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# Discussion.

The classical JMAK results have been rederived using the time cone method, which does not make use of the conceptual artifacts of extended volume and ghost nuclei that are part of many earlier derivations of the exact theory. As expected the new method gives exactly the same result, but it permits the derivation of exact results for a whole class of inhomogeneous problems. In this paper we have illustrated this with the finite specimen examples of the last section.

It should not be surprising that identical results are obtained by two exact theories for the same model, JMAK and the time cone method. What is surprising is that two different physical concepts, extended volume and expected number of nuclei in the time cone, have identical expressions, i.e. we have found that . That this equality is valid for a time dependent and can be seen by comparing equation (2.7) with the corresponding one for in the literature.[1],[11]

In recent years there have been mistaken criticisms [25][24] of the Johnson-Mehl and Avrami derivation, specifically that only nucleation in the untransformed volume should be used in computing the extended volume, that ``ghost'' nucleation should be excluded while keeping the correction for impingement. These recently ``revised'' calculations are purported to be the exact results of the same transformation model as JMAK and were shown to fit some experiments much better than JMAK at late times. However if an experiment does not fit the original JMAK we must look for physical reasons and try to create a new model that is valid for this experiment, rather than make an exact theory inexact. Because the time cone method does not rely on either of the two constructs that are part of the extended volume, the results can be viewed as an independent confirmation of the correctness of the original JMAK derivation.

The importance of an exact theory is that any disagreement between theory and experiment points to a failure of the experiment meeting one of the physical assumptions or vice versa; the construction and the use of the extended volume are not physical assumptions, but part of a technique of finding an exact solution for a particular set of physical assumptions. The agreement between the erroneous JMAK theory and experiment can only be fortuitous. Some physical assumption is not met in the experiment and understanding is advanced by tracking down the reason: nucleation may not be random or its rates may not be constant in time, growth rates may not be constant in time, grains may influence each other through diffusion or stress fields; etc. Because the time cone method does not rely on either of the two constructs that are part of the extended volume, the results can be viewed as an independent confirmation of the correctness of the original JMAK derivation.

Plotting versus is an attempt to fit the transformation rate to a JMAK theory. The slope of such curves is the exponent of time in the classical theory, sometimes called the Avrami exponent, and equals the dimension plus one. In some analyses this exponent, determined from a linear fit to experiment, is used to attempt to ascertain information about the mechanism of the transformation, or to determine time exponents of algebraic fits to either the nucleation rate [13] or growth rate.[11] This is rarely valid without some independent measurement of nucleation or growth rates. Sometimes the analysis of the data is even used for experiments in which diffusion controls the rate of growth; overlapping diffusion zones from different grains affect growth and affects the nucleation process in ways that invalidate the stochastic independence. Nonetheless in the very early stages independent nucleation and growth can be assumed. The early growth rate is assumed time dependent, proportional to . For such processes, the exponent of is interpreted as implying that all nuclei are present at the beginning, and that nucleation continues at a constant rate (unlikely at best). Many ad hoc attempts to correct for such effects have been put forth.[27]

The importance of the exact JMAK theory is for late stages; the failures of the assumptions show up only later. For the early stages, when nuclei are few and too far apart to affect each other, much simpler theoretical methods suffice, even for transformations that deviate significantly from JMAK assumptions. The exponents have some meaning during the early stages of the transformation when nucleation and growth of each grain proceeds independently of the others; when there is little impingement, corrections for it are unnecessary. The fraction transformed equals the extended volume fraction , and both equal , only when there is little transformation. Then if nucleation and growth rates can be expressed as algebraic functions of time, the time exponents will be a sum of the time exponent of the nucleation rate plus the dimension multiplied by the time exponent of the growth, regardless of whether the JMAK assumptions are met. But at late times the differences show up. For example, an exact theory for nucleation followed by diffusional growth in which depends initially on gives an Avrami exponent of for the approach to equilibrium (neglecting surface energy and hence coarsening too) in any dimension, regardless of many assumptions about the nucleation rate (e.g. whether nucleation occurs at a constant rate or all nuclei are present at ). The change of time exponent from or to at late times is without a change in mechanism.[28]

Specimen boundaries introduce a spatial inhomogeneity into the problem whose influence spreads with time. Time inhomogeneity was inherent in the original JMAK problem, and indeed the JMAK methods could easily deal with nucleation and growth rates with arbitrary time dependencies. The cone method can handle time inhomogeneity as easily as a wide variety of spatial inhomogeneities. In this paper we kept the specimen convex in order to make the cone construction depend only on growth rate and not on the specimen boundary. We have allowed the specimen to change its surface in time as well. A more general cone method can allow for an anisotropic growth rate that depends on position and time in a specimen that is not convex in space and time.[15] In this paper we have made the boundary surface inactive as a nucleation site. The surface region has less transformation, small systems transform more slowly. Often the opposite is found when the surface is a source of nuclei. This was considered approximately forty years ago[17] and will be reconsidered with the cone method in a subsequent paper.[29]

What are the essential features of when the cone method can give exact results? For the construction of the cone, it is clear that the growth rate of a grain can be dependent on time, space and the orientation of the normal, but should not depend on the presence of any other grain until impingement. Here the assumptions that lead to geometric growth and the method of characteristics have much to contribute to the theory. For computing the nucleation rate at a point can be dependent on time and the coordinates of that point (space dependent). The properties of stationary or moving specimen boundaries matter; they cause no difficulty if they are is simply places where growth in certain directions is limited.

Many relaxed assumptions are still sufficient for the required stochastic independence for calculating . The boundary simply defines the space outside the specimen where nucleation does not occur; again since nucleation can be space and time dependent, it does not matter whether the specimen boundary is stationary or moving, and whether or not it is a source of nuclei. In the JMAK assumptions a nucleation event should not depend on nucleation or growth of any other grain until the growth of some grain overruns that point. This apparent interdependence of nucleation events may have been at the heart of some of the erroneous rederivations.

For transformation for which all the JMAK assumptions are met, except for the finite size of the specimen, the cone method will provide rigorous results. But while the cone method will also permit obtaining exact results with other relaxations of the assumptions, many first order phase changes fall outside. Solidification when the matrix is fluid and the growing grains can settle, as in a casting or porphyritic mineral, is an obvious example where the JMAK results continue to be applied with doubtful validity. [26][10] On the other hand settling may be negligible in spherulites growing from a viscous polymer, or spherical eutectics growing in a thin layer of organ pipe metal. A martensitic transformation in which the various components of the stress fields of existing plates or rods (``grains'' of martensite) can both increase and decrease the nucleation of new plates is another example too far from the JMAK assumption.

Failure of some of the assumptions combined with the cone method may lead to exact bounds on some kinds of transformation. For diffusion controlled or disallowed anisotropic growth it may still be possible to construct the cone to include all points in time and space which will lead to transformation at if no grains growing from nuclei outside the cone interfered. But such interference does occur through the diffusion fields and can only lead to a reduction of the amount of transformation. The general case of anisotropic growth provides another example of such a bound. Having statistical noise in might be treated with a cone with a fuzzy surface. The method does seem to have enough flexibility to warrant reexamining my cases.

Acknowledgements

I am most grateful to many of my colleagues, especially Craig Carter, A. A. Chernov, Sam Coriell, Fern Hunt, Jeff Mcfadden, and Jean Taylor, for their generous help and valuable criticism. I am happy to acknowledge support of ARPA.

Next: References Up: THE TIME CONE METHOD Previous: Specific Geometries.

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