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

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 Delta T sub m is much smaller than the exponent A in the nucleation rate Iapproxexp(-A Delta T exp -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 exp -1 :p exp -1/3 , where p= Delta T exp -2 sub m approx30-80 is a parameter characterizing the maximum supercooling Delta T sub m . It is a weak (logarithmical) function of all the parameters including the cooling rate T. The crystal sizes are determined by the density of nuclei produced during the nucleation period and depend on the cooling rate approximately as T exp -0.5 . The theoretical predictions are found in a good agreement with experimental data for Al-Cu, Al-Si, Sn-Pb, and CaMgSi sub 2 O -CaAl sub 2 Si sub 2 O sub 8 -NaAlSi sub 3 O sub 8 .