Modelling Electroanalytical Experiments by the Integral Equation Method

This book is devoted to the theoretical modelling of transient experiments in electroanalytical chemistry, by means of the mathematical approach called the method of integral equations. In electroanalytical experiments one usually perturbs the interface studied by a particular potential-time (or current-time) signal, and observes the current-time (or potential-time) response. The goal is to identify electrochemical and non-electrochemical reactions and other physico-chemical processes occurring at the interface. Theoretical modelling of the experiments is usually based on formulating and solving relevant initial boundary value problems. In the integral equation method these are converted to integral equations that are subsequently solved (most often numerically). The advantage of the method is an increased mathematical insight into the models considered, and (frequently) the reduction of the computational cost, compared to direct numerical solutions of partial differential equations.

Electrochemical systems are generally multiphase and multicomponent systems. The various chemical species can be distributed in spatially extended phases, or localised at interfaces (for example adsorbed). The distributed species are subject to transport phenomena and homogeneous reactions. The transport is most often described by diffusion partial differential equations, but convection-diffusion equations are also in frequent use, in particular in models of dropping mercury electrodes, rotating disk electrodes, or channel and tubular electrodes. The kinetics of homogeneous reactions affect the transport equations. Anomalous diffusion transport is also known. All species may participate in heterogeneous reactions at interfaces, in particular in charge transfer reactions. The kinetics of heterogeneous reactions determine boundary conditions at the interface studied. All reactions influence the initial conditions. The dimensionality of spatial domains depends on the symmetry and coordinate systems most suitable for mathematical description. Spatial domains can be infinite, semi-infinite, or finite. Additional effects considered in electroanalytical models are the uncompensated Ohmic potential drop and double layer charging.

For a proper understanding of the integral equation method, several mathematical concepts have to be defined. The concepts include: integral and integro-differential equations; integral operators and their kernels; linearity and nonlinearity of the integral operators and integral equations; convolution kernels; kinds of integral equations (first and second kind); distinction between Fredholm and Volterra integral equations; regular, singular and weakly singular kernels (and integral equations). Techniques such as the Laplace, Fourier and Hankel transformations have to be recalled, as they are used for the conversion of initial boundary value problems into integral equations.

Electroanalytical models independent of spatial coordinates usually arise for systems in which transport phenomena can be neglected, so that the mathematical formalism reduces to equations describing the kinetics of heterogeneous reactions at the interface studied. Such equations can be ordinary differential equations or differential-algebraic equations. The conversion of such models into integral equations was not practiced in the past, but there have been several recent publications employing the conversion. Depending on the choice of unknown variables, three procedures of deriving the integral equations can be distinguished. However, the advantage of the integral equation formulation of such models (over ordinary differential or differential-algebraic equations) is debatable, because the computational cost can be greater in the case of simulations within long time intervals.

Electroanalytical models involving one-dimensional diffusion without homogeneous reactions, localised species, or other complications, are often converted into integral equations. The conversion makes use of integral concentration–production rate relationships that can be obtained analytically from the diffusion partial differential equations, separately for every species. The relationships are obtainable both for semi-infinite and finite spatial domains, assuming Cartesian, spherical or cylindrical coordinates. In a few limiting cases, corresponding to steady states, the relationships become algebraic rather than integral. Anomalous diffusion is handled in a similar way. The application of the integral equation method to all such models usually brings a reduction of the computational costs, compared to direct numerical solving of partial differential equations, because spatial dependencies are entirely eliminated. Although an extra human effort is needed to derive the integral equations, additional insights into the models can be gained.

Electroanalytical models involving one-dimensional convection-diffusion transport are similar to models involving pure diffusion, discussed in Chap. 5, in that a separate concentration–production rate relationship exists for every dynamic distributed species. However, the interfacial species production rate may generally not vanish in the initial state of the system. Integral concentration–production rate relationships are obtainable, in particular, for convection-diffusion at dropping mercury electrodes; rotating disk electrodes; and channel or tubular electrodes (provided that the Singh–Dutt approximation is used). Algebraic concentration–production rate relationships exist at steady state, for rotating disk electrodes, and channel or tubular electrodes.

There have been only few attempts to derive and solve integral equations corresponding to electroanalytical models involving two- and three-dimensional diffusion. Three approaches have been proposed for this purpose in the literature: the Cope–Tallman approach, the Mirkin–Bard approach, and the boundary integral method. The Cope–Tallman approach operates on the Laplace transforms of the concentrations and is mathematically sophisticated. It is applicable to single electrochemical reactions. The Mirkin–Bard approach is analogous to the treatment of spatially one-dimensional models, discussed in Chaps. 5 and 6 It relies on analytically obtained integral concentration–production rate relationships. The relationships involve integrals over time and space, but one spatial dimension is eliminated. The Cope–Tallman and Mirkin–Bard approaches are applicable when spatial domains are Cartesian products of single intervals. The boundary integral method allows for arbitrarily shaped spatial domains.

The treatment of homogeneous reactions by the integral equation method is generally difficult. One difficulty is that homogeneous reactions cause couplings between concentrations of various species, so that concentration–production rate relationships cannot be determined individually for every species. They must be considered jointly for all species. The simplest situation occurs when the reaction–transport partial differential equations can be decoupled by applying a certain transformation of the concentrations. If the decoupling is not possible, the derivation of the concentration–production rate relationships becomes complicated, and it has been thus far accomplished only for a few simple examples of electroanalytical models. The second difficulty is associated with homogeneous reactions subject to nonlinear kinetic equations, to which one cannot apply the Laplace transformation. Such reactions have been handled, by the integral equation method, under additional assumptions (equilibrium, steady state, the Gerischer linearisation), or by conversion to integro-differential equations.

One-dimensional models of transient electroanalytical experiments, involving simultaneously distributed and localised species, have been frequently handled by the integral equation method. This refers mostly to controlled potential, but also to controlled current experiments. One can distinguish seven procedures of deriving the integral equations or integro-differential equations for such models. The procedures lead to different but equivalent integral or integro-differential equations. There are many literature examples of electroanalytical models that assume reaction schemes involving as many as 1–6 or more reactions between distributed and localised species.

There are electroanalytical models involving additional complications such as the uncompensated Ohmic potential drop, double layer charging, or migration. Models of this kind can be handled by the integral equation method under appropriate assumptions. Relevant integral equations or integro-differential equations can be obtained by appropriate modifications of the integral equations derived in the absence of these additional complications. The assumption of equilibrium charge transfers together with the uncompensated Ohmic drop and/or double layer charging, results in certain inconsistencies of the integral or integro-differential equations obtained. Such an assumption should better be avoided. Migration can only be handled for very simple examples (binary electrolyte), or under additional simplifications, such as, for example, the linearisation of the electric potential profile.

Integral, or integro-differential equations representing electroanalytical models can rarely be solved analytically. However, in cases when analytical solutions are possible, they usually prove useful. Known analytical solution methods involve: solutions in the form of integrals, power series expansions, exponential series expansions, successive approximations, and employment of steady state approximations. Literature examples of the applications of these methods to one-dimensional models are available.

In order to obtain solutions of the integral equations arising in electroanalytical chemistry, numerical methods are often necessary. Such methods are usually based on the general idea of replacing integrals by finite sums. The methods for one-dimensional integral equations can be divided into quadrature methods, product integration methods, and methods based on discrete semi-integration. Of particular interest are product integration methods, such as the step function method and the Huber method, used in conjunction with nonuniform time grids. An adaptive version of the Huber method has also been developed, which generates solutions automatically with a prescribed accuracy. The method based on the Grünwald definition of differintegration proves slightly more accurate than the step function method, but less accurate than the Huber method. In all these methods the computational time increases quadratically with the number of grid nodes used. An alternative, approximate (degenerate) kernel method is known, for which the computational time increases linearly with the number of grid nodes, but the method works for selected kernels only, and its error cannot be fully controlled. Numerical methods applicable to two- and higher-dimensional integral equations are less developed. The Gladwell and Coen method for singular integral equations was used by Cope and Tallman. Mirkin and Bard proposed a quadrature-type method. The boundary integral method typically uses boundary elements for numerical approximation.