STOCHASTIC APPROACH TO NUCLEATION

Stochastic approach to nucleation — Basic stochastic properties of assemblies of clusters randomly distributed in space or appearing in time as a nonstationary flux of random independent events can be examined in terms of the Poisson theory [i]. See -> temporal distribution of clusters - spatial distribution of clusters. 

Nucleation — Stochastic approach to nucleation — Spatial distribution of clusters — Figure. Experimental (histograms) and theoretical (lines) distribution of the distances between first (a), second (b) and third (c) neighbor silver crystals electro chemically deposited on a glassy carbon electrode [iii-v]... 

In the absence of a fully deterministic understanding of pit initiation and pit propagation, pitting can be conveniently described by a statistical (stochastic) approach. The stochastic approach of pitting was first proposed by Shibata and Takeyama (1977), and extended by Stewart and Williams (1992), Williams et al. (1994), and Baroux (1988, 1995). Metastable pits are generally assumed to nucleate in a random manner, with a certain probability A per unit of time and area. The probability of formation of stable pits A (per unit of time and area) is then a function of A... 

The evaluation of the actual volume or the actual surface area occupied by growing clusters or by nucleation exclusion zones, respectively is a most important point in the kinetic theory of the first order phase transitions. Several authors [5.11-5.17] have considered the problem and among them the name of Avrami [5.14-5.16] seems to be the most popular one. However, the stochastic approach proposed by Kolmogoroff [5.12] is undoubtedly the most rigorous one (see e.g. the critical analysis of Belen ky [5.18]) and here we describe this approach. 

The incorporation of discreet nucleation events into models for the current density has been reviewed by Scharifker et al. [111]. The current density is found by integrating the current over a large number of nucleation sites whose distribution and growth rates depend on the electrochemical potential field and the substrate properties. The process is non-local because the presence of one nucleus affects the controlling field and influences production or growth of other nuclei. It is deterministic because microscopic variables such as the density of nuclei and their rate of formation are incorporated as parameters rather than stochastic variables. Various approaches have been taken to determine the macroscopic current density to overlapping diffusion fields of distributed nuclei under potentiostatic control. 

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

The crossover of the nucleation barrier is a stochastic process, which can be treated by considering the fluctuations in the system. It is, therefore, mandatory to study the phase separation of nanoparticles by statistical approaches. 

FIGURE 3.9 Approaches used to find the solution of the active site model of surface activity. The main distinction is made based on surface mobility. For the general case, the full interplay between on-site reactivity and extremely low COad surface diffusivity unfolds. All processes, including nucleation of active sites (rate constant kj ), forward and reverse rates of OH d formation (kf, kb), surface diffusion of COad (kdiff), and oxidative removal of COad (kox), are important for the overall kinetics. The solution for the general case, demands kinetic Monte Carlo simulations, where evolution of the system is described stochastically and positions of adsorbed COad and are relevant. Modeling is substantially simplified in the limit of... 

Two approaches can be used to model crystallization kinetics of triglycerides and fat. If the microscopic parameters can be determined, the use of microscopic models is the most appropriate, because it applies directly the theory of nucleation and growth. For example, in the case of spherulitic crystallization, kinetic parameters can be determined experimentally. Solidification can then be modeled in a detailed way with a numerical or stochastic model for the nucleation and growth of crystals. The latter kind of microscopic model is very interesting because it also gives the stereological parameters of the microstructure. Probabilistic or numerical models are easier to use, but they provide only the evolution of the latent heat or the evolution of the solid fraction in the sample. 

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