the critical radius
Pushing a boulder up a hill: before the crest it rolls back, past the crest it rolls away on its own. The critical radius, written r*, is the crest for a forming nucleus. A particle smaller than r* tends to shrink and vanish, while one larger than r* grows.
The total free-energy change of a spherical nucleus is deltaG = (4/3) pi r^3 deltaGv + 4 pi r^2 gamma, where deltaGv is the volume free-energy change (negative below the transformation temperature) and gamma is the surface energy (positive). This sum rises, peaks, then falls. Setting the slope to zero gives r* = -2 gamma / deltaGv. As you cool further below the transformation temperature, deltaGv becomes more negative, so r* shrinks: colder means the critical seed is smaller and easier to reach.
The height of the hump, deltaG*, is the nucleation barrier, and it also drops with more undercooling. This is the deep reason transformations need some undercooling to begin at all: right at the equilibrium temperature deltaGv = 0, so r* is infinite and nothing can nucleate. Heterogeneous sites reduce the effective barrier without changing r* itself.
Just 1 degree below the melting point, deltaGv is tiny, so r* is huge and few atoms ever cluster that big, so nothing freezes. Cool 100 degrees below and r* might be only a few atom diameters, so nuclei form readily.
More undercooling shrinks the critical seed and unleashes nucleation.
r* is not a fixed material constant; it depends on temperature (undercooling). Confusing r* with a single fixed particle size is a common slip.