Evaluation of the Koutecký-Koryta approximation for voltammetric currents generated by metal complex systems with various labilities

The voltammetric response of metal complex systems with various labilities is analyzed by rigorous numerical simulation with the Finite Element Method of the time-dependent concentration profiles of the different species. The ensuing exact fluxes and the corresponding currents are compared to those derived from the Koutecky-Koryta (KK) approximation which assumes a discontinuous transition in the concentration profiles from non-labile to labile behavior. The results indicate a relatively far-reaching correctness of the KK approximation in the complete kinetic range from non-labile to labile complexes, as long as the kinetic flux is computed from the effective concentration of the complex in the reaction layer. Some approximate analytical expressions for this concentration are provided. The KK approximation is shown to be applicable for any metal-to-ligand ratio, provided that the thickness of the reaction layer is expressed in terms of the ligand concentration at the electrode surface. Published in Journal of Electroanalytical Chemistry 2002, vol 526, p 10-18 DOI: 10.1016/S0022-0728(02)00745-3 reprints also to galceran@quimica.udl.cat Evaluation of Koryta-Koutecky.. 26/02/02 2


Introduction
Over the nineties there has been a substantial growth of interest in dynamic metal speciation, which not only covers the equilibrium distribution of the different metal species but also the kinetic characteristics of their interconversion [1][2][3]. Kinetically defined categories of behavior of metal complexes range from inert to dynamic, the latter comprising various degrees of lability. The different notions have been defined on the basis of the voltammetric response of systems with electroinactive complex species and electroactive uncomplexed hydrated metal ions [4][5][6][7].
The case of inert complexes is rather trivial because such complexes do not contribute at all to the metal ion reduction process. The voltammetric response is then identical to that of the mere free metal ion. The distinction between labile and non-labile complexes is much more subtle since both of them refer to systems with relatively high rates of conversion of complex species into free metal ions. Labile complexes are characterized by such high rates of dissociation/re-association that, on any relevant spatial scale, full equilibrium between complexed and free metal is maintained. Consequently, interfacial processes involving the free metal (e. g. electrochemical reduction) are then limited by coupled diffusion of complex and free metal. Non-labile complexes represent the other extreme within the dynamic range where the effective rate of dissociation is much lower than that of the diffusive supply. In that case, the rate of dissociation (a volume reaction) determines the contribution of the complex to the interfacial metal ion flux. , with the plus sign for i=M or L and the minus sign for i=ML.
where t is time, x the distance from the electrode surface and * i c the bulk concentration of i.
In case of a sufficiently large excess of ligand L the association reaction is pseudo firstorder and we define Chronoamperometric limiting current conditions are defined by which completes the formulation of the problem.
Complex systems are divided into static (or inert) and dynamic categories. The distinction is based on the values of the effective chemical rate constants k d and k a ' , relative to the effective time scale t. According to the official definitions [6], and in full agreement with the restrictions of the classical treatments [14], both non-labile and labile complex systems are subject to the condition The physical meaning is that the conversion of M into ML and vice versa is fast on the time scale considered, and the corresponding regime has been denoted as 'dynamic' [5].
It is important to bear in mind that fulfillment of condition (7) does not imply that equilibrium (1) is maintained on every relevant spatial scale contained in an experiment on time-scale t. The condition is concerned with the volume complexation reaction (1) and its fulfillment does not warrant the maintenance of equilibrium in an interfacial process of consumption of free metal ions. This feature actually forms the heart of the mere existence of the so-called reaction layer where the dissociation of ML is not fast enough to 'follow' the depletion of M. The reaction layer is defined by its thickness µ [7,10,17,18]

Concentration profiles and fluxes of M and ML
A. The non-labile regime By various definitions, non-lability is related to the rate of dissociation of the complex being so slow that depletion of ML is negligible even in the immediate vicinity of the electrode surface. Thus it is characterized by the combination of the conditions (7) and where c ML φ denotes concentration in the reaction layer.
The flux of free metal, generated in the reaction layer adjacent to the electrode surface is then simply given by which is immediately verified using (5) and the definition of µ , eqn. (8).
Conceptually, the non-labile regime is of a special nature: on the one hand it requires high rate constants k d and k a ' to fulfill condition (7) so that µ δ << M , and on the other hand k d must be so low that depletion of ML inside the reaction layer with thickness µ is not appreciable. It is this potentially conflicting set of requirements that has given rise to the differentiated definition of the notion of lability [19,20].

B. The labile regime
Labile systems obey condition (7), and have such high k d values that depletion of ML is practically complete. Thus they are characterized by kinetic fluxes that largely outweigh the diffusive fluxes J dif : 26 (12) This condition has been discussed at length in the literature [21].
The simulation reveals that approaching the electrode from the bulk solution, there is a certain region with depletion of M, together with a metal flux low enough to be "followed" by the complex so that equilibrium is maintained. This is the region where kinetics can be considered infinitely fast in the Koutecký-Koryta approximation. Closer to the electrode surface, the metal flux increases and for x less than µ , the rate of 26/02/02 9 dissociation of ML becomes limiting. The concentration of ML tends to an approximately constant, c ML φ , to reach the prescribed zero slope at the surface (see (6)). b g), we can obtain c ML 0 which can be used as a good approximation for c ML φ : -. - -. -. -.
-* * . / 0 Per60.14 (17) Expressions (16) and (17) which leads to quite a good agreement with the effective layer of disequilibration as can be seen in Fig. 7. Likewise, the kinetic contribution must also be redefined as J kin , defined as (19), is also plotted in figure 5. It is a good approximation for J M 0 in the range where there is a non-negligible contribution of the dissociation to the metal flux.
However, J kin is not a good approximation for the kinetic contribution at low k d -values when J M 0 tends to the inert value (as indicated in Fig. 5). In fact, this result is not This free metal contribution is not included in (19) and is responsible for the disparity between J M 0 and J kin defined as (19) for (14)). This concentration is of great importance for the voltammetric response of the system and it can be defined by the approximate analytical expressions (16) and (17). The analysis confirms that close to the electrode surface there is a disequilibration layer (where Q≠K) with a thickness of that of the reaction layer, even in the case of labile complexes.
The KK approximation (see eqn. (19)) is shown to be applicable for any metal-to-ligand ratio in the complex system, provided that the thickness of the reaction layer is expressed in terms of the local ligand concentration at the electrode surface (eqn. 18). In this case, however, the KK expression is a good approximation for the metal flux only The initial condition is The boundary conditions become: Linear piecewise interpolation functions have been used in the discretized weak , -. -. -. -. -. - --. -. -. -. -. -.
$ , due to the boundary condition (6). This system is iteratively solved with a Newton-like method, modified in order to avoid loss of convergence [31].