A Case Study in Inclusions Precipitate Nucleation

The volume term described above is balanced against an interfacial energy of the form 

To cast these ideas in mathematical form, we note that the total free energy in the present model may be written as 

To determine the critical radius r, we need to satisfy the condition dG r)jdr = 0. The consequence of performing the relevant differentiation and solving for the 

The model developed above serves as a convenient starting point for carrying out a dynamical analysis of the nucleation problem from the perspective of the variational principle of section 2.3.3. A nice discussion of this analysis can be found in Suo (1997). As with the two-dimensional model considered in section 2.3.3, we idealize our analysis to the case of a single particle characterized by one degree of freedom. In the present setting, we restrict our attention to spherical particles of radius r. We recall that the function which presides over our variational statement of this problem can be written genetically as 

A similar argument is made in the case of the rate of change of the bulk contribution to the free energy. In this case, we interest ourselves in the free energy when the particle has radius r (i.e. Gbutkir) = nr Eb) and when the particle has radius r - - f At. In this case, the result is 

---