Nucleation free-energy maximum

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... 

The free energy of formation of a cluster as a function of the size of a cluster of A atoms is shown in Figure 7.1. The figure shows that AG(A) increases initially, reaches a maximum AG, and then decreases with increasing A [Eq. (7.1)]. AG = f N,Tj) for a three-dimensional spherical nucleation of Ag is given by... 

AF maximum free energy barrier for nucleation, ergs... 

Consider now the nonequilibrium system when [ip < fia and the nucleation of the stable 0 phase therefore becomes possible. Under this condition, Agoes through a maximum with increasing TV as in Fig. 19.3a, and the formation of sufficiently large clusters causes the free energy of the system to decrease. Thermodynamics... 

From the texturological point of view, formation of crystallizing PMs is different because of the existence of nucleation stages at each phase transformation, usually more than one. As a result each nucleation gives a new maximum value of specific surface area, which can only decrease until the next transformation. A set of successive phase transformations including both wet and dry stages is characteristic for numerous PMs. Thus, each new phase transformation starts with the maximum possible surface area with its successive decrease directed by the necessity to decrease excess free energy. 

J is the number of nuclei formed per unit time per unit volume, No is the number of molecules of the crystallizing phase in a unit volume, v is the frequency of atomic or molecular transport at the nucleus-liquid interface, and AG is the maximum in the Gibbs free energy change for the formation of clusters at a certain critical size, 1. The nucleation rate was initially derived for condensation in vapors, where the preexponential factor is related to the gas kinetic collision frequency. In the case of nucleation from condensed phases, the frequency factor is related to the diffusion process. The value of 1 can be obtained by minimizing the free energy function with respect to the characteristic length. 

The essential problem in nucleation studies is to estimate the rate of formation of nuclei having the critical size. When r = rc, the maximum free-energy barrier that must be overcome to form a liquid drop can be found by combining Equations (355) and (356)... 

The cluster free energy, as a function of number of molecules i within the cluster, is shown for the typical case of ion-induced nucleation in Figure 11.14. The free-energy curves are in fact consistent with the kinetic point of view given above. Thus, for an appropriate value of the supersaturation, the free-energy curve shows a local minimum and a maximum for the case of ion-induced nucleation, corresponding, respectively, to the stable subcritical cluster and the unstable critical cluster. The local minimum disappears for the case of homogeneous nucleation. 

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. 

This thermodynamic approach illustrates in terms of a limiting supersaturation the energy requirements involved in nucleation. Since the majority of precipitation reactions in biological systems are likely to occur under kinetic conditions, the rate of nucleation will be of upmost importance. The rate of homogeneous nucleation Jn, can be considered as the rate at which nuclei surmount the maximum in the free energy curve of Fig. 3.1 and can be expressed as,... 

As a result, the nucleation curve (free energy change versus nucleus size) passes through a well-known single maximum point corresponding to the critical size of the nucleus (Fig. 4.14). The process is at first reversible (as expressed in Eqs. 4.237-4.241). The radius of the critical cluster can be derived from Eq. (4.246) under the condition 9 (AG/ 9r) = 0 (compared with the radius derived directly from the Kelvin equation) to be ... 

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