The double layer in the absence of specific adsorption

One of the fundamental problems in electrochemistry is the distribution of the potential and of the particles at the interface. Here we will expand on the subject of Chapter 3, and consider the interface between a metal and an electrolyte solution in the absence of specific adsorption. Until about 1980 a simple model of this interface prevailed, which was based on a particular interpretation of the interfacial capacity. The metal was assumed to be a perfect conductor in the classical sense, and hence a region of constant potential right up to the metal surface. As was pointed out in Chapter 3, the inverse capacity can be split into two terms, a Gouy-Chapman and a Helmholtz term: l/C = l/CGc + 1/CH. It was argued that these two terms pertain to two different regions in the solution: the space charge or diffuse double layer, which is already familiar to us, and the Stern or outer Helmholtz layer giving rise to the Helmholtz capacity. Since the latter does not depend on the concentration of the ions, the Stern layer was supposed to consist of a monolayer of solvent molecules adsorbed on the metal surface. The plane passing through the centers of these molecules was called the outer Helmholtz plane. Rather elaborate models were developed for the dielectric properties of this layer in order to explain Helmholtz capacity curves such as those shown in Fig. 3.3. This Gouy-Chapman-Stern model, as it was named after its main contributors, is a highly simplified model of the interface, too simple for quantitative purposes. It has been superseded by more realistic models, which account for the electronic structure of the metal, and the existence of an extended boundary layer in the solution. It is, however, still used even in current publications, and therefore every electrochemist should be familiar with it. In the remainder of this chapter we will present elements of modern double-layer theory. Two phases meet at this interface: the metal and the solution. We will consider each phase in turn.

Thermodynamics of ideal polarizable interfaces

For liquid electrodes thermodynamics offers a precise way to determine the surface charge and the surface excesses of a species. This is one of the reasons why much of the early work in electrochemistry was performed on liquid electrodes, particularly on mercury - another reason is that it is easier to generate clean liquid surfaces than clean solid surfaces. With some caveats and modifications, thermodynamic relations can also be applied to solid surfaces. We will first consider the interface between a liquid electrode and an electrolyte solution, and turn to solid electrodes later.

Preliminaries

In this chapter we introduce and discuss a number of concepts that are commonly used in the electrochemical literature and in the remainder of this book. In particular we will illuminate the relation of electrochemical concepts to those used in related disciplines. Electrochemistry has much in common with surface science, which is the study of solid surfaces in contact with a gas phase or, more commonly, with ultrahigh vacuum (uhv). A number of surface science techniques has been applied to electrochemical interfaces with great success. Conversely, surface scientists have become attracted to electrochemistry because the electrode charge (or equivalently the potential) is a useful variable which cannot be well controlled for surfaces in uhv. This has led to a laudable attempt to use similar terminologies for these two related sciences, and to introduce the concepts of the absolute scale of electrochemical potentials and the Fermi level of a redox reaction into electrochemistry. Unfortunately, there is some confusion of these terms in the literature, even though they are quite simple. Electrochemical interfaces are sometimes referred to as electrified interfaces, meaning that potential differences, charge densities, dipole moments, and electric currents occur. It is obviously important to have a precise definition of the electrostatic potential of a phase. There are two different concepts. The outer or Volta potential ψα of the phase a is the work required to bring a unit point charge from infinity to a point just outside the surface of the phase. By "just outside" we mean a position very close to the surface, but so far away that the image interaction with the phase can be ignored; in practice, that means a distance of about 10-5 — 10-3 cm from the surface. Obviously, the outer potential ψα is a measurable quantity. In contrast, the inner or Galvani potential ϕα is defined as the work required to bring a unit point charge from infinity to a point inside the phase α; so this is the electrostatic potential which is actually experienced by a charged particle inside the phase. Unfortunately, the inner potential cannot be measured: If one brings a real charged particle - as opposed to a point charge - into the phase, additional work is required due to the chemical interaction of this particle with other particles in the phase. For example, if one brings an electron into a metal, one has to do not only electrostatic work, but also work against the exchange and correlation energies.

Introduction

Electrochemistry is an old science: There is good archaeological evidence that an electrolytic cell was used by the Parthans (250 B.C. to 250 A.D.), probably for electroplating, though a proper scientific investigation of electrochemical phenomena did not start before the experiments of Volta and Galvani. The meaning and scope of electrochemical science has varied throughout the ages: For a long time it was little more than a special branch of thermodynamics; later attention turned to electrochemical kinetics. During recent decades, with the application of various surface-sensitive techniques to electrochemical systems, it has become a science of interfaces, and this, we think, is where its future lies. So in this book we use as a working definition: . . . Electrochemistry is the study of structures and processes at the interface between an electronic conductor (the electrode) and an ionic conductor (the electrolyte) or at the interface between two electrolytes. . . This definition requires some explanation. (1) By interface we denote those regions of the two adjoining phases whose properties differ significantly from those of the bulk. These interfacial regions can be quite extended, particularly in those cases where a metal or semiconducting electrode is covered by a thin film. Sometimes the term interphase is used to indicate the spatial extention. (2) It would have been more natural to restrict the definition to the interface between an electronic and an ionic conductor only, and, indeed, this is generally what we mean by the term electrochemical interface. However, the study of the interface between two immiscible electrolyte solutions is so similar that it is natural to include it under the scope of electrochemistry. Metals and semiconductors are common examples of electronic conductors, and under certain circumstances even insulators can be made electronically conducting, for example by photoexcitation. Electrolyte solutions, molten salts, and solid electrolytes are ionic conductors. Some materials have appreciable electronic and ionic conductivities, and depending on the circumstances one or the other or both may be important.

Nontraditional techniques

The traditional electrochemical techniques are based on the measurement of current and potential, and, in the case of liquid electrodes, of the surface tension. While such measurements can be very precise, they give no direct information on the microscopic structure of the electrochemical interface. In this chapter we treat several methods which can provide such information. None of them is endemic to electrochemistry; they are mostly skillful adaptations of techniques developed in other branches of physics and chemistry. The scanning tunneling microscope (STM) is an excellent device to obtain topographic images of an electrode surface . The principal part of this apparatus is a metal tip with a very fine point, which can be moved in all three directions of space with the aid of piezoelectric crystals. All but the very end of the tip is insulated from the solution in order to avoid tip currents due to unwanted electrochemical reactions. The tip is brought very close, up to a few Ångstroms, to the electrode surface. When a potential bias ΔV, usually of the order of a few hundred millivolts, is applied between the electrode and the tip, the electrons can tunnel through the thin intervening layer of solution, and a tunneling current is observed. The situation is illustrated in Fig. 15.2: A potential energy barrier exists between the tip and the substrate. Application of a bias potential shifts the two Fermi levels of the tip and of the substrate. Electrons can tunnel from the metal with the higher Fermi level through the barrier to empty states on the other metal. Roughly speaking, electrons with energies between the two Fermi levels can be transferred. A detailed calculation shows that the current is proportional to the electronic density of states at the Fermi level of the substrate. The tip is moved slowly in the yz direction parallel to the metal surface, and simultaneously the distance x from the electrode is adjusted in such a way that the tunneling current is constant (constant-current mode).

Transient techniques

The classical electrochemical methods are based on the simultaneous measurement of current and electrode potential. In simple cases the measured current is proportional to the rate of an electrochemical reaction. However, generally the concentrations of the reacting species at the interface are different from those in the bulk, since they are depleted or accumulated during the course of the reaction. So one must determine the interfacial concentrations. There are two principal ways of doing this. In the first class of methods one of the two variables, either the potential or the current, is kept constant or varied in a simple manner, the other variable is measured, and the surface concentrations are calculated by solving the transport equations under the conditions applied. In the simplest variant the overpotential or the current is stepped from zero to a constant value; the transient of the other variable is recorded and extrapolated back to the time at which the step was applied, when the interfacial concentrations were not yet depleted. In the other class of method the transport of the reacting species is enhanced by convection. If the geometry of the system is sufficiently simple, the mass transport equations can be solved, and the surface concentrations calculated. The interpretation becomes complicated if several reactions take place simultaneously. Since the measured current gives only the sum of the rate of all charge-transfer reactions, the elucidation of the reaction mechanism and the measurement of several rate constants becomes an art. A number of tricks can be used, such as complicated potential or current programs, auxiliary electrodes, etc., which work for special cases. There are several good books on the classical electrochemical techniques. Here we give a brief outline of the most important methods. We mostly restrict ourselves to the study of simple reactions, but will consider one example in which the charge-transfer reaction is preceded by a chemical reaction. The measurement of current and potential provides no direct information about the microscopic structure of the interface, though a clever experimentalist may make some inferences.

Selected experimental results for electron-transfer reactions

Innumerable experiments have been performed on both inner- and outer-sphere electron-transfer reactions. We do not review them here, but present a few results that are directly relevant to the theoretical issues raised in the preceding chapters. The Butler-Volmer equation (5.10) predicts that for |η| > kT/e0 a plot of the logarithm of the current versus the applied potential (Tafel plot) should result in a straight line, whose slope is determined by the transfer coefficient α. Because of the dual role of the transfer coefficient (see Section 5.2), it is important to verify that the transfer coefficient obtained from a Tafel plot is independent of temperature. We shall see later that proton- and ion-transfer reactions often give straight lines in Tafel plots, too, but the apparent transfer coefficient obtained from these plots can depend on the temperature, indicating that these reactions do not obey the Butler-Volmer law. In order to test the temperature independence of the transfer coefficient, Curtiss et al. investigated the kinetics of the Fe2+/Fe3+ reaction on gold in a pressurized aqueous solution of perchloric acid over a temperature range from 25° to 75°C. In the absence of trace impurities of chloride ions, this reaction proceeds via an outer sphere mechanism with a low rate constant (k0 ≈ 10-5 cm s-1 at room temperature). Figure 8.1 shows the slope of their Tafel plots, d(lni)/dη, as a function of the inverse temperature 1/T. The Butler-Volmer equation predicts a straight line of slope αe0/k, which is indeed observed. Over the investigated temperature range both the transfer coefficient and the energy of activation are constant: α = 0.425 ± 0.01 and Eact = 0.59± 0.01 eV at equilibrium, confirming the validity of the Butler-Volmer equation in the region of low overpotentials, from which the Tafel slopes were obtained. The phenomenological derivation of the Butler-Volmer equation is based on a linear expansion of the Gibbs energy of activation with respect to the applied overpotential. At large overpotentials higher-order terms are expected to contribute, and a Tafel plot should no longer be linear.

Quantum theory of electron-transfer reactions

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations are not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. To be specific we consider electron transfer from a reactant in a solution, such as [Fe(H2O)6]2+, to an acceptor, which may be a metal or semiconductor electrode, or another molecule. To obtain wavefunctions for the reactant in its reduced and oxidized state, we rely on the Born-Oppenheimer approximation, which is commonly used for the calculation of molecular properties. This approximation is based on the fact that the masses of the nuclei in a molecule are much larger than the electronic mass. Hence the motion of the nuclei is slow, while the electrons are fast and follow the nuclei almost instantaneously. The mathematical consequences will be described in the following. Let us denote by R the coordinates of all the nuclei involved, those of the central ion, its ligarids, and the surrounding solvation sphere, and by r the coordinates of all electrons.

Liquid-liquid interfaces

When we defined electrochemistry in Chapter 1, we made a special case for including the interface between two immiscible solutions (ITIES) because they show many similarities with the more usual electrochemical systems. Much of the interest in these interfaces resides in the fact that they can serve as models for membranes, but they are also interesting systems in their own right. In a certain sense the field is still in its infancy: Little is known about the structure of the interface, and most of our secure knowledge relies on thermodynamics. However, these systems pose intriguing problems. Almost all the published work is based on classical electrochemical methods based on the measurements of current, potential, and surface tension. If techniques yielding structural information (see Chapter 15) can be applied to ITIES - and at least a few optical techniques look promising - we may expect the field to grow rapidly during the next decade. Most of the liquid-liquid interfaces that have been studied involve water and an organic solvent such as nitrobenzene or 1,2-dichloroethane (1,2-DCE). Although these systems form stable interfaces, the solubility of one solvent in the other is usually quite high. For example, the solubility of water in 1,2-DCE is 0.11 M, and that of 1,2-DCE in water is 0.09 M. So each of the two liquid components is a fairly concentrated solution of one solvent in the other. It is therefore unlikely that the interface is sharp on a molecular level. We rather expect an extended region with a thickness of the order of a few solvent diameters, over which the concentrations of the two solvents change rapidly. The lower the solubility of one solvent in the other, the thinner this interfacial region should be. These expectations are supported by the indication that the dipole potentials at these interfaces seem to be small, at least near the pzc, but spectroscopic information is lacking at present. Many of the processes that are familiar from ordinary electrochemistry have an analog at ITIES; so these form a wide field of study.

Proton- and ion-transfer reactions

We consider the transfer of an ion or proton from the solution to the surface of a metal electrode; often this is accompanied by a simultaneous discharge of the transferring particle, such as by a fast electron transfer. The particle on the surface may be an adsorbate as in the reaction: . . .Cl - (sol) ⇋ Clad + e- (metal) . . . (9.1) In this case the discharge can be partial; that is, the adsorbate can carry a partial charge, as discussed in Chapter 4. Alternatively the particle can be incorporated into the electrode as in the deposition of a metal ion on an electrode of the same composition, or in the formation of an alloy. An example of the latter is the formation of an amalgam such as: . . . Zn2++2e- ⇋ Zn(Hg) . . . (9.2) The reverse process is the transfer of a particle from the electrode surface to the solution; often the particle on the surface is uncharged or partially charged, and is ionized during the transfer. Ion- and proton-transfer reactions are almost always preceded or followed by other reaction steps. We first consider only the chargetransfer step itself. Ions and protons are much heavier than electrons. While electrons can easily tunnel through layers of solution 5 to 10 Å thick, protons can tunnel only over short distances, up to about 0.5 Å, and ions do not tunnel at all at room temperature. The transfer of an ion from the solution to a metal surface can be viewed as the breaking up of the solvation cage and subsequent deposition, the reverse process as the jumping of an ion from the surface into a preformed favorable solvent configuration. In simple cases the transfer of an ion obeys a slightly modified form of the Butler-Volmer equation. Consider the transfer of an ion from the solution to the electrode. As the ion approaches the electrode surface, it loses a part of its solvation sphere, and it displaces solvent molecules from the surface; consequently its Gibbs energy increases at first.