Time-Dependent Nucleation and Growth of Crystalline Phase During a Rapid Quench into a Glassy State
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ABSTRACT Corrections to the standard expressions for the volume fraction of the crystalline phase which arise due to time-dependent nucleation effects are considered both for the isothermal (transient) and nonisothermal (quench) situations. Analytical results are tested against numerically exact solutions of the Turnbull-Fisher nucleation equations. Physical consequences of the obtained expressions are discussed. INTRODUCTION The fundamental question in the description of phase transformation kinetics via the nucleation and growth mechanisms is the determination of the volume fraction of the crystalline phase, X. The latter, according to Johnson, Mehl, Avrami and Kolmogorov (JMAK) [1] can be expressed through the "extended volume" as X = 1 - exp (-X.t). When evaluating X,,t two situations should be distinguished depending on whether one observes the phase transformation isothermally or in the course of a quench. The former situation is described by the celebrated "t 4 law" 34 = I Ut (1) 3 with Ig being the steady-state nucleation rate [2] and U the growth rate of crystallites which, for simplicity, is treated as isotropic [3]. The second situation corresponds to a continuous change in temperature with some finite rate S = -dT/dt with temperature-dependent values of IJt(T) and U(T). Here the standard expression is given by
47r XjQsjS(T)
4irTS-
dT'Int
)
rf dT"U (T")
(2)
ilt (T")Li
with Ti, being the temperature from which the quench starts, and the superscript "QSS" indicating the quasi-steady-state approximation when the nucleation rate does not depend on the quench rate, S. Equations (1) and (2) currently form the basis for theoretical description of practically every experimental study of crystallization kinetics via the nucleation and growth mechanisms. Potential limitations of these equations are due to the assumption that the nucleation rate is given by I = Ist, but are also due to the assumption of a sizeindependent growth rate which is employed in their derivation [1]. In reality, however, the growth rate, R, of a single crystallite depends on its radius, R, turning negative below the critical size, R.. Thus, the growth rate U in eqs. (1) or (2) is, in fact, the large size limit of R(R) so that the standard description is valid in situations when the characteristic size is much larger then R.. 99 Mat. Res. Soc. Symp. Proc. Vol. 398 ©1996 Materials Research Society
To proceed beyond the standard approximations in evaluation of X,,t one thus needs to account for both time-dependent nucleation and size-dependent growth effects. A numerical realization of such an approach for the case of a quench was first performed by Kelton and Greer [4,5]. Here we intend to present recent analytical results on the values of X,.t in situations when eqs.(1) or (2) loose their accuracy and to understand their sensitivity to the type of the nucleation-growth model employed. In addition, we test the results against numerical data of the type obtained in Ref. [4] and related studies. Another goal of the paper is to clarify
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