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\centerline{\blah Kinetics of Simultaneous
Transformations}\bigskip
\centerline{\large H. K. D. H. Bhadeshia}\medskip
\centerline{Department of Materials Science and Metallurgy,
University of Cambridge, U.K.} \bigskip

\x Keywords: Avrami, overall transformation kinetics,
simultaneous transformations,  impingement \bigskip

\ssec{\bf Abstract}

It is quite frequent for a number of solid--state
transformations to occur concurrently, starting from the same
parent phase. The transformations may occur at different rates, but
the  resulting competition for space or for the partitioning
of driving force between the precipitating phases can be
seminal to the development of many microstructures found in
commercial alloys. This paper reviews the overall
transformation theory available for dealing with simultaneous
transformations. The theory discussed is generally applicable, but is
illustrated with specific reference to secondary hardening
steels and structural steels.

\ssec{\bf Introduction}

There are at least three circumstances in which a  phase
might transform into more than one  product:

{\parindent=9pt\narrower \medskip
\zz{1. } The equilibrium precipitate may be
difficult to nucleate. Consequently, decomposition starts with
the formation of one or more metastable phases which are
kinetically favoured. These must eventually dissolve  as
equilibrium is approached. There are classic examples of
this in the age--hardening of aluminium alloys and in
secondary hardening steels. A recent example is the
crystallisation of metallic glass, at first to a metastable
phase [1]. In each case, the formation of the metastable phase
is accompanied by a reduction in free energy causing  
an exaggerated retardation of the stable phase.

\zz{2. } All of the product phases may be at equilibrium; \eg
the transformation of austenite into a mixture of ferrite and
graphite. However, the transformation products do not grow in
a coupled manner and may be sufficiently separated in
time to be treated as {\it sequential} rather than {\it
simultaneous.}

\zz{3. } The product phases may be coupled as in the
formation of pearlite, with a common transformation front.
\medskip}
\x It is the first case which forms the subject of this review.

\ssec{\bf Avrami Theory}

A model for a single transformation   begins with
the calculation of the nucleation and growth rates using
classical  theory, but an estimation of the volume fraction
requires impingement between particles to be taken into
account. This is generally done using the extended volume
concept of Johnson, Mehl, Avrami, and Kolmogorov [\eg 2] as
illustrated in \fagg\  (henceforth referred to as \lqq Avrami
theory"). Suppose that two particles exist  at time
$t$; a small interval $\delta t$ later, new regions marked $a$,
$b$,
$c$
\&\ $d$ are formed assuming that they are able to grow
unrestricted in extended space whether or not the region into
which they grow is already transformed. However, only those
components of $a$, $b$, $c$ \&\ $d$ which lie in previously
untransformed matrix can contribute to a change in the real
volume of the product phase (identified by the subscript \lq
1\rq) :
$$ dV_1 = (1 - {{V_1}\over {V}})dV_1^e   \numeqn $$ where  it is assumed
that  the microstructure develops randomly. The superscript
$e$ refers  to extended volume, $V_1$ is the volume of $1$
and $V$ is the total volume. Multiplying the change in
extended volume by the probability of finding untransformed
regions has the effect of excluding regions such as $b$, which
clearly cannot contribute to the real change in volume of the
product. For a random distribution of precipitated particles, this 
equation can easily be integrated to obtain the real volume fraction, 
$${{V_1}\over{V}} = 1-\exp\bl\{ -{{V^e_1}\over{V}}  \br\} \numeqn $$

\picture{extended}{164}{95}{700}{\tabtit{\figg : }  {An
illustration of the concept of extended volume. Two
precipitate particles have nucleated together and grown to a
finite size in the time $t$. New regions $c$ and $d$ are
formed as the original particles grow, but   $a$ \&\ $b$ are
new particles, of which $b$ has formed in a region which is
already transformed.}}

\ssec{\bf Simultaneous Transformations}

A simple modification for two precipitates ($1$ and $2$)
is that the  equation~\nnumeqn\ becomes  a coupled set of two
equations,  
$$
 dV_1  = \bl(1 - {{V_1+ V_2}\over {V}}\br)dV_1^e
\qquad\hbox{and}\qquad
 dV_2  = \bl(1 - {{V_1+ V_2}\over {V}}\br)dV_2^e
 \numeqn $$
The method  can be used for any number of reactions
happening together. The resulting set of equations must in
general must  be solved  numerically although a few analytical
solutions are possible for special cases which we shall now
illustrate.

\ssec{Special Cases}

For the simultaneous formation of two phases whose 
extended volumes are related linearly [3,4,5]:
$$ V_2^e = BV_1^e + C \qquad\hbox{with} \qquad B\geq 0
\qquad\hbox{and}\qquad C \geq 0\numeqn
$$ then with $v_i=V_i/V$, it can be shown [5] that 
$$v_1  = \int \exp\bl\{ - {{(1+B)V_1^e  +
C}\over{V}} 
\br\}~~{{dV_1^e}\over{V}}  \qquad \hbox{and}\qquad v_2=
Bv_1 \numeqn $$
 If the isotropic growth rate of phase~1 is $G$
and if all particles of phase~1 start growth at time $t=0$ from a fixed number
of sites $N_V$ per unit volume then $V_1^e= N_V
{{4\pi}\over{3}} G^3 t^3 $. We emphasize that the
specific assumptions made to express
$V_1^e$ can be selected at will, for example to include a
nucleation rate (details can be found in Christian [2]). On
substitution of the  extended volume in equation~\nnumeqn\
gives
$$ v_1  ={{1}\over{1+B}} \exp\{-{{C}\over{V}}\}\bl[1-\exp\bl\{ -
{{(1+B)N_V {{4\pi}\over{3}} G^3 t^3 }\over{V}} 
\br\} \br] \qquad\hbox{with}\qquad v_2=
Bv_1\numeqn $$
The term $\exp\{-C/V\}$ is the fraction of parent phase
available for transformation at
$t=0$; it arises because $1-\exp\{-C/V\}$ of  phase~2  exists prior to
commencement of the simultaneous reaction at $t=0$. Thus, $v_2$
is the additional fraction of phase~2 that forms during simultaneous reaction.
When $C=0$, equations~\nnumeqn\ reduce  to the case
considered by Robson and Bhadeshia [3]. It is emphasized
that $C\geq 0$.   A case for which $C=0$ and
$B=8$ is illustrated in \fagg.

\picture{simmod}{80}{61} {700}{\tabtit{\figg : }
{Simultaneous transformation to phases $\alpha\equiv 1$ and
$\beta\equiv 2$ with $C=0$ and $B=8$.}}

For the case where  the extended volumes are related
parabolically [5]:
$$\eqalign{ v_1  & = \exp\bl\{-{{C}\over{V}}\br\} \bl[
\sqrt{{{\pi}\over{4A}}}\exp\bl\{{{(1+B)^2}\over{4A}}\br\}
\bl(\hbox{erf}\bl\{{{1+B}\over{\sqrt{4A}}} + \sqrt A V_1^e\br\}
-\hbox{erf}\bl\{{{1+B}\over{\sqrt{4A}}}\br\} \br)
\br]
\cr
v_2  & = \exp\bl\{-{{C}\over{V}}\br\} \bl[
1 - \exp\bl\{-
{{A(V_1^e)^2 + (1+B)V_1^e}\over{V}}\br\} \br] -v_1}
\numeqn $$
The volume fractions $v_i$ again refer to the phases that form {\it
simultaneously} and hence there is a scaling factor $\exp\{-C/V\}$ which is
the fraction of parent phase available for coupled transformation to phases~1
and 2. 

\ssec{Secondary Hardening Steels}

Whereas the analytical cases are revealing, it is unlikely in
practice for the phases to be related in the way described.
This is illustrated for secondary hardening steels of the kind
used commonly in the construction of power plant [3]. The
phases interfere with each other not only by reducing the
volume available for transformation, but also by removing
solute from the matrix and
 thereby changing its composition. This change in matrix
composition affects the growth and nucleation rates of the phases.

 The calculations must allow for the simultaneous precipitation
of \mtwox, \mtwentythreecsix, \msevencthree, \msixc\ and
Laves phase. \mthreec\ is assumed to nucleate
instantaneously with the paraequilibrium
composition. Subsequent enrichment of \mthreec\ as it
approaches its equilibrium composition is accounted for.  All
the phases, except \mthreec, are assumed to form with
compositions  close to  equilibrium [3]. The
driving forces and compositions of the precipitating phases are
calculated using standard thermodynamic methods. 

The interaction between the precipitating phases is accounted
for by considering the change in the average solute level in the
matrix as each phase forms. This is frequently called the \lqq
mean field approximation". It is necessary because the
locations of precipitates are not predetermined in the
calculations.


A plot showing the predicted  variation of volume fraction of
each precipitate as  a function of time at 600$\,$\degg\ is
shown in \fagg . It is worth emphasising that there is no
prior knowledge of the actual sequence of precipitation,
since all phases are assumed to form at the same time,
albeit with different precipitation kinetics. The fitting
parameters  common to all the steels are the  site densities
and interfacial energy terms for each phase
[3]. The illustrated dissolution of metastable precipitates is a
natural consequence of changes in the matrix chemical
composition as the equilibrium state is approached. 

\picture{all}{290}{260}{400}{\tabtit{\figg : }{The predicted
evolution of precipitate volume fractions at 600$\,^\circ$C
for three power plant materials   (a)
Fe--0.15C--2.12Cr--0.9Mo--0.5Mn--0.17Ni wt\%;    (b)
Fe--0.1C--3Cr--1.5Mo--1Mn--0.1Ni--0.1V and (c)
Fe--0.11C--10.22Cr--1.42Mo--0.5Mn--0.55Ni--0.l2V--0.5Nb--0.056N
 (after [3]).}}

Consistent
with experiments, the precipitation kinetics of
\mtwentythreecsix\ are  predicted to be much slower in the
2.25Cr1Mo \ steel compared to the 10CrMoV and  3Cr1.5Mo
alloys. One contributing factor is that in the 2.25Cr1Mo \
steel a relatively  large volume fraction of
\mtwox\ and
\msevencthree\ form prior to \mtwentythreecsix. These deplete
the matrix and therefore suppress 
\mtwentythreecsix\ precipitation. The volume fraction of \mtwox\
which forms in the  10CrMoV steel is  relatively small, and
there remains a considerable excess of solute  in the matrix,
allowing \mtwentythreecsix\  to precipitate rapidly. Similarly, in
the 3Cr1.5Mo steel the volume fractions of \mtwox\   and
\msevencthree\   are insufficient to suppress
\mtwentythreecsix\  precipitation to the same extent as in the 2.25Cr1Mo \ steel.

It is even possible in this scheme to treat
precipitates nucleated at grain boundaries
separately from those nucleation at dislocations,
by taking them to be different phases in the
sense that the activation energies for nucleation
will be different. 

The computer program for doing these calculations is
available freely on the world wide web [6]. We note for the
moment, that this is as far as microstructure modelling has
progressed.  Work is in progress to calculate size distributions
by avoiding the conversion from extended to real volume. This is
reasonable when the volume fractions of precipitate phases are
small because the probability of particle impingement can be
negligible.



\ssec{\bf Ferrite, \wid\ and Pearlite}

There have been many  studies about the occurrence of \wid\
in steels as a function of the chemical composition, austenite
grain size and the cooling rate during continuous cooling
transformation. It is understood that \wid\ is
favoured in austenite with a large grain structure. This is
probably because
\wid\ is rarely found in isolation but often forms as secondary
plates growing from allotriomorphic ferrite layers. The prior
formation of allotriomorphic ferrite, which is favoured by a {\it
small} grain size, enriches the residual austenite with carbon,
so it is not surprising that a small austenite grain size
suppresses \wid. For the same reason, an increase in
cooling rate will tend to favour the formation of \wid.

These and other concepts are implicitly built into the model
based on simultaneous transformation kinetics [3--5]. This is
because allotriomorphic ferrite,
\wid\ and pearlite are allowed to grow together assuming that
thermodynamic and kinetic conditions are satisfied. Their
interactions are all taken into account during the course of
transformation. 


The reasonable overall level of agreement  between
experiment and theory is illustrated in \fagg, for   data
from [7]. In all cases where the allotriomorphic ferrite content
is underestimated, the \wid\ content is overestimated. This is
expected both because the composition of the austenite
changes when allotriomorphic ferrite forms and because its
formation changes the amount of austenite that is free to
transform to \wid.

\picture{allbodnar}{80}{77}{700}{\tabtit{\figg : } {A
comparison of the calculated volume fraction versus
experimental data reported by Bodnar and Hansen [7].}}



\fagg\ shows calculations which illustrate how the model can
be used to study the evolution of microstructure as the
sample cools [4].  The computer program for these
calculations can be obtained freely from the world wide web
[6].

All of the generally recognised trends are reproduced. The amount of
\wid\ clearly increases with the austenite grain size, and with the
cooling rate within the range considered. Bodnar and Hansen [7]
suggested also that the effect of cooling rate on the amount of \wid\
was smaller than that of the austenite grain size (for the values
considered). This is also evident in \figg.

\picture{combined1}{190}{160}{700} {\tabtit{\figg : }
{Calculated evolution of microstructure in a sample of
Fe--0.18C--0.48Si--1.15Mn as a function of the austenite
grain size and the cooling rate [4].}}

The work has been applied to the competition of
allotriomorphic and idiomorphic ferrite as a function of the
number densities of austenite grain surface and intragranular
nucleation sites [8]. One result is illustrated in \fagg\ for a
particular steel~{\it c} [8]. A reduction in the austenite grain
size should lead to a change in the balance between
allotriomorphic and idiomorphic ferrite. Data are presented for
austenite grain sizes of 150, 50 and 25
\um.  The reduction in the austenite  grain size leads to a
change from an idiomorphic to allotriomorphic ferrite
dominated transformation. Such an effect is well established
from a qualitative point of view; large austenite grain
sizes favour intragranularly nucleated transformation products
for two reasons. Firstly, the number density of grain boundary
nucleation sites decreases relative to intragranular sites as
$d_\gamma$ is increased. Secondly, grain boundary
nucleation sites are generally more potent than inclusions so
transformation commences first at the boundaries. Therefore,
assuming a constant thickness of allotriomorphic ferrite along
the austenite grain boundaries, a reduction in the austenite
grain size leads to a larger volume fraction of allotriomorphic
ferrite.

Applications of the theory of simultaneous transformations
have also been made to the estimation of microstructure in
steel weld deposits [9].



\picture{c_150_a}{125}{55}{800}{}
\picture{c_50_b}{125}{35}{800}{}
\picture{c_25_c}{125}{35}{800}{\tabtit{\figg :
}{Transformations in a particular steel~{\it c} described in
reference [8]. (a) An austenite grain size of 150 \um; (b) an
austenite grain size of 50
\um;   (c) an austenite grain size of 25 \um.}}

\ssec{\bf Deconvolution of Simultaneous Transformations}

The tempering of martensite in steels occurs by the formation of
metastable carbides before the precipitation of cementite is
completed. The tempering process is often followed 
indirectly by monitoring the change in electrical resistivity,
thermoelectric power or hardness. These parameters, when
normalised with respect to zero time and infinite time define an
overall \lqq fraction" which is taken to represent the entire set of
reactions involved on the atomic scale. 

The variation of fraction with time is frequently found to deviate
from a simple sigmoidal relationship, presumably because the curve
actually represents more than one reaction occurring at the same
time. There have been a number of attempts to
deconvolute such master curves into components due to individual
reactions.

Hanawa and Mimura [10],  following work by
Yamamoto [11], defined  relaxation times
($\tau$) which   are   related empirically to the formation or
dissolution of phases during simultaneous transformation:
$${{dy}\over{dt}} = -\bl({{1}\over{\tau_1}} +
{{1}\over{\tau_2}}\br)y + {{y_1}\over{\tau_3}}$$ where $y$ has
a value of unity when none of the excess solute is precipitated, and
zero when all of the excess is precipitated. $y_1$ and $y_2$
represent the fractions of metastable and stable precipitates
respectively, such that
$1-y=y_1+y_2$. $\tau_1$,  and $\tau_2$ are the
relaxations times for the precipitation of the metastable and stable
phases respectively, and $\tau_3$ the corresponding term for the
dissolution of the metastable phase. The relaxation times
are inversely proportional to the reaction rate constant. With
boundary conditions and certain other modifications, these
equations can be fitted to experimental data to deconvolute the
overall  $y$ versus time $t$ curve.

Luiggi and Betancourt [12--14] have followed a similar
phenomenological approach which appears mathematically to be
general in form, although it
too requires a large number of fitting parameters.

Although attempts have been made in all of these cases to deduce
parameters such as activation energies, it is difficult to
see whether these are physically meaningful. For example, because
each reaction involves nucleation and growth, there should be more
than one thermally activated process for each phase. 

\ssec{\bf Summary}

There has been  some progress in the theory for the
overall transformation kinetics simultaneous transformations. In
particular, the Johnson--Mehl--Avrami--Kolmogorov concept of
extended volume can be adapted for the case where more than
one reaction occurs at the same time. It is suggested that further
effort should focus on the treatment of: (a) the consequences, if
any, of the mean field approximation of the impingement of solute
diffusion fields; (b) a more rigorous treatment of multicomponent
effects; (c) the incorporation of coarsening as a natural
phenomenon within the overall model.

\ssec{\bf Acknowledgments}

I am grateful to Professor Alan Windle for the provision of
laboratory facilities at the University of Cambridge, to the
organisers of PTM '99 for inviting me to present this work, and to
 Joe Robson, Stephen Jones, Kazutoshi Ichikawa, Nobuhiro
Fujita and Tadashi Kasuya for valuable discussions on the subject of
simultaneous transformations over the course of many years.

\ssec{\bf References}


{\parindent=10pt \narrower \medskip \newcount\refno \refno=0
\def\ref#1#2#3#4#5{\item{\global\advance\refno by1 \the\refno ) } #1: { #2}
{\bf #3} {(#4)}  {#5}\vskip 0.5truemm} 

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