Atomistic critical nucleus

Nucleation — Atomistic theory of nucleation — Figure 1. Dependence of the nucleation work AG (ft) on the cluster size n (a) and dependence of the critical nucleus size nc on the supersaturation Ap (b) according to the atomistic nucleation theory (a schematic representation)... 

Apart from the purely thermodynamic analysis, the description of the -> electro crystallization phenomena requires special consideration of the kinetics of nucleus formation [i-v]. Accounting for the discrete character of the clusters size alteration at small dimensions the atomistic nucleation theory shows that the super saturation dependence of the stationary nucleation rate /0 is a broken straight line (Figure 2) representing the intervals of Ap within which different clusters play the role of critical nuclei. Thus, [Ap, Apn is the supersaturation interval within which the nc -atomic cluster is the critical nucleus formed with a maximal thermodynamic work AG (nc). 

Classical nucleation theory may be not well suited to describe the nucleation kinetics of diamond in CVD, since the critical nucleus size under the typical CVD conditions may be on the order of a few atoms. The surface energy contribution may cause a reverse effect on the phase stability and the GFobs free-energy of the formation of a critical nucleus may be negative, a case referred to as nonclassical nucleation. In such a case, atomistic theory should be employed as the starting point of theoretical analyses. 

Classical nucleation theory uses macroscopic properties characteristic of bulk phases, like free energies and surface tensions, for the description of small clusters These macroscopic concepts may lack physical significance for typical nucleus sizes of often a few atoms as found from experimental studies of heterogeneous nucleation. This has prompted the development of microscopic models of the kinetics of nucleation in terms of atomic interactions, attachment and detachment frequencies to clusters composed of a few atoms and with different structural configurations, as part of a general nucleation theory based on the steady state nucleation model [6]. The size of the critical nucleus follows straightforwardly in the atomistic description from the logarithmic relation between the steady state nucleation rate and the overpotential. It has been shown that at small supersaturations, the atomistic description corresponds to that of the classical theory of nucleation [7]. 

According to Eq. (45), the atomistic theory predicts a linear dependence of the nucleation rate with overpotential for constant size of the critical nucleus, as shown in Fig. 7. Also, in accordance with Eq. (45), the value of the slope doubles for mercury deposition from Hg(II) as compared with deposition from Hg(I). 

With increasing overpotentials, the number of atoms, iVc, constituting the critical nucleus becomes reduced dramatically attaining values of the order of several atoms. Macroscopic quantities, such as volume, surface, surface energies, etc. lose their physical meaning in such cases and the use of atomic forces of interaction becomes more reasonable. The atomistic approach for the calculation of the dependence of nucleation rate on supersaturation was first used by Walton, and then developed later to a general nucleation theory by Stoyanov et... 

In the same time, the condition co. loXri< means that a single atomjoins the 3-atomic cluster before its disintegration and the configuration in Figure 3c, in which each atom is connected with three bonds is relatively stable at the temperature T. In terms of the atomistic model the 2-atomic cluster is defined as a critical nucleus, the 3-atomic one is defined as the smallest stable cluster and the stationary nucleation rate is ... 

The difference between the two models, as far as such difference does exist, should be sought for in the supersaturation dependence of the critical nucleus size. In the classical model this dependence is determined by the Gibbs-Thomson equation which juxtaposes a different critical cluster to each supersaturation. An analytical expression is proposed also for the supersaturation dependence of the stationary nucleation rate. The atomistic model takes into account the discrete character of the clusters size alteration at small dimensions (see Chapter 1.4) and does not propose a simple analytical relation between c and Ap. Beside that, it accounts for the fact that a supersaturation interval and not a fixed supersaturation corresponds to each critical cluster. This changes also the shape of the supersaturation dependence of the stationary nucleation rate which, in coordinates In/ / vs. Ap, is a broken straight line (Figure 2.6) representing the intervals of... 

For a small number of Me atoms in a critical cluster, the strain of a 2D UPD Meads overlayer can be inherited by the nucleus. Using the atomistic approach [4.13], the rate of nucleation on top of a strongly compressed and internally strained 2D UPD Meads overlayer can be expressed by [4.54-4.57] (cf. Section 4.2) ... 

The striking quantitative contradiction between the classical nucleation theory and the experimental data accompanies the studies of electrochemical nucleation since the time of Thomfor and Volmer [2.29] who were the first to obtain a surprisingly low value for the size of a critical mercury nucleus on a platinum substrate. The problem has been successfully solved by the atomistic theory of the nucleation rate [2.10-2.12, 2.33, 2.62-2.66], which answers the question how to interpret the experimental data on electrochemical nucleation The next Section contains a survey of these theoretical considerations. 

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