Lippmann equation and the ideally polarizable electrode

Abstract

Publications costs assisted by Syracuse UniversityThe Lippmann equation for the ideally polarizable interface is normally derived by thermodynamics, using the Gibbs dividing surface. Therefore, the quantities appearing in the Lippmann equation can have no reference to the actual charge distribution in the interfacial region. For example, the quantity referred to as surface charge is actually a sum of surface excesses, rather than the integral of a true charge density. In this article we derive, by statistical mechanical methods, the Lippmann equation for a model at the molecular level, thus giving a precise physical definition to all quantities which appear. First, we derive the conditions for mechanical equilibrium for a system (the interface between metal and solution) in which an electric field is present, and whose properties are inhomogeneous and anisotropic. From the balance of forces, we obtain equations for the surface tension. in terms of the pressure, electric field, electric charge density, and electric polarization at each point within the system. Considering a spherically symmetric system (mercury drop), we then proceed to a direct calculation of the change in the surface tension produced by a change in the potential drop across the interface, maintaining thermal equilibrium, constant temperature, and the pressure and chemical composition in homogeneous regions (on the boundaries of the interfacial region). Since an ideally polarizable interface does not permit charge transport across it, we introduce a surface within the interface on which the charge density is always zero. This surface serves to divide the interfacial region into two parts, thus allowing the surface charge to be defined as the integral of the charge density over the metal side of the interface. Only the solution side is treated by statistical mechanics. Boltzmann distributions for charged and polarizable species (solute and solvent) are used to guarantee thermal equilibrium. The Lippmann equation is obtained (a) considering only ions and supposing a dielectric constant equal to that of vacuum and (b) considering ions and molecules in thermal equilibrium, and a dielectric constant varying from point to point and changing with field. Finally, the response of our system to an imposed alternating potential is considered. A direct calculation of the impedance shows that it behaves, in the low-frequency limit, as a pure capacitance, and that the value of this capacitance is the derivative of the previously defined surface charge density with respect to the potential drop across the interface.

A continuum theory of surface piezoelectricity for nanodielectrics

“…In the field of electrochemistry, how the classical electrocapillary Lippmann equation should be modified to account for the elastic effects on solid electrodes is still a controversial issue [2][3][4][5][6][7][8][9][10][11]. As to dielectrics, various novel surface-induced phenomena have *Corresponding author (email: yusw@mail.tsinghua.edu.cn) been observed, e.g., the emergence of dielectric 'dead layer' in nanocapacitors [12][13][14], the presence of an intrinsic conducting electron layer on the surface of insulating BaTiO 3 [15], and the possibility of polarity inversion in ZnO nanowires as indicated by recent ab initio calculations [16].…”

Potential of mean force on an ion near a wall in presence of a molecular solvent

“…In recent publications [4][5][6] it has been shown that a dipolar hard sphere fluid near a neutral hard wall exhibits an orientational structure: for instance, near the wall, an orientation parallel to the surface is favoured. At low densities the position-orientation density profile presents a long tail decreasing as the inverse cubic power of the distance from the wall and the dipoles tend to be parallel to the surface in any point of the interface [4].…”

“…At densities corresponding to the liquid state a layering effect appears in the orientational structure. For instance, at a distance corresponding to one molecular diameter from the wall the orientation normal to the surface predominates [5,6]. This fine structure of the profile only appears if an approximation more sophisticated than the MSA is used.…”

Thermodynamics and statistical mechanics of the ideally polarisable electrode Gibbs isotherm and surface tension

“…In ref. 7 it was shown that this model leads to purely capacitative behavior when the potential across the interface is varied sinusoidally and adiabatically. However, we shall replace this model below by another, which is more coherent with the statistical mechanical approach.…”