Effect of cooling rate on the critical undercooling for crystallization

Crystallization in solutions under steady cooling is considered in the case of progressive nucleation, in which new crystallites are continuously nucleated among the already growing ones. Expressions are obtained for the total volume and number of crystallites as functions of the steadily increasing undercooling. These expressions are employed for determining the dependence of the relative critical undercooling uc for crystallization on the cooling rate q. The resulting uc(q) formula reveals that in the case of crystallization by progressive nucleation, the often used semi-empirical Nyvlt equation is an approximation which contains only the linear terms in the relationship between ln uc and ln q. The results obtained are also directly applicable to overall crystallization of melts at sufficiently small undercoolings.