Potential distribution double-layer

In this appendix we provide the generalized equations for the mean-field potential and double layer interaction free energy between two surfaces having distinct but periodic nonuniform distributions. These results were taken from Ref. [78]. We denote by yL(s) the charge distribution on the left (L) surface at z = 0 and yR(s) that on the right (R) surface at z — h. As in the text the variable y represents either surface potential, T (s), or surface charge, periodic distribution represented by the Fourier expansions,... 

When two conducting phases come into contact with each other, a redistribution of charge occurs as a result of any electron energy level difference between the phases. If the two phases are metals, electrons flow from one metal to the other until the electron levels equiUbrate. When an electrode, ie, electronic conductor, is immersed in an electrolyte, ie, ionic conductor, an electrical double layer forms at the electrode—solution interface resulting from the unequal tendency for distribution of electrical charges in the two phases. Because overall electrical neutrality must be maintained, this separation of charge between the electrode and solution gives rise to a potential difference between the two phases, equal to that needed to ensure equiUbrium. 

On the electrode side of the double layer the excess charges are concentrated in the plane of the surface of the electronic conductor. On the electrolyte side of the double layer the charge distribution is quite complex. The potential drop occurs over several atomic dimensions and depends on the specific reactivity and atomic stmcture of the electrode surface and the electrolyte composition. The electrical double layer strongly influences the rate and pathway of electrode reactions. The reader is referred to several excellent discussions of the electrical double layer at the electrode—solution interface (26-28). 

Previous considerations have shown that the interface between two conducting phases is characterised by an unequal distribution of electrical charge which gives rise to an electrical double layer and to an electrical potential diflFerence. This can be illustrated by considering the transport of charge (metal ions or electrons) that occurs immediately an isolated metal is immersed in a solution of its cations ... 

If an electrolyte AB is distributed between two solvents, a and f3, there will in general be a difference of potential established across the boundary, due to the existence of a double layer ( 198). 

Figure 23. Electric potential distribution in electric double layer. HL, Helmholtz layer DL, diffuse layer.

When an electrode is in contact with an electrolyte, the interphase as a whole is electroneutral. However, electric double layers (EDLs) with a characteristic potential distribution are formed in the interphase because of a nonuniform distribution of the charged particles. 

Two types of EDL are distinguished superficial and interfacial. Superficial EDLs are located wholly within the surface layer of a single phase (e.g., an EDL caused by a nonuniform distribution of electrons in the metal, an EDL caused by orientation of the bipolar solvent molecules in the electrolyte solution, an EDL caused by specific adsorption of ions). Tfie potential drops developing in tfiese cases (the potential inside the phase relative to a point just outside) is called the surface potential of the given phase k. Interfacial EDLs have their two parts in dilferent phases the inner layer with the charge density in the metal (because of an excess or deficit of electrons in the surface layer), and the outer layer of counterions with the charge density = -Qs m in the solution (an excess of cations or anions) the potential drop caused by this double layer is called the interfacial potential... 

In the crudest approximation, the effect of the efectrical double layer on electron transfer is taken into account by introduction of the electrostatic energy -e /i of the electron in the acceptor into the free energy of the transition AF [Frumkin correction see Eq. (34.25)], so that corrected Tafel plots are obtained in the coordinates In i vs. e(E - /i). Here /i is the average electric potential at the site of location of the acceptor ion. It depends on the concentration of supporting electrolyte and is small at large concentrations. Such approach implies in fact that the reacting ion represents a probe ion (i.e., it does not disturb the electric held distribution). 

This result was taken as an experimental eonfirmation of the model developed by Sehmiekler [7]. However, it appeared somehow eontradictory with other results obtained with SECM. It was also suggested that eoneentration polarization phenomena occurring at the aqueous side are negligible as the whole potential drop is presumably developed in the benzene phase. This assumption can be qualitatively verified by evaluating a simplified expression for the potential distribution based on a back-to-back diffuse double layer [40,113],... 

From the equilibrium requirement that the chemical potential involving all ionic species be uniform throughout the phase boundary, the distribution of ions within the electrical double layer can be expressed by the Boltzmann equation ... 

Fig. 4.1 Structure of the electric double layer and electric potential distribution at (A) a metal-electrolyte solution interface, (B) a semiconductor-electrolyte solution interface and (C) an interface of two immiscible electrolyte solutions (ITIES) in the absence of specific adsorption. The region between the electrode and the outer Helmholtz plane (OHP, at the distance jc2 from the electrode) contains a layer of oriented solvent molecules while in the Verwey and Niessen model of ITIES (C) this layer is absent...

The distribution of potential in TC is practically the same as that near the flat surface if the electrolyte concentration is about 1 mol/1 [2], So the discharge of TC may be considered as that of a double electric layer formed at the flat electrode surface/electrolyte solution interface, and hence, an equivalent circuit for the TC discharge may be presented as an RC circuit, where C is the double layer capacitance and R is the electrolyte resistance. 

This potential-energy surface will change when the electrode potential is varied consequently the energy of activation will change, too. These changes will depend on the structure of the double layer, so we cannot predict the value of the transfer coefficient a unless we have a detailed model for the distribution of the potential in the double layer. There is, however, no particular reason why a should be close to 1/2. Also, a temperature dependence of the transfer coefficient is not surprising since the structure of the double layer changes with temperature. 

To minimize absorption from the solution, optical thin layer cells have been designed. The working electrode has the shape of a disc, and is mounted closely behind an IR-transparent window. For experiments in aqueous solutions the intervening layer is about 0.2 to 2 ftm thick. Since the solution layer in front of the working electrode is thin, its resistance is high this increases the time required for double-layer charging - time constants of the order of a few milliseconds or longer are common - and may create problems with a nonuniform potential distribution. 

The second term in equation (9) is the usual electrostatic term. Here vA is the valency of the unit and e is the elementary charge, and ip(z) is the electrostatic potential. This second term is the well-known contribution accounted for in the classical Poisson-Boltzmann (Gouy -Chapman) equation that describes the electric double layer. The electrostatic potential can be computed from the charge distribution, as explained below. 

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