Convolved current

The complex time dependence of the diffusion process to an inlaid disk electrode is described by Eqs. (5.10) and (5.11), which are defined in terms of the dimensionless time parameter 0. It follows that under a specific set of experimental conditions, a unique ratio of Vd / will apply and this ratio can be defined as a single parameter, p. In order to obtain the convolved current, M(t), an initial estimate for p must be inputted into the convolution algorithm. If c is known and Ml can be obtained from the limiting convolved current plateau, then it is possible to iteratively refine p based on Eq. (5.7) for a known value of Tq ... 

A commonly enconntered experimental scenario arises when an inlaid disk electrode is nsed and the electron-transfer kinetics are quasireversible. The implication in this circnmstance is that the convolved current will no longer be directly proportional to the snrface concentration in the vicinity of the standard potential as described by Eq. (5.6), because the current is kinetically limited in this potential region. However, once the diffusion limited plateau is attained, then the convolved cnrrent is once again directly proportional to the surface concentration, and Eq. (5.7), which is the limiting form of Eq. (5.6), is applicable [46, 50],... 

Fig. 5.8 (a) Cyclic voltammogram obtained from the oxidation of 4.3 mM 1 in [C2inim][N(Tf)2] using a 25 pm diameter Pt microdisk electrode with a scan rate of 50 mV s . (b) The convolved current of the voltammogram (forward sweep) shown in (a) before dashed line) and after solid line) background subtraction. Reprinted with permission from Bentley et al., Anal. Chem. 2014, 86, 2073-2081 [15]. Copyright 2014, American Chemical Society... 

Convolution is used in electrochemistry particularly as a means of converting the faradaic current /(f) into a more useful quantity. In this application, /(f) is most often convolved with 1 js/nt, this operation being identical with - semiintegration, but more complicated functions may also be used. See also -> convolution voltammetry, - deconvolution. 

Deconvolution — This operation, as its name implies, means undoing a -> convolution. If a function, say M(t), results from convolving the functions I(t) and g(t), M(t) = I(t) g(t), then I(t) may be regenerated from M(t) by convolution with another function h(t) such that h(t) g(t) = 1. Thus deconvolution is simply convolution with h(t), the complementary function of g(t). The term deconvolution is often used in electrochemistry as a synonym for -> semidifferentiation. The motive is to convert an expression proportional to a concentration difference to an expression proportional to current. 

Current in the channel in subthreshold is a function of charge carrier concentration in the channel. The best subthreshold slope which can be observed in an FET device at room temperature is 60mV/decade, which is the slope at the edge of a Fermi distribution when it is convolved with an abrupt density of states. Because organic semiconductors exhibit a gradual rise in the density of states at the channel edge and not all carriers are equally mobile, the convolution produces a shallower rise in the carrier density and the observed subthreshold slope is worse than this value. 

Ml Mass transport limited convolved/semiintegrated current... 

Fig. 13 (A) Schematic drawing of the dual-plate nano-fluidic device and simulation of the number of randomly diffusing molecules present in the device as a function of time. (B) Simulated current response (consistent with experimental data) generated by convolving the data with a measurement circuit response function and adding background noise. (C) Probability distribution of the occupancy times of a ferrocene molecule in the dual-plate electrode channel. The solid curve shows the probability that a molecule remains in the channel for a given occupancy time. The dashed curve is the probability distribution of occupancy times, with the additional condition that the molecule enters from one side of the channel and exits from the other (average transit time of (length) /6D. The shaded region denotes events too short to be individually resolved by experiment (taken from ref. 69).

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