Nucleation atomistic theory

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)... 

Atomistic theory of nucleation — The theory applies to very small clusters, the size n of which is a discrete variable and the process of nucleus formation must be described by means of atomistic considerations. Thus, the thermodynamic barrier AG ( ) that has to be overcome in order to form an n-atomic nucleus of the new phase is given by the general formula [i-v]... 

Under the specific conditions of electrochemical metal deposition, the critically sized clusters of the new phase have been found to consist of only a few atoms, where classical thermodynamic bulk quantities cannot be applied. Therefore, the original kinetic theory of Becker and Doering was further developed to an atomistic theory of nucleation. 

The atomistic theory becomes of additional significance for the transition from 2D Me phase formation in the UPD range to 3D Me phase formation in the OPD range. Experimental results obtained using modern in situ techniques with lateral atomic resolution showed that the transition phenomena can only be interpreted on the basis of atomistic approaches. The UPD surface modification turns out to be a more general phenomenon affecting not only the nucleation processes but also the growth mode and epitaxy of 3D metal phases. 

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. 

Stoyanov [6] reexamined Walton s atomistic theory of heterogeneous nucleation and derived the nucleation rate avoiding... 

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). 

Three-dimensional nucleation is still in a very early stage of development. Significant progress has, however, been evidently made by the introduction of the atomistic theory, but we are obviously still far from an adequate understanding of the processes of nucleation, cluster orientation, grain growth interaction, dendrite formation, and properties of bulk deposits. 

Milchev, A., Stoyanov, S. and Kaischew, R. (1974) Atomistic theory of electrolytic nucleation I. Thin Solid Films, 22, 255-265. 

Stoyanov, S. (1973) On the atomistic theory of nucleation rate. Thin Solid Films, 18, 91—98. 

In Chapter 1.3.3 we have already defined the frequencies of direct attachment and detachment, W+i and W.i, of single atoms to and from the /th site of the crystal surface in the case of electrocrystaUization. In equation (2.9) the quantities co+ and fii represent the frequencies of direct attachment and detachment of single atoms to and from an n-atomic cluster and were defined by Milchev, Stoyanov and Kaischew in the framework of the atomistic theory of electrochemical nucleation [2.10-2.12], Here we shall clarify the meaning of this definition as follows. 

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. 

The readers who are already acquainted with Chapter 2.1.2 should have a fairly good idea of the physical significance of equation (2.80). However, in the early seventies the exact meaning of this expression was still obscure. Though, one thing was clear an atomistic expression for the stationaiy nucleation rate could be obtained in any particular case of phase formation if the frequencies 6>+ and o). were presented as functions of the supersaturation A/i. The first result ofthis finding was the atomistic theory of electrochemical nucleation developed by Milchev, Stoyanov and Kaischew in 1974 [2.10-2.12] (see also [2.5, 2.62-2.65]). The next Section presents the basic theoretical results obtained by these authors. 

Nowadays it is quite clear that it does not make sense to speak about two different theories of the nucleation rate the classical and the atomistic one. In reality, what we do have is a general nucleation theory comprising two limiting cases. The classical model describes the nuclei by means of macroscopic physical quantities and can be used to predict the size and to evaluate the nucleation work of sufficiently large critical clusters. The atomistic model is valid in the case of high supersaturations and very active substrates when the critical nuclei are very small. Therefore the quantitative interpretation of experimental data on the stationary nucleation rate based on the atomistic theory provides valuable information on the specific properties of clusters consisting of a few building units. 

Figure 2.7 Data for the stationary nucleation rate of silver on platinum from Figure 2.4 plotted in the coordinates of the atomistic theory of electrochemical nucleation [2.63].(With...

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). 

Thin-film formation is described as a sequential process which includes nucleation, coalescence and subsequent thickness growth, whereby all states can be influenced by deposition parameters, such as temperature, pressure, gas flow rate, etc. [3,4], For physical vapour deposition (PVD) processes, significant works have been published and progess made in understanding the microstructure evolution of the films. In the atomistics of growth processes, there exists much in common bewteen CVD and PVD. Theories from PVD processes can thus be used to analyse the microstructure evolution of CVD processes [5, 6],... 

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]. 

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... 

Both aerosol modeling and more fundamental atomistic and molecular level models have been applied to this problem. Aerosol dynamics modeling has lead to a better understanding of the individual steps that comprise the formation of particles, all the way from nucleation to subsequent growth. Both molecualar orbital and reaction rate theory was used as sources of fundamental data for input to the aerosol dynamics model. A simplistic molecular dynamics computation has been used to explain the particle morphology observed. 

Formulating a criterion to discriminate between brittle and ductile response based on atomistic features of the solid has been a long sought after goal in the mechanics of solids. At the phenomenological level, theories have been developed that characterize the two types of behavior in terms of cleavage of the crystal or the nucleation and motion of dislocations[138, 139], We review here the basic elements of these notions. 

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