Nucleation cluster theory

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]... 

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)... 

Unlike the CNT for homogeneous nucleation, whose theoretical foundation has not been advanced since late 30s, the theoretical formalism of the ion-induced nucleation theory has been recently improved and extended. The critical importance of the dipole moment of condensing monomers and pre-nucleation clusters have been pointed out in the series of recent publications of Nadykto with coauthors [34-39] and, more recently, Leopold with co-authors [69]. The classical ion-induced nucleation theory has been advanced through the incorporation of the effect of the polar host vapour molecule-charged cluster interactions and some of the serious shortcomings of the original model have been successfully corrected [36]. 

The mechanisms of crystal phase formation are a key problem in materials science that has not clear comprehension still now. At present, the study of this problem is especially important in connection with the development of nanostructured materials. There are two different approaches to consideration of crystal nucleation/growth as well as crystal melting/dissolution processes [1,2], In accordance with the first approach based on the atomic-molecular theory, the individual atoms or molecules take the leading part in these processes (the role of clusters is ignored). In accordance with the second approach based on the cluster theory, these processes are carried out mainly by means of clusters. Till recently the atomic-molecular theory was generally accepted. However, today many scientific data vote for the cluster theory. The aim of this paper is to analyze the main statements of the cluster conception of crystal phase formation and as a result to consider the nature of nanocrystal. 

It is interesting to calculate the size of the critical embryo involved in nucleation. From (4.28) at —40 °C the critical radius is 11 3 A, so that the embryo contains about 190 molecules. This result has interesting implications for the various flickering cluster theories of water structure. In order that the metastable supercooled state be maintained for reasonably long times at temperatures a few degrees above the nucleation threshold, it is necessary that... 

Altman, I. S., I. E. Agranovskii, M. Choi and V A. Zagainov (2008) To the theory of homogeneous nucleation Cluster energy. Russian Journal of Physical Chemistry A, Focus on Chemistry 82, 2097-2102... 

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. 

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

In the classic nucleation theory, the free energy of forming a cluster of radius r containing n atoms or molecules is the sum of two terms ... 

The classic nucleation theory is an excellent qualitative foundation for the understanding of nucleation. It is not, however, appropriate to treat small clusters as bulk materials and to ignore the sometimes significant and diffuse interface region. This was pointed out some years ago by Cahn and Hilliard [16] and is reflected in their model for interfacial tension (see Section III-2B). 

The MD simulations provided the necessary thermodynamic information to obtain the equilibrium configurations of the films. Often the deposition process will produce films which are not in the equilibrium configuration, and then the problem is to determine the stablity of these films against changes in morphology. Here simulations can also be helpful, since data on the surface energies and chemical potentials of strained films can be used to calculate the probability of cluster nucleation, using classical nucleation theory. 

The hypothesis was extended to nucleation of hydrates from liquid water. An alternative hypothesis was proposed by Rodger [1516]. The main difference between these two sets of theories is that Rodger s hypothesis relates the initial formation process to the surface of the water, whereas the theory of Sloan and coworkers considers clusters related to soluted hydrate formers in liquid water as the primary start for joining, agglomeration, and crystal growth. The theories of Sloan and coworkers have been discussed and related to elements of the hypothesis proposed by Rodger [1043]. 

Nucleation in a pure liquid. According to the kinetic theory for pure gases and liquids, there are local fluctuations of densities, which are clusters of molecules in a gas and holes (or vapor clusters) in a liquid. Frenkel (1955) established the population distribution of such holes of phase B in a liquid of continuum phase A by Boltzmann s formula,... 

By using the classical theory of ion induced nucleation to describe the growth of radon daughters from the free activity mode to the nucleation mode, we loose information about the size of the subcritical clusters. These clusters are all lumped together between the size of a pure H2O ion cluster at 75% r.h. and the size of the critical H2O-H2SO4 cluster. The model only does keep track of the growth by condensation of the radon daughters once they arrived in the nucleation mode. 

The prediction made by the model calculations should be taken with some care for two reasons 1) H2O and H2SO4 are considered to be the condensing species, whereas other species may be active in experimental or domestic environments 2) the model uses classical nucleation theory, which is the only workable theory, but which is also to be criticized because it applies macroscopic entities to clusters that contain only a few molecules (3). 

Quite recently Raes (1985) applied the classical theory of homogenous nucleation originally developed by Bricard et al (1972) to atmospheres containing SO2, H2O and 218Po ions. Depending on the H2O and SO2 concentrations, ions could grow to a quasi-stable cluster which would evaporate upon electrical neutralization, or to a larger size which would survive neutralization. 

The "classical" theory of nucleation concentrates primarily on calculating the nucleation free energy barrier, AG. Chemical interactions are included under the form of thermodynamic quantities, such as the surface tension. A link with chemistry is made by relating the surface tension to the solubility which provides a kinetic explanation of the Ostwald Step Rule and the often observed disequilibrium conditions in natural systems. Can the chemical model be complemented and expanded by considering specific chemical interactions (surface complex formation) of the components of the cluster with the surface ... 

Nucleation rate based on the classical nucleation theory The nucleation rate is the steady-state production of critical clusters, which equals the rate at which critical clusters are produced (actually the production rate of clusters with critical number of molecules plus 1). The growth rate of a cluster can be obtained from the transition state theory, in which the growth rate is proportional to the concentration of the activated complex that can attach to the cluster. This process requires activation energy. Using this approach, Becker and Coring (1935) obtained the following equation for the nucleation rate ... 

Transient Nucleation If a liquid is cooled continuously, the liquid structure at a given temperature may not be the equilibrium structure at the temperature. Hence, the cluster distribution may not be the steady-state distribution. Depending on the cooling rate, a liquid cooled rapidly from 2000 to 1000 K may have a liquid structure that corresponds to that at 1200 K and would only slowly relax to the structure at 1000 K. Therefore, Equation 4-9 would not be applicable and the transient effect must be taken into account. Nonetheless, in light of the fact that even the steady-state nucleation theory is still inaccurate by many orders of magnitude, transient nucleation is not discussed further. 

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