Although the new phase contains a lower free energy, generating new phase
domains always brings an increase of interface free energy, which is unfavorable
in the system. Since a spherical shape has a minimum surface area, Gibbs assumed a
new phase domain as a sphere with the radius ofr, and created the classical
nucleation theory to provide a phenomenological interpretation to the free energy
change on nucleation (Gibbs 1878 ), as given by
DF¼
4 p
3r^3 Dfþ 4 pr^2 s (9.40)Here the first term represents the decrease of body free energy, and the second
term represents the increase of surface free energy brought by the emergence of
new phase. When the domain size of the new phase is relatively small, the second
term dominates the free energy contribution, which is unfavorable for the decrease
of total system free energy. Therefore, the new phase will disappear quickly. Only
when the domain of new phase becomes large enough, the first term dominates the
free energy contribution, and the total free energy begins to decrease. As a result,
there exists a free energy barrier, as illustrated in Fig.9.8. The top of free energy
barrier corresponds to the critical size of nucleus, as
r^ ¼2 s
Df(9.41)
Only when the size of new phase domains goes beyond that critical value, the
new phase can continuous to grow. At the early stage of the phase separation, the
morphology of new phases generated by a few sporadic nuclei is significantly
different from the periodic structure in the spinodal decomposition mechanism, as
illustrated in Fig.9.9. At the later stage of phase separation, the small domains of
Fig. 9.7 Illustration of the
maximum scattering intensity
keeping constant with time
evolution as the early-stage
characteristics of spinodal
decomposition of phase
separation
9.2 Kinetics of Phase Separation 177