Ultrathin layer convolution

Ultrathin layers are defined as thin layers which are sufficiently thin that no concentration gradients are established within the layer on the time scale of a voltammetric measurement. Mathematically, ultrathin layers are characterized by ordinary differential equations in time. These equations are simpler to solve than the space and time dependent partial differential equations which describe all other electrode geometries. In this paper, a method is presented which capitalizes on the mathematical simplicity of the ultrathin layer to model any arbitrary, parameterizable electrode geometry. Laplace transforms are used to find an integral relationship between the current response of the modeled geometry and the ultrathin layer current. The integral relationship can be evaluated either analytically or numerically. Any voltammetric perturbation, under either Nernstian or mass transport-controlled conditions, can be modeled. The method is demonstrated for both planar and spherical electrodes. Cyclic voltammetric responses are modeled numerically and potential step responses are modeled analytically. It is also shown that for cyclic voltammetric perturbations, the current-voltage curves for the following systems have the same functional form. That is, the curves are identical within known multiplicative constants. The functionally equivalent responses are for (1) radial diffusion to a point electrode (the polarographic curve), (2) convective transport to a rotating disk, (3) the integral of the ultrathin layer response, and (4) the convolution or semi-integration of the response for linear diffusion to a planar electrode.