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]