Abstract
A kinetic model of primary homogeneous non-isothermal crystal nucleation with transient and athermal effects is developed. For comparison, steady-state and transient isothermal nucleation rates are considered. Kinetic equation for the development of cluster size distribution provides the basis for the model. Transient effects are characterized by the longest relaxation time which increases with temperature at low and moderate undercooling. In isothermal conditions, nucleation rate is controlled by thermal mechanism; in non-isothermal conditions, there appears also athermal mechanism. Closed-form analytical formula for the development of transient cluster size distribution in single-relaxation-time approximation is derived for non-isothermal processes, as well as thermal and athermal nucleation rates and total number of nuclei produced in a cooling or heating run. The transient term contributes to isothermal nucleation kinetics the more the higher is temperature. Under non-isothermal conditions, the relaxation time contributes to the nucleation kinetics by the product with the cooling/heating rate. Considerable transient effects should be expected for the relaxation times as long as 102–105 s. Contribution of thermal nucleation to the concentration of nuclei is inversely proportional to the temperature rate, while the contribution of athermal nucleation depends on the temperature interval of cooling or heating. Our kinetic model indicates similarities in the nucleation mechanisms in polymers and metals undergoing crystallization. Example computations are presented for molten indium and a linear polymer—polyhydroxybutyrate (PHB). A low-temperature limit is predicted for the nucleation mechanism in PHB, while for indium the mechanism is active in the entire temperature range.
Highlights
Crystallization plays important role in material science and technology
The model in single-relaxation-time approximation allows to predict transient nucleation rate under isothermal and non-isothermal conditions, athermal nucleation kinetics, as well as influence of individual material and processing parameters and concentration of nuclei produced in a cooling or heating run
Transient effects are characterized by a single relaxation time
Summary
Crystallization plays important role in material science and technology. Crystal structure, degree of crystallinity, crystal size distribution, and molecular orientation strongly affect mechanical, sorptional, and optical properties of many metals, ceramics, and polymers. For a semiquantitative analysis of non-isothermal primary homogeneous nucleation, we consider an approximate single-relaxation-time model for time development of the cluster size distribution using temperature-dependent functions qstðTÞ and sðTÞ in Eq (23). With respect to cluster volume leads us to the following original formula for transient nucleation rate under non-isothermal conditions where the critical size v* and the coefficient DðvÃÞ refer to instantaneous temperature T. Replacing qstðvÞ in Eq (41) by transient cluster size distribution qtransðv; TÞ given by Eq (32), an analytical closed-form expression for athermal nucleation rate is obtained, accounting for the relaxation effects. In the nucleation considered as a quasi-static process (steady-state nucleation rate instantaneously follows varying temperature), concentration of nuclei produced at constant T_ , at the absence of transient and athermal effects, is expressed by the integral. It is evident that the thermal contribution, DNqs, is inversely proportional to the cooling/heating rate, while the athermal contribution depends only on the temperature interval (T0, T)
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