Frumkin, double-layer effects

The simplest effect of pure electrostatic ionic adsorption on electrode reaction rates of ions is the Frumkin double layer effect already been discussed in Sect. 3.5. 

The effect of the phospholipids on the rate of ion transfer has been controversial over the last years. While the early studies found a retardation effect [6-8], more recent ones reported that the rate of ion transfer is either not retarded [9,10] or even enhanced due to the presence of the monolayer [11 14]. Furthermore, the theoretical efforts to explain this effect were unsatisfactory. The retardation observed in the early studies was explained in terms of the blocking of the interfacial area by the phospholipids, and therefore was related to the size of the transferring ion and the state of the monolayer [8,15]. The enhancement observed in the following years was attributed to electrical double layer effects, but a Frumkin-type correction to the Butler Volmer (BV) equation was found unsuitable to explain the observations [11,16]. Recently, Manzanares et al. showed that the enhancement can be described by an electrical double layer correction provided that an accurate picture of the electrical double layer structure is used [17]. This theoretical approach will be the subject of Section III.C. 

The decrease of the concentration of the electroactive species with increasing potential has to be attributed to double layer effects. As first pointed out by Frumkin [58], in dilute solutions the electron transfer rate is affected by variations of the potential in the double layer in two ways. The potential in the outer Helmholtz plane, fa, is due to the extension of the double layer not identical to the potential in the solution (at the end of the double layer), so that the effective driving force of the reaction is DL — fa. Furthermore, the concentration of ionic reactants in the reaction plane, c, is influenced by electrostatic effects and differs from the concentration just outside the double layer, c0, by a Boltzmann term ... 

Frumkin effect - double layer, effect on charge transfer rate... 

Refs. [i] Frumkin A (1933) Z phys Chem A 164 121 [ii] Frumkin AN (1961) Hydrogen overvoltage and adsorption phenomena, part 1, mercury. In Delahay P (ed) Advances in electrochemistry and electrochemical engineering, vol 1. Interscience, New York [iii] Frumkin AN, Petrii OA, Nikolaeva-Ferdorovich NV (1963) Electrochim Acta 8 177 [iv] Frumkin AN, Nikolaeva-Fedorovich NV, Berezina NP, Keis KhE (1975) J Electroanal Chem 58 189 [v] Fawcett WR (1998) Double layer effects in the electrode kinetics of electron and ion transfer reactions. In Lipkowski J, RossPN (eds) Electrocatalysis. Wiley-VCH, New York, p 323... 

The static - double-layer effect has been accounted for by assuming an equilibrium ionic distribution up to the positions located close to the interface in phases w and o, respectively, presumably at the corresponding outer Helmholtz plane (-> Frumkin correction) [iii], see also -> Verwey-Niessen model. Significance of the Frumkin correction was discussed critically to show that it applies only at equilibrium, that is, in the absence of faradaic current [vi]. Instead, the dynamic Levich correction should be used if the system is not at equilibrium [vi, vii]. Theoretical description of the ion transfer has remained a matter of continuing discussion. It has not been clear whether ion transfer across ITIES is better described as an activated (Butler-Volmer) process [viii], as a mass transport (Nernst-Planck) phenomenon [ix, x], or as a combination of both [xi]. Evidence has been also provided that the Frumkin correction overestimates the effect of electric double layer [xii]. Molecular dynamics (MD) computer simulations highlighted the dynamic role of the water protrusions (fingers) and friction effects [xiii, xiv], which has been further studied theoretically [xv,xvi]. 

We now turn to the potential dependence of electrosorption of neutral molecules, considering first the model developed by Frumkin. This is a phenomenological model, which depends on considerations of the changes in the electrostatic energy of the interphase caused by adsorption. Assuming that measurements are taken in concentrated solutions of a supporting electrolyte, we can neglect diffuse-double-layer effects and focus our attention on the Helmholtz part of the double layer, considered as a parallel-plate capacitor. In the pure solvent the... 

All these findings may point to limitations of the classical Frumkin model for correction of the double-layer influence on electrode kinetics in nonaqueous solvents, although it works well in aqueous solution. In the present author s opinion these rather surprising results may follow from some kind of compensation effects. For instance, ion-pair formation in these solutions by decreasing the effective charge of the reactant could reduce the double-layer effect. 

In mixed (0.8 - x) M NaClO4 + x M NaF supporting electrolyte the electroreduction of Cd(II) was also studied by Saakes etal. [25]. The kinetic parameters were analyzed using CEE mechanism. The obtained chemical rate constants at both steps, fcg 1 and fcg 2, decreased with increasing NaF concentration. The data were corrected for nonspecific double-layer effect (Frumkin correction). The interpretation of CEE mechanism with parallel pathways connected with coexisting cadmium complexes was presented. 

When analyzing kinetic data for ion-transfer reactions, some groups used different corrections such as the Frumkin correction for the double-layer effects, or the Levich correction for the double-layer corrections under Nernst-Planck transport as proposed by Samec et al. [102]. These corrections have been thoroughly discussed by Murtomaki et al. who have clearly shown that the Frumkin correction can be used for equilibrium measurements, and the Levich correction must be used under faradaic conditions [103]. 

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). 

The Frumkin theory of the effect of the electrical double layer on the rate of the electrode reaction is a gross simplification. For example, the electrode reaction does not occur only at the outer Helmholtz plane but also at a somewhat greater distance from the electrode surface. More detailed considerations indicate, however, that Eq. (5.3.20) can still be used to describe the effect of the electrical double layer as a good approximation. 

One should note that the field in the outer part of the double layer should repel OH ions and attract hydrogen ions. This, however, is not expected to have a major effect on the kinetics (the Frumkin effect ) since the main reactant is likely to be water. 

A. N. Frumkin, Z Physikal Chem. 164A 121 (1933). Effect of the double-layer structure on the concentration dependence of areactionrate. 

The addition of the inert electrolyte affords other advantages. The most important point is that the conductivity of the solution increases (and thus the ohmic drop decreases through a decrease of the resistance of the cell, Rccw see Sect. 1.9). Moreover, the diffuse double layer narrows, being formed mainly by the ions of the inert electrolyte (with a sharp potential drop over a very short distance from the electrode surface). This makes the capacitance more reproducible and the Frumkin effects less obtrusive. Activity coefficients of the electroactive species are also less variable (and, therefore, quantities like formal potentials and rate constants), since... 

Perhaps the most striking example of the effect of the diffuse double layer on electrode kinetics was demonstrated by Frumkin in 1955,... 

Girault (lb) pointed out that the apparent potential dependence of the ET rate may be attributed to the change in concentration of the reactants near the interface rather than to activation control. This model, further developed by Schmickler (9), postulates that the rate constant is essentially potential-independent because the potential drop across the compact part of the double-layer at the ITIES is small. In this model, the ET rate dependence on the interfacial potential drop is only due to the diffuse layer effect similar to Frumkin effect at metal electrodes. 

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