Last time: Spinodal decomposition--III; Nucleation--I.
Nucleation theory: Introductory concepts
Role of elastic strain energy in nucleation
Theory for steady-state nucleation rate
Role of elastic strain energy
In supersaturated crystalline solid solutions the first particles of the new phase that form are generally coherent with the matrix, because of the lower interphase boundary energy relative to semicoherent or incoherent interfaces. However, coherent inclusions have an associated elastic strain energy that resists nucleation. This elastic strain energy is not easy to express with simple algegraic expressions, except for particularly ideal cases.
If nucleation of an incoherent particle were to occur, it would do so without attendant elastic strain energy, because the incoherent interface would act as a perfect source or sink of point defects and the successful critical nucleus, with the lowest , would be a configuration without the energy penalty of elastic strain energy.
The method for calculating the elastic strain energy of coherent inclusions was developed most elegantly by J.D. Eshelby, and it uses an idealized procedure to create the inclusion and calculate the accompanying strain energy. Eshelby's procedure consists of four steps:
The easiest class of problems of this type to solve make these assumptions: elastic isotropy, elastic homogeneity (same elastic constants in particle and matrix), and that the particle shape is an ellipsoid of revolution (semiaxes , , and ). Thus, the theory allows for a particle-shape dependence of elastic strain energy.
Two cases: Pure dilatational transformation; pure shear transformation
Equilibrium distribution of clusters. Effect of losing clusters at sizes greater than critical sizes. Concept of "diffusion equation" for steady-state distribution when clusters greater than critical size are being lost.
Heterogeneous nucleation can occur on imperfections when the nucleating particle effectively "replaces" some energetic feature of the original configuration. For instance, nucleation on a grain boundary removes grain boundary area and replaces it with interphase boundary area that would have been created anyway by formation of the new phase. Thus, prexisting imperfections effectively "catalyze" the nucleation process. We'll consider the case of heterogeneous nucleation on grain boundaries.