Current response for a single redox moiety trapped in a closed generator-collector system: the role of capacitive coupling.
A theoretical model is proposed to describe the steady-state average limiting current associated with a single redox moiety (ox or red) trapped in a closed generator-collector system along with excess supporting electrolyte. By "closed" we mean that neither solvent nor solutes can enter or leave the system. The potential difference, EOE - ERE, between the oxidizing electrode (OE) and the reducing electrode (RE) is maintained constant with the values of EOE and ERE chosen so that the operative faradaic electrode processes are very fast, i.e., red = ox + nETe(-) (kox = ∞) at the OE and ox + nETe(-) = red (kred = ∞) at the RE. Because there is only a single redox moiety the faradaic process occurs at only one electrode at a time while current at the other electrode is purely capacitive (we refer to this as capacitive coupling). We propose that a two-step process is required to transfer nETqe coulombs (qe is the absolute value of the elemental electronic charge). The first step is associated with diffusion (approximated as a random walk) of a single red moiety to the OE where it is oxidized to ox with a concomitant transfer of qstep1 (= nETqe/(1 + AOECOE/ARECRE)) coulombs; the second step is associated with the diffusion (random walk) of the newly formed single ox moiety to the RE with the concomitant transfer of qstep2 (= nETqe/(1 + ARECRE/AOECOE)) coulombs (ARE,AOE andCRE,COEare the areas (cm(2)) and differential capacitances (farads cm(-2)) of the corresponding electrodes). The total charge transferred in the two steps is nETqe(= qstep1 + qstep2). Transport of the redox moiety from one electrode to the other is accomplished by a random walk. The probability density function (pdf) and cumulative density function (CDF) for the duration of a full redox cycle are presented as the analytical solution to a 1-dimensional bounded random-walk problem (confirmed by numerical simulation). These show that tfull, the average time for the full redox cycle (step 1 + step 2), is equal to L(2)/D where L is the intraelectrode distance and D is the diffusion coefficient. The average steady-state limiting current is shown to be described by the familiar expression for a generator-collector system: ilim = (qstep1 + qstep2)/tfull = nETqe/tfull = nETqeD/L(2).
†Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973, United States.
Stephen W
SW
Feldberg
‡Department of Chemistry, University of Utah, 315 South 1400 East, Salt Lake City, Utah 84112, United States.
Martin A
MA
Edwards