Principle of Unchanging Total Concentration and Its Implications for Modeling Unsupported Transient Voltammetry

Theory of linear sweep voltammetry with diffuse charge: Unsupported electrolytes, thin films, and leaky membranes

Linear sweep and cyclic voltammetry techniques are important tools for electrochemists and have a variety of applications in engineering. Voltammetry has classically been treated with the Randles-Sevcik equation, which assumes an electroneutral supported electrolyte. In this paper, we provide a comprehensive mathematical theory of voltammetry in electrochemical cells with unsupported electrolytes and for other situations where diffuse charge effects play a role, and present analytical and simulated solutions of the time-dependent Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions for a 1:1 electrolyte and a simple reaction. Using these solutions, we construct theoretical and simulated current-voltage curves for liquid and solid thin films, membranes with fixed background charge, and cells with blocking electrodes. The full range of dimensionless parameters is considered, including the dimensionless Debye screening length (scaled to the electrode separation), Damkohler number (ratio of characteristic diffusion and reaction times), and dimensionless sweep rate (scaled to the thermal voltage per diffusion time). The analysis focuses on the coupling of Faradaic reactions and diffuse charge dynamics, although capacitive charging of the electrical double layers is also studied, for early time transients at reactive electrodes and for nonreactive blocking electrodes. Our work highlights cases where diffuse charge effects are important in the context of voltammetry, and illustrates which regimes can be approximated using simple analytical expressions and which require more careful consideration.

Decoupling of the nernst-planck and poisson equations. Application to a membrane system at overlimiting currents

This paper deals with one-dimensional stationary Nernst-Planck and Poisson (NPP) equations describing ion electrodiffusion in multicomponent solution/electrode or ion-conductive membrane systems. A general method for resolving ordinary and singularly perturbed problems with these equations is developed. This method is based on the decoupling of NPP equations that results in deduction of an equation containing only the terms with different powers of the electrical field and its derivatives. Then, the solution of this equation, analytical in several cases or numerical, is substituted into the Nernst-Planck equations for calculating the concentration profile for each ion present in the system. Different ionic species are grouped in valency classes that allows one to reduce the dimension of the original set of equations and leads to a relatively easy treatment of multi-ion systems. When applying the method developed, the main attention is paid to ion transfer at limiting and overlimiting currents, where a significant deviation from local electroneutrality occurs. The boundary conditions and different approximations are examined: the local electroneutrality (LEN) assumption and the original assumption of quasi-uniform distribution of the space charge density (QCD). The relations between the ion fluxes at limiting and overlimiting currents are discussed. In particular, attention is paid to the "exaltation" of counterion transfer toward an ion-exchange membrane by co-ion flux leaking through the membrane or generated at the membrane/solution interface. The structure of the multi-ion concentration field in a depleted diffusion boundary layer (DBL) near an ion-exchange membrane at overlimiting currents is analyzed. The presence of salt ions and hydrogen and hydroxyl ions generated in the course of the water "splitting" reaction is considered. The thickness of the DBL and its different zones, as functions of applied current density, are found by fitting experimental current-voltage curves.

Generalized principles of unchanging total concentration

We consider the transport of multiple reacting species under the continuum assumption in situations such as those that frequently arise in electroanalytical chemistry. Under certain limitations, it has been shown that the total species concentration (as defined by Oldham and Feldberg) of such a system is uniform and constant. In this work, we extend the limits of the previous analysis to enable greater applicability. This is accomplished by using either of two new dependent variables, which are generalizations of the concept of total concentration. Then, conditions are determined under which the dependent variable will be uniform and constant in time. From a practical viewpoint, the described formalism allows one to simplify the multispecies transport problem formulation by eliminating one equation from the system of governing equations.

General Theory for Migrational Voltammetry. Strong Influence of Diversity in Redox Species Diffusivities on Charge Reversal Electrode Processes

High sensitivity of the microelectrode response to the difference between the substrate and the product diffusion coefficients is predicted for the charge reversal processes. This effect is anticipated from the general theoretical model developed for the diffusional-migrational transport to microelectrodes. The model predicts the voltammetric wave heights for any type of electrode processes carried out in the presence of any number and concentration of nonelectroactive ions. It involves changes in diffusion coefficients of the redox species and assumes no homogeneous complications. Handy, analytical expressions for the limiting current and limiting potential can be derived for a system of a univalent product and univalent ions of supporting electrolyte. This case covers charge reversal processes of the following type: S(z) --> P(+/-) + ne (n + z = sgn(n), absolute value(n) > or = 2). It has been shown that under migrational conditions the change in the ratio of the product and the substrate diffusivities (D(P)/D(S)) by as little as 10% results in significant changes in the voltammetric wave height. For 2-e charge reversal processes, a 10% increase in D(P) versus D(S) leads to a drop in the voltammetric wave height of 18.3% compared to that calculated for equal diffusion coefficients. The reversed change, i.e., the 10% decrease of the D(P) value with respect to D(S), increases the voltammetric wave height by 30.5% compared to that obtained for equal diffusivities. The theoretical predictions were confronted with our recent experimental results obtained for the 2-e oxidation of sodium (6,8-diferrocenylmethylthio)octanoate, which process can be classified as the charge reversal reaction. The best fit was obtained for D(P)/D(S) equal to 0.71.

Generalized theory of steady-state voltammetry without a supporting electrolyte. Effect of product and substrate diffusion coefficient diversity

A generalized theory of the steady-state voltammetric response of a microelectrode in the absence of supporting electrolyte and for any values of diffusion coefficients of the substrate and the product of an electrode process is presented. The treatment applies to any reasonable combination of the charge numbers of the substrate, its counterion, and the product. A way to incorporate the activation polarization into the model is also demonstrated. It has been shown that the height, position, and shape of the migrational voltammogram are affected by the ratio of the product to substrate diffusivity (theta). In particular, for the electrode processes with sign retention, unequal diffusivities of electroactive species influence both characteristic points of the voltammogram (the limiting current and the half-wave potential). For charge neutralization processes (uncharged product), the changes in theta parameter are accompanied only by a shift in the half-wave potential. The most dramatic changes in the I-E relation can be observed for the charge reversal processes. In this case, a consecutive increase in theta results in the transition of the voltammogram shape from rapid exponential growth (theta < 1), through ramp shape (theta = 1), to common wave shape (theta > 1). On the basis of the expressions derived for the limiting current (exact and linearized), a possibility of the determination of the diffusion coefficient of the electrode reaction product is demonstrated. In addition, the ranges of theta where the assumption of equal diffusivities of the substrate and the product is obeyed within an insignificant error have been determined quantitatively. The theory has been experimentally verified using voltammetric oxidation of hexacyanoferrate(II).