One of the problems that faced early electrochemists interested in current-potential-time relationships in cyclic voltammetry was the complexity of the diffusion equations. Sevcik [ Coll Czech Chem Comm, 13 (1948) 349 ] derived a series approximation for the current-potential curve in CV, but cyclic voltammetry got a big boost as a mechanistic tool from the landmark series of publications by Nicholson and Shain [ Anal Chem, 36 (1964) 706 ].
Nicholson solved the x-dependence of the diffusion equation via Laplace Transforms but was then left with an integral equation for current vs. E. He numerically integrated this equation for various values of sweep rate and kinetic parameters and published ‘working curves’ for others to use without having to repeat the (then) tedious calculations.
At about the same time, Feldberg [ Anal Chem, 36 (1964) 505 ] began using direct numerical solution of the diffusion equations by finite difference to solve electrochemical mechanisms for chronopotentiometry, and then for cyclic voltammetry. A big advantage of solving the diffusion equation numerically is that even mathematically intractable mechanisms could be rapidly programmed and simulated. This work has culminated in the commercial program DigiSim® [ Anal Chem, 66 (1994) 589A ], which is available through BAS. Nearly any electrochemical mechanism can be input. The software then executes the numerical solution for the proper set of diffusion equations. The parameters for the cyclic voltammetry experiment must also be entered. It is possible to use the DigiSim software to do a least squares fit to experimental data to obtain kinetic parameters.
DigiSim simulates cyclic voltammetry done with a smooth analog ramp, or a very close approximation to it. Often, particularly at higher sweep rates, is more convenient to do Cyclic Staircase Voltammetry (CSV). In CSV, the voltage change is a step from data point to data point rather than a smooth ramp to the next data point’s voltage. The current is conveniently measured at the end of each step, just before the voltage is changed. Neither the working curves of Nicholson and Shain, nor the simulations of DigiSim, strictly apply in this case. However, a free program is available which can numerically solve a CSV experiment and obtain kinetic parameters using the COOL algorithm [ J.Phys.Chem., 90 (1986) 2761 ]. This program also numerically solves a square wave voltammetry (SWV) or a chronoamperometry (CA) experiment.
Further information about digital simulation in electrochemistry can be found in Vol 3 and Vol 19 of the Bard series, and in several texts in the Bookstore.