Nucleus, critical size

The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

The models incorporate two microscopic parameters, the site density and the critical nucleus size. A fit of experimental current transients to the models allows conclusions, for example, concerning the effect of additives on nucleation rate. Fabricus et al. found by analysis of current transients that thiourea increases the nucleation density of copper deposited on glassy carbon at low concentration, but decreases it at higher concentration [112], Schmidt et al. found that Gold nucleation on pyrolytic graphite is limited by the availability of nucleation sites [113], Nucleation density and rate were found to depend on applied potential as was the critical nucleus size. Depending on concentration, critical nuclei as small as one atom have been estimated from current transient measurements. Michailova et al. found a critical nucleus of 11 atoms for copper nucleation on platinum [114], These numbers are typical, and they are comparable to the thermodynamic critical radii [86],... 

This may continue until eventually the cluster is large enough to be thermodynamically stable (i.e., will not redissolve). However, if the cluster is smaller than the critical nucleus size, then there is the possibility that the nucleus will redissolve. The lifetime of the nucleus will then depend on its size and also on the temperature lower temperatures will slow the redissolution step. Thus lower temperature increases the chance that a subcritical nucleus will eventually grow to a stable size rather than redissolve. This kinetic stabilization of small nuclei results in a greater total density of nuclei and therefore smaller crystal size, since the total quantities of reactants are fixed. 

The free energy barrier AF and the critical nucleus size r are given by [4]... 

The definitive hydrate kinetic inhibition mechanism is not yet available. Some work suggests that the mechanism is to prevent hydrate nucleation (Kelland, 2006). However, a significant amount of evidence suggests that hydrate kinetic inhibitors inhibit the growth (Larsen et al., 1996). However, this apparent conflict is due to the definition of the size at which crystal nucleation stops and growth begins. To resolve this confusion, one may consider growth to occur after the critical nucleus size is achieved. 

Figure 4.6 shows the dependence on r for both contributions with small values of r its square is predominant and AG increases with increasing r the nucleus will stop growing and (with homogeneous nucleation) it disappears. From a certain value of r, the critical nucleus size, rk, AG decreases upon growth the nucleus is then stable and continues growing. The value of rk can be easily calculated at rk ... 

Nucleation — Atomistic theory of nucleation — Figure 1. Dependence of the nucleation work AG (ft) on the cluster size n (a) and dependence of the critical nucleus size nc on the supersaturation Ap (b) according to the atomistic nucleation theory (a schematic representation)... 

Hence, in this approximation, the critical nucleus size may be estimated from the slope of plots of the logarithm of the nucleation rate as a function of the logarithm of the supersaturation. 

The critical nucleus size is given by the values of l and that minimize AG (Equation 10-24) ... 

We are assuming As/ and Ahf don t vary much with temperature. Remember that this is just the bulk free energy (per unit volume) of the crystal, it does not include any additional free energy from those segments at the surfaces. We can now return to our expression for the critical nucleus size (thickness), which was given previously in Equation 10-25, and substituting for Ag, we obtain Equation 10-30 ... 

FIGURE 7.5 Nucleation rate as a fimction of P/P(, and also as a function of critical nucleus size. From Nielsen [25]. 

The carbon atoms arriving at the substrate surface must exceed a certain concentration at the solid-gas interface to reach and exceed the critical nucleus size. Therefore the diamond nucleation density as well as the growth rate are dependent on the relative rates of bulk and surface diffusion of carbon atoms. ° These are different for different substrates. Thus, the nucleation process needs a temperature dependent incubation time which is related to the time required to form critical size diamond clusters on the substrate surface. The nucleation rate, which is initially negligible, reaches a maximum after a certain time period and tends to zero for longer deposition times. ... 

For the initial formation of a solid phase on a substrate surface from vapor precursors through heterogeneous nucleation, as is schematically illustrated in Figure 20.2, the critical nucleus size, r, and the corresponding energy barrier, AG, are given by the following equations ... 

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