Nucleation general theory

The general theory of phase transition by crystallisation was developed by Gibbs, and later extended by Becker and Doring (1935), Avrami (1939/1941), Turnbull and Fisher (1949) and Hoffman et al. (1958/1966). The theory is based on the assumption that in super-cooled melts there occur fluctuations leading to the formation of a new phase. The phase transformation begins with the appearance of a number of very small particles of the new phase (nucleation). 

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]... 

At the beginning of the chapter it is shown that the usual models for coagulation and nucleation presented in Chapters 7 and 10 arc special cases of a more general theory for very small particles. An approximate criterion is given for determining whether nucleation or coagulation is rate controlling at the molecular level. The continuous form of the GDE is then used to derive balance equations for several moments of the size distribution function. 

The general theory of nucleation and polymerization in aqueous systems, in which silica shows some solubility, is discussed in detail in Iler s book (3). However, very little was known at the time the book was published (1979) about the polymerization of silica when Si(OH)4 is formed in nonaqueous systems. Progress made up to 1990 in the understanding of the hydrolysis and condensation of silicon alkoxides that leads to silica gels or to silica sols of large particle diameter are lucidly discussed by Brinker and Scherer (8). Brinker s chapter in this book (Chapter 18) includes a clear explanation of the difference between hydrolysis and condensation of aqueous silicates and silicon alkoxides. 

Secondary nucleation is a complex phenomena and is not well understood. A general theory for the prediction of nucleation rates does not exist. Several correlations based on the power law model have been found to explain most of the experimental data satisfactorily. The power law (from the Becker-Doering relationship) is given by... 

There is no general theory of decomposition reactions. However, a generalized a versus time curve similar to that shown in Fig. 2.9 is often observed (22). The stage A is an initial reaction, sometimes associated with the decomposition of impurities or unstable superficial material. B is an induction period that is usually regarded as terminated by the development of stable nuclei, while C is the acceleratory period of growth of such nuclei, perhaps accompanied by further nucleation, which extends to the maximum rate of reaction at D. Thereafter, the... 

Zanotto E.D. and Galhardi A., "Experimental Test of the General Theory of Transformation Kinetics Homogeneous Nucleation in a Na20-2Ca0 3Si02 Glass," J. Non-Cryst. Solids, 104, 73-80 (1988). 

There is a close link between the particular crystallization mode and the general resolution of the T-T-T or C-Tdiagrams. The time needed to crystallize a certain fraction, a, at a given temperature, T, can be conveniently obtained by the DTA/DSC measurements, see Chapter 10. On the basis of nucleation-growth theories the highest temperature to which the kinetic equation can reasonably well reproduce the T-T-T or C-T curve was estimated to be about 0.6 Tm, . There are certainly many different methods used for numerical modeling [374,380, 385], which detailed analysis [6,375] falls, however, beyond the scope of this chapter. 

The similarity of the processes of ferroelectric domain reversal to many solid-solid phase change phenomena is considerable and yet not generally recognized. Moreover, the ferroelectric analog is relatively simple and as such appeals as a potentially suitable system in which to test general theories of nucleation and growth. 

The central quantity of interest in homogeneous nucleation is the nucleation rate J, which gives the number of droplets nucleated per unit volume per unit time for a given supersaturation. The free energy barrier is the dommant factor in detenuining J J depends on it exponentially. Thus, a small difference in the different model predictions for the barrier can lead to orders of magnitude differences in J. Similarly, experimental measurements of J are sensitive to the purity of the sample and to experimental conditions such as temperature. In modem field theories, J has a general fonu... 

Emulsion Polymerization. Emulsion SBR was commercialised and produced in quantity while the theory of the mechanism was being debated. Harkins was among the earliest researchers to describe the mechanism (16) others were Mark (17) and Elory (18). The theory of emulsion polymerisation kinetics by Smith and Ewart is still vaUd, for the most part, within the framework of monomers of limited solubiUty (19). There is general agreement in the modem theory of emulsion polymerisation that the process proceeds in three distinct phases, as elucidated by Harkins (20) nucleation (initiation), growth (propagation), and completion (termination). 

Section 3.2 is an introduction to general nucleation behaviour as first used for low molecular weight materials, that is, non-polymeric substances. This topic is frequently not covered in polymer papers but the extensive work done in this area provides the framework for the theory, and indicates the limitations and problems to be expected in predicting the physical behaviour. 

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