, New York, pp. 47–109.CrossRefGoogle Scholar
[7] Bockris, J. O. M., Devanathan, M. A. V. & Muller, K. (1963) On the structure of charged interfaces. Proc. Roy. Soc. Lond. A 274 (1356), 55–79.Google Scholar
[8] Bockris, J. O. M., Reddy, A. K. N. & Gamboa-Aldeco, M. E. (2002) Modern Electrochemistry A: Fundamentals of Electrodics, 2nd ed., Vol. 2, Kluwer Academic Publishers, New York.CrossRefGoogle Scholar
[9] Bothe, D. & Dreyer, W. (2015) Continuum thermodynamics of chemically reacting fluid mixtures. Acta Mech. 226 (6), 1757–1805.Google Scholar
[10] Caginalp, G. & Fife, P. (1988) Dynamics of layered interfaces arising from phase boundaries. SIAM J. Appl. Math. 48 (3), 506–518.Google Scholar
[11] Chernenko, A. A. (1963) Theory of the passage of a DC current through a solution of a binary electrolyte. Dokl. Akad. Nauk SSSR 153, 1129.Google Scholar
[12] Cuong, N. H., D'Alkaine, C. V., Jenard, A. & Hurwitz, H. D. (1974) The surface phase at the ideal polarized mercury electrode: 2. coulostatic measurements at the Hg electrode in dilute NaF aqueous solutions at various temperatures. J. Electroanal. Chem. 51 (2), 377–393.Google Scholar
[13] Defay, D. & Sanfeld, A. (1967) The tensor of pressure in spherical and plane electrocapillary layers. Electrochim. Acta 12 (8), 913–926.CrossRefGoogle Scholar
[14] deGroot, S. R. & Mazur, P. (1984) Non-equilibrium Thermodynamics, Dover Publications Inc., New York.Google Scholar
[15] Dreyer, W., Giesselmann, J. & Kraus, C. (2014) A compressible mixture model with phase transition. Physica D 273–274, 1–13.CrossRefGoogle Scholar
[16] Dreyer, W., Guhlke, C. & Müller, R. (2015) Modeling of electrochemical double layers in thermodynamic non-equilibrium. Phys. Chem. Chem. Phys. 17, 27176–27194.CrossRefGoogle ScholarPubMed
[17] Foster, J. M., Snaith, H. J., Leijtens, T. & Richardson, G. (2014) A model for the operation of perovskite based hybrid solar cells: Formulation, analysis, and comparison to experiment. SIAM J. Appl. Math. 74 (6), 1935–1966.CrossRefGoogle Scholar
[18] Frumkin, A. (1928) Die Elektrokapillarkurve. Ergebnisse der Exakten Naturwissenschaften. Springer, Berlin, Heidelberg, Chapter 7, pp. 235–275.Google Scholar
[19] Frumkin, A. N. (1968) The Lippmann equation. Comments on part XXII of the paper “On the impedance of galvanic cells”, by Timmer, Sluyters-Rehbach and Sluyters. J. Electroanal. Chem. 18 (3), 328–329.Google Scholar
[20] Frumkin, A., Petry, O. & Damaskin, B. (1979) The notion of the electrode charge and the Lippmann equation. J. Electroanal. Chem. 27 (1), 81–100.Google Scholar
[21] Fuhrmann, J. (2015) Comparison and numerical treatment of generalised Nernst–Planck models. Comp. Phys. Comm. 196 (Supplement C), 166–178.CrossRefGoogle Scholar
[22] Gouy, L. G. (1903) Sur la fonction électrocapillaire. Ann. Chim. Phys. 29, 145–241.Google Scholar
[23] Gouy, L. G. (1906) Sur la fonction électrocapillaire. Ann. Chim. Phys. 8, 291–363.Google Scholar
[24] Gouy, L. G. (1906) Sur la fonction électrocapillaire. Ann. Chim. Phys. 9, 75–139.Google Scholar
[25] Gouy, L. G. (1910) Sur la constitution de la charge électrique à la surface d'un électrolyte. J. Phys. Théor. Appl. 9 (4), 457–468.Google Scholar
[26] Grafov, B. M. & Chernenko, A. A. (1962) Theory of direct-current flow through a binary electrolyte solution. Dokl. Akad. Nauk SSSR 146, 135.Google Scholar
[27] Grahame, D. C. (1947) The electrical double layer and the theory of electrocapillarity. Chem. Rev. 41 (3), 441–501.Google Scholar
[28] Guhlke, C. (2015) Theorie Der Elektrochemischen Grenzfläche, PhD thesis, TU-Berlin, Germany.Google Scholar
[29] Hurwitz, H. D. & D'Alkaine, C. V. (1973) The surface phase at the ideal polarized mercury electrode: I. General thermodynamic and hydrostatic considerations of ionic adsorption. J. Electroanal. Chem. 42 (1), 77–103.Google Scholar
[30] Jasper, J. J. (1972) The surface tension of pure liquid compounds. J. Phys. Chem. Ref. Data 1 (4), 841–1010.CrossRefGoogle Scholar
[31] Landstorfer, M., Guhlke, C. & Dreyer, W. (2016) Theory and structure of the metal-electrolyte interface incorporating adsorption and solvation effects. Electrochimica Acta 201, 187–219.Google Scholar
[32] Lippmann, G. (1873) Beziehungen zwischen den capillaren und elektrischen Erscheinungen. Ann. Phys. 225 (8), 546–561.Google Scholar
[33] Lippmann, G. (1875) Relations Entre les Phénomènes électriques et Capillaires, Ann. Chim. Phys. 5 (11), 494–549.Google Scholar
[34] Meixner, J. & Reik, H. G. (1959) Thermodynamik der Irreversiblen Prozesse, Vol. 3, Springer, Berlin, pp. 413–523.Google Scholar
[35] Müller, I. (1985) Thermodynamics, Interaction of Mechanics and Mathematics Series. Pitman Advanced Publishing Program, Boston.Google Scholar
[36] Newman, J. (1965) The polarized diffuse double layer. Trans. Faraday Soc. 61, 2229–2237.Google Scholar
[37] Newman, J. & Thomas-Alyea, K. E. (2004) Electrochemical Systems. John Wiley & Sons, Hoboken, New Jersey.Google Scholar
[38] Nic, M., Jirat, J. & Kosata, B. (2012) IUPAC compendium of chemical terminology (gold book) [online]. URL: goldbook.iupac.org/index.html. Release 2.3.3b, 2017/03/27.Google Scholar
[39] Pego, R. L. (1989) Front migration in the nonlinear Cahn–Hilliard equation. Proc. R. Soc. Lond. A 422 (1863), 261–278.Google Scholar
[40] Richardson, G. (2000) A multiscale approach to modelling electrochemical processes occurring across the cell membrane with application to transmission of action potentials. Math. Med. Biol. 26 (3), 201–224.Google Scholar
[41] Sanfeld, A. (1968) Introduction to the Thermodynamics of Charged and Polarized Layers, Vol. 10 of Monographs in Statistical Physics and Thermodynamics. John Wiley & Sons, London, New York, Sydney.Google Scholar
[42] Schmeiser, C. & Unterreiter, A. (1994) The derivation of analytic device models by asymptotic methods. In: Coughran, W. M., Cole, J., Lloyd, P. & White, J. K. (editors), Semiconductors: Part II, Springer, New York, pp. 343–363.CrossRefGoogle Scholar
[43] Vetter, K. J. (1967) Electrochemical Kinetics, Academic Press, New York.Google Scholar