The key factors here are the reorganization energy, the work terms, the driving force, and the overlap integral. Let us comment on each of them separately.
6.3.1 Reorganisation Energy
Let us assume that for a single electron cathode and anode transfer the reorganization ener- gies are the same. Er(c)= Er(a)= Er This is usually estimated using the Marcus formula,534
which near the electrode with account of image forces reads Er= e2 3 1 ‘Œ≠ 1 ‘ 4 3 1 2a ≠ 1 4d 4 (30) where a is the effective radius of the oxidized and reduced forms of the reactant (assumed to be the same in both cases) and d is the distance to the reaction plane; ‘Œ and ‘ are, respectively, the high and low frequency dielectric constants of the liquid. ‘Œ is due to the contribution of the polarizability of internal electrons of ions of the liquid, the quantity usually close to 2. ‘ is the ‘static’ dielectric constant; the latter stands in the quotes because, as emphasized earlier in this review, there is no such thing as static dielectric constant of ionically conducting liquid. We mean here the measured ‘intermediate’ dielectic constant due to vibrations and librations of ionic pairs, or more generally the strongly dissipating polar-phonon-like modes of the ionic liquid, the quantity of the order of 10 (see subsection 2.2.3).
This formula was the Marcus-derived535 extension of the Pekar-Marcus (formula for the reorganization energy of the electron exchange between the ions of radii a in the bulk of a dielectric medium).535–537 Er = e2 3 1 ‘Œ≠ 1 ‘ 4 31 a≠ 1 d 4 . (31)
The reorganization energy in the bulk is larger because the electric field of ions near the electrode is image-screened by the metal. Generally both formulae are expected to give rough estimates of the corresponding reorganization energies, because they do not take into account the short range structure of the liquid, neither the electric field penetration into the electrode. There were various attempts to take both effects into account through the spatial dispersion of dielectric permittivity of the liquid and the metal in various approxi- mations of the latter for these two phases, done in the context of reactions in ordinary polar liquids.538–543 These studies used a number of approximations; most importantly they are built on a linear response theory of the medium to the reorganizing charge of the reactant. The study in Ref.357 has shown, however, that the mere presence of the solute as well as the nonlinear electrostatic effects can perturb the environment and substantially distort the nononlocal electrostatics effects544,545 washing out some of the fine structural effects that are predicted by the linear response nonlocal electrostatic theory. In view of such situation the simple formulae of Marcus type remain to be a good orientation point. How much this should be applicable for RTILs? First studies for an electron transfer in the bulk546–548 and at electrodes397 has shown substantial deviations of the estimates due to the short range structure, although as an ‘order of magnitude estimate’ the Marcus formulae can still be used.
One of the sources of the deviation can be actually traced, if we go back to more general expression for the reorganization energy,549–551 which, if we do not take into account spatial dispersion of the dielectric permittivity, reads
Er= e2 31 a≠ 1 d 42 fi kBT /4h ⁄ 0 dÊIm ‘(Ê) Ê|‘(Ê)|2, (32)
where ‘(Ê) is the complex frequency dependent dielectric permittivity of the liquid. The exact Kramers-Kroenig (KK) relation expresses the integral over all frequencies with the static dielectric constant ‘, if ‘(0) = ‘
2 fi Œ ⁄ 0 dÊIm ‘(Ê) Ê|‘(Ê)|2 = 1 ≠ 1 ‘(0). (33)
For the integral in eq 33 there is an approximate KK-relation if the frequency kBT /4h lies
in the transparency band between the electronic excitations in the liquid and the vibrational modes550 and if ‘(Ê æ 0) = ‘ = const.
2 fi kBT /4h ⁄ 0 dÊIm ‘(Ê) Ê|‘(Ê)|2 = 3 1 ‘Œ≠ 1 ‘ 4 . (34)
With this equation valid, we recover expression of eq 32 for the reorganization energy. How- ever, in RTILs we do not know whether any of the two assumptions hold. In particular in view of the very slow relaxation in ionic liquid (low frequency tail of, ‘(Ê), where should we truncate the integration in eq 34? So the best strategy would be, if we have experimental data for Im ‘(Ê)|‘(Ê)|2 to keep the integral form for the reorganization energy. Even better, had
we had information not only about frequency-, but also the wave-vector-dependence of the dielectric response function Im ‘(k,Ê)|‘(k,Ê)|2 (say, from computer simulations, such as e.g. obtained
for water356 we could use then even more general equations for the reorganization energy,
Er= e2 2 fi Œ ⁄ 0 dk I 1 2 C (sin ka1)2 (ka1)2 +(sin ka2)2 (ka2)2 D ≠sin kaka 1 1 sin ka2 ka2 sin kd kd J 2 fi kBT /4h ⁄ 0 dÊIm ‘(k,Ê) |‘(k,Ê)|2 (35) drawn here for a single electron exchange between two ‘spherical ions of radii’ a1 and a2, the centers of which are separated by a distance d( a1+ a2). In the case of k-independent dielectric-response function, Im ‘(k,Ê) |‘(k,Ê)|2 ¥ Im ‘(k = 0,Ê) |‘(k = 0,Ê)|2 = Im ‘(Ê) |‘(Ê)|2 (36)
the two integrals in eq 35 are decoupled and eq 35 reduces to eq 33 (if we put there a1=
a2= a). For RTILs such assumption in the range of wave numbers k : (2fi/a), important in the integral eq 35, will unlikely be realistic. Thus eq 35 suggests an interesting scheme for future calculation of the reorganization energy and comparison of the results with the direct simulation of the latter.547,548
This expression however would only be applicable for the electron transfer between donor and acceptor ions in the bulk. How it should look for the electrochemical electron transfer
reactions at electrodes? This is generally a difficult problem, c.f. Refs.,538–541,552 which is yet to be solved. Such enterprise, however, would only make sense if such approach showed good results in the bulk; the studies in Ref.357 performed for water, suggest that this may not be the case. However, a priori we do not know, whether this would be the same for RTILs.
6.3.2 Work terms
These tell us what will be the concentrations of oxidized (for the cathodic current) and reduced (for the anodic current) species at the reaction plane, the factors also determining the currents. The probability to find there those species will depend on their charge and the potential of the plane. The latter is very much an issue of the potential distribution near the electrode, and the effects of overscreening can contribute new interesting effects here (but see a discussion below). They may be particularly important for electrodeposition reactions with a slow adsorption stage.
6.3.3 The Driving force and the potentially new form of the Frumkin correction
One of the functions of electrolyte in electrode kinetics is to provide the most compact localization of the electrostatic potential drop between the electrode and the reaction plane, to utilize the strongest drive for the reaction. In diluted electrolyte solution the potential drop between the electrode and the bulk will be spread over the diffuse double layer and only a small portion of this drop will be localized between the electrode and the reaction plane. Hence polarizing the electrode will be majorly wasted for the electrode kinetics. The correction to the expressions for the electrode current which takes into account this effect is known as the Frumkin correction.553,554 This we symbolically incorporated in the expression for V1(V ), with V1(V ) ¥ V only if total potential drop is fully localized between the electrode and the reaction plane. With the effect of overscreening in place in RTILs, one may expect to encounter something completely new, different from the case of diluted electrolytic solutions, where V1(V ) < V . Here, in the overscreening regime, V1(V ) may quite easily become substantially larger than V .
namic simulations in Ref.397The result was however negative: manifestation of overscreening in the driving force were not found. The authors concluded that the potential distribution near the electrode averaged in the lateral plane in the absence of the reactant in question is not what actually counts in the driving force. They write: “Rather we should be consid- ering the potential at the ion’s center due only to the other charged present in the system: this might be better called a Madelung potential.” This question requires further detailed investigation. If we try to translate that conclusion into the necessity to take into account what is called in electrochemistry the effect of micropotential,555 in order to corroborate the conclusion of Ref.397 we should assume that such effects should be abnormally high in strong correlated Coulomb systems, such as RTILs. Recent studies of Ref.208 has shown that electrostatic potential experienced but a solute ions in RTIL is substantially different from the one that sets in the absence of such ion. However, this difference was not to the degree that the effect of oscillating overscreening patterns fully disappear.
Last but not least, in the regime of very large electrode polarization where the lattice saturation effects (crowding) might be expected,V1(V ) Ã
Ô
V.556This can result in a current- voltage law qualitatively different (c.f.556) from the Bulmer-Volmer’s.555This is a whole new area for future investigations, where, however, experiments should first set the scene.
6.3.4 Matrix element of transition (overlap integral)
Evaluation of this quantity requires extension of modern methods of quantum chemistry with account of dense ionic environment of RTIL. In a simplified version531,532 this may take a combination of electron density functional theory and molecular dynamics, in the spirit of ab initio Car-Parinello schemes.557 Will that be worth the effort? The question is, indeed, justified as it refers to the calculation of the pre-exponential factor in the expression for the electrode current, whereas so many issues are yet unclear about the exponential factors! The importance of the overlap integrals is, however, obvious when they are vanishingly small, as then they substantially affect the scale of the electrode current. They are certainly important for tuning the optimal parameters of the electrodeposition reactions. Next, if they also become voltage dependent531,532 the matrix element of transition can contribute corrections to the current voltage plot. To our knowledge, for RTILs no detailed studies in
this direction have yet been reported.