Critical-sized clusters

The rate at which critical sized clusters are formed is very sensitive to the height of the free energy barrier (AG), or equivalent to the extent of penetration into the metastable region. As the critical cluster size becomes smaller, so does the free energy barrier that must be overcome to form the critical cluster. With increasing... 

In the classical theory of nucleatlon (1) for condensing vapors, there exists a "critical size cluster through which the passage of clusters smaller than critical to clusters larger than critical. Is rate limiting. According to this theory, the exact... 

Depending on conditions the apparent critical size cluster may, in fact, be as small as the dimer, A2. Blander and Katz (2) have pointed out some of the ambiguities in the conventional macroscopic theory for AG in terms of bulk liquid surface tensions and densities. For smaller size clusters (n 20), a molecular approach to the nucleatlon problem is clearly preferred. In this regard, Bauer and Frurlp have recently used a somewhat empirical approach to determine the cluster entropies of condensing iron (3). 

The sum in this equation includes the areas Acrit,i of all faces of the critically sized cluster with their respective surface energies. 

Under the acting overpotential, the probabilities for growth and dissolution of a critically sized cluster are equal. In other words, the nucleus of the new phase stays in equilibrium with the ambient phase at a more negative potential than the equilibrium potential of the bulk crystal. It has to be emphasized, however, that this equilibrium is metastable, and the smallest change of the cluster dimensions will result in either further growth or complete dissolution. 

The interpretation of the nucleation rate equation in terms of its overpotential dependence is rather difficult to perceive as represented in the form of eq. (4.42). It has been shown already that the products in the denominator of this equation contain the formation energy AG(iV) of clusters of class N (cf. eq. (4.47)), and that, at a given overpotential, AG has a maximum determining the critically sized cluster, N=Ncnt-The n terms in the sum of the denominator of eq. (4.42) show a maximum for the critically sized cluster. All terms other than that for this cluster can be neglected including the unity in the denominator [4.13, 4.14). Note that/is always much smaller than the impingement rate ti att.o times the adsorption sites Z<, and hence the denominator is much larger than unity. Then ... 

Under the specific conditions of electrochemical metal deposition, the critically sized clusters of the new phase have been found to consist of only a few atoms, where classical thermodynamic bulk quantities cannot be applied. Therefore, the original kinetic theory of Becker and Doering was further developed to an atomistic theory of nucleation. 

Rapid addition avoids nucleation from local supersaturation by mixing prior to formation of critical-sized clusters. 

Ortho II arrangement contains eight oxygen atoms and therefore nine copper atoms. Of course the experimental data are fit to the model. The critical size clusters are those which need to be chosen to reproduce the observed plateau widths. 

Understanding the effect of ions on nucleation would be very difficult if the ions specific chemical characteristic had a significant effect on their nucleating efficiency. Since frequently the number of molecules of the nucleating species in the critical-sized cluster is on the order of tens to hundreds of molecules, it is reasonable to assume that an ion buried near its center has little effect on the cluster s surface properties other than those due to its charge. Thus the ion creates a central force field that makes it more difficult for a molecule to evaporate than it would be from an uncharged but otherwise identical cluster. Ion self-repulsion and the typical low density of ions ensure that almost all such clusters contain only one ion. Molecules arrive at and evaporate from the ion-vapor molecule clusters. The critical cluster size for nucleation is that for which the evaporation rate of... 

This equation is an expression for the steady state homogeneous gas rate J. We note that the cut-off size g appears explicitly in Eq. (12). If the free energy of a small cluster depends on size as shown in Fig. 2, then the equilibrium concentration of critical size clusters bi is very much smaller than that of the very, very large clusters. Hence, the largest clusters will contribute very little to the sum in Eq. (12). Therefore, Eq. (12) can just as well be replaced by... 

Determining the flux of monomers to the critical nuclei (/, molecules/L sec) is a challenge. This flux has been modeled using various statistical mechanics approaches. These models estimate the flux of molecules from the solution to the surface of the critical nuclei but they include various adjustments such as the Zeldovich factor (Markov, 2003), which accounts for the possibility that some critical-sized clusters will dissolve to a smaller size instead of growing into crystallites. Presently there is no widely accepted model for J, so Eq. (9.28) is usually recast into a semi-empirical form, which combines the monomer flux and cluster surface area into a pre-exponential term (Q, crystallites/L sec). 

Rearranging Eq. (9.19) gives the concentration of critical-sized clusters. 

Surface free energy (y), reaction temperature (T), degree of supersaturation (S) and ratio of critical-sized clusters converting into stable nuclei (fo) are the important variables that must be taken into consideration in this context, based... 

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