Classical theory of nucleation

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 ... 

Classical Theory of Nucleation in a One-Component System without Strain Energy... 

Thus, the work, W(J), to form a hydrate cluster of n building units can be determined using the classical theory of nucleation. 

While the classical theory of nucleation is limited by the implicit assumptions in its derivation, it successfully predicts the nucleation behavior of a system. Inspection of the equation above clearly suggests that the nucleation rate can be experimentally controlled by the following parameters molecular or ionic transport, viscosity, supersaturation, solubility, solid-liquid interfacial tension, and temperature. 

There are several other points of view regarding the definition of the nucleus of zeolite. For example, it was suggested that some primary structural units of the framework, such as rings and basic cages, could be defined as the nucleus of zeolites and other microporous crystals. It was also proposed that the nucleus of zeolite could be defined as particles with critical size. These particles should be stable under crystallization conditions. Compared with the classical theory of nucleation from homogeneous solution, the theory developed by Pope could well explain the significant decrease of the free-energy barrier of nucleation for zeolites and other microporous compounds.[43] This... 

The classical theory of nucleation (Volmer 1939 Nielsen 1964) assumes that clusters are formed in solution by an addition mechanism... 

The classical theory of nucleation, stemming from the work of Gibbs (1948), Volmer (1939), Becker and Doring (1935) and others, is based on the condensation of a vapour to a liquid, and this treatment may be extended to... 

According to the classical theory of nucleation nuclei are bom by the successive addition of units following the formation scheme ... 

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]. 

Fig. 9.4 - A double logarithmic plot of the critical free energy, AGc, as a function of overpotential. The plot illustrates the inverse dependence of on 17 (Equation (9.12a)) predicted by the classical theory of nucleation.

The experimentally determined temperature range of the formation of microvoids in crystals with a large diameter is 1403... 1343 K (Kato et al, 1996 Itsumi, 2002). In this respect, the approximate calculations for the solution in terms of the model of point defect dynamics were performed at temperatures in the range 1403...1073 K. The computational model uses the classical theory of nucleation and formation of stable clusters and, in strict sense, represents the size distribution of clusters (microvoids) reasoning from the time process of their formation and previous history. 

Following the classical theory of nucleation, the Gibbs free energy change (AG) of crystals nucleated from a homogeneous solution can be expressed as (Wu and Nancollas, 1999) ... 

The mentioned circumstances lead to the necessity of inevitable change of the classical theory of nucleation - both in its kinetic and thermodynamic aspects. Such modifications have been made during the last decades. These modifications are analyzed in this and the next chapters. 

Classical theory of nucleation predicts there is a critical radius, or equivalently a critical number i of atoms, below which precipitates are unstable and will re-disolve into the solid solution and above which precipitates will grow, i is obtained by considering the competition between the interface free energy a and the nucleation free energy per atom AG",... 

Figure 7 Evolution of the steady-state nucleation rate J with the nominal concentration for T = 1000 K. Pull and open symbols are respectively for the set of parameters with and without order corrections on triangles and tetrahedrons. The full line corresponds to the nucleation rate predicted by classical theory of nucleation with a a = 64.1 meV and the dotted line with o

We should always bear in mind that the Classical Theory of Nucleation is, in essence, a macroscopic theory. But, at the microscopic level, such a level of description is not adequate. In the end, all observable quantities should be expressed as functions of material properties that are, themselves, unambiguously observable. 

The classical theory of nucleation was developed by Volmer and Weber, and Becker and Doring" for the condensation of a pure vapour to form a liquid. The subsequent theory-- for the liquid-solid transformation was based on this earlier work. The theory considered homogeneous nucleation, ia the formation of one phase by the aggregation of components of another phase without change of composition and without being influenced ly impurities or external surfaces. Impurity particles and external surfaces are taken into account in heterogeneous nucleation theory (Section 2.3). Modifications to the classical theory are necessary to allow for the effects of compositional changes. 

Supersaturation plays an important role in nanoprecipitation, as it determines the nucleation rate of the particles (J). According to the classical theory of nucleation, the nucleation rate of spherical nuclei can be calculated by Eq. 9.2 [31] ... 

Studying the spontaneous appearance of two-dimensional clusters on the electrode surface one obtains direct information on the average time f] needed to form a 2D nucleus at a given overpotential 7. As we have seen in Chapter 3 (equations (3.8) and (3.10)), in the case of negligible non-stationary effects the time /, equals the reciprocal stationary nucleation rate. This has been used by Budevski et al. [4.16, 4.17] to examine the overpotential dependence of the stationary rate of two-dimensional nucleation. The obtained results confirm the validity of the classical theory of nucleation on a like substrate and provide the possibility to determine the nucleation work, the size of the two-dimensional critical nucleus and the specific free edge energy at the nucleus-solution interface boundary. 

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