A kinetic treatment of glass formation V: Surface and bulk heterogeneous nucleation

An analytical method is described for calculating the detailed distribution of crystallite sizes in a supercooled liquid, and the changes in this distribution as a function of temperature (time) while the liquid is cooled from above the melting point. This method, termed the analysis of crystallization statistics, is applied to the calculation of continuous cooling curves for anorthite and o-terphenyl as representative of inorganic and organic systems. In addition to homogeneous nucleation, bulk as well as surface heterogeneous nucleation are considered. The effects of distributions of heterogeneities with contact angles between 40 and 100° as well as overall concentrations of heterogeneities between 10³ and 10⁹ cm^-3 are considered. Heterogeneities with contact angles higher than about 100° are shown not to have an effect on the critical cooling rate for typical concentrations of heterogeneities. For liquids containing distributions of heterogeneities, the nucleation behavior is dominated by small concentrations of heterogeneities having small contact angles. Theoretical log (I_vη) versus (T_r³ΔT_r²)^-1 curves have been constructed for homogeneous nucleation + heterogeneous nucleation with a single type of heterogeneity and for homogeneous nucleation + heterogeneous nucleation with the heterogeneities distributed with regard to contact angle. In the former case, the curve is composed of two linear portions; and in the latter case, the curve shows pronounced curvature. The curvature reflects a continous change in the frequency of heterogeneous nucleation. Surface heterogeneous nucleation was assumed to originate at discrete surface heterogeneities and was shown to give rise to continuous cooling curves similar to those calculated for bulk heterogeneous nucleation.