The potentiostatic transient for 3D nucleation with diffusion-controlled growth: theory and experiment for progressive nucleation

The theory of the potentiostatic transient for 3D nucleation with diffusion-controlled growth is discussed. It is shown that the theoretical model of Mirkin and Nilov [J. Electroanal. Chem., 283 (1990) 35] and Heerman and Tarallo [J. Electroanal. Chem., 470 (1999) 70] predicts too high values of the current, which becomes very apparent for high values of the site density and low values of the nucleation rate constant (progressive nucleation). For example, the model then predicts that the current in the limit of long times will be higher than the Cottrell limit by a factor of 4/3 which is physically unacceptable. Therefore, a modification to this model is proposed which is based on a careful analysis of the Kolmogorov–Avrami theorem. The “extended area” in the Kolmogorov–Avrami theorem includes contributions from “phantom nuclei” that are born inside already existing zones but do not exist physically. This is necessary to preserve the randomness of the system and allows the correct calculation of the appearance rate of the nuclei and the nucleus saturation density. The “extended current”, defined in analogy with the “extended area”, then also attributes current to the phantom nuclei. It follows that the ratio jex(t)/θex(t) which appears in the model of Mirkin and Nilov and Heerman and Tarallo does not correspond to the actual number of nuclei formed on the electrode. Therefore, the “extended quantities” in this ratio must be replaced with quantities that relate directly to the real number of clusters (this implies what is fairly obvious, that the appearance rate of the clusters must be calculated first). This makes it is possible to derive an equation that predicts correctly the current in the limits of both short and long times which is directly linked to the Na(t) vs. time relation (where Na(t) is the actual number of nuclei on the electrode). Experiments for the nucleation of silver on glassy carbon electrodes, with the simultaneous recording of both j(t) vs. time and Na(t) vs. time relations, are described. The experimental results obtained from the transients and the direct visual counting of nuclei are compared with the theoretical predictions.