Electroanalytical Chemistry and lnterfacial Electrochemistry, 43 (1973) 1-8 1 © Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands AN IMPLICIT FINITE DIFFERENCE METHOD SIMULATION OF SPECTRO-ELECTROCHEMICAL WORKING CURVES NICHOLAS WINOGRAD Department of Chemistry, Purdue University, West Lafayette, Ind. 47907 (U.S.A.) (Received July 31st, 1972) The process of digitally simulating concentration profiles for electrogenerated species has greatly expanded, in recent years, the number of chemical systems amenable to quantitative kinetic analyses. In several reviews 1' 2, details necessary to apply simulation to a variety of electrochemical problems have been developed. Procedures for setting up initial and boundary conditions as well as correcting for hydrodynamics and unusual electrode geometries are now available. Imple- mentation of digital simulation is within the grasp of any electrochemist who has access to a computer. Although a multitude of numerical methods exist for obtaining solutions to partial differential equations 3, the number of different approaches which have been applied to electrochemical problems have been limited. Aside from several attempts to obtain approximate analytical solutions4 and a single paper dealing with a modified Crank-Nicolson finite difference method 5, the Feldberg approach a has clearly been most widely employed. In this paper the concept of digitally simulating complicated diffusion~inetic partial differential equations will be extended to include an implicit method of finite difference 3' 6. The major feature of thig approach is that due to much greater stability characteristics, results identical to the Feldberg treatment are obtained in one to two orders of magnitude less computer time. This increased efficiency is particularly significant with the new common usage of laboratory computers and with the spread of simulation techniques to more and more complicated mechanistic problems where computer time is extensive7. Results of the implicit finite difference scheme will be illustrated for spectro-electrochemical investigations although extension to other types of electrochemical problems can be accomplished by using previously described operations 1. FORMULATION OF THE MODEL The heart of applying any numerical analysis to solving the diffusion equation is based on determining the concentration profiles as a function of distance, x, and time, t. To introduce the implicit finite difference approach, the model will first be developed for the simple diffusion equation for the concentration of species A as follows: = DA CA x, , /0x 2 (1) where DA is the diffusion coefficient of species A. Initial and boundary conditions