Kinetic Analysis of Nonisothermal Crystallization
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clusters follow nucleation kinetics and above which they grow with a macroscopic growth velocity[ 1]. These restrictive assumptions fail to describe the rich behavior underlying real phase transformations, however. The model presented here systematically follows the time-dependent development of clusters from the smallest sizes to macroscopic ones. Only a brief description of the essential features of the computer model can be provided here; more detail is presented elsewhere [2-5]. Time Dependent Nucleation Within the classical theory of nucleation, the cluster size distribution is generally taken to be time-independent, characterized by a steady-state nucleation rate. For metastable states that are formed by rapidly changing the temperature, however, there is frequently insufficient time to maintain this steady-state cluster distribution, resulting in a nucleation rate that can be orders of
magnitude different from that expected. With annealing time, the distribution relaxes to the steady-state one and the nucleation rate approaches its steady-state value. Such transient behavior is common (see [6]); it has even been observed in heterogeneously nucleating systems[7-8]. It is also not inconsequential; transient nucleation can dramatically effect phase formation and stability and is likely critical for glass formation in some metal alloys[3]. Nucleation is a result of the coupled evolution of clusters by the addition or loss of a single molecule at a time. The time-dependent cluster density, N,,, is obtained by solving the coupled differential equations describing cluster growth: d
= - Nfl
dtn
1,
k,_
_ [ NJC 1 - + Nýt k,+ ] n
+
(1)
Nn+l, k-+l tn1
The rate of cluster evolution is determined by the thermodynamic driving force, related to the reversible work of cluster formation, Wn, and the monomer mobilities. Assuming spherical clusters, W, = n6p, + (36n)113
2 3
/V
n 213
ai
(2)
where 5[i is the Gibbs free energy per molecule of the new phase less that of the initial phase, v is the molecular volume and a is the interfacial energy per unit area. The forward and backward rate constants, k,+ and k- respectively, are related to the diffusion coefficient, D, in the initial phase 3 6gnJ kn± - 24cK2 D exp f T= 2k,-----)
(3)
Here, 5g, is the free energy of a cluster ofn+1 molecules less that of a cluster containing n molecules, k is an atomic jump distance, cc+ = n and cc. = (n-1) (assuming a spherical cluster), T is the temperature and kB is Boltzmann's constant. The nucleation rate is defined as the flux in cluster-size space; in the most general case it is a function of both time and the cluster size at which it is measured, Iý't = N n,t kn* - N ,n.I,, k •,.-+
414
(4)
Computer Calculation A phase transition is simulated by dividing the transformation time into a series of short isothermal intervals, At, over which new nuclei appear and previously generated nuclei grow. For nonisothermal transformations, At = AT/c, where (Dis the scan rate in °C/s and AT is the temperature step size, generally 1C. T
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