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Specific Geometries.

Let us next examine some specific geometries.

1. Transformation near the planar surface of a semi-infinite solid, half plane or half line.

If we look at the transformation for points on the surface, we have exactly halved the expected number of nuclei. Thus the expected fraction untransformed at the surface is the square root of what it would be deep in the interior. For 3-d this would be

Indeed equals the as given in equations (2.14). More generally for any we compute the volume of a cone truncated by a plane parallel to its axis. If the cone is not truncated and the surface has no effect on reducing the amount of transformation. If the cone is truncated and its volume is Defining a reduced distance from the specimen surface we obtain for

which may be rewritten

If the correction factor

when Because the lead term in the correction factor has the same Avrami exponent of 4 a plot of versus will have two parallel asymptotes. At any time a plot of will show a surface zone of reduced transformation with limiting values in the square root relation.

2. Transformation at a point in an infinite thin film of thickness .

Without loss of generality we let and compute the volume of a cone truncated at and by up to two planes parallel to its axis. If (and as well) is greater than the film surfaces have no effect on reducing the transformation and equation (2.14) holds. If , but ), the cone intercepts only one surface, and the reduction in transformation is as in the semiinfinite case, equation (3.2). If we have to subtract the effects of two planar truncations and obtain an equation whose domain of validity spreads from the center of the film when and reaches the surfaces when

Although the Avrami time exponent of has disappeared when , the behavior has not quite become 2-dimensional since there remains a correction factor as seen in the remaining terms with lower time exponents.

The product can be considered the nucleation rate per unit area of film; then the time dependent factor is a correction for the true three- dimensionality of the problem. Letting , and a reduced time we obtain

While this correction factor to increases with time, its effect on decreases. (Although the lead time exponent is that of two-dimensional kinetics, the results are bracketed between the two and three dimensional JMA expressions.)

In many experiments an average over or is wanted

This is readily computed numerically from the above expressions. However since is a smooth curve, Simpson's Rule, with three points, ,

should be an adequate approximation. Until , this approximation takes on the form Thereafter equations (3.2 and 2.14) must be used.

3. Transformation after deposition of a continuously growing specimen.

To illustrate that there is no need to fix specimen boundaries in time, we consider steady continuous deposition followed by transformation. We assume steady deposition of a one phase, say a glass, since infinitely long ago, and that this phase transforms, say devitrifies, with JMA kinetics. Let the surface of the specimen be given by with , and consider the transformation at The time cone is now truncated with a sloping hyperplane When is infinite this plane becomes the base, of a right truncated cone and the result should correspond to the classic JMA. When the cone volume is infinite and transformation is complete everywhere in the specimen. (When the plane is one of the truncation planes of the half space problem.)

The volume of the truncated time cone is Because of the translational invariance we need only study for and

Something like a Lorenz time dilation by a factor of is showing up in this problem with , instead of , behaving like the velocity of light in relativity. As approaches , the intersection of the time cone with the nucleation domain diverges, and the transformation goes to the completion at the growing surface. This parallel with the Lorenz factor fails in surface nucleation in which the time dilation is [15]

The deposition of a thin film growing with a velocity beginning at from a substrate at followed by transformation, is computed with an additional truncation of the cone by the plane . There is no difficulty with this, but two expressions are obtained; one is equation(3.10) when .[15]

Next: Discussion. Up: Generalizations of the Previous: Generalizations of the
Wed Feb 14 17:48:17 EST 1996