Electrochemical Double-layer Theory

Skulason E, Karlberg GS, Rossmeisl J, Bligaard T, Greeley J, Jonsson H, Nprskov JK. 2007. Density functional theory calculations for the hydrogen evolution reaction in an electrochemical double layer on the Pt(lll) electrode. Phys Chem Chem Phys 9 3241-3250. 

The above effects are more familiar than direct contributions of the metal s components to the properties of the interface. In this chapter, we are primarily interested in the latter these contribute to M(S). The two quantities M(S) and S(M) (or 8% and S m) are easily distinguished theoretically, as the contributions to the potential difference of polarizable components of the metal and solution phases, but apparently cannot be measured individually without adducing the results of calculations or theoretical arguments. A model for the interface which ignores one of these contributions to A V may, suitably parameterized, account for experimental data, but this does not prove that the neglected contribution is not important in reality. Of course, the tradition has been to neglect the metal s contribution to properties of the interface. Recently, however, it has been possible to use modern theories of the structure of metals and metal surfaces to calculate, or, at least, estimate reliably, xM(S) and 5 (as well as discuss 8 m, which enters some theories of the interface). It is this work, and its implications for our understanding of the electrochemical double layer, that we discuss in this chapter. 

The problem of ion transfer across the interface has been treated in detail by Sato,26,27 Scully,28 and also Valand and Heus-ler,29 following the general theory of Vetter.30 Valand and Heusler assumed the same type of activation-controlled charge transfer kinetics, except that the dominant charge here is that on the O2-ions (or OH- ions) obtained by splitting water at the interface. The electrochemical double layer here is of the usual type for aqueous systems and the equilibrium p.d. is determined by the main charge transfer reaction... 

Double layer emersion continues to allow new ways of studying the electrochemical interphase. In some cases at least, the outer potential of the emersed electrode is nearly equal to the inner potential of the electrolyte. There is an intimate relation between the work function of emersed electrodes and absolute half-cell potentials. Emersion into UHV offers special insight into the emersion process and into double layer structure, partly because absolute work functions can be determined and are found to track the emersion potential with at most a constant shift. The data clearly call for answers to questions involving the most basic aspects of double layer theory, such as the role water plays in the structure and the change in of the electrode surface as the electrode goes frcm vacuum or air to solution. 

G. Pirug, H. P. Bonzel, Electrochemical double-layer modeling under idtrahigh vacuum conditions in Interfacial Electrochemistry. Theory, Experiment and Applica tions (Eds. A. Wifckowski), Marcel Dekker New York-Basel, 1999, p. 269. 

Thus summarizing, we note that at the leading order the asymptotic solution constructed is merely a combination of the locally electro-neutral solution for the bulk of the domain and of the equilibrium solution for the boundary layer, the latter being identical with that given by the equilibrium electric double layer theory (recall (1.32b)). We stress here the equilibrium structure of the boundary layer. The equilibrium within the boundary layer implies constancy of the electrochemical potential pp = lnp + ip across the boundary layer. We shall see in a moment that this feature is preserved at least up to order 0(e2) of present asymptotics as well. This clarifies the contents of the assumption of local equilibrium as applied in the locally electro-neutral descriptions. Recall that by this assumption the electrochemical potential is continuous at the surfaces of discontinuity of the electric potential and ionic concentrations, present in the locally electro-neutral formulations (see the Introduction and Chapters 3, 4). An implication of the relation between the LEN and the local equilibrium assumptions is that the breakdown of the former parallel to that of the corresponding asymptotic procedure, to be described in the following paragraphs, implies the breakdown of the local equilibrium. 

Ruckenstein and Schiby derived4 an expression for the electrochemical potential, which accounted for the hydration of ions and their finite volume. The modified Poisson-Boltzmann equation thus obtained was used to calculate the force between charged surfaces immersed in an electrolyte. It was shown that at low separation distances and high surface charges, the modified equation predicts an additional repulsion in excess to the traditional double layer theory of Deijaguin—Landau—Verwey—Overbeek. 

Redox ions in solution are subject to chaotic Brownian movement. In principle, a certain range of tunneling distances between the metal and the redox species should be taken into account in a kinetic theory. The tunneling probability decays exponentially with increasing distance between the metal and the redox ion. Only redox ions nearest to the metal surface are, therefore, taken into account. Then, the inner solvation shell of the ion contacts the Helmholtz layer. There is no penetration of the reacting system into the electrochemical double layer (See Section 4.7.2). 

Already in this Modem Aspects of Electrochemistry series the theory and status of data for electrochemical double layers (dl) were detailed, but the sections devoted to results obtained with solid electrodes and their discussion were brief. Some aspects of the dl on polycrystalline metals, such as the potential of zero charge (pzc), were described." In this series a chapter was also devoted to the metal-gas interface certainly the comparison of this interface with that at an electrode is fruitful, although the local electric field in the latter case can be varied far more easily. 

Figure 3. A small portion of the electrochemical double layer at the clay/water (electrolyte) interface is shown to depict the microscopic stmc-ture and the potential drops involved, by analogy with the metal/electrolyte interface. (Diagram from Conway, Theory and Principles of Electrode Processes, p. 26, Ronald Press, New York, 1965).

In what follows, we propose a phenomenological model of the chemisorbed radical-anion standing in the electrochemical double-layer. We shall hence detail the reasons why this chemisorbed radical anion is intrinsically unstable but most probably has a finite lifetime on the polarized metallic surface. We outline the procedure through which we expect that an order of magnitude of the lifetime of the chemisorbed radical-anion may be evaluated numerically via this model. The model potential felt by the radical-anion as it is formed on the polarized electrode is described as the sum of three terms, for which a parametrisation is proposed. One of these terms is meant to include both the surrounding solvent and the repulsion by the polarized electrode, thanks to a mean, locally uniform, electric field. In the present paper, the intensity of this uniform electric field is calibrated on the basis of a conparison between experimental Stark-Tuning shifts for CO chemisorbed on palladium surfaces in solution and Density Functional Theory calculations of field-induced vibrational shifts for CO chemisorbed on palladium clusters. The shape of the resulting model potential is then discussed. 

The reason for this state of affairs may be seen in past emphasis on surface phenomenological studies which attempted to model the metal surface as an array of surface atoms with some valences saturated by subsurface metal atoms and other valences saturated by ions or molecules making up the environment. This model led to the description of the interface in terms of the Helmholz and Guy-Chapman double layer theories, and inhibitors were visualized as interfering with the double layer structure through adsorption on the surface atoms of the metal, thereby altering the electrochemical reaction rates which are governed by the energetics of the double layer. While this model has been... 

All electrochemical processes take place in the double-layer region. Therefore, electrochemists need to know the distribution of the various particles and of the electrostatic potential and the charge near the interface in detail. After more than a century of double-layer theory, and more than 50 years after the seminal paper by Gra-hame [64], it is pertinent to ask how far we have succeeded. Of course, the following remarks are the personal view of the author. 

Fig. 1. Electric potential function, y (0, in a diffiuse charge layer of an electrochemical double layer according to the theory of Gouy-Chapman. Curves are given for Z = 8. z — 4, z = 2 and z z = ve y ve jkT, h -- xx). Dotted lines ...

The early concept of an electrochemical supercapacitor (ES) was based on the electric double-layer existing at the interface between a conductor and its contacting electrolyte solution. The electric double-layer theory was first proposed by Hermann von Helmholtz and further developed by Gouy, Chapman, Grahame, and Stem. The electric double-layer theory is the foundation of electrochemistry from which fhe electrochemical processes occurring at an electrostatic interface... 

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