Free Energy of the Cluster

P = 1 /kB T, /cB being the Boltzmann constant pt are the atom momenta. The Hamiltonian H is the sum of kinetic and potential energies (note that the potential energy takes into account the interaction between cluster atoms and the surrounding liquid). 

When calculating Z (N, P) we take into consideration that (i) on integrating over q the main contribution to (6.44) comes from the vicinities of the potential energy minima of the cluster (ii) the contributions from atoms of the surface layer and the cluster body can be separated asymptotically (iii) the SSD at N 1 is given by the expressions of types (6.38-40). In view of the above, after not complicated calculations, we obtain the approximate expression for Z(P,N) valid for great N  

Here Q ps, I2pv are the results of integration in (6.44) with respect to atom momenta of the surface layer and the cluster body, respectively Qqs and Qqv are the integration results with respect to atom coordinate deviations from equilibrium values they contain, in particular, the phonon contribution. 

Here and further the tilde denotes the cluster quantities. 

As is evident, the free energy of the cluster is the function of the random variable e with the density 

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Burton (39) has calculated properties of Ar clusters containing up to 87 atoms. He finds that the vibrational entropy per atom becomes constant for about 25 atoms. The entropy per atom for spherical face-centered cubic structures exceeds that of an infinite crystal and reaches a maximum between 19 and 43 atoms. An expression for the free energy of the cluster as a function of size was derived and shows that the excess free energy per atom increases with cluster size up to the largest clusters calculated. Although this approach yields valuable thermodynamic information on small clusters, it does not give electronic information. 

For our purposes we wish to relate the concentrations required by Eq. (2.7) to the free energies of the clusters as a function of cluster size sketched in Fig. 1. This motivates the calculations of the free energies in this work. 

A similar quantity svc can be calculated for the cluster body as well. In accordance with the definition given in [6.11], Svc represents the complexity of the structure displaying the degree of the SS uncertainty for the system with a broken ergodidty. In addition to the free energy of the cluster, we shall also introduce the effective free energy... 

As a rough approximation we can think of a cluster, or embryo, of n molecules as being made up of a core of nb molecules having the properties of the bulk solid, surrounded by rcs surface molecules (Figure 4.1). The free energy of the cluster, g(n), can thus be expressed in terms of bulk (gb) and surface (gs) molecular free energies ... 

Let the chemical potential of a metal atom in the supersaturated state be uss and that in the saturated phase /Ueq- In the case of solutions, the supersaturation is given by ln(jCss/JCeq), where Xgs and Xeq are the mole fraction of metal atoms. In considering the fate of metal atoms in the overall phase change from fluid to solid state, it is clear that some of them end up in the bulk (interior) of the crystal nuclei, while others become part of the surface. Each atom in a cluster can potentially form six intermolecular interactions. In the bulk all these interactions are satisfied, while on the surface, the atoms are in a different energetic state because they cannot realize their full interaction potential. In a cluster or nucleus containing z atoms, Zb have the properties of a bulk solid and Zs are surface atoms. The free energy of the cluster, g, can be written as the sum of the bulk and surface free energies, gb and gs. ... 

The free energy of the cluster can be written in terms of chemical potentials as follows ... 

Calculation of the equilibrium cluster concentration from Eq. (13) requires knowledge of AF(i). Note that only the first term in AF(i), namely Fi - /Fb, which is the excess free energy of the cluster relative to the bulk material, actually depends on the properties of the cluster itself. Therefore, calculation of E) - iFb is the principal problem of nucleation theory. This will be the subject of our next sections. 

Here a is the microscopic surface free energy of the cluster, ai the area of a cluster of i molecules, and t and c constants. [Note that if x is, as for spherical particles, for suitable values of r and c Eq. (48) looks like either the classical drop model or the drop model modified to include rotation, translation, etc. Therefore, Eq. (48) is, in a sense, a generalization of the classical drop model.] The Fisher drop model, Eq. (48), was originally developed to describe properties of gases very near the critical point, x and r can be obtained from critical-point indices and are found to be x = and t = 2.333. Hamill showed that c could be obtained from the density of the gas and cr from the second virial coefficient. Using the same equation for the free energy of a... 

During their initial growth stages, the clusters are likely to have structures that are more disordered than that of the bulk solid (Garten and Head, 1970). This idea has been incorporated into models that posit the formation of a disordered cluster as an initial step followed by the formation of a crystallite in a second step (Erdemir et al, 2009). These models are more complicated conceptually and quantitatively than the classical model presented here. Alternatively, Sohnel and Garside (1988) deal with this possibility by making the surface free energy of the clusters a fimction of their size. 

Clusters of soap bubbles provide examples of systems that are more complex than those of simple soap films contained by a fixed boundary. The free energy of the clusters will depend on the surface area of the soap film and the gas contained by the bubbles. 

Most likely, the atom adsorbed on the surface will diffuse until it is trapped in a defect (seed formation). If the concentration of free transition metal atoms around the seed is large enough, a nanoparticle is formed from the atoms that are one by one trapped in the potential well formed by the seed. If one wants to obtain a material with uniform distribution of transition metal atoms bound to defect sites (or heteroatom inclusions), the system should be gradually heated in such a way that all defect sites will be occupied. The last is possible only if the absolute free energy of the cluster formation (per transition metal atom) is less than the absolute free energy of binding to a vacancy site. No data are available on the entropy of these elemental reactions nevertheless, since all degrees of freedom are constrained as... 

The main point of Eq. (21) is that we can write down a microscopic expression for the equilibrium number of n-clusters if this number, which is equal to the probability of finding one cluster of size n, is much less than one. Using Eq. (6) this in turn defines an intensive Gibbs free-energy of the cluster where the reference state is the homogeneous phase ... 

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