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\title{\Large Steels}
\author{H. K. D. H. Bhadeshia}

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\hrule
Question and example class documents attached to the end of this handout. 

The book associated with this course: ``Steels", 2017, 4th edition, H. K. D. H. Bhadeshia and R. W. K. Honeycombe, available on the Tripos reference shelves in the library.

Many questions associated with this course are on: \\ 

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You may click directly on the above link on your PDF file to reach the location. The link will
take you to an automated learning system based on the edX platform courtesy of Harvard and MIT.\\

In order to optimally benefit from this system, you should attempt one set of questions after each lecture. This will test your understanding of the lecture material. Marking is fully automated. Supervisions can therefore be better exploited to deal with conceptual difficulties and for broadening the discussion.


The full set of resources associated with this course is available on: \\

\centerline{\htmladdnormallink{http://www.phase-trans/msm.cam.ac.uk/teaching.html}
{http://www.phase-trans/msm.cam.ac.uk/teaching.html}}

~\\

\hrule


\section{\Large Martensite in Steels}
\label{sec:martensite}
     The name {\em martensite}  is after  the German
scientist Martens. It was used originally to describe the hard
microconstituent found in quenched steels. Martensite remains of the
greatest technological importance in steels where it can confer an
outstanding combination of strength ($>3500\,{\rm MPa}$) and toughness
($>200\,{\rm MPa\,m^{1\over2}}$). Many materials other than steel are now
known to exhibit the same type of solid-state phase transformation, known
as a {\em martensitic transformation}, frequently also called a {\em
shear} or {\em displacive transformation}. Martensite occurs in, for
example,  nonferrous alloys, pure metals, ceramics, minerals, inorganic
compounds, solidified gases and polymers (Table~\ref{tab:martensite}). We shall review first
the experimental facts about martensite and then proceed to explain them.


\begin{table}[htbp]
\caption{The temperature $M_S$ at which
martensite first forms on cooling, and the approximate
Vickers hardness of the resulting martensite for a
number of materials.}
\begin{center}
\begin{tabular}{lcc}\midrule
{Composition } & $M_S$ / K & Hardness HV\\[3pt]\midrule
{ZrO$_2$} & 1200 & 1000 \\[3pt]
{Fe--31Ni--0.23C wt\%} & 83 & 300 \\[3pt]
{Fe--34Ni--0.22C wt\%} & $<4$ & 250\\[3pt]
{Fe--3Mn--2Si--0.4C wt\%} & 493 & 600 \\[3pt]
{Cu--15Al} & 253 & 200 \\[3pt]
{Ar--40N$_2$} & 30 & \\[3pt]\midrule
\end{tabular}
\end{center}
\label{tab:martensite}
\end{table}

\subsection{Diffusionless Character}

 Martensitic transformations are diffusionless, but
what evidence is there to support this?

Martensite can form at very low
temperatures, where diffusion, even of interstitial atoms, is not
conceivable over the time period of the experiment. Table~\ref{tab:martensite} gives values
of the highest temperature at which martensite forms in a variety of
materials; this temperature is known as the martensite--start, or $M_S$
temperature. It is obvious that although martensite {\em can} form at low
temperatures, it need not do so. Therefore, a low transformation
temperature is not sufficient evidence for diffusionless transformation.

Martensite plates can grow at speeds which approach that of sound in the
metal. In steel this can be as high as $1100{\rm\,m\,s^{-1}}$, which
compares with the fastest recorded solidification front velocity of about
$80{\rm\,m\,s^{-1}}$ in pure nickel. Such large speeds are inconsistent
with diffusion during transformation. Note that martensite need not grow
so rapidly. For example, in shape--memory alloys or in single--interface
transformations, the interface velocity is small enough to observe.

The chemical composition of martensite can be measured and shown to be
identical to that of the parent austenite. The totality of these
observations demonstrate convincingly that martensitic transformations are
diffusionless.

\subsection{The Habit Plane}

This is the interface plane between austenite and martensite as measured
on a macroscopic scale (Fig.~\ref{fig:martensite_1}), for example by using one or
two--surface crystallographic trace analysis on metallographic samples.
For unconstrained transformations this interface plane is flat, but
strain energy minimisation introduces some curvature when the
transformation is constrained by its surroundings. Nevertheless, the
macroscopic habit plane is identical for both cases, as illustrated in
Fig.~\ref{fig:martensite_1}.

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.52\linewidth}
\raisebox{-6cm}{\includegraphics[width=0.98\linewidth]{martensite_1.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.45\linewidth}
      \caption{ An illustration
of the habit plane between austenite $(\gamma)$ and martensite
$(\alpha')$.}  \label{fig:martensite_1}
    \end{minipage}
  \end{center} \end{figure} 


Steels of vastly different chemical composition can have martensite with
the same habit plane (Table~\ref{tab:habit}), and indeed, other identical
crystallographic characteristics.
\begin{table}[htbp]
\caption{ Habit plane indices for martensite. With
the exception of $\epsilon$--martensite, the quoted indices are
approximate because the habit planes are in general irrational.}
\begin{center}
\begin{tabular}{lc}\midrule
Composition /wt.\% & Approximate habit plane indices \\[3pt]\midrule
Low--alloy steels, Fe--28Ni & $\{1~1~1\}_\gamma$ \\[3pt]
Plate martensite in Fe--1.8C & $\{2~9~5\}_\gamma$ \\[3pt] 
Fe--30Ni--0.3C & $\{3~15~10\}_\gamma$ \\[3pt] 
Fe--8Cr--1C & $\{2~5~2\}_\gamma$ \\[3pt]
$\epsilon$--martensite in 18/8 stainless steel &
$\{1~1~1\}_\gamma$ \\[3pt]\midrule
\end{tabular}
\end{center}
\label{tab:habit}
\end{table}%


\subsection{Orientation Relationships}

The formation of martensite involves the coordinated movement of atoms.
It follows that the austenite and martensite lattices will be intimately
related. All martensitic transformations therefore lead to a
reproducible orientation relationship between the parent and product
lattices. It is frequently the case that a pair of corresponding
close--packed\footnote{ The body--centred cubic lattice does not
have a close--packed plane but $\{0~1~1\}_{\alpha}$ is the most densely
packed plane.} planes in the ferrite and austenite are parallel or nearly
parallel, and it is usually the case that corresponding directions within
these planes are roughly parallel:
\begin{table}[htbp]
\begin{center}
\begin{tabular}{p{6cm}}
Kurdjumov--Sachs \\ \begin{eqnarray}
 \{1~1~1\}_\gamma &||& \{0~1~1\}_{\alpha} \cr \nonumber
 \langle1~0~\overline 1\rangle_\gamma &|| &\langle 1~1~\overline 1\rangle_{\alpha}\end{eqnarray}\\[3pt]
Nishiyama--Wasserman \\ \begin{eqnarray}
 \{1~1~1\}_\gamma & || & \{0~1~1\}_{\alpha} \cr\nonumber
 \langle1~0~\overline 1\rangle_\gamma \hbox{~~~about}~5.3^\circ \hbox{~from} &&\langle
1~1~\overline 1\rangle_{\alpha} \hbox{~towards}~\langle\overline 1~1~\overline
1\rangle_{\alpha} \end{eqnarray}\\[3pt]
Greninger--Troiano \\ \begin{eqnarray} \{1~1~1\}_\gamma \hbox{~~~about}~0.2^\circ
\hbox{~from} && \{0~1~1\}_{\alpha} \nonumber\cr
 \langle1~0~\overline 1\rangle_\gamma \hbox{~~~about}~2.7^\circ \hbox{~from} &&\langle
1~1~\overline 1\rangle_{\alpha} \hbox{~towards}~\langle\overline 1~1~\overline
1\rangle_{\alpha}\end{eqnarray}\nonumber\\[3pt]
\end{tabular}
\end{center}
\end{table}%


Note that these have been stated approximately: the true relations are
irrational, meaning that the indices of the parallel planes and
directions cannot be expressed using rational numbers (the square root of
2 is not a rational number).

\subsection{Athermal Nature of Transformation}

In the vast majority of cases, the extent of reaction is found to be
virtually independent of time:
\begin{equation} 1-V_{\alpha'} = \exp\{\beta(M_S-T)\}\qquad\hbox{where}\qquad
\beta\simeq -0.011 \end{equation}
$V_{\alpha'}$ is the fraction of martensite and $T$ is a temperature
below $M_S$. This is the Koistinen and Marburger equation; notice that
time does not feature in this relation, so that the fraction of
martensite depends only on the undercooling below the martensite--start
temperature. This athermal character is a consequence of very rapid
nucleation and growth, so rapid that the time taken can in normal
circumstances be neglected.

Isothermal martensite {\em is} possible when nucleation is
hindered, although the growth rate of individual plates of martensite can
still be rapid.

\subsection{Structure of the Interface}


  Any process which contributes to the formation of 
martensite cannot rely on assistance from thermal
activation. There must therefore exist a high level of continuity
across the interface, which must be  coherent or semi-coherent. A
stress--free fully coherent interface is impossible for the
$\gamma\rightarrow \alpha'$ transformation since the lattice deformation
{\bf BR} is an invariant--line strain.  A semi--coherent interface must be
such that the interfacial dislocations can glide as the interface moves
(climb is not permitted). It follows that the Burgers vectors of the
interface dislocations must not lie in the interface plane unless the
dislocations are screw in character.


There is an additional condition for a semi--coherent interface to be
glissile. The line vectors of the interfacial dislocations must lie along
an {\em invariant--line}, {\em i.e.} a line which joins the parent and product 
crystals without any rotation or distortion. Why is that? If there
is any distortion along the dislocation line, then other dislocations are
needed to accommodate that misfit. It will then be necessary to have more
than one set of non--parallel dislocations in the interface. These
non--parallel dislocations can intersect to form jogs which render the
interface sessile.

{\em It follows that for martensitic transformation to be possible,  the
deformation which changes the parent into the product must leave one or
more lines invariant (unrotated, undistorted).} A deformation which
leaves one line invariant is called an `invariant--line strain'.

\subsection{The Shape Deformation}
     
  The passage of a slip dislocation through a crystal causes the
formation of a step where the glide plane intersects the free surface
(Fig.~\ref{fig:IPS}a,b). The passage of many such dislocations on parallel slip
planes causes macroscopic shear (Fig.~\ref{fig:IPS}c,d). Slip causes a change in
shape but not a change in the crystal structure, because the Burgers
vectors of the dislocations are also lattice vectors.

 During martensitic transformation, the pattern in which the atoms in the
parent crystal are arranged is {\em deformed} into that appropriate for
martensite,   there must be a corresponding   change in the macroscopic
shape of the crystal undergoing transformation. The dislocations
responsible for the deformation are in the $\alpha'/\gamma$ interface,
with Burgers vectors such that in addition to deformation they also cause
the change in crystal structure. The deformation is such that an initially
flat surface   becomes uniformly tilted about the line formed by the
intersection of the interface plane with the free surface. Any scratch
traversing the transformed region is similarly deflected though the
scratch remains connected at the
$\alpha'/\gamma$ interface. These observations, and others, confirm that 
the measured shape deformation is an invariant--plane strain (Fig.~\ref{fig:IPS}e--g)
with a large shear component ($\simeq 0.22$) and a small dilatational
strain ($\simeq 0.03$) directed normal to the habit plane. 


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{martensite_3.eps}
\caption{(a, b) Step caused by
the passage of a slip dislocation. (c, d) Many slip dislocations, causing
a macroscopic shear. (e) An invariant--plane strain with a uniaxial
dilatation. (f) An invariant--plane strain which is a simple shear. (g)
An invariant--plane strain which is the combined effect of a uniaxial
dilatation and a simple shear.}
\label{fig:IPS}
\end{center}
\end{figure} 

\subsection{Bain Strain}

We now consider the nature of the strain necessary to transform the 
c.c.p. lattice of $\gamma$ into the b.c.c. lattice of $\alpha'$. Such a
strain was proposed by Bain in 1924 and hence is known as the `Bain
Strain' (Fig.~\ref{fig:Bain}). There is a compression along the
$z$ axis and a uniform expansion along the $x$ and $y$ axes. 

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.62\linewidth}
\raisebox{-8cm}{\includegraphics[width=0.98\linewidth]{martensite_4.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.35\linewidth}
      \caption{The Bain strain (not all
lattice points illustrated).}  \label{fig:Bain}
    \end{minipage}
  \end{center} \end{figure} 

The deformation describing the Bain Strain is given by
\begin{equation}
{\bf B} = \begin{pmatrix}\epsilon_0 & 0 & 0 \\ 0  & \epsilon_0 & 0\\ 0 &
0 &
\epsilon_0'\end{pmatrix}\end{equation}

\begin{equation}\epsilon_0 = {{\sqrt 2 a_{\alpha'} - a_\gamma}\over{a_\gamma}} \qquad
\epsilon_0' = {{ a_{\alpha'} - a_\gamma}\over{a_\gamma}}\end{equation}
where $a_{\alpha'}$ and $a_\gamma$ are the lattice parameters of
martensite and austenite respectively. The contraction is therefore along
the $[0~0~1]_\gamma$ axis and a uniform expansion on the $(0~0~1)_\gamma$
plane.

The Bain strain implies the following orientation relationship between
the parent and product lattices:
$$
[0~0~1]_{fcc}  || [0~0~1]_{bcc} \qquad
[1~\overline 1~0]_{fcc} ||  [1~0~0]_{bcc} \qquad
[1~1~0]_{fcc} || [0~1~0]_{bcc}
$$ but in fact, the experimentally observed orientation relationships
are irrational, as discussed earlier. We shall deal with this
inconsistency later.


Temporarily neglecting the fact that the Bain orientation is
inconsistent with experiments, we proceed to examine whether the Bain
strain leaves at least one line invariant. After all, this is a
necessary condition for martensitic transformation.

In Fig.~\ref{fig:ellipse}a,b, the austenite is represented as a sphere which, as a
result of the Bain strain {\bf B}, is deformed into an ellipsoid of
revolution which represents the martensite. There are no lines which are
left undistorted or unrotated by {\bf B}. There are no lines in the
$(0~0~1)_{fcc}$ plane which are undistorted. The lines  $wx$ and $yz$
are undistorted but are rotated to the new
positions $w'x'$ and $y'z'$. Such  rotated lines are not invariant.
However, the combined effect of the Bain strain {\bf
B} and the rigid body  rotation {\bf R} is indeed an
invariant--line strain (ILS) because it brings $yz$ and $y'z'$ into
coincidence (Fig.~\ref{fig:ellipse}c). This is the reason why the observed irrational
orientation relationship  differs from that implied by the
Bain strain. The rotation required to  convert {\bf B} into an ILS
precisely corrects the Bain orientation into that which is observed
experimentally. 

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{martensite_5.eps}
\caption{(a) and (b) show the
effect of the Bain strain on austenite, which when undeformed is
represented as a sphere of diameter $wx=yz$ in three--dimensions. The
strain transforms it to an ellipsoid of revolution. (c) shows the
invariant--line strain obtained by combining the Bain strain with a rigid
body rotation through an angle $\theta$.}
\label{fig:ellipse}
\end{center}
\end{figure} 


 As can be seen from 
Fig.~\ref{fig:ellipse}c, there is no  rotation which can make {\bf B}
into an invariant--plane strain since this would require two
non--parallel invariant--lines. Thus, for the $fcc\rightarrow bcc$
transformation,  austenite cannot be transformed into martensite by a
homogeneous strain which is an IPS. And yet, the observed shape
deformation leaves the habit plane undistorted and unrotated, {\em i.e.} it is
an invariant--plane strain. 

The phenomenological theory of
martensite crystallography solves this remaining problem (Fig.~\ref{fig:phenomenological}). The
Bain strain converts the structure of the parent phase into that of the
product phase. When combined with an appropriate rigid body rotation,
the net homogeneous lattice deformation ${\bf RB}$ is an
invariant--line strain (step $a$ to $c$ in Fig.~\ref{fig:phenomenological}). However, the
observed shape deformation is an invariant--plane strain ${\bf P_1}$
(step $a$ to $b$ in Fig.~\ref{fig:phenomenological}, but this gives the wrong crystal structure.
If  a second homogeneous shear ${\bf P_2}$ is combined with
${\bf P_1}$ (step $b$ to
$c$), then the correct structure is obtained but the wrong shape since 
$${\bf P_1 P_2 = RB}$$
These discrepancies are all resolved if the shape changing effect
of ${\bf P_2}$ is cancelled macroscopically by an inhomogeneous
lattice--invariant deformation, which may be slip or twinning as
illustrated in Fig.~\ref{fig:phenomenological}. 

The theory explains all the observed
features of the martensite crystallography. The orientation
relationship is predicted by deducing the rotation needed to change
the Bain strain into  an invariant--line
strain. The habit plane does not have rational indices because the
amount of lattice--invariant deformation needed to recover the correct
the macroscopic shape is not usually rational. The theory predicts a
substructure in plates of martensite (either twins or slip steps) as is
observed experimentally. The transformation goes to all the trouble of
ensuring that the shape deformation is macroscopically an
invariant--plane strain because this reduces the strain energy when
compared with the case where the shape deformation might be an
invariant--line strain. 


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.95\textwidth]{martensite_6.eps}
\caption{The phenomenological
theory of martensite crystallography.}
\label{fig:phenomenological}
\end{center}
\end{figure} 

\subsection{Thermodynamics of Martensitic Transformations}

Martensite is not represented on phase diagrams because the latter deal
with equilibrium. Martensite deviates from equilibrium in two important
ways:

Martensite grows without diffusion, so it inherits the chemical
composition of the parent austenite. In an equilibrium transformation the
chemical elements partition into the parent and product phases in a
manner which leads to a minimisation of free energy. 

Secondly, the shape deformation associated with martensitic
transformation causes strains; the resulting strain energy has to be
accounted for before the transformation can happen. 

These deviations can be represented on a free energy plot as illustrated
in Fig.~\ref{fig:energy}.


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8cm]{martensite_7.eps}
\caption{ The distance {\bf
ac} represents the free energy decrease when austenite of composition
$\overline x$ decomposes into an equilibrium mixture of ferrite and
austenite of compositions $x^{\alpha\gamma}$ and $x^{\gamma\alpha}$
respectively. The distance {\bf ab} is the smaller decrease in free
energy when martensite forms without any composition change, taking into
account the strain associated with the transformation.}
\label{fig:energy}
\end{center}
\end{figure} 


\begin{table}[htbp]
\caption{Typical  energies associated
with martensitic transformation.}
\begin{center}
\begin{tabular}{lc}\midrule
& ${\rm J\,mol^{-1}}$ \\[3pt]\midrule
Strain energy & 600 \\[3pt]
Twin interface energy & 100 \\[3pt]
$\gamma/\alpha'$ interface energy & 1 \\[3pt]
Stored energy due to dislocations & 20 \\[3pt]\midrule
\end{tabular}
\end{center}
\label{tab:energy}
\end{table}%
The relationship with the phase diagram is illustrated in Fig.~\ref{fig:T0}.
Martensitic transformation is only possible below the $T_0'$ temperature.


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8cm]{martensite_8.eps}
\caption{Schematic
illustration of the origin of the $T_0$  curve on the phase diagram. The
$T_0'$ curve incorporates a  strain energy term for the ferrite,
illustrated on the  diagram by raising the free energy curve for ferrite by
an  appropriate quantity.}
\label{fig:T0}
\end{center}
\end{figure} 


\clearpage
\section{\Large Bainite in Steels}\label{sec:bainite}



Bainite forms by the decomposition of austenite at a temperature which is
above $M_S$ but below that at which fine pearlite forms. All bainite
forms below the $T_0$ temperature.

All time--temperature--transformation (TTT) diagrams consist essentially of
two C--curves (Fig.~\ref{fig:bainite_1}). If we focus first on the Fe--Mn--C steel with the
higher hardenability (slower rates of transformation) then the two curves
are separated. The upper C--curve represents the time
required to initiate reconstructive transformations such as ferrite or
pearlite, whereas the lower C--curve represents displacive
transformations such as bainite or Widmanst\"atten ferrite. Notice that
as the hardenability of the steel decreases, the two curves tend to
overlap so that in experiments it appears as if the TTT diagram contains
just one curve with a complicated shape, describing all the reactions.
This is not the case because it is possible to show that this is an
experimental artefact caused by the inability to detect the two C--curves
separately.

\begin{figure}[htbp]
  \begin{center}
    \begin{minipage}[t]{0.55\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{bainite_1.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.42\linewidth}
      \caption{TTT diagrams for two
steels, one of which has a high hardenability. }  \label{fig:bainite_1}
    \end{minipage}
  \end{center} \end{figure} 

A further feature to note (Fig.~\ref{fig:bainite_1}, Fe--Mn--C) is that the lower C--curve
representing displacive transformations has a flat top. This represents
the highest temperature $T_h$ at which displacive transformations may
occur. The temperature $T_h$ may equal the bainite--start temperature
$B_S$ if the hardenability is high enough, but otherwise, $T_h=W_S$ where
$W_S$ is the Widmanst\"atten ferrite start--temperature (Fig.~\ref{fig:bainite_2}). The
latter does not form in high--hardenability steels and we shall discuss
in this lecture the detailed differences between bainite and Widmanst\"atten ferrite. 

\begin{figure}[htbp]
  \begin{center}
    \begin{minipage}[t]{0.55\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{bainite_2.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.42\linewidth}
      \caption{TTT diagrams showing the different domains of transformation. }  \label{fig:bainite_2}
    \end{minipage}
  \end{center} \end{figure} 


Bainite is  a non--lamellar aggregate of carbides and
plate--shaped ferrite (Fig.~\ref{fig:bainite_3}). As we shall see later, the carbide part of
the microstructure is not essential; the carbides form as a secondary
reaction, rather as in the tempering of martensite. The ferrite plates
are each about 10$\mu$m\ long and about 0.2$\mu$m\ thick, making the individual
plates invisible in the optical microscope. 


{\em Upper bainite} consists of clusters of platelets of ferrite adjacent
to each other and in almost identical crystallographic orientation, so
that a low--angle boundary arises whenever the adjacent platelets touch.
The ferrite always has a Kurdjumov--Sachs type orientation relationship
with the austenite in which it grows.

 Elongated cementite particles usually
decorate the boundaries of these platelets, the amount and continuity of the
cementite layers depending on the carbon concentration of the steel.


\begin{figure}[htbp]
  \begin{center}
    \begin{minipage}[t]{0.52\linewidth}
\raisebox{-3cm}{\includegraphics[width=0.98\linewidth]{bainite_3.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.45\linewidth}
      \caption{Schematic
illustration of the microstructure of upper and lower bainite. }  \label{fig:bainite_3}
    \end{minipage}
  \end{center} \end{figure} 

The clusters of ferrite plates are known as `sheaves' (Fig.~\ref{fig:bainite_4}); each
sheaf is itself in the form of a wedge--shaped plate on a macroscopic
scale. The sheaves inevitably nucleate heterogeneously at austenite
grain surfaces. The cementite precipitates from the carbon--enriched
austenite between the ferrite plates; the ferrite itself is free from
carbides. Cementite precipitation from austenite can be prevented by
increasing the silicon concentration to about 1.5~wt\%; this works because
silicon is insoluble in cementite. Silicon--rich bainitic steels can have
very good toughness because of the absence of brittle cementite. 


\begin{figure}[htbp]
  \begin{center}
    \begin{minipage}[t]{0.52\linewidth}
\raisebox{-6cm}{\includegraphics[width=0.98\linewidth]{bainite_4.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.45\linewidth}
      \caption{Evolution of a bainite
sheaf as a function of time.}  \label{fig:bainite_4}
    \end{minipage}
  \end{center} \end{figure} 

\subsection{Shape Deformation}

The formation of bainite causes a deformation (Fig.~\ref{fig:bainite_5}) which is an
invariant--plane strain with a shear component of about 0.26 and a
dilatational strain normal to the habit plane of about 0.03. This is
consistent with a displacive mechanism of transformation.


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8cm]{bainite_5.eps}
\caption{Atomic force microscope
image of the displacements caused on a polished surface of austenite by
the growth of bainite. Notice the shear deformation (dark contrast) and
indeed the plastic accommodation (light contrast tapering from the ridge
of the region of dark contrast) of the shape change in the austenite
adjacent to the bainite plates.}
\label{fig:bainite_5}
\end{center}
\end{figure} 

Bainite forms at a relatively high temperature when compared with
martensite. The parent austenite is weaker at high temperatures and
cannot accommodate the large shape deformation elastically. It therefore
relaxes by plastic deformation in the region adjacent to the bainite.
This is evident in Fig.~\ref{fig:bainite_5}, but is also presented as a height scan in
Fig.~\ref{fig:bainite_6}. The effect of this plastic deformation is to stifle the growth of
bainite plates before they hit any obstacle. This is why each bainite
plate grows to a size which is often smaller than the austenite grain
size and then comes to a halt. Further transformation happens by the
formation of a new plate and this is why the sheaf morphology arises. 


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.9\textwidth]{bainite_6.eps}
\caption{(a) A perfect
invariant--plane strain surface relief effect. (b) One where plastic
relaxation of the shape change occurs in the adjacent matrix. (c,d) An
actual atomic force microscope scan across the surface relief due
to a bainite sub--unit (Swallow and Bhadeshia, 1996).}
\label{fig:bainite_6}
\end{center}
\end{figure} 

\subsection{Substitutional Alloying Elements}

These do not redistribute at all during transformation, even though
equilibrium requires them to partition between the austenite and ferrite
(Fig.~\ref{fig:bainite_7}). The ratio of substitutional to iron atoms remains constant
everywhere including across the interface. This is consistent with a
displacive mechanism of transformation and the existence of an atomic
correspondence between the austenite and bainitic ferrite. The results
exclude any mechanism which involves local equilibrium at the interface,
or solute drag effects associated with interfacial motion.


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{bainite_7.eps}
\caption{Imaging
atom--probe micrographs, taken across an austenite--bainitic
ferrite interface in a Fe--C--Si--Mn alloy. The images confirm
quantitative data (Bhadeshia and Waugh, 1982) showing the
absence of any substitutional atom diffusion during
transformation. (a) Field--ion image; (b) corresponding silicon
map; (c) corresponding carbon map; (d) corresponding iron
map.}
\label{fig:bainite_7}
\end{center}
\end{figure} 

\subsection{Interstitial Alloying Elements (C, N))}

 It appears from Fig.~\ref{fig:bainite_7}\ that the carbon has partitioned into the
austenite. It is simple to establish that martensitic transformation  is
diffusionless, by measuring the local compositions before  and after
transformation. Bainite forms at somewhat higher  temperatures where the
carbon can escape out of the plate   within a fraction of a second. Its
original composition  cannot therefore be measured directly. 

There are three possibilities. The carbon may partition 
during growth so that the ferrite may never contain any 
excess carbon. The growth may on the other hand be 
diffusionless with carbon being trapped by the advancing 
interface.  Finally, there is an intermediate case in which 
some carbon may diffuse with the remainder being trapped to 
leave the ferrite partially supersaturated. It is therefore 
much more difficult to determine the precise role of carbon 
during the growth of bainitic ferrite than in martensite. 


 Diffusionless growth requires that transformation occurs at 
a temperature below $T_0$, when the free energy of bainite 
becomes less than that of austenite of the same composition. 
A locus of the $T_0$ temperature as a function of the carbon 
concentration is called the $T_0$ curve, an example of which 
is plotted on the Fe--C phase diagram in Fig.~\ref{fig:bainite_8}. Growth 
without diffusion can only occur if the carbon concentration 
of the austenite lies to the left of the $T_0$ curve. 


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{bainite_8.eps}
\caption{Schematic illustration of
the origin of the 
$T_0$ construction on the Fe--C phase diagram. Austenite with 
a carbon concentration to the left of the $T_0$ boundary 
can in principle transform without any diffusion. 
Diffusionless transformation is thermodynamically impossible 
if the carbon concentration of the austenite exceeds the 
$T_0$ curve.}
\label{fig:bainite_8}
\end{center}
\end{figure} 

Suppose that the plate of bainite forms without diffusion, 
but that any excess carbon is soon afterwards rejected into 
the residual austenite. The next plate of bainite then has to 
grow from carbon--enriched austenite (Fig.~\ref{fig:bainite_9} a). This process 
must cease when the austenite carbon concentration reaches 
the $T_0$ curve. The reaction is said to be incomplete, since 
the austenite has not achieved its equilibrium composition 
(given by the Ae3 curve) at the point the reaction stops. If 
on the other hand, the ferrite grows with an equilibrium 
carbon concentration then the transformation should cease 
when the austenite carbon concentration reaches the Ae3 
curve. 


It is found experimentally that the transformation to bainite 
does indeed stop at the $T_0$ boundary (Fig.~\ref{fig:bainite_9} b). The balance 
of the evidence is that the growth of bainite below the $B_S$ 
temperature involves the successive nucleation and 
martensitic growth of sub--units, followed in upper bainite 
by the diffusion of carbon into the surrounding austenite.  
The possibility that a small fraction of the carbon is 
nevertheless partitioned during growth cannot entirely be 
ruled out, but there is little doubt that the  bainite is at 
first substantially supersaturated with carbon. 


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{bainite_9.eps}
\caption{(a) Illustration of
the incomplete reaction  phenomenon. During isothermal transformation, a
plate of  bainite grows without diffusion, then partitions its excess 
carbon into the residual austenite. The next plate therefore 
has to grow from carbon--enriched austenite. This process 
continues until diffusionless transformation becomes 
impossible when the austenite composition eventually reaches 
the $T_0$ boundary. (b) Experimental data showing that the 
growth of bainite stops when  the austenite carbon 
concentration reaches the $T_0$ curve (Fe--0.43C--3Mn--2.12Si 
wt.\% alloy).}
\label{fig:bainite_9}
\end{center}
\end{figure} 

These conclusions are not significantly modified when the 
strain energy of transformation is included in the analysis. 

There are two important features of bainite which can be 
shown by a variety of techniques, {\em e.g.} dilatometry, electrical 
resistivity, magnetic measurements and by metallography. 
Firstly, there is a well defined temperature ${\rm B_S}$ 
above which no bainite will form, which has been confirmed 
for a wide range of alloy steels. The amount of bainite that 
forms increases as the transformation temperature is reduced 
below the ${\rm B_S}$ temperature. The fraction increases 
during isothermal transformation as a sigmoidal function of 
time, reaching an asymptotic limit which does not change on 
prolonged heat treatment even when substantial quantities of 
austenite remain untransformed. Transformation in fact ceases 
before the austenite achieves its equilibrium composition, so 
that the effect is dubbed the ``incomplete--reaction 
phenomenon\rq\rq. 

These observations are understood when it is realised that 
growth must cease if the carbon concentration in the 
austenite reaches the $T_0$ curve of the phase diagram. Since 
this condition is met at ever increasing carbon 
concentrations when the transformation temperature is 
reduced, more bainite can form with greater undercoolings 
below ${\rm B_S}$. But the $T_0$ restriction means that 
equilibrium, when the austenite has a composition given by 
the Ae3 phase boundary, can never be reached, as observed 
experimentally. A bainite--finish temperature ${\rm B_F}$ is 
sometimes defined, but this clearly cannot have any 
fundamental significance. 

\subsection{Summary}

Bainite grows by displacive transformation; the growth is accompanied by
a shape deformation which is an invariant--plane strain with a large
shear component. The transformation is diffusionless but carbon atoms
partition into the residual austenite (or precipitate as carbides),
shortly after growth is arrested. The precipitation of carbides is
therefore a secondary event.

\newpage
\section{\Large Alloy design: strong bainite}

High-strength {bainitic} steels have not in practice been as successful as 
quenched and tempered {martensitic} steels, because the coarse cementite 
particles in {bainite} are detrimental for toughness. However, it 
is now known that the precipitation of cementite during {bainitic} 
transformation can be suppressed. This is done by alloying the steel with 
about 1.5\,wt{\%} of silicon, which has a very low solubility in cementite 
and greatly retards its growth.

\looseness-1An interesting microstructure results when this silicon-alloyed steel is 
transformed into upper {bainite}. The carbon that is rejected into the 
residual {austenite}, instead of precipitating as cementite, remains in the 
{austenite} and stabilises it down to ambient temperature. The resulting 
microstructure consists of fine plates of {bainitic} ferrite separated by 
carbon-enriched regions of {austenite} (Fig.~\ref{fig:strong_1}).

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{strong_1.eps}
\caption{Transmission electron micrograph of a mixture of 
{bainitic} ferrite and stable {austenite}. (a) Bright field image. (b) Retained 
{austenite} dark field image.}
\label{fig:strong_1}
\end{center}
\end{figure}
The potential advantages of the mixed microstructure of {bainitic} ferrite and 
{austenite} can be listed as follows:


\begin{enumerate}
\item[1.]
Cementite is responsible for initiating fracture in high-strength steels. 
Its absence is expected to make the microstructure more resistant to 
cleavage failure and void formation.


\item[2.]
The {bainitic} ferrite is almost free of carbon, which is known to 
{embrittle} {ferritic} microstructures.

\item[3.]
The microstructure derives its strength from the {ultrafine} grain size of 
the ferrite plates, which are less than 1\,$\mu$m in thickness. It is the 
thickness of these plates which determines the mean free slip distance, so 
that the effective grain size is less than a micrometer. This cannot be 
achieved by any other commercially viable process. It should be borne in 
mind that grain refinement is the only method available for simultaneously 
improving the strength and toughness of steels.

\item[4.]
The ductile films of {austenite} which are intimately dispersed between the 
plates of ferrite have a crack blunting effect. They further add to 
toughness by increasing the work of fracture as the {austenite} is induced to 
transform to {martensite} under the influence of the stress field of a 
propagating crack. This is the TRIP, or transformation-induced plasticity 
effect.

\item[5.]
The diffusion of hydrogen in {austenite} is slower than in ferrite. The 
presence of {austenite} can, therefore, improve the stress corrosion resistance 
of the microstructure.

\item[6.]
Steels with the {bainitic} ferrite and {austenite} microstructure can be 
obtained without the use of any expensive alloying additions. All that is 
required is that the silicon concentration should be large enough to 
suppress cementite.
\end{enumerate}

In spite of these appealing features, the {bainitic} ferrite/{austenite} 
microstructure does not always give the expected good combination of 
strength and~tough-ness. This is because the relatively large `{blocky}' 
regions of {austenite} between the sheaves of {bainite} (Fig.~\ref{fig:strong_2}) readily 
transform into high-carbon {martensite} under the influence of stress. This 
{untempered}, hard {martensite} {embrittles} the~steel.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{strong_2.eps}
\caption{Optical micrograph of upper {bainite} in an 
Fe--0.43C--3{Mn}--2.02Si wt{\%} showing the blocks of retained {austenite} 
between sheaves of {bainite}.}
\label{fig:strong_2}
\end{center}
\end{figure} 


The blocks of {austenite} are detrimental to toughness, and anything 
that can be done to reduce their fraction, or increase their stability to 
{martensitic} transformation, would be beneficial. Both of these effects are 
controlled by the $T^{\prime}_{0}$ curve of the phase diagram. This curve determines the composition of the {austenite} at the point 
where the reaction to {bainite} stops. By displacing the curve to larger 
carbon concentrations, both the fraction of {bainite} that can form, and the 
carbon concentration of the residual {austenite} can be increased. 
Modifications to the $T^{\prime}_{0}$ curve can be achieved by altering 
the alloy composition. It is therefore necessary to calculate the effect of 
{substitutional} solutes on the $T^{\prime}_{0}$ curve.


\subsection{The improvement in toughness}

An apparently ideal microstructure consisting of {bainitic} ferrite and 
ductile {austenite} in a Fe--3{Mn}--2.02Si--0.43C\,wt{\%} exhibits poor toughness 
because of the presence of {blocky} unstable {austenite} (Fig.~\ref{fig:strong_3}). It is 
necessary to increase the amount of {bainitic} ferrite in the microstructure 
and to increase the stability of the {austenite}. Both of these aims can be 
achieved by changing the {substitutional} solute concentration such that the 
$T^{\prime}_{0}$ curve is shifted to higher carbon concentrations ({\em i.e.} 
$T^{\prime}_{0}$ is raised at any given carbon concentration).

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{strong_3.eps}
\caption{(a) Experimentally determined impact transition curves 
showing how the toughness improves as the amount of {blocky} {austenite} is 
reduced. (b) Calculated ${\textit T}^{\prime }_{\text 0}$ curves for the Fe--C, 
Fe--{Mn}--Si--C and Fe--Ni--Si--C steels.}
\label{fig:strong_3}
\end{center}
\end{figure} 
Manganese has a large effect in depressing the $T^{\prime }_{0}$ temperature. An examination of thermodynamic data shows that one 
possibility is to replace all of the manganese with nickel (Fig.~\ref{fig:strong_3}). Thus, for a 
Fe--4Ni--2Si--0.4C\,wt{\%} (3.69Ni, 3.85Si\,at{\%}) alloy  remarkable improvement in toughness achieved by doing this, without any 
sacrifice of strength, is illustrated in Fig.~\ref{fig:strong_3}, along with the 
$T^{\prime }_{0}$ curves as calculated above.

\clearpage
\section{\Large Widmanst\"atten Ferrite}


\subsection{Morphology}


Primary Widmanst\"atten ferrite grows directly from the austenite
grain surfaces, whereas secondary Widmanst\"atten ferrite develops from any
allotriomorphic ferrite that may be present in the microstructure
(Fig.~\ref{fig:wid_1}).
Widmanst\"atten ferrite\ can form at temperatures close to the $Ae_3$ temperature and hence
can occur at very low driving forces; the undercooling needed amounts to a
free energy change of only 50$\,{\rm J\,mol^{-1}}$. This is much less
than required to sustain diffusionless transformation.


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{wid_1.eps}
\caption{Morphology of
primary and secondary Widmanst\"atten ferrite.}
\label{fig:wid_1}
\end{center}
\end{figure} 


\subsection{Shape Change}

The growth of a single plate of martensite is accompanied by an
invariant--plane strain of the type illustrated in Fig.~\ref{fig:wid_2}a. However, at
the high temperatures (low undercoolings) at which Widmanst\"atten ferrite\ grows, the
driving force is not sufficient to support the strain energy associated
with a single plate. Widmanst\"atten ferrite\ formation therefore involves the simultaneous
and adjacent cooperative growth of two plates, which are
crystallographic variants such that their shape deformations mutually
accommodate (Fig.~\ref{fig:wid_2}b). This has the effect of cancelling much of the strain
energy.


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{wid_2.eps}
\caption{(a)
A single invariant--plane strain shape deformation. (b) The combined
effect of two mutually accommodating, back--to--back IPS deformations.
(c) The morphology of two plates, with different habit plane variants,
growing together in a mutually accommodating manner.}
\label{fig:wid_2}
\end{center}
\end{figure} 

It follows that what is seen as a single plate in an optical microscope
is actually a combination of two variants, usually separated by a
low--misorientation boundary (Fig.~\ref{fig:wid_2}c). Widmanst\"atten ferrite\ has a habit plane which is
close to $\{5~5~8\}_\gamma$. Hence, the two plates $\alpha_{w1}$ and
$\alpha_{w2}$ which have different variants of this habit with the
austenite, together form the thin--wedge shaped plate which is
characteristic of Widmanst\"atten ferrite.

Because Widmanst\"atten ferrite\ forms at low undercoolings (and above the $T_0$
temperature), it is thermodynamically required that the carbon is
redistributed during growth. $\alpha_w$ therefore always has a
paraequilibrium carbon content and grows at a rate which is controlled by
the diffusion of carbon in the austenite ahead of the plate--tip. For
plates, diffusion--controlled growth can occur at a constant rate
because solute is partitioned to the sides of the plate, whereas the
growing tip can advance into fresh austenite. Since the transformation
is, nevertheless, displacive, substitutional atoms do not partition and
an atomic correspondence is maintained between the parent and product
lattices for all atoms other than carbon.

\subsection{Growth Kinetics}

For isothermal transformation during growth in which there is no partitioning of substitutional solutes, the growth rate is governed by the rate at which carbon diffuses ahead of the Widmanst\"atten ferrite plate tip. In the first approximation, the concentrations at the
interface are given by a tie-lie of the phase diagram as shown in
Fig.~\ref{fig:allot_wid}. The
diffusion flux of solute from the interface must equal the rate at
which solute is incorporated in the precipitate so that:
\begin{equation} \underbrace{(c^{\gamma\alpha} - c^{\alpha\gamma}){{\partial
z^*}\over{\partial t}}}_{\hbox{rate solute partitioned}} = 
\underbrace{-D {{\partial c}\over{\partial z}}}_{\hbox{diffusion flux
from interface}} \simeq D{{c^{\gamma\alpha}-\overline c}\over{\Delta z}}
\label{eqn:flux2}\end{equation} where $z$ is a coordinate normal to the interface with a value
$z^*$ at the position of the interface. Note that concentration gradient here is assumed to be constant but it is in general evaluated at the position of the interface $(z=z^*)$.
 
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{allot_2.eps}
\caption{Phase diagram and its
relationship to the concentration profile at the $\alpha/\gamma$
interface during diffusion-controlled growth.}
\label{fig:allot_wid}
\end{center}
\end{figure} 
The plate can be described as a parabolic cylinder in three dimensions (Fig.~\ref{fig:parabolic}), a shape which is preserved as the plate lengthens. If it is assumed that the diffusion distance $\Delta z$ is equal to the plate tip radius $r$ (Fig.~\ref{fig:parabolic}), then from equation~\ref{eqn:flux2} it follows that the lengthening rate $v_l=\partial z^*/\partial t$ is given by
\begin{equation} 
v_l\approx \frac{D}{r} \frac{c^{\gamma\alpha}-\overline c}{c^{\gamma\alpha} - c^{\alpha\gamma}}.\label{eqn:lengthening}
\end{equation}
This leads to the obvious difficulty that $v_l\rightarrow\infty$ as $r\rightarrow 0$, caused by the fact that the creation of additional interfacial area as the plate grows is neglected in the derivation.
\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.62\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{parabolic_cylinder.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.35\linewidth}
      \caption{Widmanst\"atten ferrite plate represented as a parabolic cylinder, with tip radius $r$. }  \label{fig:parabolic}
    \end{minipage}
  \end{center} \end{figure} 
Given that the change in surface area per atom added to the plate is $v_a/r$, the corresponding increase in the free energy per atom due to the creation of additional interface is $\sigma v_a/r$ where $\sigma$ is the interfacial energy per unit area and $v_a$ is the volume per atom. Therefore, the net free energy change per atom, $\Delta g_r$, as the plate grows is 
\begin{equation}
\Delta g_r=\Delta g_\infty-\frac{\sigma v_a}{r} \label{eqn:capillarity}\end{equation} where  $\Delta g_\infty$ represents the free energy change per atom, driving the transformation in the absence of interface creation. At a critical radius $r_c$, $\Delta g_r=0$ so that $\Delta g_\infty=\sigma v_a/r_c$ and $v_l=0$. Equation~\ref{eqn:capillarity} can therefore be written as
\begin{equation}
\Delta g_r=\frac{\sigma v_a}{r_c} -\frac{\sigma v_a}{r} \qquad \hbox{or}\qquad \frac{\Delta g_r}{\Delta g_\infty}= 1- \frac{r_c}{r}\end{equation}
The velocity scales with the driving force when the latter is small, so equation~\ref{eqn:lengthening} can be rewritten to account for the interface creation as follows:
\begin{equation} 
v_l\approx \frac{D}{r} \biggl(\frac{c^{\gamma\alpha}-\overline c}{c^{\gamma\alpha} - c^{\alpha\gamma}} \biggr) \times\biggl(1- \frac{r_c}{r}\biggr) \label{eqn:lengthening2}\end{equation}
The accounting for interfacial energy in this manner is known as the {\em capillarity effect} \citep{Christian:1975} which governs the equilibrium between a curved particle and the matrix. Fig.~\ref{fig:rates} shows how the lengthening rate now goes through a maximum, and it is often assumed that the plate picks a radius consistent with the maximum growth rate. This is approximately consistent with experimental measurements as shown in Fig.~\ref{fig:rates}b \citep{Bhadeshia:1985b}.


\begin{figure}[htbp]
\centering
\subfigure(a){
\includegraphics[width=0.45\textwidth]{capillarity.eps}}
\subfigure(b){
\includegraphics[width=0.45\textwidth]{lengthening.eps}}
\caption{(a) Influence of interface curvature on plate lengthening rate. (b) Comparison of measured versus calculated lengthening rates.}
\label{fig:rates}
\end{figure}


\newpage
\subsection{Summary}

\begin{itemize}
\item  An atomic correspondence is maintained for substitutional
solutes, consistent with a displacive transformation mechanism.

\item Ferrite has a paraequilibrium carbon content during growth
which occurs at a constant rate, controlled by the diffusion of carbon
in the austenite ahead of the plate tip.

\item Growth involves the simultaneous and cooperative formation of
a pair of adjacent, self--accommodating plates of Widmanst\"atten ferrite.
\end{itemize}

\newpage
\section{\Large Allotriomorphic Ferrite}

An allotriomorph has a shape which does not reflect its internal
crystalline symmetry. This is because it tends to nucleate at the
austenite grain surfaces, forming layers which follow the grain boundary
contours (Fig.~\ref{fig:allot_1}).

An idiomorph on the other hand, has a shape which reflects the symmetry
of the crystal as embedded in the austenite. Idiomorphs nucleate without
contact with the austenite grain surfaces; they tend to nucleate
heterogeneously on non--metallic inclusions present in the steel.


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{allot_1.eps}
\caption{Allotriomorphic
\&\ idiomorphic ferrite.}
\label{fig:allot_1}
\end{center}
\end{figure} 

These are both true diffusional transformations, {\em i.e., } there is no
atomic correspondence between the parent and product crystals, there is
no invariant--plane strain shape change accompanying transformation, the
growth rate is either diffusion--controlled, interface--controlled or
mixed. Thermal activation is necessary for transformation, which can
therefore only occur at high homologous temperatures.

The $\alpha/\gamma$ interface need not in this case be glissile; the motion
of the interface involves diffusion and is not conservative.


\subsection{Revision of Diffusion--Controlled Growth in Fe--C }


The ferrite has a different chemical composition from the austenite in which
it grows. We shall assume that the growth of ferrite ($\alpha$) is
controlled by the diffusion of carbon in the austenite ($\gamma$) ahead of
the interface. 

As the ferrite grows, so does the extent of its diffusion field.
This retards growth because the solute then has to diffuse over
ever larger distances. As we will prove, the thickness of the ferrite
increases with the square root of time, {\em i.e.,}  the growth rate slows
down as time increases. We will assume in our derivation that the
concentration gradient in the matrix is constant, and that the far--field
concentration $\overline c$ never changes ({\em i.e.,} the matrix is semi--infinite
normal to the advancing interface). This is to simplify the mathematics
without loosing any of the insight into the problem.

For isothermal transformation in a binary alloy, the concentrations at the
interface are given by a tie--lie of the phase diagram as shown in
Fig.~\ref{fig:allot_2}. The
diffusion flux of solute from the interface must equal the rate at
which solute is incorporated in the precipitate so that:
\begin{equation} \underbrace{(c^{\gamma\alpha} - c^{\alpha\gamma}){{\partial
z^*}\over{\partial t}}}_{\hbox{rate solute partitioned}} = 
\underbrace{-D {{\partial c}\over{\partial z}}}_{\hbox{diffusion flux
from interface}} \simeq - D{{\overline c-c^{\gamma\alpha}}\over{\Delta z}}
\label{eqn:flux}\end{equation} where $z$ is a coordinate normal to the interface with a value
$z^*$ at the position of the interface. Note that the concentration
gradient is evaluated at the position of the interface $(z=z^*)$.
 
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{allot_2.eps}
\caption{Phase diagram and its
relationship to the concentration profile at the ferrite/austenite
interface during diffusion--controlled growth.}
\label{fig:allot_2}
\end{center}
\end{figure} 

A second equation can be derived by considering the overall conservation
of mass:
\begin{equation} (c^{\alpha\gamma}-\overline c)z^* =  {1\over 2}(\overline c -
c^{\gamma\alpha})\Delta z\end{equation} On combining these expressions to eliminate
$\Delta z$ we get:
\begin{equation} {{\partial z^*}\over{\partial t}} =
{{D(\overline c-c^{\gamma\alpha})^2}\over{2z^*(c^{\alpha\gamma}
 -c^{\gamma\alpha})(c^{\alpha\gamma}-\overline c)}}\label{eqn:velocity}\end{equation} It follows that \begin{equation}
z^*\propto
\sqrt{Dt}\nonumber\end{equation}


\subsection{Thermodynamics of Irreversible Processes}

Thermodynamics generally deals with measurable properties of
materials, formulated on the  basis of equilibrium. Thus, 
properties such as entropy and free energy are, on an appropriate
scale, static and time--invariant during equilibrium.  There are 
other parameters  not relevant to the
discussion of equilibrium: thermal conductivity,  diffusivity and
viscosity, but which are interesting because they can describe a
second kind of time--independence, that of the steady--state. Thus, the  concentration profile does not change
during steady--state diffusion, even though energy is being
dissipated by the diffusion. 

The thermodynamics of irreversible
processes  deals with systems which are not at equilibrium but are
nevertheless {\em stationary}.  The theory in effect uses
thermodynamics to deal with {\em kinetic} phenomena.
There is nevertheless, a  distinction 
between the thermodynamics of irreversible processes and
kinetics. The former applies strictly to the
steady--state, whereas there is no such restriction on kinetic
theory.

\subsubsection{Reversibility}

 A process whose direction can be changed by an
infinitesimal alteration in the external  conditions is
called reversible. Consider the example illustrated in Fig.~\ref{fig:pressure_volume}, which
deals with the response of an ideal gas contained at uniform pressure
within a cylinder, any change being achieved by the motion of the
piston. For any starting point on the $P/V$ curve, if the application of
an infinitesimal force causes the piston to move slowly to an adjacent
position still on the curve, then the process is reversible since
energy has not been dissipated. The removal of the infinitesimal force
will cause the system to revert to its original state.

On the other hand, if there is friction during the motion of the
piston, then deviations occur from the $P/V$ curve as
illustrated by the cycle   in
Fig.~\ref{fig:pressure_volume}. An infinitesimal force cannot move the piston because energy is
dissipated due to friction (as given by the area within the cycle).
Such a process, {\em which involves the dissipation of energy}, is
classified as irreversible with respect to an infinitesimal change in
the external conditions. 
\begin{figure}[htbp]
  \begin{center}
    \begin{minipage}[t]{0.42\linewidth}
\raisebox{-5.5cm}{\includegraphics[width=0.98\linewidth]{pressure_volume.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.55\linewidth}
      \caption{The curve
represents the variation in pressure within the cylinder as the
volume of the ideal gas is altered by positioning the
frictionless piston. The cycle represents the dissipation of
energy when the motion of the piston causes friction. }  \label{fig:pressure_volume}
    \end{minipage}
  \end{center} \end{figure} 

More generally, reversibility means that it is possible to pass 
from one state to another without appreciable deviation from 
equilibrium. Real processes are not reversible so equilibrium
thermodynamics can only be used approximately, though the same
thermodynamics defines whether or not a process can occur spontaneously
without ambiguity.

For irreversible processes the {\em equations} of classical 
thermodynamics become {\em inequalities}. For example, at 
the equilibrium melting temperature, the free energies of the
pure liquid and solid are identical ($G_{liquid} = G_{solid}$)
but not so below that temperature ($G_{liquid} > G_{solid}$). Such
inequalities are much more difficult to deal with though they
indicate the natural direction of  change. For steady--state
processes however, the thermodynamic framework for irreversible
processes as developed by Onsager is particularly useful in
approximating relationships even though the system is not at
equilibrium.

\subsubsection{The Linear Laws}

At equilibrium there is no change in entropy or free 
energy. An irreversible process dissipates energy and entropy is
created continuously. In the example illustrated in Fig.~\ref{fig:pressure_volume}, the
dissipation was due to friction; diffusion ahead of a moving interface
is dissipative. The rate at which energy is dissipated is the product of
the  temperature and the rate of entropy production ({\em i.e.}
$T\sigma$) with:
\begin{equation}T \sigma = J X \label{eqn:production}\end{equation} 
where $J$ is a generalised flux of some 
kind, and $X$ a generalised force. In the case of an electrical
current, the heat  dissipation   is the product of  the current
($J$) and the electromotive force ($X$).

As long as the flux--force sets can be expressed as in
equation~\ref{eqn:production}, the flux must naturally depend  in some way on the
force. It may then be written as a function
$J\{X\}$ of the force
$X$. At equilibrium, the force  is zero. If  $J\{X\}$ is expanded in a
Taylor series about  equilibrium ($X=0$), we get 
\begin{eqnarray}
J\{X\} & =& \sum^\infty_0 a_nX^n \cr
& =& J\{0\} + J'\{0\}{X\over{1!}} + J''\{0\}{X^2\over{2!}} \ldots
\label{eqn:linear}\end{eqnarray} 
Note that $J\{0\} = 0$ since that represents equilibrium. 
If the high order terms are neglected then we see that 
$$J \propto X. $$
This is a key result from the theory, that
the  forces  and their conjugate  fluxes  are linearly related
$(J\propto X)$ whenever the dissipation can be written as in
equation~\ref{eqn:linear}, at least when the deviations from equilibrium are
not large. 

Consider for example, a closed system in which a quantity ${\rm d}H$ of heat is transferred in a time interval ${\rm d}t$ across an area $A$ in a direction $z$ normal to that area, from a region at temperature $T_{\rm h}$ to a lower temperature $T_{\rm \ell}$. The receiving part increases its entropy by ${\rm d}H/T_{\rm \ell}$ whereas the depleted region experiences a reduction ${\rm d}H/T_{\rm H}$, so that the change in entropy is
\begin{equation}
{\rm d} S= {\rm d}H\biggl(\frac{1}{T_{\rm \ell}}- \frac{1}{T_{\rm h}}\biggr). \nonumber 
\end{equation}
The rate of entropy production per unit volume is therefore 
\begin{equation}
\sigma=\frac{1}{V}\frac{{\rm d} S}{{\rm d}t}= \frac{1}{V}\frac{{\rm d}H}{{\rm d}t} \biggl(\frac{1}{T_{\rm \ell}}- \frac{1}{T_{\rm h}}\biggr). \label{eqn:heatflux} 
\end{equation}
The flux of heat $J$ is defined as $A^{-1}{\rm d}H/{\rm d}t$ so Equation~\ref{eqn:heatflux} becomes
\begin{equation} \sigma= J \frac{A}{V} \biggl(\frac{1}{T_{\rm \ell}}- \frac{1}{T_{\rm h}}\biggr) \equiv J\biggl(-\frac{1}{T^2}\biggr)\frac{{\rm d}T}{{\rm d}z}  \qquad\hbox{or} \qquad T\sigma= \underbrace{J}_{\hbox{flux}} \underbrace{\biggl(-\frac{1}{T}\biggr)\frac{{\rm d}T}{{\rm d}z}}_{\hbox{force}} \nonumber\end{equation}

Examples of forces and fluxes in the context of the
present theory are listed in Table~\ref{tab:linear}.
\begin{table}[htbp]
\caption{Examples of forces and their conjugate
fluxes. $z$ is distance, $\phi$ is the electrical potential, and
$\mu$ is a chemical potential.}
\begin{center}
\begin{tabular}{ll}\midrule
Force & Flux \\[3pt]\midrule
$-{{\partial \phi}\over{\partial z}}$ &
Electrical Current\\[3pt]
$-{1\over T}{{\partial T}\over{\partial z}}$ & Heat flux \\[3pt]
$-{{\partial \mu_i}\over{\partial z}}$ & Diffusion flux \\[3pt]
Stress & Strain rate \\[3pt]\midrule
\end{tabular}
\end{center}
\label{tab:linear}
\end{table}%

\subsubsection{Multiple Irreversible Processes}

 There are many circumstances in which a number of irreversible
processes occur together. In a ternary Fe--Mn--C alloy, the
diffusion flux of carbon depends not only on the gradient of carbon,
but also on that of manganese. Thus, a uniform distribution of
carbon will tend to become inhomogeneous in the presence of a
manganese concentration gradient. Similarly, the flux of heat
may not depend on the temperature gradient alone; heat can be
driven also by an electromotive force (Peltier
effect)\footnote{In the Peltier effect, the two junctions of
a thermocouple are kept at the same temperature but the passage
of an electrical current causes one of the junctions to absorb
heat and the other to liberate the same quantity of heat. This
Peltier heat is found to be proportional to the current.}.
Electromigration involves diffusion driven by an electromotive
force. When there is more then one dissipative process, the total
energy dissipation rate can still be written
\begin{equation}T\sigma = \sum_i J_i X_i. \label{eqn:multiple}\end{equation}

In general, if there is more than one irreversible process 
occurring, it is found {\em experimentally} that each flow $J_i$ is 
related not only to its conjugate force $X_i$, but also is 
related linearly to all other forces present. Thus,
\begin{equation}J_i = M_{ij} X_j \label{eqn:dissipation}\end{equation} with $i,~j = 1,2,3\ldots $. 
Therefore, a given flux depends on all the forces causing the
dissipation of energy.

\subsubsection{Onsager Reciprocal Relations}

Equilibrium in real systems is always dynamic on a microscopic scale.
It seems obvious  that to maintain equilibrium under these dynamic
conditions, a process and  its reverse must occur at the same rate on
the microscopic scale. The consequence  is that provided
the forces and fluxes are chosen from the dissipation equation and are
independent,
$M_{ij} = M_{ji}$. This is known as the Onsager theorem, or the Onsager
reciprocal relations. It applies to systems near equilibrium when the
properties of interest have even parity, and assuming that the fluxes
and their corresponding forces are independent. 
An exception occurs with magnetic fields in which
case there is a sign difference $M_{ij} = - M_{ji}$. 
    



\subsection{Ternary Steels}


Consider now a ternary steel, say Fe--Mn--C. It would be necessary to
satisfy two equations of the form of equation~1, simultaneously, for each
of the solutes:
\begin{eqnarray}
 (c_1^{\gamma \alpha} - c_1^{\alpha \gamma})v &=& 
-D_{1} \nabla c_1\cr
 (c_2^{\gamma \alpha} - 
c_2^{\alpha \gamma})v &=& -D_{2} \nabla c_2 \label{eqn:coupled}\end{eqnarray}
where the subscripts refer to the solutes (1 for carbon and 2 for Mn).
The interface velocity $v$ is the ${\partial
z^*}/{\partial t}$ in equation~\ref{eqn:velocity}.

  Because $D_{1}\gg D_{2},$ these equations cannot in 
general be simultaneously satisfied for the tie--line passing 
through the alloy composition 
$\overline{c}_1,\overline{c}_2.$ It is, however, possible to 
choose other tie--lines which satisfy equation~\ref{eqn:coupled}. If the tie--line 
is such that $c_1^{\gamma \alpha}=\overline{c}_1$ ({\em e.g.} line 
{\em cd} for alloy A of Fig.~\ref{fig:allot_3}a), then $\nabla c_1$  will become 
very small, the driving force for carbon diffusion in effect 
being reduced, so that the flux of carbon atoms is forced to 
slow down to a rate consistent with the diffusion of manganese. 
Ferrite forming by this mechanism is said to grow by a \lq 
Partitioning, Local Equilibrium' (or PLE) mechanism, in 
recognition of the fact that $c_2^{\alpha \gamma}$ can 
 differ significantly from $\overline{c}_2$, giving 
considerable partitioning and long--range diffusion of manganese into the
austenite.

 
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{allot_3.eps}
\caption{Schematic isothermal
sections of the Fe--Mn--C  system, illustrating ferrite growth
occurring with local  equilibrium at the $\alpha / \gamma $ interface.
(a) Growth at  low supersaturations (P--LE) with bulk redistribution of
manganese, (b)  growth at high supersaturations (NP--LE) with negligible 
partitioning of manganese during transformation. The bulk alloy 
compositions are designated by the symbol $\bullet$ in each case. }
\label{fig:allot_3}
\end{center}
\end{figure} 

  An alternative choice of tie--line could allow $c_2^{\alpha 
\gamma}\rightarrow \overline{c}_2$ ({\em e.g.} line {\it cd} for the alloy 
of Fig.~\ref{fig:allot_3}b), so that $\nabla c_2$ is drastically increased 
since only very small amounts of Mn are partitioned into the 
austenite. The flux of manganese atoms at the interface correspondingly 
increases and manganese diffusion can then keep pace with that of
carbon,  satisfying the mass conservation conditions of
equation~\ref{eqn:coupled}. The growth of ferrite in this manner is said to occur
by a \lq Negligible  Partitioning, Local Equilibrium' (or NPLE) mechanism,
in  recognition of the fact that the manganese content of the ferrite 
approximately equals $\overline{c}_2$, so that little if any 
manganese partitions into austenite. 

What circumstances determine whether growth follows the PLE or NPLE mode?
Fig.~\ref{fig:allot_4}\ shows the Fe--Mn--C phase diagram, now divided into domains where
either PLE or NPLE is possible but not both. The domains are obtained by
drawing right--handed triangles on each tie--line in the $\alpha+\gamma$
phase field and joining up all the vertices. For example, prove to
yourself that if you attempt to define NPLE conditions in the PLE domain,
then the tie--line determining interface compositions will incorrectly
show that both austenite and ferrite contain less carbon than $\overline
c_1$, a circumstance which is physically impossible.


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{allot_4.eps}
\caption{Regions of the
two--phase field where either PLE or NPLE modes of transformation are
possible.}
\label{fig:allot_4}
\end{center}
\end{figure} 


Paraequilibrium is a constrained equilibrium. It occurs at
temperatures where the diffusion of substitutional solutes
is not possible within the time scale of the experiment.
Nevertheless, interstitials may remain highly mobile. Thus,
in a steel, manganese does not partition between the ferrite
and austenite, but subject to that constraint, the carbon
redistributes until it has the same chemical potential in
both phases. 

Therefore, the tie--lines in the phase diagram (Fig.~\ref{fig:allot_5}) are all
virtually parallel to the carbon axis, since Mn does not
partition between ferrite and austenite.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{allot_5.eps}
\caption{A paraequilibrium phase
diagram.}
\label{fig:allot_5}
\end{center}
\end{figure} 

Prove to yourself that in an isothermal section of the ternary
phase diagram, the paraequilibrium phase boundaries must lie within the
equilibrium phase boundaries as illustrated in Fig.~\ref{fig:allot_6}.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.9\textwidth]{allot_6.eps}
\caption{The paraequilibrium
phase field lies within the equilibrium field. The tie--lines illustrated
are for equilibrium.}
\label{fig:allot_6}
\end{center}
\end{figure} 

\newpage
\section{\Large Pearlite}

A  colony of pearlite when viewed in three dimensions consists of an interpenetrating bicrystal of ferrite and cementite. In planar sections the phases appear as lamellae which grow at a common front with the austenite. Cementite $(\theta)$ is rich in carbon whereas ferrite $(\alpha)$ accommodates very little when it is in equilibrium with either cementite or austenite $(\gamma)$. It is therefore necessary for carbon to be redistributed at the transformation front.  This can happen either by diffusion in the austenite in a direction  parallel to the transformation front (Fig.~\ref{fig:colony}). 

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.32\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{colony.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.65\linewidth}
      \caption{Diffusion flux parallel to the advancing interface. $S$ is the interlamellar spacing. }  \label{fig:colony}
    \end{minipage}
  \end{center} \end{figure} 

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.32\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{phase_diagram.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.65\linewidth}
      \caption{Phase diagram with extrapolated phase boundaries to identify the concentrations in the austenite which is in equilibrium with cementite or ferrite.}  \label{fig:phase_diagram}
    \end{minipage}
  \end{center} \end{figure} 

The diffusion distance parallel to the interface can be approximated as $aS$ where $a$ is a constant and $S$ is the interlamellar spacing. By analogy with equation~\ref{eqn:flux}, it follows that the rate at which solute is absorbed by the cementite must equal the amount arriving there by diffusion, so that 
\begin{equation}v(c^\theta - c^{\gamma\theta}) = D \frac{c^{\gamma\alpha}-c^{\gamma\theta}}{aS}\label{eqn:grow}\end{equation} where $v$ is the speed of the growth front, $D$ is the diffusivity of carbon in austenite and the concentration terms are self--explanatory.

However, there is an additional process which consumes energy, the creation of cementite/ferrite interfaces within the pearlite colony.  The minimum value of interlamellar spacing possible is a critical spacing $S_C=2\sigma^{\alpha\theta}/\Delta G$ where $\sigma^{\alpha\theta}$ is the interfacial energy per unit area and $\Delta G$ is the magnitude of the driving force for transformation in Joules per unit volume. Growth ceases when $S=S_C$. To allow for the energy consumed in the process of interface creation, equation~\ref{eqn:grow} is modified by a term $1-[S_C/S]$ as follows:
\begin{equation}v = \frac{D}{aSS_C} \frac{c^{\gamma\alpha}-c^{\gamma\theta}}{(c^\theta - c^{\gamma\theta})}\biggl(1-\frac{S_C}{S}\biggr)\label{eqn:grow2}\end{equation}
We now need to specify the value that $S$ will adopt during growth, and one assumption is that the spacing will correspond to that consistent with the maximum growth rate, {\em i.e.} when $S=2S_C$. 

The theory presented here is necessarily oversimplified and uses a number of unnecessary assumptions; more rigorous models are available but get extremely complex as soon as interfacial diffusion and the influence of solutes other than carbon is taken into consideration. 

\clearpage
\section{\Large Overall Transformation Kinetics}

\subsection{Isothermal Transformation}

To model transformation it is obviously necessary to calculate the
nucleation and growth rates, but an estimation of the volume fraction
requires {\em impingement} between particles to be taken into account. 


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{extended.eps}
\caption{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.}
\label{fig:extended}
\end{center}
\end{figure} 


This
is done using the extended volume concept of Kolmogorov, Johnson,
Mehl and Avrami. Referring to Fig.~\ref{fig:extended}, 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 ($\alpha$) :
\begin{equation} dV^\alpha = \biggl(1 - {{V^\alpha}\over {V}}\biggr)dV^\alpha_e  \end{equation}
where  it is assumed that  the microstructure develops at random. The
subscript $e$ refers  to extended volume, $V^\alpha$ is the volume of
$\alpha$ 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, 
\begin{equation}{{V^\alpha}\over{V}} = 1-\exp\biggl\{ -{{V_e^\alpha}\over{V}}  \biggr\} \label{eqn:one}
\end{equation}


The extended volume $V_e^\alpha$ is straightforward to calculate using
nucleation and growth models and neglecting completely any impingement
effects. Consider a simple case where the $\alpha$ grows isotropically at a
constant rate $G$ and where the nucleation rate per unit volume,
$I_V$. The volume of a particle nucleated at time $t=\tau$ (Fig.~\ref{fig:incubation}) is given
by
\begin{equation}v_\tau = {4\over3}\pi G^3 (t-\tau)^3 \end{equation}
The change in extended volume over the interval
$\tau$ and $\tau+d\tau$ is 
\begin{equation}dV_e^\alpha = {4\over3}\pi G^3 (t-\tau)^3 \times I_V \times V \times
d\tau \end{equation}
 On substituting into equation~\ref{eqn:one} and writing
$\xi=V^\alpha/V$, we get
\begin{eqnarray} 
& dV^\alpha = \biggl(1 - {{V^\alpha}\over {V}}\biggr){4\over3}\pi G^3
(t-\tau)^3  I_V V~~ d\tau  \nonumber \\
\hbox{so that}\qquad &  -\ln\{1-\xi\} ={4\over3}\pi G^3 I_V \int_0^t
(t-\tau)^3~d\tau \nonumber\\
\hbox{and}\qquad  & \xi  = 1-\exp\{-\pi G^3I_Vt^4/3\}
\end{eqnarray}


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{incubation.eps}
\caption{An illustration of
the incubation time $\tau$ for each particle.}
\label{fig:incubation}
\end{center}
\end{figure} 

 This equation has been
derived for the specific assumptions of random nucleation, a constant
nucleation rate and a constant growth rate. There are different
possibilities but they often reduce to the general form:
\begin{equation}\xi  =  1 - \exp\{-k_At^n\} \label{eqn:two} \end{equation}
where $k_A$ and $n$ characterise the 
reaction as a function of time, temperature and other
variables. The values of $k_A$ and $n$ can be obtained from experimental
data by plotting $\ln(-\ln\{1-\xi\})$ versus $\ln\{t\}$. The specific
values of $k_A$ and $n$ depend on the nature of nucleation and growth.
Clearly, a constant nucleation and growth rate leads to a time exponent
$n=4$, but if it is assumed that the particles all begin growth
instantaneously from a fixed number density of sites (i.e., nucleation is not
needed) the $n=3$ when the growth rate is constant. There are other
scenarios and the values of the Avrami parameters are not necessarily
unambiguous in the sense that the same exponent can represent two
different mechanisms. 


The form of
equation~\ref{eqn:two} is illustrated in Fig.~\ref{fig:kinetics}. Note that the effect of
temperature is to alter the thermodynamic driving force for
transformation, to alter diffusion coefficients and to influence any
other thermally activated processes. The effect of manganese is via its
influence on the stability of the parent and product phases.


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.9\textwidth]{kinetics.eps}
\caption{The calculated
influence of (a) transformation temperature and (b) manganese
concentration on the kinetics of the bainite reaction (Singh, 1998).
Bainite is a particular kind of solid--state phase transformation that
occurs in steels.}
\label{fig:kinetics}
\end{center}
\end{figure} 


The results of many isothermal transformation curves such as the ones
illustrated in
Fig.~\ref{fig:kinetics} can be plotted on at time--temperature--transformation diagram as
illustrated in Fig.~\ref{fig:TTT}. The curves typically have a $C$ shape because the
driving force for transformation is small at high temperatures whereas the
diffusion coefficient is small at low temperatures. There is an optimum
combination of these two parameters at intermediate temperatures, giving
a maximum in the rate of reaction. The curve marked {\em start}
corresponds to a detectable limit of transformation (e.g., 5\%), and that
marked finish corresponds to say 95\%\ transformation.


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{TTT.eps}
\caption{A
time--temperature--transformation (TTT) diagram.}
\label{fig:TTT}
\end{center}
\end{figure} 

\clearpage
\section{\Large TRIP Steels}

The steels developed to exploit the properties obtained when the
{martensite} reaction occurs during plastic deformation are known as 
\textit{transformation-induced plasticity} (TRIP) steels \citep{Zackay:1970}. The way in which they enhance the strength and uniform ductility of the steel is discussed later, but suffice it to say that a significant constituent of the microstructure must be austenite which is capable of transforming into martensite under the influence of an applied stress. Some of the original studies were conducted on alloys rich in solutes in order to preserve the austenite to ambient temperature; this can be expensive, but is a good starting point for the discussion of the TRIP effect.

As pointed out in section~\ref{sec:martensite}, martensitic transformation is also a deformation, described accurately as an invariant-plane strain with a shear $s$ on the habit plane, and a dilatation $\delta$ normal to that plane,  Fig.~\ref{fig:sheared}. Given the orthonormal coordinate system Z defined by the the unit vectors ${\bf z}_1$ parallel to the direction of shear,  and ${\bf z}_3$ normal to the habit plane, the deformation matrix {\bf P} becomes:
\begin{equation}
\begin{pmatrix}{\rm Z}&{\bf P}&{\rm Z}\end{pmatrix} = \begin{pmatrix}1 & 0 & s \\ 0 & 1 & 0 \\0 & 0 & 1+\delta \end{pmatrix}.
\end{equation} 
The effect of the deformation on any vector {\bf u} to produce a resultant vector {\bf v} is then given by 
\begin{equation}
\begin{pmatrix}{\rm Z}&{\bf P}&{\rm Z}\end{pmatrix}[{\rm Z};{\bf u}] = [{\rm Z};{\bf v}] \label{eqn:vector}
\end{equation}
The two vectors will not in general be parallel, but a comparison of the magnitudes gives an impression of the strain expected due to martensitic transformation. 
\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.40\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{sheared.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.57\linewidth}
      \caption{Coordinate axes for the derivation of the deformation matrix representing martensitic transformation ($\alpha'$) from austenite ($\gamma$). }  \label{fig:sheared}
    \end{minipage}
  \end{center} \end{figure} 
This calculation of strain  is along a specific direction {\bf u}, due to the shape deformation associated with the formation of a plate of martensite. In dealing with TRIP steels, the problem needs to be posed somewhat differently, i.e., what is the strain along a particular direction when a stress is applied to induce martensitic transformation in an otherwise stable austenite. It is necessary therefore to consider the thermodynamics of stress-affected martensitic transformation. Fig.~\ref{fig:MS} shows that in the absence of external stress ($\sigma=0$), the martensite-start temperature $M_S$ is defined by the temperature at which the free energy change $\Delta G^{\gamma\alpha}$ when austenite decomposes into ferrite of the same composition, reaches a critical value $\Delta G_{M_S}$.  

In contrast, when transformation occurs under the influence of a stress, the latter interacts with the shape deformation and the interaction energy $U$ is given by \citep{Patel:1953} to be:
\begin{equation}
U=\tau_\circ s + \sigma_N \delta \label{eqn:U}
\end{equation}
where $\tau_\circ$ is the shear stress on the habit plane and $\sigma_N$ the stress normal to that plane. Notice that the strains involved are plastic, so the interaction energy is given simply by the product of the stress and strain, rather than half that value as is sometimes assumed on the basis of elastic strains. If the stress is such that it favours the formation of martensite then  $U$ supplements $\Delta G^{\gamma\alpha}$ and the martensite-start temperature is raised to $M_S^\sigma$ which is above ambient temperature, so that stress-induced martensitic transformation becomes feasible (Fig.~\ref{fig:MS}).

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{MS.eps}
\caption{Plot of the chemical driving force for martensitic transformation against temperature, with $\sigma$ representing the applied stress.}
%:
\label{fig:MS}
\end{center}
\end{figure} 
Each single crystal of austenite can in principle transform into 24 different crystallographic variants of martensite. Each of these variants is associated with a particular value of $U$ depending on its orientation relative to the applied stress. Therefore, those variants with the largest interaction with the stress, i.e., which transform in a manner that relieves the stress, are favoured. This process is known as {\em variant selection} so that stress-induced martensite results in a biased microstructure with reduced variety. This is illustrated in Fig.~\ref{fig:variant}, where the martensite is generated by applying a tensile stress to polycrystalline metastable austenite, resulting in plates which are approximately at 45$^\circ$ to the tensile axis.

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.52\linewidth}
\raisebox{-5cm}{\includegraphics[width=0.98\linewidth]{selection.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.45\linewidth}
      \caption{A non-random distribution of martensite habit-plane orientations produced by stress-induced martensitic transformation at a temperature between $M_S$ and $M_S^\sigma$. The sample is polycrystalline austenite. }  \label{fig:variant}
    \end{minipage}
  \end{center} \end{figure} 

Assuming that a tensile stress $\sigma_1$ is applied, inclined at an angle $\theta$ to the  habit plane normal, with the stress axis in the plane containing ${\bf z_1}$ and ${\bf z_3}$, then from equation~\ref{eqn:U} and the Mohr's circle representation in Fig~\ref{fig:Mohr},
\begin{eqnarray}
U&=&\underbrace{\frac{\sigma_1}{2}\sin 2\theta}_{\tau_\circ} \times s + \underbrace{ \frac{\sigma_1}{2}(1+\cos 2\theta)}_{\sigma_N} \times \delta \nonumber \\
\frac{dU}{d\theta} &=&\frac{\sigma_1}{2}[2s\cos2\theta -2\delta\sin 2\theta] 
\end{eqnarray}
Setting the differential to zero gives the maximum value of $U$ at $\tan 2\theta_{\rm max}  = s/\delta$, which for typical values of $s=0.26$ and $\delta=0.03$ gives $\theta_{\rm max}  = 41.7^\circ$. Given that there are 24 habit plane orientations within a single austenite grain, it is likely that one close to this value will form first, hence explaining the observation of aligned plates in Fig.~\ref{fig:Mohr} even though the sample has many orientations of austenite grains.
\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.52\linewidth}
\raisebox{-5cm}{\includegraphics[width=0.98\linewidth]{Mohr.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.45\linewidth}
      \caption{Mohr's circle representation of the shear and normal stresses on a habit plane normal inclined at $\theta$ to the tensile stress $\sigma_1$. }  \label{fig:Mohr}
    \end{minipage}
  \end{center} \end{figure} 
The tensile axis can be represented as a unit vector
\begin{equation}
[{\rm Z};{\bf u}]=[\sin\theta_{\rm max} ~~~0~~~\cos\theta_{\rm max}] \nonumber \end{equation}
and using $\theta=\theta_{\rm max}$, the elongation obtained along the tensile axis when a single crystal of austenite transforms completely into the most favoured orientation of martensite is, using equation~\ref{eqn:vector}, given by:
\begin{eqnarray}
[{\rm Z};{\bf v}] &= & [(\sin\theta_{\rm max} + s\cos\theta_{\rm max} )~~0~~(1+\delta)\cos\theta_{\rm max} ] \nonumber\\
\hbox{elongation\,\%} & = & \biggl(\frac{\vert{\bf v}\vert}{\vert{\bf u}\vert} -1 \biggr)\times 100 \nonumber \label{eqn:elongation}
\end{eqnarray}
This assumes that {\bf v} is parallel to  {\bf u} and a correction would be needed if there is a relative rotation.
The elongation due to phase transformation is therefore  calculated is found to be 15\% \citep{Bhadeshia:trip}. This impressive value of elongation due to phase transformation alone supplements that due to ordinary dislocation plasticity, which can be a significant boon to the design of strong steels which usually suffer from early plastic instabilities. However, steels which are fully austenitic at ambient temperature can, in the context of iron-based alloys, be expensive.

\subsection{TRIP-Assisted Steels}

One way of producing cheap austenite is to stabilise it with carbon, but excessive carbon can harm other important engineering properties such as the ability to use the steel in a welded state. But an ingenious method involves a low-carbon steel ($\approx 0.15$\,wt\%\ C), which is first transformed into about 70\% of allotriomorphic ferrite and the remaining austenite cooled to partly transform it into bainitic ferrite. Both of these transformations leave the residual austenite enriched in carbon to a concentration in excess of 1.2\,wt\%, thus leaving about 15\% of retained austenite in the final microstructure.  Since bainite is the last phase to form, the carbon concentration of the austenite is limited by the $T_0$ condition described in section~\ref{sec:bainite}.

It is because the steel is not fully austenitic, that it is referred to as {\em TRIP-assisted} and typically has a lean composition Fe--0.15C--1.5Si--1.5Mn\,wt\%. The silicon serves to prevent the precipitation of cementite from the carbon-enriched austenite, and the manganese enhances the hardenability required to implement heat treatments suitable for mass production. The microstructure and typical properties are illustrated in Fig.~\ref{fig:Jacques}, which also shows how the retained austenite transforms during the course of deformation.  

\begin{figure}[htbp]
\centering
\subfigure(a){
\includegraphics[width=0.45\textwidth]{Jacques_HighSi.eps}}
\subfigure(b){
\includegraphics[width=0.45\textwidth]{TRIP_tensile.eps}}
\caption{TRIP-assisted steel. (a) Typical final microstructure  (micrograph courtesy of P. Jacques). (b) True stress versus true strain, and the stress-assisted decomposition of retained austenite [data from \citep{Jacques:2004}].}
\label{fig:Jacques}
\end{figure}

The fraction of retained austenite in a TRIP-assisted steel is $V_\gamma=0.20$ (Fig.~\ref{fig:Jacques}, in which case the maximum elongation to be expected if it all transforms into the favoured crystallographic variant of martensite is, using equation~\ref{eqn:elongation} scaled with $V_\gamma$ given by $0.15\times 0.2=0.03 $. The transformation strain therefore makes a negligible contribution to the overall elongation which is about 25\% as shown in Fig.~\ref{fig:Jacques}b. The major contribution from stress-affected transformation is not via the shape deformation of the martensite, but through a composite effect \citep{Bhadeshia:trip}. The production of hard, untempered martensite increases the work hardening rate and hence delays the onset of plastic instabilities. This is why TRIP-assisted steels have a large uniform elongation.

\subsection{$\delta$--TRIP Steel}

In this alloying concept, $\delta$--ferrite which forms during solidification is stabilised by specific aluminium additions, and substitutes for the 70\% of allotriomorphic ferrite in conventional TRIP-assisted steels \citep{Chatterjee:2007}. The advantage in doing this is that the $\delta$-ferrite can never be fully removed from the microstructure so that a  a fully martensitic structure cannot be produced in the heat--affected zone of a resistance spot weld. Another benefit is that the aluminium substitutes for the role of  silicon in suppressing cementite precipitation. The difficulty with silicon is that during hot processing, it forms a low-melting temperature oxide known as fayalite which adheres to the steel, making it difficult to remove other disfiguring oxides from the surface of the steel during de-scaling operations. Silicon can therefore lead to a reduction in the quality of the steel surface, and cause problems during coating processes. The chemical composition of the steel is typically Fe-0.4C-0.2Si-1Mn-3Al\,wt\% and the microstructures obtained directly from casting and following processing are illustrated in Fig.~\ref{fig:delta}.

\begin{figure}[htbp]
\centering
\subfigure(a){
\includegraphics[width=0.45\textwidth]{delta_AC_om_2.eps}}
\subfigure(b){
\includegraphics[width=0.45\textwidth]{trans_8.eps}}\\
\subfigure(c){
\includegraphics[width=0.65\textwidth]{properties.eps}}
\caption{$\delta$-TRIP steel. (a) Optical microstructure of the as-cast condition showing the $\delta$-ferrite dendrites and martensite between the dendrite arms \citep{Chatterjee:2007}. (b) The structure after processing to produce bainite and retained austenite \citep{Yi:2011e}. (c) The properties obtained (in red) compared against a range of established automotive steels.}
\label{fig:delta}
\end{figure}


\clearpage
\section{\Large TWIP Steels}

There are three essential modes by which steels can be permanently
deformed at ambient temperature, without recourse to diffusion.
Individual dislocations whose Burgers vectors correspond to lattice
vectors can glide, leading to a change in shape without altering the
crystal structure or volume. In contrast, a {displacive} transformation
({\em e.g.} {martensite} or {bainite}) results not only in a plastic strain, but
also a change of crystal structure and density; this is the phenomenon
exploited in the TRIP steels. 

The third mode of deformation is mechanical twinning, in which the
crystal structure of the steel is preserved but the twinned region is
{reoriented} in the process. Mechanical twinning results in a much larger
shear strain $s\,{=}\,1{\kern-.5pt/\kern-.5pt}\sqrt{2}$, compared with {displacive} transformations
where $s$ is typically 0.25. There is a particular class of
extraordinarily ductile alloys of iron, known as the {TWIP} steels, which
exploit mechanical twinning to achieve their properties. ``TWIP" stands for twinning-induced plasticity.

{TWIP} stands for twinning-induced plasticity. The alloys are {austenitic}
and remain so during mechanical deformation, but the material is able to
accommodate strain via both the glide of individual dislocations and
through mechanical twinning on the
$\{1~1~1\}_{\gamma}\,{\langle}\,1~1~\overline{2}\,{\rangle}_{\gamma}$ system. The
alloys typically contain a large amount of manganese, some aluminium and
silicon ({\em e.g.} Fe-25{Mn}-3Si-3Al\,wt\%) with carbon and nitrogen present
essentially as impurities. Larger concentrations of carbon may be added to enhance strength. At high manganese concentrations, there is a
tendency for the {austenite} to transform into $\epsilon$-{martensite}
(hexagonal close packed) during deformation. $\epsilon$-{martensite} can
form by the dissociation of a perfect
${a}/{2}\,{\langle}\,0~1~1\,{\rangle}_{\gamma}$ dislocation into {Shockley}
{partials} on a close packed $\{1~1~\overline{1}\}_{\gamma}$ plane, with
a fault between the {partials}. This faulted region represents a
three layer thick plate of $\epsilon$-{martensite}. A reduction in the
fault energy therefore favours the formation of this kind of {martensite}.
The addition of aluminium counters this because it raises the stacking
fault energy of the {austenite}. Silicon has the opposite effect of
reducing the stacking fault energy, but like aluminium, it leads to a
reduction in the density of the steel; the combination of Al and Si at
the concentrations used typically reduces the overall density from some
7.8\,g\,cm$^{-3}$ to about 7.3\,g\,cm$^{-3}$.

The alloys have a rather low yield strength at 
200-300\,{MN}\,m$^{-2}$ but the\break ultimate tensile strength can be much higher, in excess
of 1100\,{MN}\,m$^{-2}$. This is because the strain-hardening
coefficient is large, resulting in a great deal of uniform elongation,
and a total elongation of some 60-95\%. The effect of mechanical
twinning is two-fold. The twins add to plasticity, but they also have a
powerful effect in increasing the work-hardening rate by subdividing the
{untwinned} {austenite} into finer regions (Fig.~\ref{fig:TWIP}). 


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8cm]{f10-17-H8084.eps}
\caption{(a) Typical stress-strain curve for a {TWIP} steel. (b) Optical
microstructure of a {TWIP} steel following deformation, showing profuse
twinning (image and data courtesy of {Frommeyer}, G., Br{\"ux}, U. and
{Neumann}, P).}
\label{fig:TWIP}
\end{center}
\end{figure} 

One major advantage of {TWIP} steels is that they are {austenitic} and they
maintain attractive properties at cryogenic 
temperatures (${-}$150$^{\circ}$C) and high strain rates, e.g.,
10$^{3}$\,s$^{-1}$.
They therefore have great potential in enhancing the safety of
automobiles by absorbing energy during crashes. 



\clearpage
\section{\Large Mitigation of Residual Stress}

Residual stress is that which remains in a body which is stationary and at
equilibrium with its surroundings. It can be very detrimental to the
performance of a material or the life of a component.

Jones and Alberry conducted an elegant series of 
experiments to illustrate the role of transformations
 on the development of residual stress in steels. 
Using  bainitic, martensitic and stable austenitic steels, 
they demonstrated that transformation plasticity during the 
cooling of a uniaxially constrained sample from the 
austenite phase field, acts to relieve the build up of 
thermal stress as the sample cools. By contrast, the 
non-transforming austenitic steel exhibited a monotonic 
increase  in residual stress with decreasing temperature, 
as might be expected from the thermal contraction of a 
constrained sample. When the steels transformed to bainite or
martensite, the  transformation strain compensated for any thermal 
contraction strains that arose during cooling. Significant 
residual stresses were therefore found to build up only after 
transformation was completed, and the specimens approached 
ambient temperature (Fig.~\ref{fig:residual_1}).


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{residual_1.eps}
\caption{(a) The axial macro
stress that develops in uniaxially constrained samples during cooling of 
a martensitic (9Cr1Mo), bainitic ($2{1\over 2}$Cr1Mo) and  austenitic
steel (AISI 316). Also  plotted are some
experimental data for the yield strength of austenite in a low-alloy
steel. (b)
Interpretation of the Alberry and Jones  experiments. The thermal
expansion coefficient of austenite  is much larger than that of
ferrite.
}
\label{fig:residual_1}
\end{center}
\end{figure} 

The experiments contain other revealing features. The 
thermal expansion coefficient of austenite $(1.8 \times
10^{-6}\,{\rm K^{-1}})$ is much larger  than that of ferrite $(1.18
\times 10^{-6}\,{\rm K^{-1}})$, and yet,
the slope of the line prior to  transformation is smaller when compared
with that after  transformation is completed (Fig.~\ref{fig:residual_1}). This is because
the  austenite deforms plastically; its yield strength at high 
temperatures is reduced so much that the sample is unable to 
accommodate the contraction strains elastically. Thus, the high
temperature austenite part of each curve is virtually a plot of the
yield strength as a function of temperature, as is evident from the
comparison versus the actual yield strength data also plotted on
Fig.~\ref{fig:residual_1}a. 

In the region of the stress/temperature curve where 
transformation happens, the interpretation of experimental 
data of the kind illustrated in Fig.~\ref{fig:residual_1} is difficult.  
In the case of displacive transformations, the shape change due to
transformation has a shear component  which is much larger than the
dilatational term. This will give rise to significant intergranular
microstresses, part of which will be relaxed plastically. This
shear component will on average cancel out in a fine  grained polycrystalline
sample containing plates in many  orientations so that the average type II
microstress component will be zero. However, the very nature of the stress
effect  is to favour the formation of selected variants], in which  case 
the shear component rapidly begins to dominate the  transformation plasticity.
Fig.~\ref{fig:residual_1}a shows that the stress can temporarily change sign as the sample cools.
This is because the stress selected variants continue to grow preferentially
until transformation is exhausted.


Notice that if transformation is 
completed at a higher temperature, then the ultimate level 
of stress at ambient temperature is larger, since the fully 
ferritic sample contracts over a larger temperature range. 
To reduce the residual stress level at ambient temperature 
requires the design of alloys with low transformation 
temperatures. The sort of high strength welding alloys used 
for making submarine hulls tend to have very low 
transformation temperatures ($< 250^\circ$C). This fact may be 
fortuitous, but such alloys should be less susceptible to 
cracking induced by the development of macro residual stresses. Fig.~\ref{fig:residual_2}
illustrates one kind of   distortion found in welds, measured in terms
of the angle $\theta$ through which the unconstrained plates rotate
during the cooling to ambient temperature. It has now been demonstrated that the use of appropriate martensitic weld metal can dramatically reduce the distortion.


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{residual_2.eps}
\caption{ An illustration
of the distortion caused when a pair of coplanar plates are welded
together and the joint is then allowed to cool to ambient
temperature.}
\label{fig:residual_2}
\end{center}
\end{figure} 


\clearpage
\section{\Large Bulk Nanostructured Steel}

A nanostructured material is here defined as one containing an exceptionally large density of strong interfaces, rather than one which simply contains a minor fraction of features such as precipitates, which are small in size. The desire for such materials in the engineering context comes from the expectation of novel mechanical properties, particularly the stress that can safely be  tolerated in service. It is difficult to invent such materials  because any design must address three basic issues: 
\begin{description}
\item[(i)] it should ideally be possible to make samples which are large in all dimensions, not simply wires or thin sheets;
\item[(ii)] there are commercially available steels in which the distance between interfaces is of the order of 250--100~nm. The novelty is in approaching a structural scale in polycrystalline metals which is an order of magnitude smaller.
\item[(iii)] The material concerned must be cheap to produce if it is not to be limited to niche applications. A good standard for an affordable material is that its cost must be similar to that of bottled water when considering weight or volume.
\end{description}

These are formidable challenges and the process of design can begin with a consideration of how strength can be achieved. The  long--range periodicity which is assumed to typify the crystalline state is  in practice  punctuated by defects, some of which make a significant contribution to configurational entropy and hence can exist in thermodynamic equilibrium. This kind of entropy scales with the number of entities (atoms) in the crystal and hence it is only possible to approximate perfection in small crystals.  Such crystals can be strong because in the absence of defects, deformation must occur by the wholesale glide of planes over each other, rather than by the propagation of discontinuities such as dislocations. Micrometer sized crystals of pure iron have achieved strength levels in excess of 10~GPa, although in principle the strength  can exceed 20~GPa. The crystals becomes weaker as they are made larger, both because of the thermodynamically stabilised defects and also accidents of growth. This is fundamentally why the impressive mechanical properties of carbon {\em nanotubes} are not maintained (and indeed, should not be expected to be maintained) when the tubes become long. 

An alternative method for manufacturing sizeable strong materials is to introduce large numbers of defects such as interfaces or dislocations, which interfere with the ordinary mechanisms of slip or twinning.  The defects can be introduced by deformation. Techniques which involve severe plastic deformation are limited in the shape of the final product and in the quantity that can be produced at reasonable cost. Examples include fine nanostructured--wire with a strength in excess of 5~GPa; metals subjected to equal--channel angular extrusion in which redundant work is used in order to achieve large plastic strains whilst maintaining the external shape of the object being deformed. Accumulative roll--bonding involves the repeated rolling and folding of sheet material in order to accomplish strain increments without thinning the sample entering the rolls; the process is suited for large scale production but does not lead to particularly fine grains, which tend to be closer to micrometers than nanometers  in size.

Thermomechanical processing is particularly suited to the large scale production of fine--grained steels by phase transformation from the deformed parent austenite. However, the minimum ferrite grain size achieved in practice is about 1\,$\mu$m, partly because the speed of production and the thickness of the steel leads to recalescence during transformation, and hence prevents the achievement of the large undercoolings needed to refine the grain size. 

There are a couple of further difficulties. The ductility decreases sharply as the grain size in a polycrystalline metal is reduced. Secondly, there is often a requirement for rapid heat--treatment which becomes impractical for large components.

A recent development seems to avoid all of these difficulties, and meets the criteria outlined in the opening paragraph of this chapter. A nanostructure has been achieved in large lumps of steel by  phase transformation, with the design of the steel based on an understanding of the atomic mechanisms of crystal growth in the solid state.

\subsection{Alloy design}

Suppose we now use  theory to estimate the lowest temperature at which bainite can form.  Such calculations are illustrated in Fig.~\ref{fig:design}a, which shows for an example steel, how the $B_S$ and $M_S$ temperatures vary as a function of the carbon concentration. There appears to be no lower limit to the temperature at which bainite can be generated. On the other hand, the rate at which bainite forms slows down dramatically as the transformation temperature is reduced,  Fig.~\ref{fig:design}b. It may take hundreds or thousands of years to generate bainite at room temperature. For practical purposes, a transformation time of tens of days is reasonable, corresponding to a carbon concentration of about 1 \,wt\%, in which case bainite can be generated at a temperature as low as 125$^\circ$C, which is so low that the diffusion distance of an iron atom is an inconceivable 10$^{-17}\,$m over the time scale of the experiment! 


\begin{figure}[htbp]
\centering
\subfigure(a){
\includegraphics[width=0.45\textwidth]{Figure1a.eps}}
\subfigure(b){
\includegraphics[width=0.45\textwidth]{Figure1b.eps}}
\caption{(a) Calculated transformation start temperatures in Fe-2Si-3Mn steel as a function of the carbon concentration. (b) The calculated time required to initiate bainite.}
\label{fig:design}
\end{figure} 



A steel designed on this basis was manufactured and characterised;  Fig.~\ref{fig:micrographs} shows the structure obtained following isothermal transformation at 200$^\circ$C, consisting of platelets of bainitic ferrite only 200--400 \AA\ thick,  with intervening regions of the parent austenite $(\gamma)$. This {\em retained} austenite is important because when it undergoes stress or strain--induced martensitic transformation, it enhances the work--hardening capacity of the material, thereby avoiding the usual problem of fine--grained metals where ductility diminishes as the grain size is reduced. 


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=11cm]{Figure2b.eps}
\caption{Fe-0.98C-1.46Si-1.89Mn-0.26Mo-1.26Cr-0.09V wt\%, transformed at 200$^\circ$C for 15 days. Transmission electron micrograph.}
\label{fig:micrographs}
\end{center}
\end{figure}


The bainite obtained by low--temperature transformation is harder than ever achieved, with values in excess of  700~HV. Some strength, ductility and toughness data are illustrated in Fig.~\ref{fig:properties}. The simple heat treatment involves the austenitisation of  a chunk of steel  (at say 950$^\circ$C), followed by a gentle transfer into an oven at the low temperature (say 200$^\circ$C) to be held there for ten days or so. There is no rapid cooling -- residual stresses are avoided. The size of the sample can be large because the time taken to reach 200$^\circ$C from the austenitisation temperature is much less than that required to initiate bainite. This is an important commercial advantage.


\begin{figure}[htbp]
\centering
\subfigure(a){
\includegraphics[width=0.42\textwidth]{strength.eps}}
\subfigure(b){
\includegraphics[width=0.42\textwidth]{ductility_toughness.eps}}
\caption{Some mechanical properties of two superbainitic steels. (a) The ultimate tensile strength (UTS) and 0.2\% proof strength as a function of the volume fraction of bainitic ferrite ($V_b$) divided by the ferrite platelet thickness $t$. (b) Ductility (points and curve) and toughness $K_{IC}$ represented as crosses.}
\label{fig:properties}
\end{figure}


\clearpage

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