A phase can be thermodynamically favored without appearing instantly. Most transformations require two kinetic steps: nucleation of small regions of the new phase and growth of nuclei that survive.

Competition in a nucleus

For a spherical nucleus of radius $r$, a simple free-energy model is

$$\Delta G(r)=4\pi r^2\gamma+\frac{4}{3}\pi r^3\Delta g_v,$$

where $\gamma>0$ is interfacial energy and $\Delta g_v<0$ is the bulk free-energy change per volume when the new phase is favored.

The surface term penalizes small nuclei; the volume term favors sufficiently large ones. Setting $d\Delta G/dr=0$ gives the critical radius

$$r^*=\frac{2\gamma}{|\Delta g_v|}.$$

Nuclei smaller than $r^*$ tend to disappear; larger ones can lower free energy by growing.

Real materials often nucleate heterogeneously at grain boundaries, surfaces, inclusions or defects because those sites reduce the interfacial-energy penalty.

Growth can be transport limited

Once a stable nucleus forms, its growth may require atoms to diffuse or heat to be removed. A phase diagram tells which state is favored at equilibrium, but nucleation barriers and transport rates determine whether that state is reached on the available time scale.

This distinction explains why cooling rate and prior microstructure can produce very different structures from the same composition.