Vitaly A. Shneidman

Crystallization kinetics and the JMAK equation

Abstract In the present work the application of the Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory for the calculation of the volume fraction crystallized is discussed for several particular cases of isothermal transformations. In particular, the following three situations, for which the JMAK theory requires extensions, are considered: (1) finite size effects and non-uniform nucleation, (2) anisotropic particle formation, and (3) transient nucleation. We present new equations which describe these three situations. In general, we find that anisotropic particle formation, finite size effects and non-uniform nucleation lead to a reduction of the crystallization rate. Furthermore, transformations which produce anisotropic particles are characterized by reduced values of the Avrami exponents. Finally, we demonstrate that corrections to the JMAK t4 law arising from time dependent nucleation must include size-dependent growth effects to obtain a logically consistent result.

Induction time in transient nucleation theory

We derive an exact expression for the induction time associated with the Zeldovich nucleation equation in terms of rapidly convergent integrals. Also, we consider an asymptotic approximation of the exact expression which is obtained in the limit of a high nucleation barrier and derive explicit formulas for several widely employed nucleation models. We demonstrate that analytical results are in excellent agreement with numerical results of the present and previous studies.

On the applicability of the classical nucleation theory in an Ising system

Large-scale dynamic Monte Carlo simulations of a lattice gas on a 2000×2000 square lattice with a Glauber-type spin flip dynamics were performed. The results are discussed in the light of classical nucleation theory (CNT) which can be fully specified for the problem due to the availability of exact values for the interfacial energy of a large nucleus, known from the Onsager solution. Several alternative (field-theoretic or nonclassical) descriptions were also considered. Special attention was paid to the pre-exponential in the cluster distribution function and to the finite-size corrections to the interfacial energies which are required in order to comply with observations. If taken literally, the CNT produces large errors when predicting either the cluster distribution function or the nucleation rate. However, at intermediate temperatures (up to 0.7 Tc) the correspondence can be substantially improved by considering the low-temperature properties of small clusters and adjusting the pre-exponential. At hi...

The fast cooling/heating rate effects in devitrification of glasses. II. Crystallization kinetics

An analytical description of devitrification kinetics induced by time-dependent nucleation and growth of crystallites during a quench-heating cycle is proposed. Relevant experimental situations include differential thermal analysis (DTA) or differential scanning calorimetry (DSC). The proposed description involves very few assumptions regarding the temperature dependence of various kinetic parameters, but rather employs measured values of growth rates. It is shown that the conventional description of nucleation based on the steady-state approximation, as a rule, is inapplicable for the description of the DTA/DSC experiment for experimentally reasonable quench/heating rates. The latter is confirmed by the analysis of available experimental data on o-terphenyl and lithium disilicate.

The effects of transient nucleation and size-dependent growth rate on phase transformation kinetics

Abstract The problem of the kinetics of new phase formation by a nucleation and growth mechanism is considered. A correction to the classical t 4 law for the total volume fraction transformed which results from non-constant nucleation and growth rates is derived. The main qualitative difference from previous treatments is that the effects of transient nucleation and size-dependent growth rate are shown to be intimately related. This relation results in a much larger correction than anticipated. An elementary treatment of the problem is presented, and then the results are confirmed by rigorous analytical and numerical evaluation of the third moment of the distribution function.

Transient nucleation induction time from the birth–death equations

For the set of finite‐difference equations of Becker–Doring an exact formula for the induction time, which is expressed in terms of rapidly convergent sums, is presented. The form of the result is particularly amenable for analytical study, and the latter is carried out to obtain approximations of the exact expression in a rigorous manner and to assess its sensitivity to the choice of the nucleation model. The induction time, tind, is found to be governed by two main nucleation parameters, Φ*/kT, the normalized barrier height, and g*, the number of molecules in the critical cluster. The ratio of these two parameters provides an assessment of the importance of discreteness effects. We study the exact expression in both the continuous (g*→∞) and the asymptotic (Φ*/kT→∞) limits. Asymptotic results for tind are compared with those previously reported from simulation studies as well as with tind obtained numerically from the exact expression in the present study. Also, the accuracy of the Zeldovich equation, w...

Crystallization of rapidly heated amorphous solids

Abstract Traditional theoretical approaches to the description of a non-isothermal phase transformation view its initial stage-nucleation - as a quasi-steady-state (QSS) process. These approaches are approximate, however, and may fail completely even for quite moderate heating (cooling) rates due to very large internal relaxation times of the amorphous state. A simple way to adjust the QSS approximation to account for strong time-dependent nucleation effects during rapid heating of amorphous solids with no quenched in nuclei is proposed. In combination with the approximate expression for the growth of nucleated particles, this leads to explicit closed formulas for the rate of phase transformation and the volume fraction transformed. Results are tested by comparison with numerical solutions of the time-dependent Becker-Doring nucleation equation, and it is demonstrated that the present method provides highly accurate corrections to the QSS approximation.

Scaling properties of induction times in heterogeneous nucleation

In the asymptotic limit of a high nucleation barrier we obtain the heterogeneous‐to‐homogeneous induction time ratio as a function of the contact angle. The result is in excellent agreement with the data of numerical simulations by Greer et al. [J. Cryst. Growth 99, 38 (1990)]. An expression to describe individual (nonreduced) induction times is also obtained.

Theory of time‐dependent nucleation and growth during a rapid quench

Phase transformation via the nucleation and growth mechanism during a rapid cooling (quench) of a glass‐forming liquid or melt is considered. Traditional approaches here are based on the quasi‐steady‐state (QSS) approximation for nucleation and the assumption of a size‐independent growth. However, the QSS approach becomes invalid if the dimensionless rate of the barrier change, n=−τ∂(W*/kT)/∂t (τ is the inner time scale of the nucleation process) is not vanishingly small. For such ‘‘strongly time‐dependent’’ situations an asymptotic (singular perturbation) technique of matching the time‐ and size‐dependent nucleation and growth solutions is elaborated, and an explicit expression to describe the distribution function of large particles is derived. Formally, the results can be reproduced by the QSS approximation with the steady‐state nucleation rate multiplied by an n‐dependent factor. Analytical treatment is tested against numerically exact solutions of the nucleation (Becker–Doring) and growth equations.

Transient nucleation distributions and fluxes at intermediate times and sizes

General interpolating expressions, valid for near- and arbitrary overcritical sizes of clusters, are proposed for the nucleation fluxes and distributions. Results are expressed in terms of the deterministic growth rates, and are characterized by a non-Gaussian dependence on the size of nuclei. In a sense, the proposed approximations combine the positive aspects of the parabolic model by Trinkaus and Yoo [Philos. Mag. A 55, 269 (1987)] and of the boundary layer (“matched asymptotic”) solution earlier described by the author [Sov. Phys. Tech. Phys. 32, 76 (1987); 33, 1338 (1988)]. Specifications of the general results are made for several mainstream nucleation models via selection of appropriate growth rates. Examples include surface- and diffusion-limited nucleation in the continuous (Zeldovich–Frenkel) and discrete (Becker–Doring, Turnbull–Fisher) versions of the nucleation equation.