Crystal nucleation in a glass during relaxation well below Tg.

Until quite recently, in almost all papers on crystal nucleation in glass-forming substances, it was assumed that nucleation proceeds in a completely relaxed supercooled liquid and, hence, at constant values of the critical parameters determining the nucleation rate for any given set of temperature, pressure, and composition. Here, we analyze the validity of this hypothesis for a model system by studying nucleation in a lithium silicate glass treated for very long times (up to 250 days) in deeply supercooled states, reaching 60 K below the laboratory glass transition temperature, Tg. At all temperatures in the considered range, T < Tg, we observed an enormous difference between the experimental number of nucleated crystals, N(t), and its theoretically expected value computed by assuming the metastable state of the relaxing glass has been reached. Analyzing the origin of this discrepancy, we confirmed that the key parameters determining the nucleation rates change with time as a result of the glass relaxation process. Finally, we demonstrate that, for temperatures below 683 K, this particular glass almost fully crystallizes prior to reaching the ultimate steady-state nucleation regime (e.g., at 663 K, it would take 176 years for the glass to reach 99% crystallization, while 2600 years would be needed for complete relaxation). This comprehensive study proves that structural relaxation strongly affects crystal nucleation in deeply supercooled states at temperatures well below Tg; hence, this phenomenon has to be accounted for in any crystal nucleation model.

Crystallization of Supercooled Liquids: Self-Consistency Correction of the Steady-State Nucleation Rate.

Crystal nucleation can be described by a set of kinetic equations that appropriately account for both the thermodynamic and kinetic factors governing this process. The mathematical analysis of this set of equations allows one to formulate analytical expressions for the basic characteristics of nucleation, i.e., the steady-state nucleation rate and the steady-state cluster-size distribution. These two quantities depend on the work of formation, Δ G ( n ) = - n Δ µ + γ n 2 / 3 , of crystal clusters of size n and, in particular, on the work of critical cluster formation, Δ G ( n c ) . The first term in the expression for Δ G ( n ) describes changes in the bulk contributions (expressed by the chemical potential difference, Δ µ ) to the Gibbs free energy caused by cluster formation, whereas the second one reflects surface contributions (expressed by the surface tension, σ : γ = Ω d 0 2 σ , Ω = 4 π ( 3 / 4 π ) 2 / 3 , where d 0 is a parameter describing the size of the particles in the liquid undergoing crystallization), n is the number of particles (atoms or molecules) in a crystallite, and n = n c defines the size of the critical crystallite, corresponding to the maximum (in general, a saddle point) of the Gibbs free energy, G. The work of cluster formation is commonly identified with the difference between the Gibbs free energy of a system containing a cluster with n particles and the homogeneous initial state. For the formation of a "cluster" of size n = 1 , no work is required. However, the commonly used relation for Δ G ( n ) given above leads to a finite value for n = 1 . By this reason, for a correct determination of the work of cluster formation, a self-consistency correction should be introduced employing instead of Δ G ( n ) an expression of the form Δ G ˜ ( n ) = Δ G ( n ) - Δ G ( 1 ) . Such self-consistency correction is usually omitted assuming that the inequality Δ G ( n ) ≫ Δ G ( 1 ) holds. In the present paper, we show that: (i) This inequality is frequently not fulfilled in crystal nucleation processes. (ii) The form and the results of the numerical solution of the set of kinetic equations are not affected by self-consistency corrections. However, (iii) the predictions of the analytical relations for the steady-state nucleation rate and the steady-state cluster-size distribution differ considerably in dependence of whether such correction is introduced or not. In particular, neglecting the self-consistency correction overestimates the work of critical cluster formation and leads, consequently, to far too low theoretical values for the steady-state nucleation rates. For the system studied here as a typical example (lithium disilicate, Li 2 O · 2 SiO 2 ), the resulting deviations from the correct values may reach 20 orders of magnitude. Consequently, neglecting self-consistency corrections may result in severe errors in the interpretation of experimental data if, as it is usually done, the analytical relations for the steady-state nucleation rate or the steady-state cluster-size distribution are employed for their determination.

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

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 an 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. Advancing the methods of theoretical analysis of elastic stress effects on nucleation, 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 of 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. If these effects are properly accounted for, then CNT appropriately describes both the initial, the intermediate, and the final states of crystal nucleation.

Diffusion coefficients for crystal nucleation and growth in deeply undercooled glass-forming liquids.

We calculate, employing the classical theory of nucleation and growth, the effective diffusion coefficients controlling crystal nucleation of nanosize clusters and the subsequent growth of micron-size crystals at very deep undercoolings, below and above Tg, using experimental nucleation and growth data obtained for stoichiometric Li2O.2SiO2 and Na2O.2CaO.3SiO2 glasses. The results show significant differences in the magnitude and temperature dependence of these kinetic coefficients. We explain this difference showing that the composition and/or structure of the nucleating critical clusters deviate from those of the stable crystalline phase. These results for diffusion coefficients corroborate our previous conclusion for the same glasses, based on different experiments, and support the view that, even for the so-called case of stoichiometric (polymorphic) crystallization, the nucleating phase may have a different composition and/or structure as compared to the parent glass and the evolving macroscopic crystalline phase. This finding gives a key to explain the discrepancies between calculated (by classical nucleation theory) and experimentally observed nucleation rates in these systems, in particular, and in deeply undercooled glass-forming liquids, in general.

Pressure dependence of viscosity.

We reanalyze the pressure dependence of viscosity of liquids of constant composition under isothermal conditions. Based exclusively on very general considerations concerning the relationship between viscosity and "free volume," we show that, at moderate values of pressure, viscosity increases, as a rule, with increasing pressure, provided the liquid is in stable or metastable (undercooled) equilibrium states. However, even if the behavior of the viscosity is governed by free volume effects, deviations from a positive pressure dependence are possible, when the liquid's thermal expansion coefficient is negative. We derive an equation that allows one to quantitatively determine the pressure dependence of viscosity, which requires, in the simplest case, only the knowledge of the temperature dependence of viscosity at constant pressure, the thermal expansion coefficient, and the isothermal compressibility of the liquid. As an example, the negative pressure dependence of water in the range of temperatures 0-4 degrees C and of several silicate liquids, such as albite, jadeite, dacite, basalts, etc., could be explained in such a way. Other glass-forming liquids initially (for moderate pressures) show a positive pressure dependence of viscosity that changes to a negative one when subjected to high (approximately GPa) isostatic pressure. A detailed analysis of water and already mentioned silicate melts at GPa pressures shows that, in addition to free volume effects, other pressure induced structural transformations may have to be accounted for in a variety of cases. By this reason, the theoretical analysis is extended (i) in order to describe the pressure dependence of viscosity for systems that are in frozen-in thermodynamic nonequilibrium states (glasses, i.e., undercooled liquids below the glass transition temperature Tg) and (ii) to systems which undergo, in addition to variations of the free volume, pressure induced changes of other structural parameters. In such cases a decrease of viscosity with increasing pressure may occur, in principle, even if the thermal expansion coefficient is positive. In this way, the present analysis grants a general tool to estimate the pressure dependence of viscosity and supposedly settles the controversy in the current literature.

Dynamics of first-order phase transitions in multicomponent systems: a new theoretical approach.

In the theoretical description of nucleation-growth processes, currently Gibbs's classical thermodynamic theory of heterogeneous systems is predominantly employed for the description of the properties of the clusters. However, Gibbs's approach does not make it possible to describe, in general, the properties of critical clusters (determining the rate of nucleation) in a sufficiently correct way. Moreover, Gibbs's approach is restricted by its applicability to thermodynamic equilibrium states exclusively. For this reason, it does not give a theoretically founded prescription for a determination of the possible states of clusters of sub- and supercritical sizes in dependence on supersaturation and size of the clusters. In order to overcome these shortcomings, in recent years a generalization of Gibbs's classical approach has been developed and employed for the description of nucleation processes. This generalization of Gibbs' classical method leads, for a variety of different applications, to dependencies of the work of critical cluster formation on supersaturation, which are qualitatively and widely even quantitatively in agreement with density-functional computations. The theoretical methods and results are summarized in the first part of the present paper. They are then extended for the first time to a description of processes of growth of single clusters and ensembles of clusters. In order to fulfill this task, a new method for determination of the parameters of sub- and supercritical clusters is developed. It turns out as the result of the analysis that a variety of thermodynamic and kinetic parameters, determining cluster growth, become dependent on cluster size as well. The results are illustrated for a model system (segregation in regular solutions) and applied to the interpretation of experimental results on segregation processes in solutions and crystallization processes in glass-forming melts. It is shown that the newly developed approach resolves a variety of problems in the interpretation of experimental data on the kinetics of phase formation processes which could not be given a satisfactory explanation so far.