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# Introduction.

The Kolmogorov [1], Johnson-Mehl [2], and Avrami[5][4][3] (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[10][9][8][7][6][5]. These assumptions are met approximately for a large number of first order phase 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 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 and and integrated this expression over time. Jackson ([12] focussed on calculating the probability that a point in one dimension is not transformed at a time 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 ,[13][1] 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),[14][2][1] and allowing for some form of growth anisotropy (Some of the limitations on the functional form of anisotropy growth rates were correctly recognized,[9][1] but the Kolmogorov's quantitative treatment was incorrect.). Kinetics during heating and cooling have been treated exactly by assuming that both the nucleation and growth rates depend on a time dependent temperature, in a specimen which is assumed isothermal at each instant. Such a theory for idealized cooling or heating experiments is widely used both for research purposes and for practical industrial applications.

Some sort of uniformly random spatial aspects (also called homogeneous in the statistics papers) of the nucleation seemed central to obtaining exact results in the original theory.[7] To avoid confusion in this paper, homogeneous nucleation and heterogeneous nucleation on randomly dispersed sites in a volume will be called ``volume'' nucleation, heterogeneous nucleation on surfaces and lines will be called ``surface'' and ``line'' nucleation. Homogeneous will denote uniform probability. Spatial homogeneity is met by the assumption of an infinite system with spatially uniform nucleation rates. But it can be easily shown that the theory can be extended to cases of surface or line nucleation if certain symmetries are present; each point on a surface or curve, to which the nucleation is confined, must have identical surroundings and nucleation rates. Thus, an exact theory can be made for nucleation on surfaces, such as planes, spheres, and on curves, such as lines, circles, and even helixes.[15] In an unpublished appendix to their original paper Johnson and Mehl gave an exact solution for a spherical specimen in which all nucleation was random but confined to the sphere surface.[16] In this finite domain example, exact results are obtainable because there is a stochastic homogeneity to the heterogeneous nucleation sites, and a related homogeneity in the same spherical boundary to the specimen that stops the growth. (Every grain nucleating at the same time has the same statistical shapes; the same chances of impingement with other grains and with the specimen surface.) It is apparent that the method Johnson and Mehl used would work equally well on nucleation confined to a single spherical surface in an infinite domain with inward and outward growth, or to outward growth from nuclei on the surface of a single void. This aspect of the theory permitted adaptation to random nucleation confined to a planar surface, either within an infinite space or bounding a half space, or to a line within an infinite space or at the apex of an infinite wedge, and these exact results were used in an approximate theory of grain boundary nucleated reactions.[17]

When there is a surface bounding the specimen domain the stochastic method of correcting the multiple counting in the extended volumes used by Johnson and Mehl fails for volume nucleation. Nonetheless good approximate solutions have been found for certain ratios of the parameters and for many specimen geometries that are accurate at short or long times. For example, at times that are long compared to the time it takes for a grain to grow across the thickness of a thin wire or thin film, one or two-dimensional JMAKs result becomes respectively valid approximations. Results for other geometries, such as spheres with volume nucleation, have been obtained by simulation[18] and for time dependent nucleation and growth.[19] Many of these geometries can be easily handled analytically by the theory presented in this paper.

Finite specimen geometries result in a spatial stochastic inhomogeneity. But since the nucleation is already inhomogeneous in time,[7] the insistence on spatial homogeneity would seem to be unwarranted. In this paper we explore an alternate way of derivation that allows for exact results under a much greater set of assumptions, including most cases of specimen geometry more complex than thin films and wires, and even cases where the specimen geometry changes with time as in the case of a growing film. To illustrate the method for dealing with problems in specimen geometry, we will keep other factors in this paper simple; nucleation and growth rates are kept constant in space, but not in time, and the growth rates is assumed isotropic as well. Results are expected to depend on position, which can be averaged to compare with experiments that measure averages or with results for infinite uniform systems. We will see that many approximations in the literature were unnecessary and some clearly wrong.

To demonstrate this method we begin with a rederivation of the classical JMAK result for volume fraction transformed in any dimension in an infinite specimen, and then extend the method for finite specimens. In a subsequent paper we will also include cases of spatial inhomogeneity, where nucleation and growth rates are functions of position as well as time and where growth rates can have a limited anisotropy.

Next: A rederivation of Up: THE TIME CONE METHOD Previous: THE TIME CONE METHOD

cahn@CTCMS.nist.gov
Wed Feb 14 17:48:17 EST 1996