Modeling of a voltammetric experiment in a limiting diffusion space

Abstract

A voltammetric experiment confined in a limiting diffusion space is analyzed theoretically governed by conventional or time-anomalous factional diffusion under conditions of cyclic and square-wave voltammetry. The solution for conventional diffusion is derived by means of the Jacobi theta function $\Theta \left( {{a^2}/{\pi^2}t} \right)\left( {a = L{D^{ - 1/2}}} \right.$, where L is the thickness of the finite diffusion space, D is the diffusion coefficient, and t is the time of the experiment) and compared with the solution frequently used in the literature expressed in the form Θ(a ⁻² t). For L → ∞, the present solution converges to the one for the semi-infinite diffusion, thus being of a general applicability for both finite and semi-infinite diffusion. Hence, the mathematical model for simulation of both cyclic and square-wave voltammetric experiment provides significant advances in terms of simulation time and accuracy compared to the previous model based on the modified step-function method Mirčeski (J Phys Chem B 108:13719, 2004). For the fractional diffusion experiment, the solution is derived by combining an infinite series and the Wright function $\phi \left( { - \alpha /2,\alpha /2; - 2a{\xi^{ - 1/2}}{t^{ - \alpha /2}}} \right)$, where α is the time fractional parameter ranging over the interval $0 < \alpha < {1}$, and ξ = 1 s^1−α is the auxiliary constant. The voltammetric properties of the experiment controlled by fractional diffusion are comparable for both finite and semi-infinite diffusion.