Free energy of cluster

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.

Figure 4-4 Effect of supersaturation on free energy of cluster formation. 

The cluster nucleation is the process of random walks in m, while the crystallite nucleation is the diffusion process in the one-dimensional size space. The probabilities of transitions from the vertex to either of the adjacent vertices depend on the free energy of clusters in different structure states. 

FIGURE 11.14 Ion-induced nucleation (a) free energy of cluster formation in homogeneous nucleation (b) free energy of cluster formation in ion-induced nucleation. Smax and 5max are the maximum values of the saturation ratio for which no barrier to nucleation exists in homogeneous and ion-indued nucleation, respectively. 

FIGURE 10.10 Free energy of cluster formation, AG(nA, B), for binary nucleation. (a) Schematic diagram of saddle point in the AG surface, (b) AG surface for H2SO4-H2O system at 298 K. 

The emission parameter, is calculated, taking into account contribution of matrix frustration to the free energy of cluster distributions in binary alloys (Lepinoux 2006,2009). This effect is typical for concentrated alloys, when Frenkel s model of ideal cluster gas (Frenkel 1955) is not valid and it is necessary to consider the interaction between the clusters by means of so-called exclusion volume, i.e., the number of forbidden atomic sites (or volume normalized by the atomic volume) to a -mer by an n-mer. In our paper it is suggested to take into account the frustration effect empirically by the use of the thermodynamic free energy expression from CALPHAD (Andersson and Sundman 1987) with the correction suggested by Bonny et al. (2010). 

AF g) is free energy of cluster formation in the unstressed and relaxed system. Positive elastic free energy of chain deformation, > 0, and en-... 

Now, as usual, we split the free energy of cluster formation into three parts ... 

Figure 1. Free energy of cluster formation plotted vs. n, the number of molecules in the cluster.

For a spherical nucleus, the energy barrier to nucleation, AG , is simply related to the volume free energy of cluster formation and to the energy required for the formation of a new surface ... 

From left to right, the information provided in this table includes temperature, Zimmerman code, number of individual structures in cluster, total probality (X), p ), and free energy of cluster... 

In the classic nucleation theory, the free energy of forming a cluster of radius r containing n atoms or molecules is the sum of two terms ... 

The simplest approach to understanding the reduced melting point in nanocrystals relies on a simple thennodynamic model which considers the volume and surface as separate components. Wliether solid or melted, a nanocrystal surface contains atoms which are not bound to interior atoms. This raises the net free energy of the system because of the positive surface free energy, but the energetic cost of the surface is higher for a solid cluster than for a liquid cluster. Thus the free-energy difference between the two phases of a nanocrystal becomes smaller as the cluster size... 

In the CVM, the free energy of a given alloy is approximated in terms of probabihties for a selected set of finite clusters. The largest cluster explicitly considered in the free energy functional specifies the level of the approximation. The common practice for an fcc-based system is the tetrahedron approximation [26] in which nearest neighbor tetrahedron cluster is taken as the largest cluster. Hence, within the tetrahedron approximation, the free energy expression, F,is symbolically expressed as... 

Fig. 3.2. Variation of free energy of a cluster of molecules on a surface as a function of the number of molecules, at a fixed supercooling. N is the number of molecules at which the free energy is a maximum. Any cluster larger than Ns is stable. The dashed curves show the contributions from the increase in surface area and the decrease in bulk free energy. Increasing the supercooling shifts all curves towards the origin and decreases the height of the maximum...

Asthagiri, D. Pratt, L. R. Ashbaugh, H. S., Absolute hydration free energies of ions, ion-water clusters, and quasi-chemical theory, J. Chem. Phys. 2003,119, 2702-2708... 

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