  The minimum free-energy path (MFEP) is the most probable route of the
nucleation process on the multidimensional free-energy surface. In this study,
the phase-field equation is used as a mathematical tool to deduce the minimum
free-energy path (MFEP) of homogeneous nucleation. We use a simple
square-gradient free-energy functional with a quartic local free-energy
function as an example and study the time evolution of a single nucleus placed
within a metastable environment. The time integration of the phase-field
equation is performed using the numerically efficient cell-dynamics method. By
monitoring the evolution of the size of the nucleus and the free energy of the
system simultaneously, we can easily deduce the free-energy barrier as a
function of the size of the sub- and the super-critical nucleus along the MFEP.

Materials Science