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

Abstract

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 KolmogorovAvrami 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 j ex ðtÞ=h ex ðtÞ 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 N a ðtÞ vs. time relation (where N a ðtÞ 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 jðtÞ vs. time and N a ðtÞ 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.

Three-dimensional nucleation with diffusion-controlled growth: Simulation of hierarchical diffusion zones overlap

“…For N 0 /A ) 1 nucleation is progressive and as shown in Fig. 7a and b, the currents obtained from simulations of hierarchical overlap approach those predicted by the MLMH conjecture [19], however attaining higher maximum currents and approaching faster the Cottrell limit corresponding to planar diffusion to the entire electrode surface. The SM formulation of the standard model does not consider the spread in birth times of diffusion zones occurring during progressive nucleation, thus the corresponding currents remain invariably below the Cottrell limit at all times.…”

“…In 2004 Matthijs et al (MLMH) [19] considered the ''phantom nuclei" (cf. [10][11][12]) needed to recover the random distribution required for proper application of the KJMA theory to progressive nucleation.…”

“…A and N 0 were found from the SM model solving a system of nonlinear equations using the coordinates of the current maximum [6], while in the case of the HML [17] and HT [15] models the kinetic parameters were obtained from non-linear fitting of the whole data sets to the theoretical equations, using the Levenberg-Marquardt algorithm. Values obtained from the MLMH model have not been included in this particular analysis because the cubic polynomial developed as approximate solution [19] holds only before the current maximum, a time span in general too short for comparison with other models.…”

A complementary survey of staircase voltammetry with metal ion deposition on macroelectrodes

“…(17) and (25), v and q depend on the potential scan rate. However, as we have just seen, the change in these parameters has no significant impact on g 0 ; that is why the zerocrossing is almost common to all curves performed with different scan rates and same DE s .…”

“…Under this condition, the phenomenon can be treated as in chronoamperometry except that the nucleation constant is likely to vary with time. With the conventional wisdom [11][12][13][14][15][16][17], it is assumed that around a growing hemispherical nucleus created at t = s from the beginning of nucleation at t = t 1 , because of the ingestion of active sites, a circular area free of other nuclei develops which surface is proportional to t-s. This area was identified to an equivalent area needed to sustain the growth of the hemispherical nuclei by planar diffusion.…”