Activation free energy of nucleation

In the initial phase of nucleation, the positive surface term prevails. The addition of new monomers to the cluster causes an increase in the free energy. As the volume grows faster than the surface, the favorable volume term starts to dominate the free energy expression when the nuclei exceed a certain critical size (oc)-The free energy of the nuclei with the critical size is called the activation free energy of nucleation. From this point stable growth of the cluster occurs. Every cluster below this size can be dissolved because the interactions are not enough to hold the molecules in the cluster. 

The activation free energy of nucleation, AG was evaluated using the Fisher-Turnbull equation [8],... 

Table 3 Activation Free Energy of Nucleation as a Function of Temperature and Degree of Supercooling for the First Hydrogenation Process...

One of the key outcomes of this theory is the concept of critical size which must be achieved by an assembly of molecules in order to be stabilized by further growth. The higher the operating level of supersaturation the smaller is this size (typically a few tens of molecules). Now, in Fig. 2.9 the supersaturation with respect to II is simply Go - Gn and is lower than Go - Gj for structure I. However it can now be seen that if for a particular solution composition the critical size is lower for II than for I then the activation free energy for nucleation is lower and kinetics will favour form II. Ultimately form II will have to transform to form I, a process that we discuss later. Overall we can say that the probability that a particular form i will appear is given by... 

Fisher-Turnbull Analysis The activation free energy for nucleation, Gc, may be found from the Fisher-Turnbull equation given in Equation 1. The term in the second exponential of Equation 1 is often given as Gc/kT. Combination of Equations (1) and (3) allows development of the following equation ... 

Fig. 3.9 A, B. Schematieai representation of the free energy states and the activation energy barriers for mineralization. Pathway A, crystallization of a mineral phase from pure and impure solution (with no major structural modifications) pathway B, formation of a crystalline mineral from intermediate phases of different crystal structure. AGn, free energy of nucleation AGg, free energy of growth AGx, free, energy of phase transformation...

Plastic shear on the (100) [001] system in a lamella can also be initiated in a more general case by the nucleation of a screw-dislocation half loop from the narrow XA face of the lamella of Fig. 9.17. This process has been considered rigorously using the variational boundary-integral method developed by Xu and Ortiz (1993). The problem of interest here has also been solved by Xu and Zhang (2003), giving a stress dependence of the activation free energy of this mode shown in Fig. 9.19 as the upper curve. It is of the form... 

Having described the equilibrium structure and thermodynamics of the vapor condensate we then re-examine homogeneous nucleation theory. This combination of thermodynamics and rate kinetics, in which the free energy of formation is treated as an activation energy in a monomer addition reaction, contains the assumption that equilibrium thermodynamic functions can be applied to a continuum of non-equilibrium states. For the purpose of elucidating the effects of the removal of the usual approximations, we retain this assumption and calculate a radially dependent free energy of formation. Ve find, that by removing the conventional assumptions, the presumed thermodynamic barrier to nucleation is absent. 

The existence of the phase boundary between the solid and liquid phase complicates matters, since a phase boundary is associated with an increase in free energy of the system which must be offset by the overall loss of free energy. For this reason the magnitudes of the activated barriers are dependent on the size (i.e. the surface to volume ratio of the new phase) of the supramolecular assembly (crystal nucleus). This was recognized in 1939 by Volmer in his development of the kinetic theory of nucleation from homogeneous solutions and remains our best model today (Volmer 1939). 

Nucleation Rate. The value of AGmax is often considered to be an activation free energy for nucleus formation. Moreover, it is generally assumed that there is also a certain barrier to incorporation of a molecule into an embryo, owing to diffusional resistance this is usually expressed in a molar activation free energy for transport AG. Classical nucleation theory then stipulates that the nucleation rate (number of nuclei formed per unit time per unit volume) will be given by... 

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