\input texp 

\def\oaa{\epsilon_{AA}}
\def\obb{\epsilon_{BB}}
\def\oab{\epsilon_{AB}}
{\it Annual Lecture, Royal Microscopical Society, 2000}\vskip 1cm


\title{Mechanically Alloyed Metals}
\centerline{H. K. D. H. Bhadeshia}
\medskip
\centerline{ University of Cambridge}
\centerline{ Department of Materials Science and Metallurgy}
\centerline{ Pembroke Street, Cambridge CB2 3QZ, U.K.}

\bigskip 
\doublespace

\sec{ABSTRACT}

{\parindent=20 pt  \narrower \medskip
\x Mechanical alloying involves the severe deformation of mixtures of
powders until they form the most intimate of atomic solutions.  Inert oxides can also be
introduced to form a uniform dispersion of fine particles which strengthen the
consolidated product. Large quantities of iron and nickel--base alloys with unusual
properties are produced commercially using this process. The theory describing the way in
which the powders evolve into a solution is reviewed. There are some fundamental
constraints which dictate how the microstructure must change during mechanical alloying
for the process to be at all viable. The strange recrystallisation behaviour of the alloys can be
understood if it is assumed that unlike normal metals, the grains  in the mechanically
alloyed sample are not topologically independent.

\medskip
}

\sec{INTRODUCTION}

An alloy can be created without melting, by violently deforming mixtures of different 
powders, \fagg, [1--4].  Inert oxides can, using this technique, be introduced uniformly into
the microstructure. The dispersion--strengthened alloyed powders are then consolidated
using hot--isostatic pressing and extrusion, to produce a solid with a very fine grain
structure.   Heat treatment then induces recrystallisation, either into a coarse columnar
grain structure or into a fine, equiaxed set of grains. Columnar grains occur for two reasons:
the oxide particles tend to become aligned along the extrusion direction, making that a
favoured growth direction. Alternatively, and in the absence of particle alignment, columnar
growth can be stimulated by recrystallising in a temperature gradient; the latter may be a
stationary gradient or one which moves along the sample, as in zone annealing. The
columnar microstructure is desirable in applications where the resistance to creep
deformation is paramount.


\picture{MA}{153}{241}{400}{\tabtit{\figg : }{The manufacture of mechanically alloyed
metals for engineering applications. The elemental powders/master--alloys/oxides are
milled together to produce solid solutions with uniform dispersions of oxide
particles. This powder is  consolidated and the resulting
material heat--treated to achieve a coarse, directional grain
structure.}}

The chemical compositions of some of the commercial alloys produced using this method are
listed in \tablaa. They all contain chromium and/or aluminium for corrosion and oxidation
resistance, and yttrium or titanium oxides for creep strength. Yttrium oxide cannot be
introduced into either iron or nickel by any method other than mechanical alloying;
indeed, this was the motivation for the original work by Benjamin [1].


\midinsert \thicksize=1pt \thinsize=0.8pt 
\tablewidth=6.2truein \begintable 
{Fe--base}| C & Cr & Al & Mo & Ti & N &  & ${\rm Ti_2O_3}$  & ${\rm Y_2O_3}$  & Fe \cr
{MA957} | 0.01 & 14.0 & --  & 0.3 & 1.0 & 0.012 & & -- & 0.27  & Balance \nr
{DT2203Y05}|   & 13.0 & -- & 1.5 & 2.2 & & & -- & 0.5  & Balance \nr
{ODM 331}  |   & 13.0 & 3.0 & 1.5 & 0.6 & & & -- & 0.5 & Balance \nr
{ODM 751}  |   & 16.5 & 4.5 & 1.5 & 0.6 & & & -- & 0.5 & Balance \nr
{ODM 061}  |   & 20.0 & 6.0 & 1.5 & 0.6 & & & -- & 0.5 & Balance \nr
{MA956} | 0.01 & 20.0 & 4.5 &  -- & 0.5 & 0.045 & & -- & 0.50 & Balance \nr
{PM2000} | $< 0.04$   & 20.0 & 5.5 &  & 0.5 & & & -- & 0.5 & Balance\nr 
{PM2010} | $< 0.04$   & 20.0 & 5.5 &  & 0.5 & & & -- & 1.0 & Balance\nr 
{DT} |     & 13.0 & --  & 1.5 & 2.9 & & & 1.8 & -- & Balance\nr 
{DY} |     & 13.0 & --  & 1.5 & 2.2 & & & 0.9 & 0.5 & Balance\nr 
{} | & & & & & & & & &  \cr
{Ni--Base} |C & Cr & Al  & Ti & W & Fe & N & Total O & ${\rm Y_2O_3}$ &  Ni  \cr
{MA6000} | 0.06 & 15.0 & 4.5 & 2.3 & 3.9 & 1.5 & 0.2 &  0.57  &
1.1 & Balance \nr 
{MA760}|   0.06  & 19.5 & 6.0 & --    & 3.4 &
1.2 & 0.3 & 0.6 & 1.0 & Balance \nr 
{MA758 $^{\dag}$} | 0.05 & 30.0 & 0.3 & --  & 0.5 & -- & -- & 0.37  & 0.6 & Balance \nr
{PM1000 $^{\dag}$} |  &  20.0 & 0.3 & 0.5 &  & 3.0  &  &  &  0.6 & Balance
\endtable \tabtit {\tablee : }{ Compositions (wt\% ) of some 
typical alloys. $^{\dag}$ MA758 and PM1000 are nickel base mechanical 
alloys without $\gamma'$ strengthening. The compositions of 
ODM061,  DT and DY are from Regle [5], as are the 
nitrogen data for MA956 and MA957. The compositions of  PM2000
and PM2010 are from Krautwasser \et [6].} \endinsert 


\sec{MICROSTRUCTURE}

Immediately after the mechanical alloying process, the 
powders have a grain size which can be as fine as
1--2~nm locally [7]. This is hardly surprising given the extent of the deformation during  
mechanical alloying, with true strains of the order of 9, equivalent to stretching a unit
length by a factor of 8000. The consolidation process involves hot extrusion and rolling at
temperatures of about 1000\degg, which causes  recrystallisation into to a sub--micron
grain size (\fagg).  It is known that during the course of  consolidation, the material may 
dynamically recrystallise several times [7]. It should be emphasised that the sub--micron
grains illustrated in \figg\ are not low--misorientation cell structures, but true grains with
large relative misorientations [8]. Subsequent
heat--treatment   leads to primary recrystallisation into a very
coarse grained  microstructure whose dimensions may be comparable to those of the
sample (\figg b). 

\picture{submicron}{100}{100}{500}{\tabtit{\figg :
}{ (a) Transmission electron micrographs showing the sub--micron
grain structure of mechanically alloyed and consolidated
iron--base MA956 alloy. The micrograph is a section
normal to the extrusion direction.  (b) Optical micrograph showing the coarse, columnar
recrystallisation grain structure resulting from heat treatment at temperatures as high as
1400$\,^\circ$C. }}

The grains in \figg a are elongated because the hot--rolling leaves a microstructure with a
residual deformation, with a dislocation density of about  $10^{15}\,$m$^{-2}$  [9];
although this is large, it is not particularly high when compared with dislocation densities
found in conventional steel martensitic microstructures [10]. The vast majority of the
stored energy of about 55~${\rm J\, mol^{-1}}$ in the material is due to the very fine
grain size [8].

\sec{CHEMICAL STRUCTURE}


The intense deformation associated with mechanical alloying 
can force atoms into positions where they may not prefer to be at 
equilibrium. The atomic structure of solid solutions in commercially important
metals formed  by the mechanical alloying process has been studied using field ion
microscopy and the atom--probe [11].


A solution which is homogeneous will nevertheless exhibit 
concentration differences of increasing magnitude as the 
size of the region which is chemically analysed decreases 
[12,13]. These are random fluctuations which obey the laws of 
stochastic processes, and represent the real distribution of 
atoms in the solution.  These equilibrium variations cannot usually 
be observed directly because of the lack of spatial 
resolution and noise in the usual microanalytical techniques. 
The fluctuations only become apparent when the resolution 
of chemical analysis falls to less than about a thousand 
atoms block. The atom probe technique collects the 
experimental data on an atom by atom basis.  The atom by atom data can be presented at
any  block size. 

\fagg\ illustrates the variation in the iron and chromium 
concentrations in fifty atom blocks, of the ferrite in {\it 
MA956}.  There are real fluctuations but further 
analysis is needed to show whether they are beyond what is 
expected in homogeneous solutions


\picture{clusters}{165}{90}{400}{\tabtit{\figg : }{The variation in the iron and
chromium concentrations of 50 atom samples of  MA956  [11].}}


For a random solution, the distribution of concentrations 
should be binomial since the fluctuations are random; any 
significant deviations from the binomial distribution would 
indicate either the clustering of like--atoms or the ordering 
of unlike pairs. 

The frequency distribution is obtained by plotting the total 
number of composition blocks with a given number of atoms of 
a specified element against the concentration. \fagg\ shows 
that the experimental distributions are essentially identical 
to the calculated binomial distributions, indicating that the 
solutions are random. 

The atom probe data can be analysed further if it is 
assumed, fairly reasonably, that the successive atoms picked up by the 
mass spectrometer were near neighbour atoms in the sample. 
Successive atoms which are identical then represent bonds 
between like atoms \etc so that pair probabilities used in 
solid solution theory can be measured experimentally. These 
data can be compared against calculated pair probabilities.
Thus, in a {\it random} $A-B$ solution, the probability 
$p_{AB}$ of finding $A-B$ or $B-A$ bonds is 
given by $p_{AB} = 2x_Ax_B$ where $x_i$ is the atom fraction 
of element $i$. Similarly, $p_{AA} = x_A^2$ and  $p_{BB} = 
x_B^2$. \tablaa\ shows the excellent agreement between the experimentally 
measured pair probabilities and those 
calculated assuming a random distribution of atoms.


\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=5.2truein \begintable
{Alloy} | Element | $p_{AA}$ & $p_{AB}$ & $p_{BB}$ & $N$  \cr
{\it MA956} | Cr (Measured)   | 0.638 & 0.324 & 0.038 & 12168 \nr
{\it MA956} | Cr (Calculated) | 0.637 & 0.322 & 0.041 &  \cr
{\it MA956} | Al (Measured)   | 0.808 & 0.184 & 0.008 & 12168 \nr
{\it MA956} | Al (Calculated)  | 0.833 & 0.160 & 0.008 &  
\endtable \tabtit {\tablee: }{ Pair probability analysis. $B$ 
is the solute element (such as Cr or Al) whereas $A$ 
represents the remainder of atoms. $N$ represents 
the total number of atoms included in the analysis. The 
calculations assume a random solution.}
\endinsert


\picture{atomprobe}{245}{270}{400}{\tabtit{\figg : }{Frequency distribution curves for
iron, chromium and aluminium in mechanically alloyed MA956,   [11].}}

This does not mean 
that the solutions are thermodynamically ideal, but rather that the 
alloy preparation method which involves intense deformation 
forces a random dispersal of atoms. Indeed, Fe--Cr solutions are known to deviate
significantly  from ideality, with a tendency for like atoms to cluster 
[14,15]. Thus, it can be concluded that the alloy is in a 
mechanically homogenised nonequilibrium state, and that 
prolonged annealing at low temperatures should lead to, for 
example, the clustering of chromium atoms.

\sec{Solution Formation}

Normal thermodynamic theory for solutions begins with the mixing of
component atoms. In mechanical alloying, however, the solution is prepared by first mixing
together lumps of the components, each of which might contain many millions of identical
atoms. We examine here the way in which a solution evolves from these large
lumps into an intimate mixture of different kinds of atoms [16].  It will be shown later that
this leads to interesting outcomes which have implications on how we interpret the
mechanical alloying process.

Consider the pure components $A$ and $B$  with molar free energies
$\mu^o_A$ and $\mu^o_B$ respectively.  If the components are initially in
the form of powders then the average free energy of such a mixture of 
powders is simply: 
$$ G\{\hbox{mixture}\} = (1-x)\mu^o_A + x\mu^o_B \numeqn $$
where $x$ is the mole fraction of $B$. It is assumed that the powder particles
are so large that the $A$ and
$B$ atoms do not \lqq feel" each other's presence via interatomic
forces between unlike atoms. It is also assumed that the number of ways in
which the mixture of powder particles can be arranged is not sufficiently
different from unity to give a significant contribution to a configurational
entropy of mixing. Thus, a blend of powders which obeys equation~\nnumeqn\
is called a {\it mechanical mixture}. It has a free energy that is simply a
weighted mean of the components, as illustrated  in \fagg a
for a mean composition $x$.


\picture{thermo}{376}{269}{300}{\tabtit{\figg : }{(a) The free energy of a mechanical
mixture, where the mean free energy is simply the weighted mean of the components. (b)
The free energy of an ideal atomic solution is always lower than that of a mechanical
mixture due to configurational entropy.}}

In contrast to a mechanical mixture, a {\it solution} is conventionally taken to describe a
mixture of atoms or molecules. There will in general be an enthalpy change associated with
the change in near neighbour bonds. Because the total number of ways in which the \lqq
particles" can arrange is now very large, there will always be a significant contribution from
the entropy of mixing, even when the enthalpy of mixing is zero.  The free energy of the
solution is therefore different from that of the mechanical mixture, as illustrated in \figg b.
The difference in the free energy between these two states of the components is the free
energy of mixing $\Delta G_M$, the essential term in all thermodynamic models for
solutions.

Whereas mechanical mixtures  and  atomic or molecular solutions
are familiar in all of the natural sciences, the intermediate states have only
recently been addressed [16]. The problem is illustrated in
\fagg\ which shows the division of particles into ever smaller particles until an atomic
solution is achieved. At what point in the size scale do
these mixtures of particles begin to exhibit solution--like behaviour? 

\picture{evolution}{189}{266}{300}{\tabtit{\figg : }{Schematic illustration of the
evolution of an atomic solution by the progressive reduction in the size of different
particles, a process akin to mechanical alloying.}}

To answer this question we shall assume first that there is no enthalpy of mixing. The
problem then reduces to one of finding the configurational entropy of mixtures of lumps as
opposed to atoms. Suppose that there are $m_A$ atoms per powder particle of $A$, and
$m_B$ atoms per particle of $B$; the powders are then mixed in a
proportion which gives an average mole fraction $x$ of $B$.  

There is only one configuration when the heaps of pure
powders are separate. When the powders are mixed at random, the number
of possible configurations for a mole of atoms becomes:
$$ {{\bigl(N_a ([1-x]/m_A +
x/m_B)\bigr)!}\over{(N_a[1-x]/m_A)!~~(N_a x/m_B)!}}
\numeqn $$
where $N_a$ is Avogadro's number. The numerator in equation~\nnumeqn\ is
the total number of particles and the denominator the product of the factorials
of the $A$ and $B$ particles respectively. Using the Boltzmann equation and Stirling's
approximation, the molar entropy of mixing becomes [16]:
$$\eqalign{
 {{\Delta S_M}\over{kN_a}} = &
{{(1-x)m_B+xm_A}\over{m_Am_B}}
\ln\bl\{ N_a{{(1-x)m_B+xm_A}\over{m_Am_B}} \br\} \cr
& - {{1-x}\over{m_A}}\ln\bl\{ {{N_a(1-x)}\over{m_A}}\br\} \cr
& - {{x}\over{m_B}}\ln\bl\{ {{N_a x}\over{m_B}}\br\}
} \numeqn $$
subject to the condition that the number of particles remains
integral and non--zero. As a check, it is easy to show that this equation
reduces to the familiar $$\Delta S_M = -kN_a [(1-x)\ln\{1-x\} + x\ln\{x\}]$$
when $m_A=m_B=1$.

Naturally, the largest reduction in free energy occurs when the
particle sizes are atomic. \fagg\ shows
the molar free energy of mixing for a case where the average
composition is equiatomic assuming that only configurational entropy
contributes to the free energy of mixing. An equiatomic composition maximises
configurational entropy. When it is considered that phase changes often occur
at appreciable rates when the accompanying reduction in free energy is just
10$\,{\rm J\,mol^{-1}}$, \figg\ shows that the entropy of mixing cannot be
ignored when the particle size is less than a few hundreds of atoms. In
commercial practice, powder metallurgically produced particles are typically
100~\um\ in size, in which case the entropy of mixing can be 
neglected entirely, though for the case illustrated, solution--like behaviour occurs when the
particle size is about $10^2$ atoms. 

\picture{particlesize}{143}{112}{400}{\tabtit{\figg : }{The molar
Gibbs free energy of mixing, $\Delta G_M = -T\Delta S_M$, for a
binary alloy, as a function of the particle size when all the particles
are of uniform size in a mixture whose average composition is
equiatomic. $T=1000$~K.}}

\sec{Enthalpy and Interfacial Energy}

The enthalpy of mixing will not in general be zero as was assumed above.  The binding
energy is the change in energy as the distance between a pair
of atoms is decreased from infinity to an equilibrium separation, which for a pair of
$A$ atoms is written  $-2\oaa$.   From standard theory for atomic solutions, the
molar enthalpy of mixing is  given by:
$$ \Delta H_M  \simeq N_a z(1-x)x\omega \qquad
\hbox{where}\qquad
\omega  = \epsilon_{AA} + \epsilon_{BB} - 2 \epsilon_{AB}  \numeqn $$
where $z$ is a coordination number.

However, for particles which are not monatomic, only those atoms at the interface
between the $A$ and $B$ particles will feel the influence of the unlike atoms. It follows
that the enthalpy of mixing is not given by equation~\nnumeqn, but rather by
$$\Delta H_M = zN_a \omega ~~ 2\delta S_V  ~~  x(1-x) \numeqn $$
where $S_V$ is the amount of $A-B$ interfacial area per unit volume and $2\delta$ is the
thickness of the interface, where $\delta$ is a monolayer of atoms.

A further enthalpy contribution,  which does not occur in conventional solution theory, is
the 
 structural component of the  interfacial energy  per unit area, $\sigma$:
$$\Delta H_I = V_m S_V\sigma
\numeqn $$ where $V_m$ is the molar volume.

Both of these equations contain the term $S_V$, which increases rapidly as the
inverse of the particle size $m$. The model predicts that {\it
solution formation is impossible} because the cost due to interfaces overwhelms any gain
from binding energies or entropy. And yet, solutions do form, so there must be a
mechanism to reduce interfacial energy as the particles are divided. The mechanism is the
reverse of that associated with precipitation (\fagg). A small precipitate can be coherent
but the coherency strains become intolerable as it grows. Similarly, during mechanical
alloying it is conceivable that the particles must gain in coherence as their size diminishes.
The milling process involves fracture and welding of the attrited particles so only those
welds which lead to coherence might succeed.

\picture{coherence}{289}{105}{400}{\tabtit{\figg : }{The change in coherence as a
function of particle size. The lines represent lattice planes which are continuous
at the matrix/precipitate interface during coherence, but sometimes terminate
in dislocations for the incoherent state. Precipitation occurs in the sequence
a$\rightarrow$c whereas mechanical alloying is predicted to lead to a gain in coherence in
the sequence c$\rightarrow$a.}}

Another unexpected result is obtained on incorporating a function which allows the
interfacial energy to decrease as the particle size becomes finer during mechanical alloying. 
Thermodynamic barriers are discovered to the formation of a solution by the mechanical
alloying process, \fagg [16]. When the enthalpy of mixing is either zero or negative, there is
a single barrier whose height depends on the competition between the reduction in free
energy due to mixing and the increase in interfacial energy as the particles become finer
until coherence sets in. When the atoms tend to cluster, there is a possibility of two
barriers, the one at smaller size arising from the fact that atoms are being forced to mix
during mechanical alloying.

\picture{barrier}{289}{135}{400}{\tabtit{\figg : }{Thermodynamic barriers to solution
formation. (a) Case where the enthalpy of mixing is negative, \ie unlike atoms attract. (b)
Case where there is a tendency to cluster with a positive enthalpy of mixing. After [16].}}

\sec{RECRYSTALLISATION TEMPERATURE} 

One of the most intriguing features of the alloys discussed here is the fact that
recrystallisation  occurs at exceptionally high homologous temperatures, of the  order of 0.9
of the melting temperature ($T_M$). This  contrasts with ordinary
cold--deformed metals which recrystallise readily at about 0.6 $T_M$, even
though the mechanically alloyed variants  contain more stored
energy (\tablaa).


\midinsert \thicksize=1pt \thinsize=0.8pt 
\tablewidth=5.8truein \begintable 
{Alloy}| Stored Energy  \nr
{}     | J g$^{-1}$   \cr
{MA957} | 1.0        \nr
{MA956} | 0.4        \nr
{MA956 sheet} |$\simeq 0.4^{\dag}$ \nr
{ }     |             \nr
{MA6000} | 0.6         \nr
{MA760 } | 1.0      \nr
{MA758 } | 0.3    
\endtable \tabtit {\tablee : }{  Enthalpy of Recrystallisation
[17--20]. 
$^{\dag}$: in MA956 sheet, the stored energy is released over 
a relatively large range of temperatures and is difficult to 
measure accurately. For MA758 the stored energy is small and 
recrystallisation occurs close to the melting point making 
it difficult to measure.} \endinsert 


Early work on mechanically alloyed ODS nickel--base superalloys 
[21] attributed the high recrystallisation temperatures
the presence of  $\gamma'$ precipitates.
However, there are alloys for which  the $\gamma'$
dissolution temperature is  below that at  which
recrystallisation occurs [22-24]. Furthermore, the iron--base
alloys do  not contain any $\gamma'$ and yet also recrystallise at
similarly  high temperatures. 

It has been speculated [25] that
recrystallisation occurs when the grain boundary mobility  rises suddenly when
solute drag is overcome at high temperatures. This is inconsistent with
the fact that the recrystallisation temperature can be reduced
by many hundreds of Kelvin by a slight additional inhomogeneous deformation  [18,26].

The fine particles of  yttrium oxide may interfere with recrystallisation but this does not
explain why the limiting grain size following recrystallisation is enormous. In any case,
recrystallisation is found to be insensitive to the
overall pinning force  [27]. 


Almost all of these difficulties are resolved when nucleation 
is considered in detail [8,27,28].  It  turns out that the
activation energy for {\it nucleation} is very large. This is  because the alloys have an
unusually small grain size prior  to recrystallisation. Recrystallisation nucleates by the
bowing of grain boundaries, a process which for  conventional alloys is
straightforward since the distance  between grain boundary
junctions is usually larger than that  between other strong pinning
points. With the sub--micrometer grain  size of mechanically
alloyed metals, the grain junctions themselves  act as severe
pinning lines for grain boundary bowing (\fagg). It is  easy to
demonstrate that this should lead to an enormous  activation
energy for the nucleation of recrystallisation,  many orders of
magnitude larger than the activation energy associated with self--diffusion  [8,27,28]. The
activation energy can be  reduced dramatically if just a few  grains happen to be slightly
larger than others (either  because adjacent grains are similarly orientated or because  of
local variations due to the uncertainties in the  mechanical alloying process). 


\picture{finegrain}{173}{120}{700}{\tabtit {\figg : }{The
nucleation of recrystallisation occurs by the formation of a
grain  boundary bulge. This can occur with less constraint
when the grain junctions are spaced at distances greater than
the critical bulge size. With the ultra--fine grains of
mechanically alloyed metals, the grain junctions are themselves
pinning points, making it very difficult to form large enough
bulges. }}



\sec{SUMMARY}

\x Commercial mechanically alloyed metals are fascinating in that they have helped reveal
many new phenomena, some of which have yet to be investigated experimentally. Amongst
the latter is the prediction that there is one or more barriers to the formation of a solid
solution by a process in which the component particles are successively refined in size. A
second prediction, which could be verified using detailed microscopy, is that there must be
a gain in coherence as the mixture of powders approaches an atomic solution. One
problem which appears to have been solved is the strange recrystallisation behaviour; the
ultrafine and uniform grains of the starting microstructure do not behave independently
and hence prevent recrystallisation until temperatures close to melting. 

\sec{ACKNOWLEDGMENTS}

\x I am particularly grateful to the Royal Microscopical Society and to its staff for the
invitation to present this lecture. I would like to thank Adebayo Badmos, Carlos Capdevila,
Andy Jones and Ulrich Miller  for helpful discussions over a period of many years, and
Professor Alan Windle for the provision of laboratory facilities  at the University of
Cambridge.

\vfill\eject\sec{REFERENCES}

{\parindent=10pt \narrower \medskip
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{#6}\vskip 0.5truemm} 

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\medskip}
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\parskip=3.0mm \parindent=0pt

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\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=6.2truein \begintable
Phase Mixture in Fe--0.2C--1.5Mn wt.\%\ at 300 K \hfill |
Stored Energy / ${\rm J\,mol^{-1}}$ \cr
1. Ferrite, graphite \&\ cementite \hfill | 0 \nr
2. Ferrite \&\ cementite \hfill | 70 \nr
3. Paraequilibrium ferrite \&\ paraequilibrium cementite
\hfill | 385 \nr 4. Bainite and paraequilibrium cementite
\hfill | 785 \nr
5. Martensite \hfill | 1214 \cr
6. Mechanically alloyed ODS metal \hfill | 55
\endtable \tabtit{\tablee : } {The stored energy as a
function of microstructure, relative to the standard
state defined as a mixture of ferrite, cementite and
graphite. The phases in cases 1 and 2
involve a partitioning of all elements so as to minimise free
energy. In cases 3--5 the iron and substitutional solutes are
configurationally frozen (for martensite even the interstitial elements
are frozen). Case 6 refers to an iron--base mechanically alloyed
oxide--dispersion strengthened sample which to my knowledge is the
highest  reported stored energy prior to recrystallisation [8].}\endinsert
