Energy of cluster formation

both terms are functions of the size N of the cluster. The first term increases linearly with N and, being negative, makes the process of deposition possible. The second term, connected with the creation of the new surface, is 

For small clusters, 0(N), being a parabolic function of N, dominates in eq. (4.16). The formation of clusters, in this case, is connected with an increase of the Gibbs energy and can proceed as an energy fluctuation process only. 

Let us now consider a crystalline cluster of size N formed on a substrate. The excess energy term is connected mainly to the new interfaces. This is strictly valid for macroscopic crystals only as discussed later in Section 4.2. From eq. (4.16) with eq, (4.6) one obtains  

A differentiation of eq. (4.17) with respect to N is only possible if a relation between A,-, Aj, and N exists. This is the case when a given arbitrary geometrical form is considered. The surface area of any given 3D geometrical form is related to its volume by A = BV, where B is a constant depending on the geometry. For a sphere, evidently, B - 36n and for a cube B = 6. With A = 57 and V = 7mW, the maximum of AG(N) is found at 

As seen from eqs. (4.18) and (4.19), the size and energy of formation of the nucleus is a function of the overvoltage, but depends also on the geometrical form of the cluster. This is accounted for by the geometrical factor B and is implemented in the average surface energy j. 

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

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. 

Above expression is valid when the clusters form at random sites within the bulk of original phase (so-called homogeneous nucleation). In the case of heterogeneous nucleation, the clusters are formed on foreign surfaces placed within the system (impurities, walls of the vessel, wafers or certain base materials - e g. nanofibers) and the necessary energy of clusters formation on flat surface reads (see, e g. [4]) ... 

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 ... 

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