homogeneous nucleation
Imagine a perfectly clean glass of pure water with no dust and no scratches: ice has to start from nothing but the water's own random jiggling clumping together by luck. Homogeneous nucleation is new-phase formation uniformly throughout the parent phase, with no help from any foreign surface.
Because the whole surface of each tiny nucleus must be created from scratch, the energy barrier deltaG* is large. It only becomes surmountable at large undercooling, where deltaGv is strongly negative. For pure metals, homogeneous nucleation of the solid from the melt typically needs hundreds of degrees of undercooling (for example pure iron about 295 degrees C, pure nickel about 319 degrees C below their melting points), far more than you ever see in a foundry.
Truly homogeneous nucleation almost never governs real solidification or solid-state transformation, because real materials are full of impurities, boundaries, and container walls that offer easier heterogeneous sites. Homogeneous nucleation matters mainly as the clean theoretical baseline against which the easier heterogeneous case is measured.
Tiny, ultra-clean droplets of liquid metal can be undercooled by hundreds of degrees before they freeze homogeneously, a laboratory curiosity that reveals just how large the true nucleation barrier is.
The barrier is enormous; only huge undercooling overcomes it without help.
Homogeneous nucleation is mostly an idealization; assuming it governs a real casting will predict far too much undercooling.