Free energy nucleus formation

Since Jo and C are proportional to the diffusion coefficient (D) and activation free energy for formation of a critical nucleus (AG ), respectively, it is concluded that the Z dependence of J is mainly determined by the diffusion process of the polymer chain and not by the formation process of a critical nucleus. 

There is a point at which these aggregates reach a critical size of minimum stability r and the free energy of formation AG is a maximum. Further addition of material to the critical nucleus decreases the free energy and produces a stable growing nucleus. The nucleation rate is the product of the concentration of critical nuclei N given by... 

Pit formation. If we consider a dissolution nucleus at a screw dislocation intersecting the surface which consists of a cylindrical hole of radius r, one atom layer deep (a), then the free energy of formation of this nucleus will be composed of a volume energy, surface energy, and elastic strain energy term, respectively, as follows ... 

Figure 7.1. Free energy of formation of a cluster as a function of size N (a cluster of N atoms) the size of the critical cluster (nucleus).

For a spherical nucleus of radius r, the free energy of formation of a crystal from the liquid phase, follows analogous to Eq. (1.20) as... 

The critical radius diverges inversely with supercooling. Substitution of this result for into Eq. (1.38) leads to the free energy of formation of critical spherical nucleus as... 

AGe(jge free energy of formation for an edge-on primary nucleus... 

Fig. 2 Gibbs free energy of formation of a nucleus as a function of its size...

The free energy of formation for a flat-on nucleus attached to a substrate,... 

According to Gibbs [4] and Vohner [5], the free energy of formation of a spherical nucleus, AG, is given by the sum of two contributions (i) a positive surface energy term AGg which increases with increase in the radius of the nucleus r and (ii) a negative contribution due to the appearance of a new phase, which also increases with increases in r. 

It is clear from Equations (9.1) to (9.4) that the free energy of formation of a nucleus and the critical radius r, above which the cluster formation grows spontaneously, depend on two main parameters, namely a and (S/S ), both of which are influenced by the presence of surfactants, a is influenced in a direct way by the adsorption of surfactant onto the surface of the nucleus this adsorption lowers y and this in turn reduces r and AG in other words, spontaneous cluster formation will occur at a smaller critical radius. In addition, surfactant adsorption stabilises the nuclei against any flocculation. The presence of micelles in solution also affects the processes ofnucleation and growth, both directly and indirectly. For example, the micelles can act as nuclei on which growth may occur, and may also solubilize the molecules of the material this can affect the relative supersaturation and, in turn, may have an effect on nucleation and growth. 

The discussion above applies to heterogeneous nucleation on flat surfaces. For atmospheric particles the curvature of the surface complicates the situation. Fletcher (1958) showed that the free energy of formation AG of an embryo of critical radius r on a spherical nucleus of radius R is given by... 

The effect of particle-matrix interfacial free energy is often overlooked but is particularly important in the nucleation and coarsening of internal oxides. Consider the classical nucleation problem of forming a spherical nucleus. If strain is neglected, the free energy of formation of a nucleus of radius r is given by Equation (5.31),... 

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