Cluster, critical size

The last step, the formation of metal, takes place if in the reducing atmosphere (MX excess) a critical cluster size is reached (e.g., by heating the solution to more than 100°C), thus forming a giant cluster whose metal core is nearly identical to the bulk material and which can disproportionate to metal and Mm species without any further significant change of its bondings. 

Consider the formation of hemispherical nuclei of mercury on a graphite electrode. The intefacial tension of mercury with aqueous solutions is about 426 mN m-1. From Eq. (10.16) calculate the critical cluster sizes for 7 = —10, —100, —200 mV. Take z — 1 and ignore the interaction energy of the base of the hemisphere with the substrate. 

Figure 4-1 Extra Gibbs free energy of clusters as a function of cluster radius. The critical cluster size is when the extra free energy reached the maximum. A5m c = 56 J/K/mol, Vc = 46 cm /mol, a = 0.3 J/m, Te = melting temperature = 1600 K, and system temperature = 1500 K. AGc mX ASm ( (r—Tg) = —5600 J/mol. The radius of the critical cluster is r = 2aVg/(AGm c) = (2) (0.3) (46 x 10 )/5600m = 4.93 nm. The Gibbs free energy of the critical cluster relative to the melt is AG = (16/3)7tCT /(AGm c/l g)2 = 3.05 x lO i J.

And the free energy of the critical cluster is still Equation 4-4. If the cluster is not spherical (e.g., the cluster could be a cube, or some specific crystalline shape), then the specific relations between i and cluster volume and surface area are necessary to derive the critical cluster size. 

The mechanisms of droplet (or liquid germs) formation from a supersaturated vapour phase is still the subject of many investigations. After giving a brief account of the classical theory [64], which, as shown above, provides a simple method for estimating the energy barrier to overcome before effective nucleation is started, and permits the estimation of the critical cluster size, a complementary approach will be presented. 

Clusters grow until a critical cluster size (nucleus) is reached that is energetically favorable to sustain growth... 

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... 

Using Equations 3.3a and b, Englezos et al. (1987a) calculated the critical radius of methane hydrate to be 30-170 A. In comparison, critical cluster sizes using classical nucleation theory are estimated at around 32 A (Larson and Garside, 1986), while computer simulations predict critical sizes to be around 14.5 A (Baez and Clancy, 1994 Westacott and Rodger, 1998 Radhakrishnan and Trout, 2002). 

On the other hand, if the rate of formation of a doublet is much smaller than its rate of dissociation, the doublet is unstable. It will be shown later that the unstable doublets reach a dynamic equilibrium with the singlets in an extremely short time (of the order of the time scale of dissociation). The equilibrium concentration of these unstable doublets is small and depends upon the relative magnitudes of the rates of formation and dissociation. Since the dissociation rate of a doublet decreases rather dramatically with increasing particle size (because of the rapid increase in the depth of the interaction potential well with increasing particle size), there exists a critical particle size above which the coagulated particle pair is stable, i.e., its rate of formation is much greater than its rate of dissociation. This critical particle size is analogous to the critical cluster size... 

When the formation free energy is plotted against the number of molecules, in a one-component vapor the critical cluster size is the location of the maximum of... 

This is the equation used to interpret the depletion measurements. For reactions that involve the desorption of part of the reaction products, such as dissociative chemisorption of CO2 or dehydrogenation of HjS, collisional stabilization may not be as important since reaction products can take away the excess energy. For this case there may be no critical cluster size for reaction. 

Thus far, the nucleation equations have been derived with the assumption that the homogeneous interfacial energy, relationship between surface tension and droplet size, but thus far the theoretical predictions of this relationship and the experimental nucleation data do not seem to be in accord (Walton, 1969). 

Equation (13.5) shows the critical cluster size decreases with increase of the relative supersaturation S or a reduction of a by the addition of surfactants. This explains why a high supersaturation and/or addition of surfactants favours the formation of small particles. A large S pushes the critical cluster size N to smaller values and simultaneously lowers the activation barrier, as illustrated in Figure 13.2, which shows the variation of AG with radius at increasing S. 

Figure 4. Deviations AT = Tn-Tp of the average cluster temperatures (Tavg upper curves) curves) and the so-called by the authors local equilibrium cluster temperatures (Tc lower curves) from the bath temperatures in dependence on the droplet size n for two realizations of the molecular dynamics simulations of Wedekind et aV The vertical lines specify the location of the critical cluster sizes.

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