An analytical model for constitutional supercooling-driven grain formation and grain size prediction

Being able to predict the grain formation process and attendant grain size has been a central topic in solidification. Such an analytical model is presented for constitutional supercooling (CS)-driven grain formation with several simplifications. The model links the nucleation of new grains to the growth of a larger neighbouring grain. The average grain size (d¯) is thus determined by two components: the minimum growth (rcs) necessary to establish sufficient CS (ΔTn  ) for nucleating new grains, and the spatial mean distance (a¯) to the most potent available nucleants. Both spherical and planar growth fronts are considered, covering growth curvatures from small to infinite. Two distinct fundamental approaches are used, which result in identical descriptions of d¯, where d¯=a¯+D·ΔTn/(v·Q) (D is the diffusion coefficient, v is the growth velocity, Q is the growth restriction factor). The model is compared with literature data produced under various conditions and demonstrated on aluminium alloys as an example.