Lippmann Equation Research Articles

Comments on the thermodynamics of solid electrodes

The thermodynamics of solid electrodes are discussed in light of the recent measurements of surface stress. Interfacial tension and surface stress are not even approximately equal, and they generally exhibit a different dependence on the electrode potential. The variation of the interfacial strain with potential is small so that the Lippmann equation for a solid is practically the same as for a liquid electrode. Changes in the interfacial tension can be obtained by integrating the charge density over the potential.

Surface tension effects in tunneling microscopy

Induced electric charge lowers the surface tension of the substrate below the tunneling tip and creates a sharp surface tension gradient in its vicinity. For mercury as the substrate, the random waving is amplified by the gradient of the surface tension, resulting in observable mechanical ‘‘waving’’ of the surface that increases with the applied bias voltage. Its amplitude reaches a minimum at the potential of zero charge. In agreement with the Gibbs–Lippmann equation these effects are mitigated by the presence of adsorbed films but may lead to translocation of adsorbates by the biased tunneling tip.

Effects of Interfacial Tension in Tunneling Microscopy. 2. Experimental Observation of Surface Instability

It has been found that low-frequency waving observed during tunneling microscopy with a stationary tip on liquid mercury is voltage dependent. This phenomenon can be explained by the Gibbs−Lippmann equation, which describes electrostatically induced changes in surface tension of mercury/electrolyte interfaces. The presence of adsorbing compounds overshadows the electrostatic effects, in complete agreement with the Gibbs−Lippmann model. These observations indicate that low bias voltages are required for maximum image resolution when performing tunneling microscopy at liquid surfaces.

Quantitative Surface Stress Measurements on Au(111) Electrodes by Scanning Tunneling Microscopy

Using macroscopic cantilevers of evaporated Au(111) films on glass substrates in conjunction with scanning tunneling microscopy, we performed quantitative surface stress measurements of copper underpotential deposition in bromide and chloride acid solutions. Upon differentiation of the stress data with respect to potential, definite signatures of the corresponding voltammograms were obtained, in contrast to previous contentions of the applicability of the Lippmann equation to solid electrodes which predicts such signatures only for the second derivative of the stress.

A new interferometric method to investigate surface energies at solid electrodes

A new method is described to determine changes Δγ of the specific surface energy of solids from the elastic deformation of a circular plate using Kösters laser interferometry. It was confirmed experimentally that the deformation is the same for normal pressure and for lateral stress. A support of the plate was used which exerts almost no forces on the plate thus ensuring maximum sensitivity of the method. The error of the light path of two beams reflected at a distance of about 10 mm was ±0.2 nm corresponding to a sensitivity of Δγ = ±3 mN m −1. Simultaneous measurements of the changes of specific surface energy, mass and charge for the adsorption of KF and the underpotential deposition of Pb, Tl and Zn on polycrystalline gold are compared to theoretical predictions. In addition to the changes of superficial energy predicted by the Lippmann equation contributions from surface stress are observed which probably originate from structural changes in the surface.

Liquid Motion Induced by Electrocapillary Action at Solid Metal-Liquid Interfaces

It is shown that cyclic mechanical displacements are induced in a variety of aqueous solutions and organic liquids when alternating voltages are applied across the interfaces between them and solid metal electrodes. The displacements, in the picometer-nanometer range, are proportional to the applied voltage and are determined using an interferometer technique which detects the movement of a mirror floating on the free surface of the liquid above the interfaces. The displacements, particularly in the case of a stainless steel-glycerol interface, increase considerably when steady bias voltages are applied. The experiments are in agreement with the predictions of the Lippmann electrocapillary equation for interfaces and complement those where stresses or strains generated in the solid electrode rather than the liquid side of the interfaces have been measured. Since similar displacements have been measured at solid as well as liquid contacts to metal electrodes, it is concluded that the Lippmann equation predicts quasi-piezoelectric phenomena at all condensed matter interfaces.

Electrokinetic properties of metal-dielectric interfaces

Using an optical interferometric method capable of determining mechanical displacements in the nm to pm range, measurements are reported of displacements resulting from electrically induced strains in a variety of metal-dielectric interfaces. The dielectrics include aqueous electrolytes, glycol, glycerol, silicone oil and polyethylene. The displacements are shown to conform to the predictions of the Lippmann equation. The implications of these measurements are discussed and are believed to be particularly important in the initiation of failure in polymeric insulation at high voltage. The results also suggest possible device applications based on the electromechanical, piezoelectric properties of these interfaces.

Surface energies of solid electrodes

Theoretical relations are derived for the determination of the surface energy of solid electrodes on quartz oscillators from the bending of the quartz plate or from the associated changes in the oscillator frequency. The dependence of the charges for the reversible adsorption of a certain mass and the corresponding changes in superficial energy measured for the underpotential deposition of lead on polycrystalline gold at zero strain is in good agreement with the thermodynamic prediction using the Lippmann equation. According to the general Lippmann equation, the superficial energy is expected to depend on elastic strain and surface stress. This contribution was dominant for the underpotential deposition of thallium on gold at a strain of ca. 10 −4. The compressive volume stress produced by dissolving hydrogen at constant volume in a heterogeneous mixture of α- and β-palladium was used to estimate a reaction volume for the exchange of hydrogen between the two phases. The electrochemical dissolution rate of hydrogen into palladium increased with tensile stress and decreased with compressive stress.

Two-dimensional models for an electrode with adsorption sites

Through a new method, the following model is solved exactly in the framework of classical equilibrium statistical mechanics of two-dimensional Coulomb systems, for the special valueГ=2 of the coupling constant: the mobile charges of a one-component plasma are attracted by a line of equidistant sticky adsorption sites embedded in a background, the density of which varies in the direction orthogonal to the line. First the general expressions are given for the densities and correlation functions of nonadsorbed and adsorbed particles. Then these results are used to investigate two models of electrodes with localized adsorption: the externally charged hard wall and the impermeable polarized membrane. In each case the influence of the adsorption upon macroscopic features is studied: the potential drop across the interface, the contact theorem, and the Lippmann equation, which involves the surface free energy.

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

A definition of the ideally polarisable electrode which allows a general statistical mechanical treatment is given, comparisons with the usual thermodynamic approach being made. It is shown that, with this definition, the system behaves as a pure capacitance to an imposed alternating potential of sufficiently low frequency. The equilibrium states of the system are described by the grand canonical distribution. Thermodynamic functions, given in terms of the grand canonical partition function, can be computed in molecular terms. In particular, general expressions for the surface tension and the pressure are obtained, and compared to those proposed by various authors. The separation of the pressure into electrical and non-electrical parts is analyzed. Expressions are given for the electrochemical potentials of charged and of dipolar species. With these microscopic definitions, the Gibbs adsorption isotherms and the Lippmann equation are derived. For several simple models, including that of Gouy and Chapman for an ionic solution and a metal electrode, these relations are explicity considered. The inadmissability, in a coherent model, of reducing the solvent to a medium of fixed dielectric constant, is emphasized.

Statistical mechanical derivation of the Lippmann equation. The dielectric constant

Statistical mechanical derivation of the Lippmann equation. The dielectric constant

Surface Stress Curves for Gold

Changes in surface stress of gold ribbon electrodes as a function of potential were measured in various electrolytes with an extensometer instrument. Electrocapillary‐type curves were obtained for which the change in surface stress can be equated to change in surface tension. As for mercury, the potential of the surface stress maximum for gold is independent of pH. The observed anion adsorption order on gold Was , ,. Experimental surface stress data and charge passed gave approximate agreement with the Lippmann equation in the electric double layer regions Formation of oxide film significantly lowered the surface stress.

Mécanique statistique de l'interface idéalement polarisée à l'équilibre — équation de Lippmann

In the context of a statistical mechanical treatment, it is demonstrated that any model of the electrochemical interface implies a model for the concentration — concentration correlation functions. Using a very simple model, we derive the Lippmann equation, thus making explicit its significance on the microscopic level.

The lippmann equation and the characterization of black lipid films

An optically black lipid film in aqueous media approximates closely to an ideally polarizable system and, in this respect, can be described by a form of the Lippmann equation. This equation is derived and integrated assuming the film capacitance to be independent of applied potential. Measurements of the contact angle and of the bulk interfacial tension are used to find accurate values of the film tension and it is shown that, for a range of lipid films, the integrated equation is obeyed for potential differences up to approximately 100 mV. Various ways are then described in which the Lippmann equation may be used to find film properties not readily accessible by other methods. Thus, for phospholipid films the accurate measurement of both film and bulk interface tensions presents difficulties, but these can be overcome if the specific capacitance is known. In this way the specific free energy of thinning of the film may be obtained. Conversely, for films in media of low conductivity or in systems in which film area measurements are difficult the determination of the specific capacitance is not readily achieved. A knowledge of the tension of the Plateau-Gibbs border together with the contact angle at different applied potentials, however, circumvents this problem. Finally it is pointed out that the use of the Lippmann equation provides an independent means of finding the interfacial tension between bulk oil and water phases and hence of checking the accuracy of the nonabsolute direct methods, such as the drop-volume technique, which have to be employed in such systems.

Lippmann equation and the ideally polarizable electrode

Lippmann equation and the ideally polarizable electrode

A theory of electrophoresis of charged mercury drops in aqueous electrolyte solution

We describe a theory of electrophoresis of ideally polarizable, charged spherical mercury drops in aqueous electrolyte, which is based on the work of Henry, Overbeek and Booth. This is compared with the theory of Frumkin and Levich, who make use of the Lippmann equation, and also of a boundary condition of charge conservation which involves a surface divergence of the tangential current density. It is shown that this conservation condition may be derived from three-dimensional continuity equations of ion flow in the theory of Overbeek and Booth, provided κa ⪢ 1, where a is the drop radius and 1 κ the diffuse layer thickness. Our electrical terms in the conditions of continuity in tangential and normal stresses across the mercury-aqueous solution interface differ from those of Frumkin and Levich. In place of the Lippmann equation, we adopt Booth's use of the Maxwell electrostatic stress tensor. A key step in our theory is the introduction of a polarization term in the potential distribution, which is analogous to the relaxation term for dielectric particles. It is shown that this term implies a significant departure from superposition of the equilibrium electric field and applied field, resulting in a considerable displacement in the surface charge density on the mercury drop. A consequence is a large electrical force acting on the mercury drop, which is the driving force responsible for an electrophoretic mobility, exceeding that on a dielectric particle by orders of magnitudes. Thus the polarization term dominates the magnitude of this mobility. Our theory becomes identical with that of Frumkin and Levich in the limiting case where κa ⪢ 1 and surface potentials are in the Debye-Huckel range.

The notion of the electrode charge and the Lippmann equation

The notion of the charge has been considered for the case of an ideally polarized electrode. It has been proved necessary to distinguish between the total charge, which figures in thermodynamic relations, and the free charge, which can be determined only in terms of a certain electric double-layer model. A definition of the total charge is given, equally applicable to ideally polarized and reversible electrodes, as the amount of electricity to be supplied to the electrode to keep the electrode potential constant when its surface is increased by unity and the composition of the bulk phases of the system is maintained constant. The total charge thus determined satisfies in all cases the Lippmann equation. Expressions are given for the Lippmann equation for reversible redox systems, as exemplified by platinum-hydrogen and amalgam-thallium electrodes. It has been shown that in such systems two kinds of electrocapillary curves can be obtained, depending on which chemical potential is held constant: that of the oxidised or that of the reduced component. The number of characteristic electrocapillary curves in the general case of a reversible redox system has been shown to be equal to that of the independent variables in the Nernst equation expressing the conditions of electrochemical equilibrium of the system. The results obtained have been used for the interpretation of the electrocapillary dependences observed under polarographic conditions.

The notion of the electrode charge and the Lippmann equation

Summary The notion of the charge has been considered for the case of an ideally polarized electrode. It has been proved necessary to distinguish between the total charge, which figures in thermodynamic relations, and the free charge, which can be determined only in terms of a certain electric double-layer model. A definition of the total charge is given, equally applicable to ideally polarized and reversible electrodes, as the amount of electricity to be supplied to the electrode to keep the electrode potential constant when its surface is increased by unity and the composition of the bulk phases of the system is maintained constant. The total charge thus determined satisfies in all cases the Lippmann equation. Expressions are given for the Lippmann equation for reversible redox systems, as exemplified by platinum-hydrogen and amalgam-thallium electrodes. It has been shown that in such systems two kinds of electrocapillary curves can be obtained, depending on which chemical potential is held constant: that of the oxidised or that of the reduced component. The number of characteristic electrocapillary curves in the general case of a reversible redox system has been shown to be equal to that of the independent variables in the Nernst equation expressing the conditions of electrochemical equilibrium of the system. The results obtained have been used for the interpretation of the electrocapillary dependences observed under polarographic conditions.

The lippmann equation. Comments on part XXII of the paper “On the impedance of galvanic cells”, by Timmer, Sluyters-Rehbach and Sluyters

The lippmann equation. Comments on part XXII of the paper “On the impedance of galvanic cells”, by Timmer, Sluyters-Rehbach and Sluyters

The Lippmann equation. Comments on part XXII of the paper “On the impedance of galvanic cells”, by Timmer, Sluyters-Rehbach and Sluyters

The Lippmann equation. Comments on part XXII of the paper “On the impedance of galvanic cells”, by Timmer, Sluyters-Rehbach and Sluyters