Analytical solutions for the diffusive mass transfer at cylindrical and hollow-cylindrical electrodes with reflective and transmissive boundary conditions

Abstract

• Unified theory for the diffusive mass transfer at cylindrical electrodes. • Analytical solutions are infinite series involving Bessel functions. • Semi-Infinite diffusion results in improper integral representations. • Analytical solutions confirm numerical results obtained from Talbot’s NILT method. The theoretical treatment of cyclic voltammetry or chronoamperometry at cylindrical electrodes by means of convolutive modeling requires an a priori knowledge of the time-dependent mass transfer functions and of the diffusive flux of the electrochemically active species. In this paper, analytical solutions for both of these quantities are derived for cylindrical and hollow-cylindrical electrodes with reflective and transmissive boundaries by means of Laplace transformation techniques. Furthermore, explicit equations for the concentration profiles of a Cottrellian potential step experiment are provided. All of these expressions are given in terms of infinite series involving Bessel functions of the first and of the second kind. It is demonstrated that the summation of only a few terms of these infinite series is usually sufficient to accurately compute the desired time-dependent mass-transfer function or the current. This renders the numerical inversion of Laplace transformations, utilized so far, obsolete and represents a mathematical supplement to the recent theory.