Batch crystallization under continuous cooling: analytical solution for diffusion limited crystal growth

Abstract

The kinetic equations of nucleation and crystal growth in a multicomponent system undergoing continuous cooling are solved for asymptotically slow cooling rates meaning that the maximum supercooling ΔTm is much smaller than the exponent A in the nucleation rate I exp(−AT−2). The growth rate is controlled by chemical diffusion so that it depends not only on the supercooling but also on the crystal size. The periods of metastable cooling, nucleation and relaxation are related as 1:p−1:p−1/3 where p = AΔT−2m 30–80 is a parameter characterizing the maximum supercooling Tm. It is a weak (logarithmical) function of all the parameters including the cooling rate Ṫ. The crystal sizes are determined by the density of nuclei produced during the nucleation period and depend on the cooling rate approximately as Ṫ−0.5. The theoretical predictions are found in a good agreement with experimental data for AlCu, AlSi, SnPb, and CaMgSi2O6CaAl2Si2O8NaAlSi3O8.