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\title{\bf {THE TIME CONE METHOD FOR NUCLEATION AND GROWTH KINETICS ON
A FINITE DOMAIN.}}

%\bigskip
\author{John W. Cahn\\
Materials Sci. and Eng. Lab.\\
NIST\\
Gaithersburg, MD 20899} 
\date{}
\newcommand{\Oc}{{\Omega_c}}
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\noindent




\newcommand{\ds}{\displaystyle}
\maketitle

\begin{abstract}
The Kolmogorov-Johnson-Mehl-Avrami theory is an exact statistical
solution for the expected fraction transformed in a nucleation and
growth reaction in an infinite specimen, when nucleation is random in
the untransformed volume and the radial growth rate after nucleation
is constant until impingement.  Many of these restrictive assumptions
are introduced to facilitate the use of statistics.  The introduction
of "phantom nuclei" and "extended volumes" are constructs that permit
exact estimates of the fraction transformed.  An alternative, the time
cone method, is presented that does not make use of either of these
constructs.  The method permits obtaining exact closed form solutions
for any specimen that is convex in time and space, and for nucleation
rates and growth rates that are both time and position dependent. 
Certain types of growth anisotropies can be included.  The expected
fraction transformed is position and time dependent.  Expressions for
transformation kinetics in simple specimen geometries such as plates
and growing films are given, and are shown to reduce to expected
formulas in certain limits.
\end{abstract}

\setcounter{section}{0}


\section{Introduction.}
\setcounter{equation}{0}

The Kolmogorov \cite{K}, Johnson-Mehl \cite{JM}, and
Avrami\cite{A1,A2,A3} (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\cite{A3,Mei,Mi,Gil1,Gil2,Bel}.  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. \cite{Review}

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 $X_e$ 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 (\cite{Jackson} 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.\cite{Mei} 

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 $0$,\cite{K,Hull} confining nucleation to a set of
randomly placed points, \cite{A2} letting the growth
rate of a grain depend on time (but not on the time that the grain
nucleated),\cite{K,JM,Weinberg} and allowing for some form of growth
anisotropy (Some of the limitations on the functional form of
anisotropy
growth rates were correctly recognized,\cite{K,Gil2} 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.\cite{Mi} 
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.\cite{C1}  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.\cite{JMapp}  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.\cite{C}  

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\cite{Levine} and for time dependent nucleation
and growth.\cite{Kelton}  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,\cite{Mi} 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.  

\section{A rederivation of the classical JMAK results for volume
fraction transformed.}
\setcounter{equation}{0}
We assume that the system is an infinite line, planar surface, or
a three or higher dimensional volume, in which there is no
transformation at time 0, a constant (or time dependent) nucleation
rate in the untransformed volume after time 0, and that each
nucleus grows radially with a constant (or time dependent) rate
until growth ceases at each point of impingement.  The Johnson and
Mehl method,\cite{JM} focusses on exact statistical estimates of
the volume fraction transformed (which is $1-P$, where $P$ is
the fraction untransformed) by calculating the volumes transformed
as if there were no impingement and no reduction in nucleation rate
due to the reduction in $P$ and then making exact statistical
estimates for the corrections for impingement and the reduction in
nucleation rate.  In this section we calculate directly the
estimate $P$ that a point $\x$ in the volume will be
untransformed at time $\y$.  This is done by computing the
probability that no nuclei formed at all earlier times $\tau$ and
positions $\w$, that could have grown to reach point $\x$ at
$\y$.  (It has to be none; if there are some, it does not matter
how many.)  

We do this in three steps: 

1. It will be convenient to combine
a d-dimensional space with time into a single space, and define
a d+1 dimensional space $\Omega$ and a point $\omega $
in this space.  In three dimensions, $\omega= (x,y,z;t) = (\x;t).$ 
We then construct for the point $(\x; \y)$ a domain in this space
that is the set of all points and earlier times $(\w; \tau)$ that
would
have caused transformation at $(\x; \y)$ had nucleation occurred. 
This domain will be a subset of $\Omega$ and will be designated by
$\Oc$.  Since its shape is a cone, it will be termed an ``influence
cone'' or ``time cone.''  Its surface
will be called an ``event horizon.''

2. Integrating the nucleation rate over the time cone to obtain
the number of nuclei $<N>_c$ expected in $\Oc$.

3. Making use of the stochastic independence of the nucleation
events to estimate $P$
\begin{equation} \label{Poisson}
 P = e^{-<N>_c}.
\end{equation}

\subsection{The time cone $\Oc$ when $V = V(t)$}

At time $\y$ the radius of a grain that nucleated at time $\tau$ is
\begin{equation} \label{radius}
R(\y,\tau) = \int_{\tau}^{t}V(\tp)d\tp. 
\end{equation}
If the magnitude of the grain
radius $R$ exceeds the distance between $\w$ and $\x$, 
\begin{equation} \label{tdCone}
R(\y,\tau)^2 - | \x-\w|^2 \ge 0,  
\end{equation}
the point $\bf x$ will have transformed before time $\y$.  The time
cone is the set of all points that satisfy this inequality.

The surface of the time cone, which is the event horizon are the
nuclei points $(\x;t)$ that grow as grains that
reach $\x$ exactly at time $t$.  They are given by the points in
$\Omega$ for which the inequality (\ref {tdCone}) is an equality. 
This allows the introduction of the method of characteristic applied
to geometric growth \cite{F,CTH,TCH} into the time cone theory.  Such
growth theories are based on the time $\y(\x)$ of arrival of a moving
surface at a point $\x$.  Thus $\y$ is a function of $\x$. The surface
at any time $\y$ is a level surface of $\y(\x)$.  If the surface is
smooth, the direction of the gradient of $\y$ is parallel to  the
normal to the surface,$\n,$ and its magnitude is $1/V.$  Thus $ \n =
\nabla
\y /|\nabla \y|$, and $|\nabla \y| = |1/V|$.  In a subsequent paper,
we will use characteristics to construct time cones when $V$ is a
function of
$\x,\y$ and $\n$ with only a few restrictions, and derive relations
for the kinetics for specimens that are anisotropic and inhomogeneous
in space and time.  One crucial corollary of the theory is that it can
identify the assumptions necessary so that nuclei from points outside
the time cone never grow into the time cone.  \cite{C1}

If $V$ is constant, equation(\ref{tdCone}) becomes the equation of the
points in a cone, 
\begin{equation} \label{Cone}
V^2(\y-\tau)^2 - | \x-\w|^2 \ge 0.
\end{equation}
Compare this equation with one for the time horizon for the point
$(\x; t)$ in the theory of relativity, $c^2(t-\tau)^2 -|\x -
\w|^2 = 0,$ where $c$ is the velocity of light.
At any time $\tau$ the spaces in which any nucleation event will
affect $(\x,\y)$
are thus:  in 1-d, a line segment of length $2V(\y-\tau);$ in 2-d
a circle with radius $V(\y-\tau);$ in 3-d a sphere with radius
$V(\y-\tau)$; etc., all centered on $\x$, and increasing in size
the further back in time we go.  
With  growth rates that depend on time, the event horizon is a
conical surface of revolution in this space with apex at
$(\x; \y)$ with its axis parallel to the time axis, 
and given by
\begin{equation} \label{Coneset}
\Oc(\x; t) = \{(\w; \tau): R(\y,\tau)^2- |\x-\w|^2 \ge 0\}. 
\end{equation}
The time cone can be considered unbounded in the negative time
direction, to times before the onset of nucleation, if a value is
assigned to $V(t)$.  Because $\alpha = 0$ there, the results do not
depend on what is chosen for $V$.  

\subsection{The expected number of nuclei in $\Oc$}

If the nucleation rate is given by $\alpha(\x; t)$ (number of
nuclei 
expected in unit time and unit d-dimensional ``volume'') the expected
number of nuclei in the time cone is the integral of $\alpha$ over
the time cone in space and time.
\begin{equation} \label{nuclint} 
<N>_c =<N>_c(\x; \y) = \int_{\Oc} \alpha(\w; \tau)d\omega
\end{equation}

When $\alpha = \alpha (t)$ and $V = V(t)$ for $t \ge 0$ and $0$ for
negative time, equation (\ref{nuclint} can be integrated over space.
For  3-dimensions, this gives
\begin{equation} \label{fulltimedep}
<N>_c (t) = \frac{4\pi}{3} \int_0^t \alpha(\tp) R(t,\tp)^3 d\tp.
\end{equation}
This form is useful for heating and cooling experiments.

Equation (\ref {fulltimedep}) simplifies 
even more when $\alpha$ is constant for positive time. 
Then $<N>_c$ becomes $\alpha$ multiplied by the ``nucleation'' volume
$\|\Omega_N\|$, the volume in space-time of the cone where (and when)
there is nucleation, or equivalently the
volume of the intersection of the cone $\Oc$ with the constant
nucleation rate domain of $\Omega$.  Thus when $\alpha$ is constant
for positive time

\begin{equation} \label{nuclcnst}
<N>_c =  \alpha \|{\Omega_N}\|.
\end{equation}  

For the classical JMAK theory this intersection of the cone with the
nucleation space
is a right truncated cone, whose volume 
 $\|{\Omega_N}\|$
is the height $\y$ multiplied by the base $B$ at $t=0$ divided by
$(d+1).$

\begin{equation} \label{classxe}
<N>_c =  \alpha B \y/(d+1).
\end{equation}  

The bases $B$ and the $\|{\Omega_N}\|$  are respectively in 1, 2 and 3
spatial dimensions: $2Vt$ and $Vt^2$; $\pi V^2t^2$
and $\frac{\pi}{3}V^2t^3$; and $\frac{4\pi}{3}V^3t^3$ and
$\frac{\pi}{3}V^3t^4$.  

\subsection{The probability, $P(\x,\y)$, that $\x$ is
untransformed at $\y$.}

The postulated stochastic independence of the nucleation events
make them Poisson distributed.\cite{Poi}  As a result 
\begin{equation} \label{po}
P(\x; \y) = e^{-<N>_c}
\end{equation}
This well-known result can be demonstrated in two ways:  

1. By considering how $P$ is changed when there is a change
$d\omega$ in $\Oc$, 
\begin{equation} 
-dP  = P \cdot \alpha d\omega,
\end{equation}
which is readily integrated to give 
\begin{equation} 
log P = -\int_{\Oc} \alpha d\omega = -<N>_c.
\end{equation}

2. Alternatively we can use the Poisson distribution for finding n
events when the expectation is $N,$
\begin{equation} 
p(n,N) = N^n e^{-N}/n!.
\end{equation}
Evaluated at $n=0$ this gives $p(0,N) = e^{-N},$ which can be
identified
with $P$.


Equation (\ref{po}) combined with equation (\ref{nuclint}), or when
appropriate (\ref{fulltimedep}) or (\ref{nuclcnst}), are the basic
equations of this paper.
When $\alpha$ and $V$ are constant, equation (\ref{nuclint}) may be
replaced with equation (\ref{classxe}), which is identical to the
classical JMAK results for any dimension.  In 3 dimensions this result
is
\begin{equation} \label{JMeq}
 P(\x; t) = e^{-\frac{\pi}{3} \alpha V^3 t^4}.
\end{equation}

\section{Generalizations of the theory to finite convex specimens.}
\setcounter{equation}{0}
The goal of this paper is to explore how the time cone theory can
be made exact when the specimen is finite and convex and when $\alpha$
and $V$ are functions of time alone.  We let the
time cone for the point $(\x; \y)$ continue to be the set of all
points in time and space $(\w; \tau)$, including points outside the
specimen, that, had nucleation occurred
there and then, would have resulted in transformation at the point
$(\x; \y),$ if there were no interference from other grains. 
Extending the cone beyond the specimen boundary is tantamount to
assuming that the grains that might have nucleated outside the
specimen can grow into the specimen, which causes no problem if the
nucleation rate is set to $0$ outside the specimen and at times before
the experiment began. By
assuming that the specimen is convex in shape (and time), we insure
that there is no interference from the specimen boundary from the
nucleation event at $(\w;\tau)$ to the point $\x$ at time $\y$. 
With this assumption, time cones for finite specimens remain exactly
as for an
infinite specimen, equation (\ref {tdCone}); 
if V is constant, a cone in space-time with sides
sloping by $1/V$, and  if V is time dependent a conical surface of
revolution with sides sloping by 1/V(t).  We let the nucleation
rate be whatever it is in the specimen and identically $0$ outside
the specimen.  We then calculate the expected number of nuclei in
the time cone by integrating the nucleation rate as before in equation
(\ref{nuclint}).  If the
nucleation rate is constant in the specimen, the expected number of
nuclei remains simply proportional to the ``volume'' of the
intersection $\|{\Omega_N}\|$
of the nucleation time period in the space of the specimen with the
time cone.  $\|{\Omega_N}\|$ will be a
function of the position $(\bf x ;t)$ of the apex, and the time period
of the nucleation and the geometry of the specimen.  The expected
fraction untransformed will then be given by equation \ref{po}.

\subsection{Specific Geometries.}

Let us next examine some specific geometries.  

{\bf 1. Transformation near the planar surface $z = 0$ 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 
\begin{equation} \label{surf}
 P(0;t) = e^{-\frac{\pi}{6} \alpha V^3 t^4}.
\end{equation}
Indeed $P(0;t)$ equals the $\sqrt {P(\x; t)}$ as given in equations
(\ref{JMeq}).
More generally for any $(z,t)$ we compute the volume of a cone
truncated by a plane parallel to its axis.  If $z \ge Vt$ the cone
is not truncated and the surface has no effect on reducing the
amount of transformation.  If $z<V\y$ the cone is truncated and
its volume is $(\pi/6V)[(Vt)^4+2z(Vt)^3-2z^3Vt+z^4].$ Defining a
reduced distance from the specimen surface $Z = z/Vt$ we obtain for
$0 \le Z \le 1$
\begin{equation} \label{semi}
 P(Z;t) =  e^{-\frac{\pi}{6} \alpha V^3 t^4[1+2Z-2Z^3+Z^4]}, 
\end{equation}
which may be rewritten
\begin{equation} 
%\label{}
 P(Z;t) = f_1(Z;t) e^{-\frac{\pi}{3}\alpha V^3 t^4},
\end{equation}
If $0 \le Z <1$ the correction factor 
\begin{equation} 
%\label{}
f_1 = e^{\frac{\pi}{6}[1-2Z+2Z^3-Z^4] \alpha V^3 t^4};
\end{equation} 
when $Z \ge 1, f_1 = 1.$  
Because the lead term in the correction factor has the same Avrami
exponent of 4 a plot of $log\ log P(z)$ versus $log t$ will have two
parallel asymptotes.  At any time a plot of $log P$ will show a
surface zone of reduced transformation with limiting values in the
square root relation.

{\bf 2. Transformation at a point $z$ in an infinite thin film of
thickness $\zeta$.  }

Without loss of generality we let $0 \le z \le \zeta/2,$  
and compute the
volume of a cone truncated at $t=0$ and by up to two planes parallel
to its axis.  If $z$ (and $\zeta - z$ as well) is greater than $Vt$
the
film surfaces have no effect on reducing the transformation and
equation (\ref{JMeq}) holds.  If  
$z<Vt$, but $(1-z) > Vt$), the cone intercepts
only one surface, and the reduction in
transformation is as
in the semiinfinite case, equation (\ref{semi}).  If $ (1-z) < Vt$  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 $Vt = \zeta/2$ and reaches the surfaces when $Vt = \zeta$
\begin{equation} \label{filmomega}
\|\Omega_N\| = \frac{\pi}{3}V[\zeta(Vt)^3 - (z^3+(\zeta-z)^3)Vt + (z^4
+(\zeta-z)^4)/2]. 
\end{equation}
%Because this expression no longer contains a $t^4$ term, 
Although the Avrami time exponent of $4$ has disappeared
when $Vt = \zeta$, the behavior has not quite become 2-dimensional
since there remains a correction factor as seen in the remaining terms
with lower time exponents. 
\begin{equation} \label{filmp}
 P(z;t,\zeta) = f_2(z;t) e^{-\frac{\pi}{3} \alpha \zeta V^2 t^3}.
\end{equation}
The product $(\alpha\zeta)$ can be considered the nucleation rate per
unit area of film; then
the time dependent factor $f_2$ is a correction for the true three-
dimensionality of the problem. Letting $X_1
= z/\zeta$, $X_1 + X_2 = 1,$ and a reduced time $\theta = Vt/\zeta >1$
we obtain
\begin{equation} 
%\label{}
 f_2 = e^{\frac{\pi}{3}[\alpha\zeta^4/V][(X_1^3 +X_2^3)\theta +( X_1^3
+X_2^3)/2]}.
\end{equation}
While this correction factor to $P$ increases with
time, its effect on $log P$ 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 $z$ or $P$ is wanted
\begin{equation} 
%\label{}
<P> = \frac{1}{\zeta}\int_0^\zeta P(z)dz.
\end{equation}
This is readily computed numerically from the above expressions. 
However since $P(z)$ is a smooth curve, Simpson's Rule, with three
points, $z = 0, \zeta/2, \zeta$,
\begin{equation} \label{apx}
<P> = \frac{1}{3}(P(0) + 2P(\zeta/2))
\end{equation} 
should be an adequate approximation.  Until $t = \zeta/2V$, this
approximation takes on the form $$<P> = \frac{1}{3}((1 +
e^{-\frac{\pi}{6} \alpha V^3 t^4})^2 - 1).$$
Thereafter equations (\ref{semi} and \ref{JMeq}) must be used.

{\bf 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 $ z_s = V_s \tau$ with
$V_s>V = \beta V_s$, and consider the transformation at $z \le
V_s\tau.$    The time cone is now
truncated with a sloping hyperplane $ z = V_s \tau.$  When $V_s$ is
infinite this plane becomes the base, $\tau = const = 0,$ of a right
truncated
cone and the result should correspond to the classic JMA.  When $V \ge
V_s$ the cone volume is infinite and transformation is complete
everywhere in the specimen.  (When $V_s = 0$ the plane is one of the
truncation planes of the half space problem.)  

The volume of the truncated time cone is $(\frac{\pi}{3}) V^3 (t-
z/V_s)^4/(1-\beta^2)^2.$
Because of the translational invariance we need only study $z= 0$ for
$t \ge 0,$
and 
\begin{equation} \label{Lorenz}
 P(0;t) = e^{-\frac{\pi}{3} \alpha V^3
(\frac{t}{\sqrt{1-\beta^2}})^4}.
\end{equation}
Something like a Lorenz time dilation by a factor of
$\sqrt{1-(V/V_s)^2}$ is showing up in this problem with $V_s$, instead
of $V$, behaving like the velocity of light in relativity.  As $V$
approaches
$V_s$, 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
$(1-(V/V_s)^2)^{3/4}.$\cite{C1}

The deposition of a thin film growing with a velocity $V_s$ beginning
at $t=0$ from a substrate at $z=0$ followed by transformation, is
computed with an additional truncation of the cone by the plane
$z=0$.  There is no difficulty with this, but two expressions are
obtained; one is equation(\ref{Lorenz}) when $ z \ge V_st$.\cite{C1}

\section{Discussion.}
\setcounter{equation}{0}
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 $X_e = <N_c>$.  That this
equality is valid for a time dependent $\alpha$ and $V$ can be seen by
comparing equation (\ref{fulltimedep}) with the corresponding one for
$X_e$ in the literature.\cite{K},\cite{Review}

In recent years there have been mistaken criticisms
\cite{Baram1,Baram5} 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 $log\ log P$ versus $log \y$ 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 \cite{Hull} or growth rate.\cite{Review} 
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 $t^{-1/2}$.
For such processes, the exponent of $3/2$ is
interpreted as implying that all nuclei are present at the beginning,
and $5/2$ that nucleation continues at a constant rate (unlikely at
best).  Many ad hoc attempts to correct for such effects have been put
forth.\cite{Mora}

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 $1 - P$ equals the extended volume fraction
$X_e$, and both equal $log P$, 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 $V$ depends initially on $t^{-1/2}$ gives an Avrami exponent
of $1$ 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 $t=0$).  The change of time exponent from $5/2$ or $3/2$ 
to $1$ at late times is without a change in mechanism.\cite{Ham}

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.\cite{C1}  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\cite{C} and will be
reconsidered with the cone method in a subsequent paper.\cite{C3}

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 $<N_c>$ 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 $P$.  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. \cite{Bel,Review2} 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 $(\x,\y)$ 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 $V(t)$  might be treated with a cone with a fuzzy
surface.  The method does seem to have enough flexibility to warrant
reexamining my cases.

\bigskip
{\bf 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.



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