Effects of Glass Transition and Structural Relaxation on Crystal Nucleation: Theoretical Description and Model Analysis

Abstract

In the application of classical nucleation theory (CNT) and all other theoretical models of crystallization of liquids and glasses it is always assumed that nucleation proceeds only after the supercooled liquid or the glass have completed structural relaxation processes towards the metastable equilibrium state. Only employing such assumption, the thermodynamic driving force of crystallization and the surface tension can be determined in the way it is commonly performed. The present paper is devoted to the theoretical treatment of a different situation when nucleation proceeds concomitantly with structural relaxation. To treat the nucleation kinetics theoretically for such cases, we need adequate expressions for the thermodynamic driving force and the surface tension accounting for the contributions caused by the deviation of the supercooled liquid from metastable equilibrium. In the present paper, such relations are derived. They are expressed via deviations of structural order parameters from their equilibrium values. Relaxation processes result in changes of the structural order parameters with time. As a consequence, the thermodynamic driving force and surface tension, and basic characteristics of crystal nucleation, such as the work of critical cluster formation and the steady-state nucleation rate, also become time-dependent. We show that this scenario may be realized in the vicinity and below the glass transition temperature, and it may occur only if diffusion (controlling nucleation) and viscosity (controlling the alpha-relaxation process) in the liquid decouple. Analytical estimates are illustrated and confirmed by numerical computations for a model system. The theory is successfully applied to the interpretation of experimental data. Several further consequences of this newly developed theoretical treatment are discussed in detail. In line with our previous investigations, we reconfirm that only when the characteristic times of structural relaxation are of similar order of magnitude or longer than the characteristic times of crystal nucleation, elastic stresses evolving in nucleation may significantly affect this process. Completing the analysis of elastic stress effects, for the first time expressions are derived for the dependence of the surface tension of critical crystallites on elastic stresses. As the result, a comprehensive theoretical description of crystal nucleation accounting appropriately for the effects of deviations of the liquid from the metastable states and of relaxation on crystal nucleation of glass-forming liquids, including the effect of simultaneous stress evolution and stress relaxation on nucleation, is now available. As one if its applications, this theoretical treatment provides a new tool for the explanation of the low-temperature anomaly in nucleation in silicate and polymer glasses (the so-called \breakdown" of CNT at temperatures below the temperature of the maximum steady-state nucleation rate). We show that this anomaly results from much more complex features of crystal nucleation in glasses caused by deviations from metastable equilibrium (resulting in changes of the thermodynamic driving force, the surface tension, and the work of critical cluster formation, in the necessity to account of structural relaxation and stress effects) than assumed so far.

Further evidence against the alleged failure of the classical nucleation theory below the glass transition range

“…Moreover, and most significant is that they are also affected by the long relaxation times of glasses at temperatures below T g , as demonstrated in refs. [22,23]. Furthermore, τ is very short (from a few minutes to seconds) for temperatures above T g , which makes its determination subjected to enormous errors.…”

“…In addition to those two papers, new publications have arisen, drawing attention to the generalized assumption in applying CNT and other theoretical crystallization models that nucleation proceeds only after the glass has completed the structural relaxation process toward the metastable supercooled liquid (SCL) state. In this regard, Schmelzer et al 22 proposed a new hypothesis and provided a theoretical treatment of a different situation when nucleation proceeds concomitantly with structural relaxation. Such theoretical model is significant for the present work since it offers a plausible explanation to the very long times required before nucleation reaches a steady state, as reported in refs.…”

“…The hypothesis and model proposed in ref. [22] were tested by Fokin et al, 23 using a Li 2 Si 2 O 5 glass as a model glass‐forming substance. Their nucleation experiments were carried out for very extended times (up to about 2,200 h), at temperatures ~20 K below the laboratory T g .…”

“…On the whole, the theoretical model proposed in ref. [22] and the experimental results and analyses presented in ref. [23] give strong support to the effect of glass relaxation on crystal nucleation and confirm the results of Cassar et al 15 and Xia et al 16 that the alleged “breakdown” of the CNT at temperatures close and below T g results from the improper use of non‐steady‐state nucleation rates.…”

Statistical Approach to Crystal Nucleation in Glass-Forming Liquids

“…Time-lag effects and effects of finite times of growth of the clusters to experimentally measurable sizes we consider as important for the case of determination of the probability of occurrence of the first critical cluster in heating starting from a low temperature where the rate of crystal nucleation tends to zero due to low values of the diffusion coefficient governing nucleation, respectively, very large values of the viscosity of the liquid. Here, a variety of problems have to be solved in order to arrive at appropriate relations for determining the characteristic parameters of crystal nucleation [ 55 , 56 , 57 ]. As it seems, in order to interpret experimental data in such case, a theoretical treatment of nucleation based on the solution of the set of kinetic equations for the average rate of crystal nucleation and growth (see, e.g., in [ 11 , 12 , 46 , 58 ]) is preferable.…”