Temperature Dependence of Nucleation Rate

Jo and Ng are strongly dependent on the distribution of active sites and their thermally stability. The integration of Eq. (19.8) gives the following equation with the initial condition N = 0 at t = Tq, where Tq is the mean time to build up a critical nucleus (induction time). 

Homogeneous nucleation (sporadic) is often expressed by the following equation proposed by Turnbull and Fisher [60]. 

is generally expressed as Kj= n 7e 7 /4H, . An application to nucleation rate based on Eq. (19.11) causes that a molecular transport term is 

It is interesting to note that the above results might be corresponding to spinodal decomposition during the melt crystallization. When poly(ethylene terephthalate) is quenched down quickly to the setup crystallization temper- 

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The preexponential. A, is related to the movement of atoms or molecules to and from the nucleus. This expression does not account for diffusion effects in transporting atoms across the nucleus-liquid interface. To explain the observed temperature dependence of nucleation rate (Fig. 1), we include an energy barrier for diffusion, Q ... 

Temperature dependence of nucleation rate and of crystal growth rate. 

Fig. 19.12. Temperature dependence of nucleation rate for PESU with M = 9,150 [59]. Open circle is the nominal nucleation rate reduced by the initial view area and open triangle is the real nucleation rate reduced by the real effective area during crystallization. Solid line is the best fitting by Arrhenius expression of the molecular transport term, respectively...

The theory shows that the most important variables affecting the rates at which primary nucleation occur are interfacial energy esurf, temperature T, and supersaturation a. The high-order dependence of nucleation rate on these three variables, especially supersaturation, is important because, as shown by an examination of Eq. (16), a small change in any of the three variables could produce an enormous change in nucleation rate. Such behavior gives rise to the often observed phenomenon of having a clear... 

This is half the value determined for primary nucleation. But, as in primary nucle-ation, the fold period goes as 1/tsT. In other words, it increases with decreasing undercooling (smaller AT). The temperature dependence of the rate of secondary nucleation has a different dependence on temperature, however. 

An expression for the temperature dependence of the rate of secondary nucleation can be obtained in a similar maimer to the procedure used in the treatment of primary nucleation. The expression for Vmin is substituted into the expression for the free energy, in this case the free energy of laying down a stem, AGacm, to obtain AG the free energy of a secondary nucleus that has the critical size. The rate of secondary nucleation can then be obtained from Equation 10-37 ... 

Figure 2 Temperature dependence of nucleation density measured with an ellipsometric monitor. Closed circles and solid line show values for 1000 W microwave power, open, circles and broken line show values for 1400 W microwave power. Other deposition conditions 5 vol.% CO/Hj, flow rate of 100 seem, and pressure of 50 torr. Using CO as reactive gas led to diamond films containing hardly any non-diamond phases.1 1 (Reproduced with permission.)...

Determination of nucleation rate curves by traditional methods is discussed in Chapter 2. These studies are usually very time-consuming and tedious. A faster, easier method for determination of a pseudo-nucleation rate curve was developed by Marotta, et al, based on DSC measurements of nucleated glasses. Their method is also based directly on the JMA equation, with the assumption that the temperature dependence of the rate is given by an Arrhenian equation. The final expression derived using their assumptions indicates that ... 

The temperature dependencies of both nucleation of a new phase and its rate of growth result in a strong temperature dependence of transformation rate. 

Fig. 2.7 show the dependence of nucleation rate and crystal growth on temperature. It is recognized that, on the one hand, high nucleation rates and, on the other hand, an overlap between both curves that is as small as possible are necessary to obtain as many nuclei as possible in the parent glass, on which crystals can later on grow at higher temperatures. 

The previous discussion of this section centered on the temperature dependences of nucleation, growth, and transformation rates. The time dependence of rate (which is often termed the kinetics of a transformation) is also an important consideration, especially in the heat treatment of materials. Also, because many transformations of interest to materials scientists and engineers involve only solid phases, we devote the following discussion to the kinetics of solid-state transformations. 

These studies seem to indicate that, for structureless particles, it is most important to understand the dependence of nucleation rate coefficients on cluster size for very small clusters. At the low temperatures appropriate for argon nucleation, the decay rate coefficient for excited clusters for clusters larger than seven or eight monomers becomes essentially zero, and the capture cross section for this size cluster apparently increases very slowly with n. These facts should make it very easy to compute steady-state nucleation rates for argon, provided similar information is available for the rate coefficients for the "quenching" reactions of equation (2), since it may not be necessary to use trajectories to calculate any of these rate coefficients for clusters larger than ten or twelve monomers in size. 

The growth rate of many crystals is often observed to depend upon temperature in a manner consistent with nucleation theories. If measurements are made on growth from solutions of different concentrations then, at equivalent thicknesses, the dependence of growth rate upon concentration may be determined. Equations (3.16) and (3.17) can be used to predict the concentration dependence of this nucleation approach. 

The temperature dependence of the spreading rate is generally small and can often be neglected against that of the nucleation rate (but see Sect. 3.6.3). 

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