Semioperations and Convolutions in Voltammetry

Abstract

It is often useful to transform a measured signal into a form that is more easily interpreted. Thus, in voltammetry, where the measurement is of a current responding to a perturbation of the electrode potential, the complications arising from diffusional transport may be readily alleviated through the use of semioperators or convolutions. The use of semiintegration enables the voltammetric current to be transformed into an alternative signal that, in the case of a planar electrode, is linearly related to the concentrations of reactant and product at the electrode surface. Numerous advantages of semiintegration have been widely demonstrated. However, semiintegration is but a special case of the more general procedure of convolution. Convolving the voltammetric current with specifically designed functions of time allows the benefits of semiintegration to be applied to other electrode geometries and other experimental circumstances. The prime motive for semiintegration or convolution is to gain access to information about concentrations at the electrode surface. This information opens the door to the measurement of such quantities as bulk concentrations, rate constants, standard potentials, and diffusivities, as well as suggesting mechanisms and enabling the calibration of electrodes.

Advantage of Fractional Calculus Based Hybrid‐Theoretical‐Computational‐Experimental Approach for Alternating Current Voltammetry

“…This special instrumentation of Fourier transformed ac voltammetry has been used to understand many electrochemical phenomena [26][27][28][29][30][31][32][33][34][35]. In parallel, several other mathematical approaches have been initiated with some basic assumptions to explore ac voltammetry by series functions [10], Bessel functions [36][37][38], convolutions [39] and semi-integrals [40][41][42][43][44][45][46][47][48][49][50][51][52]. However, the popularity of ac voltammetry has not met the levels of CV and EIS due to the comparatively complex theoretical modeling and analysis of the experimental data.…”

“…In voltammetry, i t ð Þ is proportional to the gradient (J x¼0;t ) of the concentration of analyte at electrode surface (x = 0) at time t, but actually i t ð Þ holds the history of variation of J x¼0;t with t. Therefore, i t ð Þ should be represented by the fractional calculus of a function related to the concentration of analyte at the interface. The idea of fractional calculus in voltammetry was introduced in almost similar times in different names by Oldham's group as semidifferintegration where 'semi' denotes ' 1 2 ' [39,42,45,49,51,52] and Saveant's group as convolution integrals [26,[65][66][67][68]. The differintegration term represents a combined differentiation and integration operation frequently encountered in the fractional calculus [55].…”