Nucleation rate prediction

A difiiculty with this mechanism is the small nucleation rate predicted (1). Surfaces of a crystal with low vapor pressure have very few clusters and two-dimensional nucleation is almost impossible. Indeed, dislocation-free crystals can often remain in a metastable equilibrium with a supersaturated vapor for long periods of time. Nucleation can be induced by resorting to a vapor with a very large supersaturation, but this often has undesirable side effects. Instabilities in the interface shape result in a degradation of the quality and uniformity of crystalline material. 

The extraordinary dependence of the H2S04-H20 nucleation rate on temperature, relative humidity, and H2S04 concentration means that the inevitable uncertainties in atmospheric measurements produce as much uncertainty in nucleation rate predictions as do potential uncertainties in the theory. This steep variation, although a liability for predicting the nucleation rate, is an asset for estimating atmospheric conditions when... 

Table 2. The Correction Factors TlTe, S) to the Nucleation Rates Predicted by Classical Nuclea-tion Theory as Calculated Purely Microscopically by Burton ...

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

As a follow-up to Problem 2, the observed nucleation rate for mercury vapor at 400 K is 1000-fold less than predicted by Eq. IX-9. The effect may be attributed to a lowered surface tension of the critical nuclei involved. Calculate this surface tension. 

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... 

Unfortunately, both primary nucleation parameters cannot be predicted a priori as yet and in practice the nucleation rate must be measured and correlated empirically for each system. 

The predicted transient supersaturation levels (and corresponding nucleation rates) are also shown in Figure 7.3. These considerations predict that the high levels in the early period of natural cooling can be avoided by controlled cooling. 

The molecular weight (M) dependence of the steady (stationary) primary nucleation rate (I) of polymers has been an important unresolved problem. The purpose of this section is to present a power law of molecular weight of I of PE, I oc M-H, where H is a constant which depends on materials and phases [20,33,34]. It will be shown that the self-diffusion process of chain molecules controls the Mn dependence of I, while the critical nucleation process does not. It will be concluded that a topological process, such as chain sliding diffusion and entanglement, assumes the most important role in nucleation mechanisms of polymers, as was predicted in the chain sliding diffusion theory of Hikosaka [14,15]. 

Thus, the source terms for each environment S(c) and Sk ((/)) will be closed. Of particular interest are the local nucleation rates /(c ). As discussed in Wang and Fox (2004), due to poor micromixing the local nucleation rates can be much larger than those predicted by the average concentrations /((c)). This results in a rapid increase in the local particle number density mo due to the creation of a very large number of nuclei. As discussed below, this will have significant consequences on the local rate of aggregation. 

As discussed in section 2.4.4 the coordinating ability of a solvent will often affect the rate of nucleation and crystal growth differently between two polymorphs. This can be used as an effective means of process control and information on solvent effects can often be obtained from polymorph screening experiments. There are no theoretical methods available at the present time which accurately predict the effect of solvents on nucleation rates in the industrial environment. 

The prediction of transformation diagrams after Bhadeshia (1982). Later work by Bhadeshia (1982) noted that the approach of Kirkaldy et al. (1978) could not predict the appearance of the bay in the experimentally observed TTT diagrams of many steels, and he proposed that the onset of transformation was governed by nucleation. He considered that the time period before the onset of a detectable amount of isodiermal transformation, r, could be reasonably defined as the incubation period, r necessary to establish a steady-state nucleation rate. The following expression for r, was then utilised... 

It is remarkable that the predictions of classical nucleation theory without any consideration of polymer connectivity are borne out in experiments. At higher supercooling, deviations are expected because of temperature dependence of the nucleation rate prefactor. 

Heterogeneous reactions that require nucleation. Quantitative prediction of the rates of these reactions is not available because nucleation has not been quantified well. Examples include the following. 

The above qualitative predictions are consistent with experiments. However, in terms of absolute nucleation rate. Equation 4-9 usually predicts too low a rate by many orders of magnitude (see below). 

Failure of the Classical Nucleation Theory There are several suggested explanations for the failure of the classical nucleation theory to quantitatively predict the nucleation rate, including the following ... 

Equation 7 shows the exponential variation of the homogeneous nucleation rate with an increasing supersaturation ratio. Equation 8, coming from the model of Strickland-Constable [59], establishes that a linear relation exists between growth rate and supersaturation ratio. The direct comparison of Nr and Gr is difficult because of their different units. However, Fig. 7, although only schematic, shows the variation of these two rates versus S, predicting that for a certain value of S the nucleation rate will become preponderant on the growth rate. Moreover, it also shows, that, in the case of preponderance... 

Other computer simulations were made to test the classical theory. Recently, Ford and Vehkamaki, through a Monte-Carlo simulation, have identified fhe critical clusters (clusters of such a size that growth and decay probabilities become equal) [66]. The size and internal energy of the critical cluster, for different values of temperature and chemical potential, were used, together with nucleation theorems [66,67], to predict the behaviour of the nucleation rate as a function of these parameters. The plots for (i) the critical size as a function of chemical potential, (ii) the nucleation rate as a function of chemical potential and (iii) the nucleation rate as a function of temperature, suitably fit the predictions of classical theory [66]. 

---