The Time Cone method for Nucleation and Growth Kinetics on a Finite Domain
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Introduction.
The Kolmogorov [1], Johnson-Mehl [2], and Avrami [3, 4, 5] (JMAK) theory of nucleation and growth reactions is the earliest example of an exactly solvable, although phenomenological and stochastic, kinetic model of a first order phase transformation. The model assumes an infinite specimen that is untransformed at time 0. It posits a given stochastic rate (number per unit untransformed volume per unit time) of creation of point nuclei that are randomly distributed in the remaining untransformed space. A grain is assumed to grow radially at a given constant rate from the moment of creation of each nucleus until impingement with other growing grains. Growth ceases at all points of impingement. Because of the statistical homogeneity, exact solutions were given for many aspects of the phase transformation; among them the volume fraction transformed as a function of time, the number of grains (equal to the number of nuclei), the unimpinged and impinged surface area, the length of triple junctions and the number of point junctions of four grains in the final structure[5, 6, 7, 8, 9, 10]. These assumptions are met approximately for a large number of first order phase
425 Mat. Res. Soc. Symp. Proc. Vol. 398 0 1996 Materials Research Society
transformations. The theory has been widely used. Its simplicity has also led to extensions for first order transitions where the theory is clearly not even approximately applicable. [11] At least three general methods of deriving results are in the literature. Johnson and Mehl and Avrami focussed on computing the fraction transformed. This was done by calculating an extended volume fraction transformed X. by assuming continued nucleation in the entire volume and unimpeded intergrowth, ignoring impingement, and making an exact statistical correction for the multiply counted transformed regions. Kolmogorov examined the probability that a point that had not transformed would transform in the time interval between t and t + dt and integrated this expression over time. Jackson [12] focussed on calculating the probability that a point x in one dimension is not transformed at a time t by computing the probability that no nucleus had formed earlier that would have led to transformation at that point. The method he used will be extended to finite specimens in this paper. Meijering also mentions this method briefly and did not make much use of it.[6] Activities on extending the theory began with the original articles. The requirements for an exact theory permitted relaxing many of the assumptions with little modification of the theory; letting nucleation rates to be time dependent, which includes assuming that nuclei are present at time 0,[1, 13] confining nucleation to a set of randomly placed points, [4] letting the growth rate of a grain depend on time (but not on the time that the grain nucleated),[1, 2, 14] and allowing for some form of growth anisotropy (Some of the limitations on the functional form of anisotropy growth rates were correctly recognized,[1, 9] but the Ko
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