Laws of nucleation

Several laws of nucleation have been developed [28—31] on the assumption that local fluctuations of energy of the crystal at preferred sites are sufficient to overcome the barriers to the production of a stable particle of product. The following kinetic representations have been discussed. 

In principle, there is also the possibility that concurrent reactions, with different kinetics, may proceed at more than one type of site or at different crystal faces. 

Single-step nucleation, (ii) above, requires the unsatisfactory assumption that the generation of a single molecule (atom, ion-pair, etc.) of product constitutes the establishment of a nucleus. (It would seem to be more realistic to regard this as the outcome of several distinct chemical steps.) The mathematical treatment expressing the probability of the occurrence of this unimolecular process is 

When AGn is large, ft is small. The exponential may be expanded, as far as the cubic term, viz. 

It is difficult to envisage what is meant by the individual interactions which together constitute the several steps required in nucleation under 

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Instantaneous nucleation occurs for this model when l< is large, so that all possible nuclei are formed at the onset of reaction // = at virtually zero time and no further nuclei are generated during the subsequent reaction. When is small, the rate of nucleation is approximately constant as reaction proceeds, because the number of sites, N - N), undergoes little change. This is known as the linear law of nucleation-. 

This results in an acceleratory nucleation process known as the power law of nucleation of the form ... 

Single and multiple potential step experiments demonstrated that the macrokinetics of the formation of the phy-sisorbed uracil film represents a first-order phase transition and follows the exponential law of nucleation (cf. Eq. (34)) in combination with surface diffusion-controlled growth [183]. In situ STM [20, 478, 479] and time-resolved SEIRAS studies [475] suggest that these processes are strongly related to the formation/breaking of uracil-water and water-water hydrogen bonds within the Helmholtz region. 

This equation is a convolution integral that is composed of a law of nucleation ... 

N denotes the number of active (growing) nuclei. The time y represents the time the nucleus got activated. The exponent m gives the dimension of nuclei growth. The law of nucleation can be postulated in various ways, such as unimolecular decay law. The left-hand side of the equation origins from Avrami s treatment for the nuclei overly. It gives the relation between the extended rate of conversion and the true rate of conversion. The pre-exponential coefficient includes several constants grouped together. 

Bagdassarian s model for the formation of growth nuclei by product accumulation in successive steps is thus capable of explaining the general power law of nucleation... 

Nucleation of a solid phase starting from another solid phase, the first inevitable stage, plays a part that is very important but is very difficult to study. General information of the laws of nucleation, which were first established for the condensation of a vapor, but for which we can observe that they also apply to crystallization starting from the liquid solutions or the manufacture of a solid from another one, justifies the position of this chapter. We can often argue on the case of the appearance of the solid phase starting from a fluid and make the transpositions with the case, which interests us here, of the nucleation of a solid starting from another sohd. 

Ultimately, if we stick to the simple laws of nucleation, we will be satisfied with the law at constant specific frequency, the exponential law of Avrami [8.64], or possibly with the law power [8.55] or [8.70]. We will study in Chapter 11 heterogenous reactions with surface nucleation using a constant specific frequency. In Appendix A. 9, we will use the various laws to discuss reactions with nucleation in the bulk. 

Remark.- It will be noted that the last two laws of nucleation selected, and in particular [A.9.10], consider that nucleation occurs far from equilibrium since they assume that the opposite reaction of the rate-determining step is negligible. On the other hand, equation [A.9.9] can be used for conditions close to equilibrium by... 

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