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\title{{\normalsize Edward de Mille Campbell Memorial Lecture}\\[4pt]Mechanism of the bainite transformation:  a turning point}

\author{H. K. D. H. Bhadeshia}
\makenomenclature
\begin{document} 
\maketitle

\centerline{Queen Mary University of London}
\centerline{School of Materials Science and Engineering}
\centerline{Mile End Road, London, h.bhadeshia@qmul.ac.uk}

\nomenclature[ga]{$\upalpha$}{Ferrite}
\nomenclature[gc]{$\upgamma$}{Austenite}

\section*{Abstract} 

Some aspects of the theory for iron and its alloys  have a chequered history, witness the $\upbeta$-iron controversy, the notion that molecules can move through iron, and not least, the mechanism of the bainite transformation. There was a period in which bainite was designated a non-lamellar eutectoid with no participation of shear, even though the deformation was there for all to see.  We have come a long way since, so much so, that calculations have led the creation of new steels that are now commercial, including  the world's first bulk nanostructured steel, and transportation infrastructure that has seen more than a billion gross-tonnes of traffic. My goal in this lecture is to highlight the depth of quantitative understanding and to resolve a few oddities. This does not minimise the wonderful experiments that have flourished, not just in establishing reality, but  in stimulating discoveries such as the tetragonality of the crystal structure of bainitic ferrite, emulating the discovery made originally by Fink and Campbell of asymmetric martensite, exactly 100 years ago. 				      

\printnomenclature[2cm]
\section{Introduction}

It is a privilege to present this Edward de Mille Campbell Memorial Lecture, the 99th in an inspiring series. There is a legacy of publications. I mention one in particular, by Morris Cohen \cite{Cohen:1949} in 1948 on the subject of retained austenite, that I studied prior to  my adventures at  Cambridge.  Having met 12 of the previous 98 speakers, and being familiar with the work of many others, I am humbled to be in that distinguished company.

 A quote from the then President of AIME  \cite{Raymond:1893} highlights Professor Campbell's resilience, noting that even after losing his sight in a laboratory accident, his work was still `remarkable for brilliancy and accuracy.' The first paper reporting the observation of the asymmetry of the martensite lattice, by Fink and Campbell \cite{Fink:1926}, was published posthumously.  Edgar C. Bain expressed admiration for that work, which supported his proposal for the pure, homogeneous deformation that can alter the lattice of austenite into that of Fe-C martensite. In writing my own lecture, I pay homage to both Campbell and Bain, because it is now known that similar asymmetry is a revealing feature of bainite, which arguably is the most addictive of all the structures of steel. The purpose here is to explore closure between credible  experiments and theory. 

Theory in principle can help design steels that are fit for purpose. Given historical data on iron and its alloys, is it possible to substitute machine learning for this task? This kind of learning serves us well in dealing with ill-understood complexity \cite{Bhadeshia:neural}, but it comes with the baggage of billions of empirical coefficients that are not easy to fathom.  Anderson \cite{Anderson:1977} made a telling comment during his 1977 Nobel lecture: {\em $\ldots$ the perfect computation simply reproduces Nature, does not explain her.}\footnote{I recommend a lovely book about Anderson, `A mind over matter', by Andrew Zangwill \cite{Zangwill:2021}.} 

In this paper, I shall not delve on the past, except in context. It is now accepted that the transformation is displacive, either  explicitly  \cite[p.4556, ][]{Stahlkrantz:2021}, or implicitly in the definition of carbon trapping   \cite{Benrabah:2024b,Dai:2024}. To question the mechanism of the lattice change therefore seems redundant, except when probing the role of phenomena such as the diffusional-displacive hypothesis. There remain difficulties that form the focus of this paper. Topics not covered to respect space considerations, are  generously explained elsewhere   \cite{Bhadeshia:bainite3,Bhadeshia:2021}, including the seminal role of mechanical stabilisation, the approach to equilibrium, structural length-scales, secondary carbide  reactions, the transition from upper to lower bainite, and well-established metallographic details. 


\section{Thermodynamics}

As always, it is wise to begin with a framework capable of handling an Avogadro's number of atoms, and where equalities are impossible to deny. Martensitic transformation is demonstrably diffusionless. Bainite forms at higher temperatures, auto-tempers during transformation, allowing some of the excess carbon to partition into the surrounding austenite. Its {\em original} composition cannot be measured directly although vestiges of that original supersaturated state persist in the form of carbon concentrations way above equilibrium. Analogous redistribution of carbon was observed as long ago as 1960 by Matas and Hehemann when they tempered mixtures of residual austenite and supersaturated martensite \cite{Matas:1960}. The phenomenon forms the basis of the `quench and partitioning' steels  \cite{Speer:2003} that are more formable than most TRIP-assisted automotive steels  \cite{Matlock:2006}. It was Hehemann who first suggested that bainite grows like martensite and then partitions some of its carbon \cite{Hehemann:1970}.


To achieve diffusionless growth, transformation must occur below the thermodynamic $T_0$ temperature, where the free energy of bainitic ferrite is less than that of austenite with an identical composition (Figure~\ref{fig:thermodynamics}). Another way of expressing this condition is that the concentration of carbon in austenite must fall  to the left of the $T_0$ curve for the transformation to be composition invariant. The $T_0$ condition modified to account for stored energy of the transformation product is labelled $T_0'$. In Figure~\ref{fig:thermodynamics}, a change from austenite of  composition `a' to ferrite `b' of identical composition is not spontaneously possible because it  increases free energy, while a similar `c' to `d' transition would reduce free energy, albeit to a smaller extent than if both phases adopted their equilibrium compositions.

\nomenclature[rx]{$x^{\upalpha\upgamma}$}{Concentration in $\upalpha$ that is in equilibrium or constrained equilibrium with $\upgamma$}
\nomenclature[rx]{$x^{\gamma\upalpha}$}{Concentration in $\upgamma$ that is in equilibrium or constrained equilibrium with $\upalpha$}
\nomenclature[rx]{$x_{T_0}$}{C in $\upgamma$, beyond which transformation without composition change impossible}
\nomenclature[rx]{$x^{\upgamma}$}{Carbon concentration in austenite}
\nomenclature[rx]{$\overline x$}{Average carbon concentration in alloy}
\nomenclature[ga]{$\upalpha_{\rm b}$}{Bainitic ferrite}
\nomenclature[ga]{$\upalpha_{\rm W}$}{Widmanst\"atten ferrite}
\nomenclature[ga]{$\upalpha'$}{Martensite}
\nomenclature[rvc]{$V_{\rm V}$}{Volume fraction}

\begin{figure}[ht!]
\centering
\raisebox{-5.3cm}{\subfigure[]{
\includegraphics[width=0.45\textwidth]{tzero.eps}}} \hfill
\begin{minipage}[t]{0.5\linewidth}
\subfigure[]{
\includegraphics[width=0.7\linewidth]{mechanism_b.eps}}\\
\subfigure[]{
\includegraphics[width=0.9\linewidth]{mechanism_a.eps}}
\end{minipage}
\caption{(a) Definition of the $T_0$ curve. $x^{\upalpha\upgamma}$ is the carbon concentration in ferrite that is in equilibrium with austenite, and $x^{\upgamma\upalpha}$; $x_{T_0}$ is the carbon concentration in austenite that cannot be exceeded if transformation occurs without a composition change. (b) Schematic illustration of the incomplete reaction phenomenon. (c) Experimental data showing that the growth of bainite stops when the  austenite carbon concentration reaches the $T_0'$ curve (Fe-0.43C-3{Mn}-2.12Si\,wt{\%} alloy) \cite{Bhadeshia:1980}. }
\label{fig:thermodynamics}
\end{figure}

If a plate of bainite forms without diffusion in a steel of average composition $\overline x$, but the difference ($x^{\upalpha_{\rm b}}-x^{\upalpha\upgamma}$) is then partitioned into the residual austenite, then successive plates must grow from C-enriched austenite (Figure~\ref{fig:thermodynamics}b). Such a reaction must cease when the concentration in austenite reaches $x^{\upgamma}=x_{T_0'}$. The maximum  fraction of bainite possible at complete partitioning is then:
\begin{equation} V_{\rm V}^{\upalpha_{\rm b}} \approx \frac{x_{T_0'}-\overline x}{x_{T_0'}-x^{\upalpha\upgamma}} \label{eqn:frac}\end{equation}
at the chosen transformation temperature and it is assumed that carbon remains in solid solution. The quantity $x_{T_0'}$ increases as the transformation temperature is reduced, allowing greater fractions of bainitic ferrite at lower transformation temperatures, beginning with zero  when $ x_{T_0'}=\overline x$ to some finite quantity when $x_{T_0'}>\overline x $. This explains the  {\em incomplete reaction phenomenon} when the nucleation of bainite is not a limiting condition. But the quantity of bainite never reaches equilibrium since in general, $ x_{T_0'}\ll x^{\upgamma\upalpha}$ for $T<B_{\rm S}$; holding at the transformation temperature for very long periods (43 days, 478\,\celsius, \cite{Bhadeshia:1981c}) can induce the lethargic growth of ferrite tentacles from the original $\upalpha_{\rm b}/\upgamma$ interface by a reconstructive mechanism. 

Equation~\ref{eqn:frac} is used routinely  in the  design and processing (mechanical and thermal) of bainitic steels. Examples from the last two years include \cite{Rana:2024,Ueji:2024,Yadav:2025,Peng:2025,Maisuradze:2025,Tolouei:2025,Castro:2025,Wojtacha:2025}, and from additively manufactured bainitic components \cite{Franceschi:2024,Guo:2024}. Older applications are reviewed in \cite{Bhadeshia:bainite3} and an undergraduate description  of the numerical  design can be found in  \cite[pp.448-460][]{Bhadeshia:steels5}.

\nomenclature[rx]{$x_{Ae_3'}$}{Carbon concentration at $(\upgamma+\upalpha)/\upgamma$ paraequilibrium phase boundary}
\nomenclature[rx]{$x_{Ae_3}$}{Carbon concentration at $(\upgamma+\upalpha)/\upgamma$ equilibrium phase boundary}

A nuance to the interpretation of the incomplete reaction is that the carbon in the austenite is heterogeneously distributed, first shown by Matas and Hehemann  using X-ray diffraction \cite{Matas:1961}, by high-resolution measurements \cite{Waugh:1982} and later demonstrated {\em in situ} during transformation using synchrotron X-rays \cite{Stone:2008b}. The heterogeneous distribution can be demonstrated using disarmingly simple optical microscopy, where austenite in the vicinity of bainite is retained but not away from it \cite[Fig.~3.5b,][]{Bhadeshia:bainite3}. This in part is due to the fast rate of transformation relative to the characteristic time for homogenisation,  but also because some regions of austenite become isolated by the sheaf structure of bainite. These isolated regions cannot transform once $x^\upgamma=x_{T_0'}$ but can continue to accumulate carbon partitioned from adjacent ferrite \cite{Waugh:1982}. Therefore, high-resolution measurements of  $x^\upgamma$, using the atom probe, show a range of concentrations with the scatter being greater than the uncertainty of measurements \cite{Waugh:1982b}. To take account of strain energy of $\approx 400\,{\rm J\,mol^{-1}}$ due to the shape deformation accompanying the growth of bainite, the $T_0$ condition is modified to $T_0'$. However, we shall focus here on $T_0$ because there will be accommodation between different crystallographic variants of bainite sheaves \cite{Pancholi:2008,Hou:2024} as the volume fraction increases, which results in a reduction of strain energy.

Evidence confirms that $\upalpha_{\rm b}$ growth ceases when composition-invariant transformation becomes thermodynamically impossible (Fig.~\ref{fig:thermodynamics}c). The most recent data confirming this are from  \cite{Lin:2022,Pushkareva:2023}. 
Figure~\ref{fig:T0_total} is a representation that enables  a comparison across a large number of published experiments,  where a coupled pair of red and black crosses correspond to an individual study defined by alloy composition and isothermal transformation temperature from an initially fully austenitic state. It is extraordinary that the average value of $x^\upgamma/x_{T_0}=1$ over some 150 experiments, while $x^{\upgamma}\ll x_{Ae_3'}$. This is  overwhelming evidence that $\upalpha_{\rm b}$ forms without any diffusion, with  subsequent partitioning of carbon. $Ae_3'$ is the $(\upalpha+\upgamma)/\upgamma$ paraequilibrium phase boundary.

The trend for the two ratios to increase with $x^{\upgamma}$ is because as the fraction of $\upalpha_{\rm b}$  increases, there is a greater tendency  that regions of austenite become isolated. These regions cannot transform once the $x_{T_0}$ is reached, but can continue to accumulate carbon from adjacent bainitic ferrite.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.75\textwidth]{T0_total.eps}
\caption{Comparison of a large number of experiments, all of which have different transformation temperatures and alloy compositions; the observed scatter is a reflection of these parameters, rather than stochastic errors. Such a comparison is facilitated by plotting ratios, where $x_{T_0}$ and $x_{Ae_3'}$ are calculated for each point. Each pair of points (red \&\ black) represents a separate experiment, and the dashed lines represent the average of some 150 experiments. Data compiled from \cite{Bhadeshia:1980,Waugh:1982,Yang:1987b,Wang:1992b,Chupatanakul:2006b,Caballero:2009,Caballero:2011b,Liu:2013,Liu:2015e,Hasan:2020,Lin:2022,Pushkareva:2023,Watanabe:2024,Peng:2025}, covering the period 1980-2025. Apologies if the coverage is not comprehensive (time limitations). }
\label{fig:T0_total}
\end{center}
\end{figure} 

\nomenclature[rbc]{$B_{\rm S}$}{Temperature below which bainite can form}
\nomenclature[rwc]{$W_{\rm S}$}{Temperature below which Widmanst\"atten ferrite can form}
\nomenclature[rtc]{$T_0$}{Temperature below which austenite can transform without a composition change}
 


\nomenclature[rvc]{$V_{\rm V_{max}}^{\upalpha_{\rm b}}$}{Maximum possible fraction of bainite}

Assuming that carbon  partitioning occurs  as an after-event, rather like the `quench and partitioning' technology for automotive martensitic steels \cite{Speer:2003}, and that the partitioned carbon remains in  solution in $\upgamma$, the maximum fraction of residual austenite at any temperature is given by $1-V_{\rm V}^{\upalpha_{\rm b}}$ with its carbon concentration $x^\upgamma=x_{T_0'}$ (Equation~\ref{eqn:frac}). This forms the basis of austenite mechanical stability control for the optimisation of mechanical properties \cite{Bhadeshiaedmonds:1983a,Bhadeshiaedmonds:1983b,Yang:1987b,Wang:1992b,Kutsov:1999,Chang:1999,Sugimoto:2000b,Hase:2004,mateo:2005,Fugarolas:2006,Chupatanakul:2006b,Soliman:2007,Caballero:2008,Caballero:2008d,Lawrynowicz:2012,Mateo:2014,Garcia:2014,Samanta:2016,Soliman:2016,Hasan:2020,Tanaka:2021}. The quench and partitioning process involves partial transformation to martensite, followed by heating above $M_{\rm S}$ to allow the excess carbon to partition into the austenite; this process has now been applied to bainite \cite{Li:2025}.

\subsection{Distribution of carbon in $\upgamma$, before \&\ after transformation}

Our early time-resolved X-ray study indicated that the austenite develops two lattice parameters just before transformation into bainite \cite{Babu:2007}, possibly consistent with Klier's  hypothesis that the austenite degenerates into carbon-rich and carbon-poor regions prior to the onset of the bainite reaction \cite{Klier:1944}. However, the resolution of these experiments was not good, so when repeated using the Grenoble synchrotron facility, the  austenite was proven to remain uniform in composition until the onset of $\upalpha_{\rm b}$ \cite{Stone:2008b}. This has been verified independently  using neutron diffraction \cite{Koo:2009}. 

It is expected, and found \cite{Waugh:1982,Caballero:2009}, that the carbon in the {\em residual}  austenite following partial transformation to bainite  will not be homogeneously distributed. The austenite becomes geometrically partitioned into films and blocks that hinder homogenisation, with inadequate time available to homogenise by diffusion. This is vivid  in an optical micrograph that shows how a sheaf of bainitic ferrite encloses retained austenite in its perimeter, but martensite surrounds it \cite[Fig.\,3.5b, ][]{Bhadeshia:bainite3}. The block-like regions of $\upgamma$ contain less carbon than the trapped-films so are unstable to stress-induced martensitic transformation, leading to a remarkable relationship between $T_0$ and toughness  \cite{Bhadeshiaedmonds:1983a,Bhadeshiaedmonds:1983b}. 


\section{Bainitic transformation as a deformation}

There is a shear strain of $s\approx 0.26$ parallel to the habit plane of an individual $\upalpha_{\rm b}$ platelet  and a uniaxial expansion of $\zeta\approx 0.03$ normal to that plane, together with some plastic relaxation in the adjacent austenite  \cite{Swallow:1996}. Sandvik observed using a transmission electron microscope that bainite plates displaced $\upgamma$-twin boundaries \cite{Sandvik:1982}, proving that the deformation is not an artefact of surface observations. There are many more macroscopic characterisations of the shape deformation due to sheaves of bainite  \cite{Ko:1952,Speich:1958}.

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{thermoelastic_b.eps}
\caption{The  $\upalpha_{\rm b}/\upgamma$ interface bows elastically between obstacles created by collision with other platelets,  \cite{Chang:1995b}. The arrow illustrates one such bowing event at  the $\upalpha_{\rm b}/\upgamma$ interface.}
\label{fig:bowing}
\end{center}
\end{figure} 

\nomenclature[rz]{$z_\ell$}{Lengthening rate of plate-shaped transformation product}
\nomenclature[rz]{$z_{\rm t}$}{Thickness of plate}
\nomenclature[rec]{$E_{\rm s}$}{Shear modulus of austenite}
\nomenclature[rgc]{$G_{\rm V}^{\rm e}$}{Strain energy per unit volume}
\nomenclature[rs]{$s$}{Shear strain component of invariant-plane strain shape deformation}
\nomenclature[gm]{$\nu$}{Poisson's ratio}
\nomenclature[gf]{$\zeta$}{Dilatational strain component of invariant-plane strain shape deformation}
\nomenclature[rgc]{$\Delta G^{\upalpha\upgamma}$}{$G^\upalpha-G^\upgamma$}

The shape deformation is an invariant-plane strain \cite{Bilby:1956}.  Eshelby derived the strain energy of a constrained transformation for the case where the phases remain coherent \cite{Eshelby:1957}, and Christian \cite{Christian:1958} used the method to derive the strain energy {\em per unit volume} $G_{\rm V}^{\rm e}$ due to an invariant-plane strain where the plate thickness $z_{\rm t}$ is much less than its length $z_\ell$:
\begin{equation}
\frac{1-\nu}{E_{\rm s}}G_{\rm V}^{\rm e} = \underbrace{ \frac{\pi z_{\rm t}}{4 z_{\rm \ell}} \zeta^2}_{\hbox{due to volume change}} + \underbrace{\frac{\pi}{8}(2-\nu)\frac{z_{\rm t}}{z_{\rm \ell}}s^2}_{\hbox{due to shear}} \label{eqn:elastic_energy}
\end{equation} 
where $\nu$ and $E_{\rm s}$ are the Poisson's ratio and shear modulus of austenite, respectively. The thin-plate shape of a displacive phase transformation is not an assumption, rather a requirement that minimises strain energy. This strain energy should feature prominently in the thermodynamic framework but frequently is neglected  \cite{Hillert:1960,Dai:2024,Benrabah:2023,Yamanaka:2006} even when the  lengthening measurements rely on the surface relief produced by the growing plate \cite{Wan:2013}. In those cases, any theory assumes a plate shape as an unjustified assumption.

For an elastically accommodated plate, or at a stage where the degree of plastic accommodation of the shape change is small, there is a  balance to be achieved between the driving force for transformation $\vert \Delta G^{\upalpha\upgamma}=G^\upalpha-G^\upgamma\vert$ and $G_{\rm V}^{\rm e}$. A plate that has stopped lengthening because of impingement with say $\upgamma/\upgamma$ boundaries, would continue to thicken until that balance is achieved. Figure~\ref{fig:bowing} shows that the vertical platelet of bainite that is impinged by others,   bows between the points of contact, rather like a dislocation that encounters precipitate arrays.  Accommodation twinning can also be induced in similar circumstances  when inclined $\upalpha_{\rm b}$ platelets collide \cite{Chang:1995b}.


\section{Transformation interface}

\nomenclature[rbc]{\bf B}{Deformation matrix for Bain strain}
\nomenclature[rrc]{\bf R}{Matrix representation of rigid body rotation}
\nomenclature[rpc]{\bf P}{Matrix representation of invariant-plane strain}
\nomenclature[rsc]{\bf S}{Deformation matrix}

The interface responsible for displacive transformation must be  glissile,  illustrated schematically in Figure~\ref{fig:coherency}. It consists of monoatomic-height coherency dislocations that can climb and glide conservatively but are associated with intense stress fields. Anticoherency dislocations  help accommodate the long-range strain fields. In classical terminology, the latter  dislocations accomplish the inhomogeneous lattice-invariant shear {\bf P}$_1$ \cite{Olson:1979b}. The Bain strain {\bf B}, rigid body rotation {\bf R} and homogeneous shear {\bf P}$_2$ that change the crystal structure and make the net deformation an invariant-line strain,  are achieved by the translation of the coherency dislocations which may be regarded as a distribution of transformation dislocations that model the elastic field \cite[p.1050,][]{Christian:2003}.  {\bf P}$_1$ helps macroscopically cancel the effect of {\bf P}$_2$ so that the total deformation is macroscopically an invariant-plane strain.  The deformation due to martensite is, therefore, a shear parallel to, and dilatation normal to the habit plane, resulting in substantial strain energy ($\approx  700\,{\rm J\,mol^{-1}})$ when constrained. 

The total Burgers vector content {\bf b}$_{\rm t}$ crossing a probe-vector {\bf p} in the $\upalpha/\upgamma$ interface  is then given by \cite{Bilby:1955,Frank:1950,Christian:1975}
\begin{equation} [\upgamma;{\bf b}_{\rm t} ]=\{{\bf I}-(\upgamma~{\rm S}~\upgamma)^{-1}\}[\upgamma;{\bf p}]\end{equation}
where $(\upgamma~{\rm S}~\upgamma)$ is the deformation that converts the bicrystal into a single crystal with the elimination of the interface.


 \begin{figure}[htp!]
\centering
\subfigure[]{
\includegraphics[width=0.45\textwidth]{coherency_dislocations.eps}}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{terrace.eps}}
\caption{(a) Schematic structure of an interface that causes displacive transformation, for a fully enclosed plate which has a curved interface. (b) Schematic structure of atomic step  and terrace structure of a flat, irrational interface that is does not enclose the transformation product.}
\label{fig:coherency}
\end{figure}

An alternative description (`topological model') deals with a single, macroscopically planar, irrational interface consisting of steps and terraces (Figure~\ref{fig:coherency}b). Another name for the step is a {\em disconnection}, a  line defect that exists at the edge of a terrace and separates two terraces of different heights; it is characterised by its step height and Burgers vector. But it is not a `disconnection' in the colloquial meaning of the word, but rather an elastic connection between the two lattices. Because its motion produces the structural  change, it is a  transformation dislocation. There are specific conditions that must be satisfied for this interface to move conservatively \cite{Ma:2007}.  The coherency strains at the terrace plane must be relieved by dislocation arrays, a 2nd set of line defects, that achieve the lattice-invariant deformation while ensuring that the intersection of the two sets of line defects does not impede the glissile character of the interface. This condition is identical to that from the crystallographic theory of martensite, that dictates the presence of  an invariant line in the interface, with the dislocations that accomplish {\bf P}$_1$ lying parallel to that line. They can  either  be screw dislocations or with Burgers vectors that do not lie in the plane of the interface, so that  they are able to glide conservatively. However,  for these conditions to be relevant, the orientation relationship assumed must be such that an invariant line can exist in the interface.

For a general interface (not necessarily glissile) the relatively coherent terraces have a continuously distributed Burgers vector content,  which in continuum terminology is a distribution of infinitesimal dislocations. The disconnection's Burgers vector accounts for accumulated misfit that the terraces cannot accommodate, effectively mopping up the distributed misfit. Models like these are used in schematic illustrations of the $\upalpha_{\rm b}/\upgamma$ interface  \cite{Miyamoto:2020,Benrabah:2023} and so it will be necessary to return to this topic later while discussing specific models.


\section{`Diffusional-displacive' transformation} \label{diffusional-displacive}

Plates can be precipitated with the same shape deformation as martensite, yet inherit a different substitutional solute content relative to the parent phase, for example \cite{Muddle:1994},
\begin{equation}[{\rm Ag}+x{\rm Al}] \rightarrow {\rm Ag_2Al} + [{\rm Ag}+y{\rm Al}] \qquad y<x\end{equation}
where the square brackets indicate a solid solution. Such a transformation has been labelled {\em diffusional-displacive}. The precipitates have a coherent or semi-coherent interface that permits a lattice correspondence to be maintained between the parent and product phases. The diffusion that facilitates the composition change requires atoms to move readily over the sites predicted by the lattice correspondence without creating additional sites or destroying existing sites by vacancy migration \cite{Christian:1997}. Any other diffusion would locally modify the shape change.

The mechanism naturally requires atomic mobility, and when constrained would have a strain energy term given by Equation~\ref{eqn:elastic_energy}. Bainitic ferrite has the same Fe/X atom ratio as the austenite from which it grows (X stands for a substitutional solute). Independent atom probe experiments demonstrate the lack of partitioning of substitutional solutes, which show no tendency to segregate to the $\upalpha_{\rm b}/\upgamma$ interface \cite{Bach:1980,Waugh:1982,Waugh:1982b,Josefsson:1988}. The first reported experiment was on Fe-1.4C-2.8Cr-1.6Mo-0.6Si-0.5V-0.3Mn\,at.\%\ steel transformed in part to bainite during cooling from 435$\to 400$\,\celsius\ before quenching \cite{Bach:1980}. The atom-probe profiles across an $\upalpha_{\rm b}/\upalpha'$ interface are illustrated in Figure~\ref{fig:Bach}. Even these early results do not suggest any partitioning of substitutional solutes nor of any segregation or desegregation in the $\approx 2$\,nm wide interfacial region, when the 95\%\ confidence limits are accounted for. The true width of the interface is unlikely to be 2\,nm,  but probably is an artefact of the spatial resolution of the composition measurements.

\begin{figure}[ht!]
\centering
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Bach_Mo.eps}}\hfill
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Bach_Cr.eps}}
\caption{Atom probe profiles across $\upalpha_{\rm b}/\upalpha'$ interface, based on a total of some 250-600 detected ions. The martensite is on the right of the interface in each diagram, with the marked interfacial region was identified originally on the basis of the carbon concentration profile,  not reported here for brevity, but worth noting that there was no segregation of carbon detected at the interface. The $\pm2\sigma$ bars are the 95\%\ confidence limits on the individual measurements. (a) Molybdenum. (b) Chromium. Selected data from Bach et al. \cite{Bach:1980}.}
\label{fig:Bach}
\end{figure}

Later work in which the $\upalpha_{\rm b}/\upgamma$ interface was preserved confirmed that the substitutional atoms are configurationally frozen during transformation, but also a surprising result that the carbon concentration in the bainitic ferrite was much greater than expected from the paraequilibrium condition \cite{Waugh:1982,Waugh:1982b}. This was attributed to carbon locked-in at dislocations in the $\upalpha_{\rm b}$, but further more detailed imaging established that  that large concentrations of carbon remain in {\em solid-solution} \cite{Pereloma:2012,Caballero:2012b}.  Many more high-resolution experiments have demonstrated that substitutional solutes do not partition during the bainite transformation, nor do they segregate to the transformation interface, Table~\ref{tab:atom}. Even carbon does not segregate to that interface, perhaps reflecting the extent of coherency of the $\upalpha/\upgamma$ boundary.

\clearpage
\begin{threeparttable}[!ht]
\caption{Chemical compositions (wt\%) of steels where bainite has been studied using the atom probe, listed in chronological order. For exact compositions including trace elements, see original reference. } 
\begin{center}
\begin{tabular}{cccccccccr}
\toprule
C & Mn & Si & Ni & Mo & Cr & Co & Al & Other & Reference \\[3pt]
\midrule
0.30 & 0.30 & 0.30 & & 2.76 & 2.62 & & & & \cite{Bach:1980,Bach:1981}\\[3pt]
0.39 & & 2.05 & 4.08 & & & & & & \cite{Waugh:1982} \\[3pt]
0.43 &3.0  & 2.02 &  & & & & & & \cite{Bhadeshia:1982b} \\[3pt]
0.08 & 0.61 & 0.15 &0.05 & 0.89 & 2.26 & & & & \cite{Josefsson:1987,Josefsson:1988,Josefsson:1989b} \\[3pt]
0.43 &2.8  & 2.2 &  & & & & & & \cite{Stark:1990} \\[3pt]
0.33 &2.8  & 2.16 &  & 0.52 & & & & & \cite{Stark:1990} \\[3pt]
0.75 & 1.95 & 1.63 & & 0.28 & 1.48 & & 0.01 & 0.1V & \cite{Peet:atom} \\[3pt]
0.98 & 1.89 & 1.46 & 0.26 & 1.26 & & & & 0.09V & \cite{Caballero:2007,Caballero:2008c,Caballero:2010} \\[3pt]
0.21 & 1.55 & 1.55 & & & & & & & \cite{Pereloma:2007} \\[3pt]
0.21 & 1.48 & 1.50 & &0.2 & & & & 0.036Nb & \cite{Pereloma:2007} \\[3pt]
0.22 & 1.52 & 1.19 & & & & & 0.57 & 0.037Nb & \cite{Pereloma:2007} \\[3pt]
0.21 & 1.52 & 1.18 & & 0.29& & & 0.57 & 0.036Nb & \cite{Pereloma:2007} \\[3pt]
0.83 & 1.98 & 1.46 & & 0.24 & 1.02 & 1.54 & & & \cite{Caballero:2010}\\[3pt]
0.78 & 1.95 & 1.49 & & 0.24 & 0.97 & 1.60 & 0.99 & &  \cite{Caballero:2010}\\[3pt]
0.79 & 1.98 & 1.51 & 0.1 & 0.24 & 0.98 & 1.58 & 1.06 & 0.1Cu & \cite{Pereloma:2012} \\[3pt]
0.71 & 1.52 & 1.00 & & & & & 1.04 & & \cite{Seol:2012} \\[3pt]
0.70 & 1.52 & 0.52 & & & & & 1.46  & & \cite{Seol:2012} \\[3pt]
0.71 & 1.50 & 1.50 & & & & & 0.50  & & \cite{Seol:2012}\footnotemark[1] \\[3pt]
0.22 & 2.47 & 1.51 & & & & & 0.50  & & \cite{Hofer:2015,Hofer:2015c,Hofer:2016} \\[3pt]
0.22 & 2.2 & 1.8 & & 0.2 & & &  0.50  & & \cite{Pushkareva:2023} \\[3pt]

\bottomrule
\end{tabular}
 \begin{tablenotes}
    \item[1] Unlike other $\upgamma/\upalpha_{\rm b}$ interfaces, Seol et al. \cite{Seol:2012} found one case (of many) in this specific alloy with minor Mn and Si peaks, potentially at a bainitic ferrite/segregated-austenite grain boundary junction.
  \end{tablenotes}
\end{center}
\label{tab:atom}
\end{threeparttable}%

\subsection{Atomic mobility}

Bainite forms at temperatures where atomic mobility is extremely limited. This low mobility makes solute drag or the directed movement of atoms during a diffusional-displacive transformation impossible to achieve.

Chemically-ordered tetrataenite (FeNi $P4/mmm$) forms below 321\,\celsius\ from a disordered Fe-Ni solid solution ($Fm\overline 3m$). It has been observed in meteorites that cool at extraordinarily slow rates \cite{Lyons:2019} but never has been produced on Earth by annealing the disordered solution. This is because it is estimated that it would take $10^4$\,years for an atom of nickel to execute one atomic jump  at   300\,\celsius\ \cite{Scorzelli:1997}. 

Meteorites also reveal Ni-rich $\upgamma$ `rims' due to partitioning from  $\upalpha$, the thickness of which is $\approx 10\,\upmu$m when cooled at  250\,\celsius\,per million years, but 100\,$\upmu$m when $\dot T= 1$\,\celsius\,per million years  \cite{Yang:1997c}. The thicknesses  scale approximately with the square root of the time ($t\propto 1/\dot T$), consistent with classical diffusion. 

Bainite has been obtained at 125\,\celsius\ where the diffusion distance of an iron atom is an inconceivable $10^{-17}$\,m over the time scale of the experiment  \cite{Bhadeshia:2010c}. At 478\,\celsius, the bainite reaction is 6000 times faster than the rate  at which reconstructive ferrite can grow \cite{Bhadeshia:1982e}. While there may be practical difficulties, there is no theoretical lower-limit to the temperature at which bainite can form \cite{Bhadeshia:hatfield};  a 100 year experiment is in progress to generate bainite at room temperature, scheduled to finish in  2104 \cite{100year}. 

Christian's tenet stands true, that account must be taken of the lack of any solute atom mobility at the temperatures of bainite formation \cite[p.1096,][]{Christian:2003}.

Even cementite that forms during the bainite transformation, grows without any change in the X/Fe ratio. This is similar to what is found when cementite generated by the tempering of martensite is characterised at a high enough resolution \cite{Babu:1993}. The reason is the same as that for martensite or bainitic ferrite, that inadequate atomic mobility prevents any partitioning of substitutional solutes. This is the sole reason why cementite does not precipitate from austenite in high-Si steels, because it would be forced to inherit silicon, which is anathema to cementite \cite{Kozeschnik:2008}. Figure~\ref{fig:Hofer} shows that there is no perceptible change in the X/Fe ratio across the $\upalpha/\uptheta/\upalpha$ interface.  Carbides too can grow by a paraequilibrium displacive transformation mechanism in those circumstances \cite[pp.419-444,][]{Bhadeshia:2021}.


\begin{figure}[H]
\begin{center}
\includegraphics[width=0.6\textwidth]{Hofer_2015.eps}
\caption{Fe-0.22C-1.5Si-2.5Mn\,wt\%\ steel transformed isothermally to bainite at 350\,\celsius.  Atom-probe scan across a cementite particle, the location of which is identified by the two vertical lines where between which the concentration of carbon is in excess of 10\,at.\%. Ferrite is on either side of the cementite. Data from Hofer et al. \cite{Hofer:2015}, replotted to show the Si/Fe and Mn/Fe atom ratios. }
\label{fig:Hofer}
\end{center}
\end{figure} 



\section{Other `limiting' conditions}

\nomenclature[rxc]{X}{Substitutional solute}
\nomenclature[rz]{$z$}{Distance coordinate}
\nomenclature[rz]{$z^*$}{Position of interface on $z$}

The $T_0$ thermodynamic condition is absolute if any transformation is to proceed without a composition change. We discuss here other proposals to explain the incomplete reaction phenomenon.

\subsection{Negligible partitioning}
The observed uniform X/Fe ratio across the $\upalpha_{\rm b}/\upgamma$ interface is sometimes explained using the negligible partitioning local equilibrium (`NPLE') mode of transformation \cite{Hillert:1994}. The quantitative form of this is by Coates \cite{Coates:1972,Coates:1973b}, who recognised difficulties, and it is worth emphasising that the associated concentration perturbations have  never been verified experimentally.

If $z$ is defined as a coordinate normal to the interface such that it is zero at the origin of the ferrite, and $z=z^*$ at the position of the $\upalpha/\upgamma$ interface, then NPLE at the point of {\em infinitesimal} transformation of an alloy of mean composition $\overline x$, is characterised as follows:
\begin{equation} \underbrace{\biggl(\frac{x^\upalpha_{\rm X}}{x^\upalpha_{\rm Fe}}\biggr)_{z< z^*}\approx \biggl(\frac{x^\upgamma_{\rm X}}{x^\upgamma_{\rm Fe}}\biggr)_{z>z^*}}_{\hbox{far field}} \qquad \underbrace{\biggl(\frac{x^\upgamma_{\rm X}}{x^\upgamma_{\rm Fe}}\biggr)_{z=z^*}  = \biggl(\frac{x^{\upgamma\upalpha}_{\rm X}}{x^{\upgamma\upalpha}_{\rm Fe}}\biggr)}_{\hbox{at interface}}\qquad \underbrace{
\biggl(\frac{x^\upalpha_{\rm X}}{x^\upalpha_{\rm Fe}}\biggr)_{z\leq z^*}  
= \biggl(\frac{x^{\upalpha\upgamma}_{\rm X}}{x^{\upalpha\upgamma}_{\rm Fe}}\biggr) \approx \biggl(\frac{\overline x_{\rm X}}{\overline x_{\rm Fe}}\biggr)}_{\hbox{in ferrite}}
 \end{equation} 
where $x$ is a mole fraction and $x^{\upgamma\upalpha}$ is the mole fraction in $\upgamma$ that is in equilibrium with $\upalpha$, so local equilibrium is maintained there because carbon partitions to the extent that its chemical potential is uniform everywhere. 

The far-field term ensures that there is no macroscopic evidence that substitutional solutes partition between the phases, because the tie-line that governs interface compositions is such that $\overline x_{\rm X} \approx x_{\rm X}^{\upalpha\upgamma}$, though $x^{\upgamma\upalpha}_{\rm X}\neq \overline x_{\rm X}$. A sharp concentration spike ranging from $x_{\rm X}^{\upgamma\upalpha}\to \overline x_{\rm X}$ (positive if X partitions into $\upgamma$) exists in the austenite at $z\geq z^*$, the existence of which has never been verified. 

The calculated extent of the diffusion field of substitutional solutes can become so small as to lose meaning, as noted originally by Coates \cite{Coates:1972}.  Zhang et al. \cite{Zhang:2012} found the calculated width of the spike of manganese in $\upgamma$ during the NPLE growth of ferrite at 775\,\celsius\ to be only 0.03\,nm and Xia et al. for transformations at 675-700\,\celsius\ calculated a width of  0.002-0.04\,nm, clearly physically impossible.  When the popular software {\em DICTRA} \cite{Borgenstam:2000} is used to estimate the concentration spike in the Fe-Mn-0.16C\,wt\%\ low-alloy steel transformed to ferrite at 650\,\celsius, with a step size of 5\,nm in $\upgamma$, the peak concentration of manganese is just a single point that defines the concentration profile in the austenite, so the actual gradient might be even greater in magnitude  \cite{Bloder:2025}, Figure~\ref{fig:DICTRA}.


\begin{figure}[H]
\begin{center}
\includegraphics[width=0.3\textwidth]{Bloder_profile.eps}
\caption{Discrete-calculations (points) representing the manganese concentration profile across an $\upalpha/\upgamma$ interface, during NPLE mode of transformation, using {\em DICTRA}. Data courtesy of Bernhard Bloder.}
\label{fig:DICTRA}
\end{center}
\end{figure} 

Sharp variations in concentration have a cost that is expressed in terms of a gradient energy term, which has been shown to be so large as to rule out the NPLE mechanism \cite{Bhadeshia:2016b,Bhadeshia:2021,Bhadeshia:steels5}. In any event, a plot of the NPLE $x^{\upgamma}_{\rm C}$ curve versus $T$  cannot be used as a {\em limiting} condition, because the tie lines in the Fe-C-X phase diagram shift until equilibrium (as opposed to local equilibrium) is achieved. The $T_0'$ curve, on the other hand, is an inviolable thermodynamic limit. 

Some \cite{Liu:2013,Liu:2015e,Dai:2018} have adopted a different approach in calculating the fraction of $\upalpha$ that can form by the NPLE mechanism. In Figure~\ref{fig:NPLE-PLE}, a horizontal construction line defines the concentrations $x_{\rm oe}$ and $x_{\rm ne}$. If it is {\em assumed} that bainite grows by the NPLE mode, the limiting carbon concentration in austenite at the point where such transformation becomes impossible, was identified as $x_{\rm ne}$, yielding
\begin{equation}V_{\rm V,max}^{\upalpha_{\rm b}}= \frac{x_{\rm ne}-\overline x}{x_{\rm ne}-x_{\rm oe}} . \label{eqn:NPLE-PLE} \end{equation}
The average composition of the steel is plotted as the dot on Figure~\ref{fig:NPLE-PLE}. If the mass balance between the elements is neglected, then when transformation begins by the NPLE mechanism, the dot migrates horizontally on the dashed line, to represent  the evolving composition of the austenite. As more $\upalpha$ forms, it translates to greater $x_{\rm C}^\upgamma$ until this equals $x_{\rm ne}$. Further transformation by the NPLE mechanism cannot occur, and at that point, $x^\upgamma \neq x_{\rm ne}$, but rather, $x^\upgamma=x^{\upgamma\upalpha}$ as defined by a tie line below the green tie-line, until the controlling tie line actually passes through the original alloy composition. Equation~\ref{eqn:NPLE-PLE}  is therefore incorrect. 



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.45\textwidth]{FeMnC.eps}
\caption{Adapted isothermal section of Fe-Mn-C phase diagram, original by Uhrenius \cite{Uhrenius:1978}.  The red line partitions the $\upalpha+\upgamma$ phase field into domains where NPLE or PLE conditions can exclusively apply.  The green tie-lines define sets of ferrite and austenite compositions that are in local equilibrium. The dot represents the alloy composition. }
\label{fig:NPLE-PLE}
\end{center}
\end{figure} 

Research on a Fe-0.3N-1Mn\,wt\%\  transformed to bainite at 500\,\celsius\ \cite{Sato:2024} revealed an apparently unexplained phenomenon. Figure~\ref{fig:Sato_2024} shows two composition profiles across a martensite (prior austenite)/bainitic ferrite interface.  Manganese did not partition when transformed for $t=10$\,s, but there was a redistribution when $t=60$\,s. It was concluded that the initial paraequilibrium state transitions into the  NPLE or PLE modes of growth. The concentration profile within the ferrite at 60\,s contradicts  {\em growth} because it should be flat, and in addition, $x_{\rm Mn,I}^\upgamma \ll x_{\rm Mn}^{\upgamma\upalpha}$ to be consistent with local equilibrium at the interface. 

The observations indicate the partitioning of manganese after a plate that is arrested in its growth, but continues to be held at 500\,\celsius. Arrested growth is a feature of bainite and its sub-unit mechanism, given plastic relaxation that mechanically stabilises the adjacent austenite \cite[228-235, ][]{Bhadeshia:bainite3}.  The manganese depletion in the ferrite extends $\approx 10$\,nm which is greater than an estimate based on  $z=\sqrt{2D_{\rm Mn}^\upalpha t}\approx 2\,$nm, but the intense dislocation tangles created at the interface in both $\upgamma$ and $\upalpha_{\rm b}$ \cite{Bhadeshia:1979} may  explain this discrepancy. The experiment may represent a classic example of post-transformation diffusion.


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.4\textwidth]{sato_2024.eps}
\caption{Manganese concentration profiles across $\upgamma/\upalpha_{\rm b}$ interface; the austenite transformed into $\upalpha'$ on quenching from the transformation temperature. The alloy has an overall composition Fe-0.3N-1Mn\,wt\%\ but the nitrogen profiles are omitted for clarity, though the information is used to identify the position of the interface (vertical dashed lines).  The equilibrium manganese concentration $x^{\upgamma\upalpha}_{\rm Mn}$ is calculated using  commercial software \cite{MTDATA}. Data from Sato et al. \cite{Sato:2024}.}
\label{fig:Sato_2024}
\end{center}
\end{figure} 




\subsection{The $\mathbb{W}B_{\rm S}$ line} \label{wbs}

\nomenclature[rw]{$\mathbb{W}B_{\rm S}$}{So-called universal start temperature}
\nomenclature[ra]{$a$}{Empirical constant }
\nomenclature[rw]{$w_i$}{Weight percent of element $i$}

An interpretation of the transformation-start temperatures relies on a limiting function $\mathbb{W}B_{\rm S}$ defined by Hillert \cite{Hillert:1960} whose plate lengthening rate ($v_\ell$) equation (Equation~2 of \cite{Hillert:1960}) underestimated some measured data. So a variable empirical term $a$  was introduced to amend the supersaturation to $(x^{\upgamma\upalpha}_{\rm C}-\overline x_{\rm C}- a)$ allowing fit to experimental data.   The resulting $\mathbb{W}B_{\rm S}$ line on a plot of temperature versus the carbon concentration is designated as the limiting composition of austenite, at which the growth rate of the plate equals zero:
\begin{equation} \mathbb{W}B_{\rm S}\,/\,^\circ{\rm C} = 850-260w_{\rm c}-78w_{\rm Mn} -33w_{\rm Ni} -70w_{\rm Cr}-75w_{\rm Mo} -61w_{\rm Si} \label{eqn:Hillert} \end{equation} where $w_i$ represents the concentration in weight percent of the element identified in the subscript. Since this is an empirical fit, it need not account for features such as transformation strain, nucleation, microstructural detail, nor a justification of the plate shape of the ferrite. Plate growth  stops when $x^\upgamma_{\rm C}=x_{\mathbb{W}B_{\rm S}}$. 

A later adaptation expresses the deviation from local equilibrium  in terms of a thermodynamic driving force to define a $\mathbb{W}B_{\rm S}$ function but still using the original Equation~\ref{eqn:Hillert}.  The work explicitly ruled out any physical interpretation of the `barrier' that  keeps the reaction from reaching equilibrium \cite{Leach:2018}. This makes it difficult to probe, but some general comments can be made:
\begin{enumerate}
\item $\mathbb{W}B_{\rm S}$ sets a permanent deviation from equilibrium that is interpreted to stop plate growth dead. Why this applies  only  to plates is not obvious, but it nevertheless is taken in its unmodified form \cite{Hillert:1960} to alternatively interpret the incomplete reaction phenomenon, e.g., \cite{Hillert:2009,Leach:2018,Watanabe:2024}. 


\item Analysis of  the lengthening rate of Widmanst\"atten ferrite plates using Trivedi's exact solution (Equation~9, \cite{Trivedi:1970}) for the diffusion problem that allows the composition to vary along the interface of the plate shape-preserving solution, that accounts for a changing curvature at the interface and associated capillarity effects,  as well as the concentration-dependent diffusion of carbon in austenite \cite{Smith:1953b,Bhadeshia:1981b}, shows no empirical deviation from equilibrium ($a=0$) \cite{Bhadeshia:1985b} other than that due to capillarity. Indeed, the calculated growth rates are slightly greater than expected, Figure~\ref{fig:plate_v}. Earlier models for plate lengthening, such as those used by  \cite{Hillert:1960} were appropriate for their time, but the more rigorous Trivedi solution  provides a better representation.\footnote{Christian has a fascinating  assessment of plate growth solutions \cite[p.497,][]{Christian:1975}.}  The logical conclusion is that the $\mathbb{W}B_{\rm S}$ function  has an invalid foundation and should be discarded. 

In a review  of plate growth models, Hillert in 1975 \cite{Hillert:1975} stated that {\em the previous conclusion regarding friction at the interface was the result of inadequacies in the theoretical treatment}, but the model by Leach et al. \cite{Leach:2018} and others  still rely on Equation~\ref{eqn:Hillert}.

\nomenclature[gr]{$\sigma$}{Interfacial energy per unit area}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.35\textwidth]{plate_v.eps}
\caption{Measured versus calculated lengthening rates of plates of Widmanst\"atten ferrite. The calculations are based on the Zener maximum velocity hypothesis and the $\upalpha_{\rm w}/\upgamma$ interfacial energy per unit area is taken to be 0.2\,J\,m$^{-2}$. Details in \cite{Bhadeshia:1985b}; experimental data  due to \cite{Hillert:1960,Townsend:1968,Simonen:1973}.}
\label{fig:plate_v}
\end{center}
\end{figure} 

\end{enumerate}

\subsection{Solute drag, substitutional solutes}

Substantive observations reveal the absence of elemental segregation at the $\upalpha_{\rm b}/\upgamma$ transformation interface, Section~\ref{diffusional-displacive}.  `Solute drag' nevertheless often is invoked to interpret imprecise data. The drag formally can be defined as a dissipation of free energy due to chemical segregation within the  $\upalpha_{\rm b}/\upgamma$ interface, which may attenuate the mobility of the boundary.  

Consider a boundary of thickness $z_{\rm b}$. The drag theories either predict \cite{Cahn:1962b}, or are designed \cite{Hillert:1969,Hillert:1976} so the  solute concentration at the trailing edge of the interface $(z=z^*-z_{\rm b})$, during steady-state motion, always  is equal to $\overline x$ as if the boundary did not exist. The composition in a region $z\geq z^*$, roughly extending the width of the boundary is $x<\overline x$, making diffusion  necessary in the perfect parent phase, Figure~\ref{fig:drag}a.

\nomenclature[rgc]{$G_{\rm S}$ }{Segregation energy (of solute to interface)}

Calculations in which the segregation energies $G_{\rm S}$  of a variety of solutes were estimated using first-principles applied to a $\Sigma5$ austenite grain boundary, and then used  to estimate the free energy needed to overcome solute drag using a model by \cite{Purdy:1995}, are illustrated in Figure~\ref{fig:drag}b  \cite{Bloder:2024}. The calculations are limited to elevated temperatures but the $G_{\rm S}$ values refer to 0\,K because of the first principles method used to derive them. In principle, the tendency for boundary segregation becomes more intense at lower temperatures, so the calculations for elevated temperatures can be regarded as upper limits. More relevant is the fact that at lower temperatures, Figure~\ref{fig:drag}b suggests that the solute drag effect becomes insignificant. The solutes there are Nb, Mn and Ti but we know from other first principles calculations that that  Cr, Mo have similar $G_{\rm S}$ values \cite{Scheiber:2024}. However, Mo is not found to segregate to ferrite-plate/$\upgamma$ interfaces, during transformation for 60\,s at $T>B_{\rm S}\{{\rm calculated}\}$  temperature \cite{Miyamoto:2019,MUCG}. The main conclusion, assuming that solute drag is relevant to a displacive transformation, is that even if exists, it is not significant. For example, it is not implemented in \cite{Benrabah:2023,Benrabah:2024,Dai:2024} for this very reason.
\begin{figure}[hbt!]
\centering
\subfigure[]{
\includegraphics[width=0.25\textwidth]{Bloder2_2024.eps}}\hskip 1cm
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Bloder_2024.eps}}
\caption{(a) Schematic illustration of the distribution of solute that is present in a moving grain boundary. The motion is along $+z$ and the position of the interface of thickness $z_{\rm b}$ is defined by $z^*$. The excess concentration is that which exceeds $\overline x$. (b) Calculations of the free energy required to overcome solute drag. Selected data from \cite{Bloder:2024}, where the assumptions necessary to obtain absolute (rather than dimensionless) quantities are detailed. }
\label{fig:drag}
\end{figure}


\subsection{Carbon in $\upalpha_{\rm b}$} \label{carbon-alpha}

There are extraordinary vestiges of diffusionless transformation to $\upalpha_{\rm b}$. The carbon concentration in bainitic ferrite is far in excess of equilibrium, thought originally to reside in dislocations \cite{Waugh:1982,Waugh:1982b,Stark:1988,Peet:atom,Bruna:2005,Caballero:2007,Pereloma:2007,Caballero:2014b}.\footnote{Vasudevan et al. \cite{Vasudevan:1958} using chemical analysis revealed supersaturated bainitic ferrite as long ago as 1958.} Later work Pereloma et al. and Caballero et al. showed that it is in fact in  solid solution  \cite{Pereloma:2012,Caballero:2012b,Pushkareva:2023},  Figure~\ref{fig:bainite_supersaturation}a and persists there even after 250\,h at 200\,\celsius, so it is not mobility that stops the carbon from partitioning into the residual-$\upgamma$. 
\begin{figure}[ht!]
\centering
\subfigure(a){
\includegraphics[width=0.4\textwidth]{Caballero_supersaturation.eps}}\hskip 1cm
\subfigure(b){
\includegraphics[width=0.4\textwidth]{BCT.eps}}
\caption{(a) Carbon dissolved in $\upalpha_{\rm b}$ during holding at  200\,\celsius\ \cite{Mateo:2015}. (b) Fe-C phase diagram for the equilibrium between body-centred tetragonal $\upalpha$ and $\upgamma$  \cite{Jang:2012b}.}
\label{fig:bainite_supersaturation}
\end{figure}

 The reluctance of the carbon to partition from $\upalpha_{\rm b}^{\rm bct}$ is because the Bain strain leaves it in just one of the three sub-lattices of the interstices, making the unit cell tetragonal; the residual tetragonality is verified experimentally \cite{Smith:2013,Dutta:2014,Dutta:2014b,Smith:2015,Mateo:2015,Mirzayev:2017,Xiong:2018,Singh:2025}. The relevant equilibrium in these circumstances is between $\upalpha_{\rm b}^{\rm bct}$ and $\upgamma$, not that between cubic-ferrite and $\upgamma$. Figure~\ref{fig:bainite_supersaturation}b shows a calculated Fe-C diagram for the $\upalpha^{\rm bct}/\upgamma$ equilibrium \cite{Jang:2012b}; the solubility of carbon is increased significantly, explaining the large concentration that persists within $\upalpha_{\rm b}^{\rm bct}$.

Alternatively,  carbon atoms have been suggested to associate with host vacancies in the $\upalpha_{\rm b}$, thus compromising their mobility \cite{Rementeria:2020}, but in fact the fraction of those atoms that pair with host vacancies is negligibly small  \cite{McLellan:1988c}, \cite[pp.117-119, ][]{Bhadeshia:2021}. 
\section{Plate shape}

\nomenclature[rl]{$\ell$}{Length of plate}
\nomenclature[rec]{$E$}{Young's modulus}

The thin-plate shape of bainite is entirely due to the transformation strain, after all, idiomorphic ferrite which grows with a reproducible orientation relationship with the austenite does not adopt a plate shape because the anisotropy of $\sigma^{\upgamma\upalpha}$ interfacial energy is not sufficient for the equilibrium shape to be a plate. Many phase field simulations of ferrite plates have to assume unrealistic  $\sigma^{\upgamma\upalpha}$ anisotropy, where the plate tip is set at an admittedly arbitrary $1\,{\rm J\,m^{-2}}$ relative to other interface orientations. It is for this reason that Cottura et al. considered that the levels of interfacial-energy anisotropy used to generate the shape are unrealistic, so elastic energy must play a role \cite{Cottura:2014}; the subject has been reviewed \cite{Bhadeshia:2021}. The sub-unit morphology illustrated elsewhere \cite[Fig.\,2.3,][]{Bhadeshia:bainite3} is generated because the plastic relaxation of the shape deformation caused by bainite, creates sufficient dislocation debris in the adjacent austenite to stop growth, which is followed up by further platelets generated in the same orientation \cite{Chatterjee:2006}. 

Measurements have shown that a particular form of bainite generated at very low temperatures has a shape deformation with a shear strains as large as $s\approx 0.46$ \cite{Peet:2011b}. The individual  platelets are then as thin as 20\,nm  \cite{Bhadeshia:2010c}.


\section{Partial supersaturation}
\subsection{Olson et al.}


It is feasible in principle that only some of the carbon partitions during transformation, the rest being trapped by the advancing interface with an increase its chemical potential, hence the term {\rm trapped}. The diffusion field in the austenite ahead of the interface would be subject to random perturbations, in which case, it would adjust to lower free energy states. In these circumstances, there is nothing to stabilise growth with partial supersaturation from collapsing towards equilibrium. To model {\em stable} growth in which $x^\upalpha>x^{\upalpha\upgamma}$ requires  further interface response functions, in addition to that governing the motion of the diffusion field; three such functions are needed to solve for $x^\upalpha-x^{\upalpha/\upgamma}$, the composition of the austenite at the interface $x^\upgamma_{\rm I}\neq x^{\upgamma\upalpha}$, and the lengthening rate for a plate ($v_\ell$). 

Consider, for simplicity, just two interface response functions, the velocity as a function of the diffusion of carbon in the austenite ahead of the interface, and secondly, interface mobility. Figure~\ref{fig:dissipations} shows how the available driving force $\Delta G$ is partitioned into the dissipations $G_{\rm dd}$ and $G_{\rm id}$, for the diffusion and mobility functions respectively, such that when these quantities are set and substituted into the functions 
\begin{equation}v_\ell\{\rm diffusion\}=\xi\{{\mathit G}_{\rm dd}\}\qquad\hbox{ and}\qquad  v_\ell\{\rm mobility\}=\psi\{{\mathit G}_{\rm id}\},\end{equation} the outcome is $v_\ell\{\rm diffusion\}= {\mathit v}_\ell\{\rm mobility\}$.

\begin{figure}[ht!]
\centering
\subfigure[]{
\includegraphics[width=0.55\textwidth]{dissipations.eps}}\hfill
\subfigure[]{
\raisebox{8mm}{\includegraphics[width=0.35\textwidth]{dissipations2.eps}}}
\caption{(a) Constant temperature free-energy curves showing the quantities $G_{\rm dd}$, $G_{\rm id}$ for the case where the interface compositions are as illustrated in (b). Note that the chemical free energy change $\Delta G$ in this illustration does not include stored energy terms due to elastic strains or interface energy. $x_{\rm m}^\upgamma<x^{\upgamma\upalpha}$ is the concentration of carbon at the interface in austenite when $G_{\rm id}=0$ with the ferrite containing $x^\upalpha>x^{\upalpha\upgamma}$ of carbon.}
\label{fig:dissipations}
\end{figure}

\nomenclature[rgc]{$G_{\rm dd}$}{Free energy dissipated in diffusion}
\nomenclature[rgc]{$G_{\rm id}$}{Free energy dissipated in transfer of atoms across sharp interface}
\nomenclature[rp]{$p$}{P\'eclet number}
\nomenclature[rk]{$k_{\rm p}$}{Partition coefficient}
\nomenclature[rk]{$k_{\rm e}$}{Equilibrium partition coefficient}
\nomenclature[gk]{$\lambda$}{Interatomic distance}
\nomenclature[rx]{$x^\upgamma_{\rm I}$}{Carbon in austenite at interface with ferrite, $x^\upgamma_{\rm I}\leq x^{\upgamma\upalpha}$}
\nomenclature[rdc]{$\overline D_{\rm C}$}{Weighted average of concentration-dependent diffusion coefficient of carbon}
\nomenclature[rqc]{$Q$}{Activation energy which is a function of interfacial driving force}
\nomenclature[rgc]{$\Delta G_{\rm mech}$}{Mechanical barrier to interface motion}
\nomenclature[gr]{$\sigma_{\rm SS}$}{Solid-solution strengthening}
\nomenclature[gr]{$\hat\sigma$}{Back stress experienced by interface due to defects in austenite}
\nomenclature[rmc]{M}{Taylor factor}
\nomenclature[rm]{$m_{\rm Schmidt}$}{Schmid factor}
\nomenclature[rb]{$b$}{Magnitude of Burgers vector}
\nomenclature[rq]{$\rho$}{Dislocation density}
\nomenclature[gkc]{$\Lambda$}{Spacing between disconnection line-defects in interface}
\nomenclature[rdc]{$D^{\mathbb I}_{\rm C}$}{Diffusion coefficient of carbon across and within interface}
\nomenclature[rgc]{$G_{\rm SB}$}{Stored energy of bainite}
\nomenclature[rgc]{$G_{\rm SW}$}{Stored energy of Widmanst\"atten ferrite}
\nomenclature[rgc]{$G_{\rm N}$}{Minimum driving force needed to nucleate $\upalpha_{\rm b}$ or $\upalpha_{\rm W}$}
\nomenclature[rmc]{$M_{\rm S}$}{Martensite-start temperature}
\nomenclature[rgc]{$G^{\upalpha'}_{\rm N}$}{Driving force needed to nucleate martensite}

The three interface response functions needed to solve the three unknowns, all of which must lead to the same $v_\ell$, are the diffusion equation, the interface mobility determined by the structural change across the interface, and another associated with solute trapping.  The functions are as follows, with details  in the original work by Olson et al.  \cite{Olson:1990}:
\begin{subequations}\begin{align}
&\hbox{interface mobility:}& v_\ell = v_\circ \exp\biggl\{-\frac{Q^*}{kT} \biggr\} \label{eqn:Olson_1}\\
&\hbox{diffusion field:}&  \frac{\overline x - x^\upgamma_{\rm I}}{x^\upalpha-x^\upgamma_{\rm I}} = (\pi p)^{0.5} \exp\{p\} {\rm erfc}\{p^{0.5}\}\quad\hbox{where} \quad p=\frac{v_\ell r}{2\overline{D}_{\rm C}^\upgamma}  \label{eqn:Olson_2} \\
&\hbox{solute trapping:}& v_\ell = \frac{D\{x_{\rm I}\}}{\lambda} \biggl(\frac{k_{\rm p} - k_{\rm e}}{1-k_{\rm p}} \biggr) 
\label{eqn:Olson}\end{align}\end{subequations}

The velocity  $v_\circ$ is 30\,${\rm m\,s^{-1}}$, and $Q^*$ is an activation energy that is a function of the net interfacial driving force.  For the diffusion field,  $r$ is the radius of the plate tip, and $\overline D_{\rm C}$ is an integrated average diffusivity given by Trivedi. The Aziz solute-trapping function is used, where the partition coefficient $k_{\rm p}=x^\upalpha/ x^\upgamma_{\rm I}$ and $k_{\rm e}$ is the equilibrium partitioning coefficient; $\lambda$ is the inter-site distance.   Figure~\ref{fig:mech}a (red line) shows just one of the many results obtained \cite{Olson:1990}. The method does not rely on the selection of plate-tip radius that maximises the diffusion field velocity. The supersaturation increases with undercooling until the onset of martensitic transformation, at the correct martensite-start temperature.  The other data in Figure~\ref{fig:mech} are discussed in later sections.

\begin{figure}[!htp]
\begin{center}
\includegraphics[width=0.5\textwidth]{mech2.eps}
\caption{The normalised supersaturation of a ferrite plate versus the temperature for Fe-C alloys. The red curve is from Olson et al. for Fe-0.4C\,wt\%\ steel, where the martensite-start temperature is correctly predicted when $x^\upalpha_{\rm C}=\overline x_{\rm C}$  \cite{Olson:1990}. The black curves are from Benrabah et al.\cite{Benrabah:2024} for thee steels including Fe-0.4C\,wt\%,  with the vertical lines on each curve representing the temperature at which martensite is expected, but not obtained for reasons elaborated in the text.}
\label{fig:mech}
\end{center}
\end{figure} 



\subsection{Benrabah et al.}



A recent alternative model by Benrabah et al. \cite{Benrabah:2023,Benrabah:2024}  proposes a flat irrational interface with an atomic step and terrace structure, that does not enclose the growing plate. This open structure allows for continuous growth and is assumed to be displacive since carbon is trapped and  iron diffusion is slow $D_{\rm C}\gg D_{\rm Fe}$. The model neglects strain energy and does not justify the plate shape. Furthermore, it uses just one interface response function, an outdated method for calculating the diffusion field velocity \cite[p.765,][]{Hillert:1957}, instead of the more relevant Trivedi function. Interface mobility  (Equation~\ref{eqn:Olson_1}) is neglected, effectively setting $v_\ell=v_\circ$ in the interface mobility function. Therefore, the driving force $\Delta G_{\rm id}$ (Figure~\ref{fig:dissipations}) becomes available and is defined as the `chemical driving force for growth'. This is not sufficient to solve for the three unknowns ($x^\upalpha-x^{\upalpha/\upgamma}$, $x^\upgamma_{\rm I}\neq x^{\gamma\alpha}$, and $v_\ell$) so the Zener maximum velocity criterion is used to fix the velocity as:
\begin{equation}v_\ell= \frac{D_{\rm C}^{\upgamma}}{2r} \frac{x^{\upgamma}_{\rm I, C}-\overline x_{\rm C}}{\overline x_{\rm C}-x_{\rm I, C}^{\upalpha}} \qquad\qquad \hbox{{\em cf.} equation~\ref{eqn:Olson_2} }\label{eqn:1960} \end{equation} where $r$ is a plate tip radius of curvature. This approach has the contradiction that the advancing plate tip is curved (with radius $r$), but the structure of the irrational interface is taken to be flat. In Equation~\ref{eqn:1960}, $v_\ell\rightarrow \infty$ as $x_{\rm I, C}^{\upalpha}\rightarrow \overline x_{\rm C}$ rendering it unsuitable on its own to represent martensitic transformation; nor is a maximum velocity criterion applicable for martensite because it has its origin in the capillarity effect associated with the partitioning of carbon. The model therefore is unable to predict the onset of martensite (Figure~\ref{fig:mech}), even though it is argued that a `high supersaturation' solution exists \cite{Benrabah:2024}.

A `barrier' $\Delta G_{\rm mech}$ is introduced which the interface must overcome as it translates into the austenite, as its structure interacts with lattice dislocations, as well as solute hardening in the austenite ($\sigma_{\rm SS}$).  This resistance is expressed as a back-stress, $\hat\sigma$, which is calculated as a shear resistance $\tau\propto \sqrt{\rho^\upgamma}$ where $\rho^\upgamma=10^{13}\,{\rm m}^{-2}$ ,  the Taylor factor $M\approx 3$ converting it into $\hat\sigma$. 
  
  The back-stress is defined as follows:
 \begin{equation}\hat\sigma= 0.15 M\,E_{\rm s}\,b\,\sqrt\rho +\sigma_{\rm SS} \label{eqn:back-stress} \end{equation}
 with  $b=2\times10^{-10}\,$m,\footnote{Corrected from the original, see Appendix}; the shear modulus  is $E_{\rm s}\approx 76.171-0.0345(T-273)\,$GPa where $T$ is in Kelvin \cite{Fukuhara:1993}.  
 In this way, the barrier is estimated as
\begin{equation} \Delta G_{\rm mech}=V_{\rm m} \hat\sigma\,b\,m_{\rm Schmid} \Lambda^{-1} \label{eqn:mech} \end{equation}
where the partial molar volume $V_{\rm m}=7.09\times10^{-6}\,{\rm m^3\,mol^{-1}}$. The purpose of the Schmidt factor (0.5) is  to convert $\hat\sigma$ into a resolved shear stress, though how its value is justified is not clear since there is no crystallography in the analysis. The interaction between the disconnection and $\upgamma$-dislocation strictly is given by a Peach-Kohler vector equation involving a stress tensor, line vectors and Burgers vectors \cite[p.91][]{Hirth:1992}. Since there is no crystallography in the model, all these parameters that depend on the structure of the interface are missing, as is the justification for the undefined plane of the interface. The barrier is analogous to the constant $a$ used in \cite{Hillert:1960} so it is not surprising that Benrabah et al. \cite{Benrabah:2023} found Equation~\ref{eqn:1960} to under-predict $v_\ell$ as in the original work but the `mechanical barrier' purports to give a physical interpretation to the empirical constant that is based on outdated plate-lengthening theory  \cite{Bhadeshia:1985b}.  

Figure~\ref{fig:mech-2} shows the original results for $\Delta G_{\rm mech}$ and the data essentially reproduced here (dashed line) for the case where $x_{\rm I}^\upgamma\approx x^{\upgamma\upalpha}$. 
 
\begin{figure}[!htp]
\begin{center}
\includegraphics[width=0.5\textwidth]{mech.eps}
\caption{The barrier to interface motion due to Benrabah et al. \cite{Benrabah:2023} and a reproduction of their results by the present author, with caveats in the Appendix. As described in the text, the curves are for the case where the composition of the austenite at the interface is the equilibrium value. }
\label{fig:mech-2}
\end{center}
\end{figure} 


The proposed interface model presents a conceptual difficulty due to its inherent stress field. The combination of localised disconnection stress and distributed terrace stress forms a dipole-like structure, causing their long-range stress fields to cancel each other out. This self-balancing configuration means the back-stress exerted by the interface on a distant dislocation is significantly weaker, as the stress field decays rapidly (as $1/z^2$) instead of the slower $1/z$ decay seen in simpler models.

The model's derivation of the mechanical resistance, $\Delta G_{\rm mech}$, is flawed because it assumes a significant interaction between uncompensated disconnections and distant dislocations, which exaggerates the magnitude of the resistance, Figure~\ref{fig:mech-2}.

 This discrepancy can also be seen by calculating the  interfacial energy $\sigma^{\upgamma\upalpha}$, based on the model's structure. Using the assumed values \cite{Benrabah:2023} for the Burgers vector $b=2\times 10^{-10}\,$m, and disconnection spacing $\Lambda=1.5\times10^{-9}$\,m, and $E_{\rm s}^\upgamma=80$\,GPa, the interfacial energy per unit area is estimated as:
\begin{equation}\sigma^{\upgamma\upalpha}\approx \frac{E_{\rm s}b^2}{\Lambda} = 2.23\,{\rm J\,m^{-2}},\end{equation}
where  core energy is neglected, the same assumption as \cite{Benrabah:2023}. This value is an order of magnitude larger than the  value of $0.23\,{\rm J\,m^{-2}}$ applied in the calculations \cite{Hillert:1960,Benrabah:2023}. The reason for this overestimation is that the model fails to account for the fact that the stress fields of the disconnections and terraces compensate. The compensating interaction in $\upalpha'/\upgamma$ interfaces reduces the extent of the stress field to about the distance between between the anticoherency dislocations, with a realistic $\sigma^{\upgamma\upalpha} \approx 0.2\,{\rm J\,m^{-2}}$ \cite{Olson:1982c}.


For reasons already discussed, the model fails to predict the onset of martensitic transformation. Figure~\ref{fig:mech}a shows that in all three cases illustrated, the supersaturation $x^\upalpha/\overline x \ll 1$ at $M_{\rm S}$, and this is without considering the strain energy due to the shape deformation accompanying transformation. Because the analysis has less than three interface response functions, it permits the relationship between $v_\ell$ and $x^\upalpha/\overline x$ to be estimated at a given temperature, but does not assign  these quantities uniquely. Figure~\ref{fig:mech-2} shows how the `mechanical barrier' increases dramatically as the transformation temperature is reduced; it was possible to essentially reproduce the result of \cite{Benrabah:2023}. The sensitivity to temperature has is origins in $x^{\upgamma\upalpha}$ which increases from just 0.3C\,wt\%\ at 800\,\celsius, to 5.2C\,wt\%\ at 200\,\celsius, causing a large increase in $\sigma_{\rm SS}$.

\subsection{Dai et al.}

A recent model dismisses the strain energy from the invariant-plane strain shape deformation as being `extremely small' at low bainite fractions, but this overlooks the fact that growth rate depends on the strain energy {\em per unit volume} of bainite \cite{Dai:2024}.   Data for the interface carbon concentration ($x_{\rm I}^\upgamma$) or its relationship to the ferrite carbon concentration ($x^\upalpha_{\rm C}$) are not provided. The model cannot predict martensite-start temperatures (Figure~\ref{fig:Dai:2024}), and the total calculated energy dissipation is far greater than the actual driving force for diffusionless growth by a displacive mechanism.

\begin{figure}[htb!]
\begin{center}
\includegraphics[width=0.5\textwidth]{Dai_2024.eps}
\caption{Carbon inherited by $\upalpha_{\rm b}$ as a function of composition (wt\%) in Fe-C steels and temperature. The calculated $x_{\rm C}^{\upalpha}$ is much less than $\overline x_{\rm C}$ at the $M_{\rm S}$ temperature for the 0.33C\,wt\%\ steel and the other two steels have very small $x^\upalpha_{\rm C}$ compared with their average carbon concentrations which indicate that $\overline x_{\rm C}$ is unlikely to be reached if the calculations are extended to lower temperatures. Selected data from Dai et al. \cite{Dai:2024}.}
\label{fig:Dai:2024}
\end{center}
\end{figure} 

The dissipations listed include capillarity, diffusion within the interface, interfacial friction and strain energy, but the latter two quantities are set to zero.

The model has additional flaws. It uses an outdated solution for diffusion-controlled growth  and only resolves the relationship between interface velocity and supersaturation by applying the  maximum velocity criterion.

A significant inconsistency lies in its treatment of the interface. The model defines the interface as a domain of finite thickness ($2\delta$) where carbon diffuses {\em against} a chemical potential gradient without interacting with the interface itself. Furthermore, the model's derived carbon diffusivity within the interface ($D^{\mathbb I}_{\rm C}$) is inconsistent with its own equations, as seen in the mismatch between the velocity equations for interface diffusion and austenite diffusion: 
\begin{align} v_{\mathbb I} =\frac{ D^{\mathbb I}_{\rm C}}{RT} \frac{\mu_{\rm C}^\upalpha-\mu_{\rm C}^{\upgamma/\upalpha}}{2\delta} \qquad v_\ell &= \frac{D_{\rm C}^{\upgamma}}{2r} \frac{x^{\upgamma\upalpha}_{\rm C}-\overline x_{\rm C}}{\overline x_{\rm C}-x_{\rm C}^{\upalpha\upgamma}} \nonumber \\
\quad\hbox{and since }\quad v_{\mathbb I}=v_\ell,\quad& \frac{D^{\mathbb I}_{\rm C}}{D_{\rm C}^{\upgamma}} =\frac{\delta\,RT}{r} \frac{x^{\upgamma/\upalpha}_{\rm C}-\overline x_{\rm C}}{(\overline x_{\rm C}-x^\upalpha_{\rm C})(\mu_{\rm C}^\upalpha-\mu_{\rm C}^{\upgamma/\upalpha})} 
\end{align}
but the definition used is $D^{\mathbb I}_{\rm C}=\sqrt{D_{\rm C}^{\upgamma,x_{\rm C}=0}\times D_{\rm C}^{\upalpha}}$. Because the gradients of chemical potential and concentration were set to be linear within the interface, $D^{\mathbb I}_{\rm C}$ had to be related to that in austenite at zero concentration, presumably to avoid non-linear behaviour due to the concentration dependence of $D^{\upgamma}_{\rm C}$  diffusion. 

A phase change occurs with a net free energy reduction, even if compositions are not at equilibrium. The  assumption that carbon can migrate against a chemical potential gradient within the interface is not physically tenable.

Furthermore, the  claim that the bainite-austenite interfacial energy decreases with temperature due to carbon segregation, reaching zero just above room temperature, is inconsistent with first principles calculations that show $\sigma^{\upalpha/\upgamma}=2.3\,{\rm J\,m^{-2}}$ at 0\,K \cite{Wang:2023h}. It is odd that the carbon supersaturation ($x^\upalpha_{\rm C}$) goes through a minimum as a function of temperature for the Fe-1.49\,wt\%\ steel (Figure~\ref{fig:Dai:2024}).


\section{Nucleation}
Evidence-backed information on the nucleation of bainite is scarce, but the following observations do fall into that category. 

By inspecting the driving forces available when Widmanst\"atten ferrite ($\upalpha_{\rm W}$) or bainite form, it can be concluded that both evolve from the same nucleus by a mechanism in which the activation energy is a linear function of the driving force $\Delta G$, with a required partitioning of carbon during the process. The nucleus then evolves into $\upalpha_{\rm W}$ or $\upalpha_{\rm b}$ depending on the driving force available for growth, once the surface-to-volume ratio no longer features significantly \cite{Bhadeshia:1981}. 

The linear dependence of the activation energy of nucleation on $\Delta G$ for bainite has its origins in a dislocation dissociation mechanism that was proposed by Olson and Cohen \cite{Olson:1976} in the context of martensite. 

A combination of crystallography and three-dimensional observations have shown that $\upalpha_{\rm b}$ has a huge preference for nucleation at the faces of $\upgamma_1$-$\upgamma_2$ boundaries rather than at the edges where more than two grains meet \cite{Jimbo:2023}. This is because a mechanism based on the dissociation of (in this case interfacial) dislocations  would be feasible at grain faces, explaining why identically-oriented plates of bainite form at the same $\upgamma$-grain face.

Bainite in welding alloys nucleates intragranularly on particular inclusions present in the weld deposit \cite[pp.251-285, ][]{Bhadeshia:bainite3}. Whether the mechanism is an enhanced driving force due to solute depletion around the inclusion, or as the inclusion absorbs manganese, or due to some sort of lattice matching, depends on the inclusion type. In another context, direct observations have shown that bainite can grow from allotriomorphic ferrite that is suitably oriented \cite{Babu:1991}.

\section{Continuity of $\upalpha_{\rm W}$, $\upalpha_{\rm b}$, $\upalpha'$}

A comprehensive tabulation of the {\em observed} differences between Widmanst\"atten ferrite ($\upalpha_{\rm W}$), $\upalpha_{\rm b}$ and martensite ($\upalpha'$) is available in \cite[p.530, ][]{Bhadeshia:2021} where details can be found. Very briefly, these are all displacive transformations that exhibit an invariant-plane strain shape deformation, and the substitutional-to-iron atom ratio remains unchanged from the austenite. Martensite is diffusionless at all stages of its formation, $\upalpha_{\rm b}$ grows without diffusion but some of the excess carbon can partition into $\upgamma$ or precipitate as carbides depending on composition and conditions; it can form only below $T_0$, which is not a limitation for $\upalpha_{\rm W}$ that grows at a paraequilibrium, carbon diffusion-controlled rate in {\em steels}; it can also form in interstitial-free iron alloys  \cite[Chapter~7, Table~7.1, ][]{Bhadeshia:2021}. All this is quantitatively expressed \cite{Bhadeshia:1981,MAP} in a generic manner. A previously unknown concept was predicted and validated, that not all three transformations are possible in all alloys.

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{rationalisation_2.eps}
\caption{A pair of free energy curves ($\Delta G_{\rm m}$ and $\Delta G^{\upgamma\upalpha}$) for each of a low (A), medium (B) and high (C) alloy steel. The intersections of these curves with the functions defining nucleation and growth yield the transformation-start temperatures Widmanst\"atten ferrite {\Large$\color{blue}\bullet$}, bainite {\Large$\color{green}\bullet$} and martensite {\Large$\color{red}\bullet$} \cite{Bhadeshia:1981}. They reveal that not all transformations are possible in all steels. The inset shows the three-dimensional dislocation dissociation that is an embryo for the nucleation of $\upalpha_{\rm W}$, $\upalpha_{\rm b}$ and $\upalpha'$.}
\label{fig:rationalisation}
\end{center}
\end{figure} 

The nucleation of martensite, bainite and martensite all involve a complex dissociation of dislocations in austenite, which can be shown to lead to a linear dependence of the activation energy on driving force \cite{Olson:1976}. In the case of martensite, nucleation is diffusionless but it has been shown that the nucleation of $\upalpha_{\rm W}$ or $\upalpha_{\rm b}$ must involve the partitioning of carbon. In Figure~\ref{fig:rationalisation}, $\Delta G_{\rm m}$ is the maximum  free energy available for paraequilibrium nucleation and $\Delta G^{\upgamma\upalpha}$ the free energy change for growth without a composition change. The function $G_{\rm N}$ that is designated as universal, defines the free energy change needed to nucleate $\upalpha_{\rm W}$ or $\upalpha_{\rm b}$, whereas $\Delta G_{\rm N}$ is that for the diffusionless nucleation of martensite. Widmanst\"atten ferrite grows by a paraequilibrium displacive mechanism at a rate controlled by the diffusion of carbon in austenite. This concept has been used in  the design of low-carbon nanostructured bainite \cite{Yang:2008}.

The terms $G_{\rm SW}$,  $G_{\rm SB}$ represent the stored energies  of Widmanst\"atten ferrite and bainite respectively, that must be exceeded. Their origin is in the invariant-plane strain shape deformation as the plates grow, with   $G_{\rm SW}<G_{\rm SB}$ because pairs of $\upalpha_{\rm W}$ plates grow in a back-to-back, self-accommodating manner because of the low driving force available at high temperatures. This is why $\upalpha_{\rm W}$-containing microstructures are coarse because pairs of plates have to nucleate simultaneously in the correct formation (Figure~\ref{fig:Caballero_2009}a,b). Martensite also has a large stored energy, some 700\,${\rm J\,mol^{-1}}$, but usually, the free energy needed to nucleate it ($\Delta G^{\upalpha'}_{\rm N}$) is greater. 


\begin{align}
&\begin{rcases}
\Delta G_{\rm m}& < G_{\rm N} \\ \Delta G^{\upgamma\rightarrow\upalpha+\upgamma'} &< G_{\rm SW} \\
\end{rcases} &\hbox{defines onset of $\upalpha_{\rm W}$} \end{align}
\begin{align}
&\begin{rcases}
\Delta G_{\rm m}\qquad~~~& < G_{\rm N}\\
\Delta G^{\upgamma\upalpha}\qquad& < G_{\rm SB}  \\
\end{rcases} & \hbox{defines onset of $\upalpha_{\rm b}$} \end{align}
\begin{align}
\Delta G^{\upgamma\upalpha}\qquad\quad& < G_{\rm N}^{\upalpha'} 
 & \hbox{~~defines onset of $\upalpha'$} \end{align}

Steel~A exhibits all three transformation products, whereas in Steel~B, $\upalpha_{\rm W}$ is missing because at the temperature where nucleation becomes possible, bainite can form. Steel~C only can transform into martensite. This kind of behaviour is observed in practice.

These calculations can be conducted on freely available software accessible at the materials algorithms project website \cite{MAP}. Figure~\ref{fig:Caballero_2009}c shows calculated transformation temperatures and phase boundaries using this procedure. Most interesting is the fact that the bainite reaction stops at the $T_0$ boundary but as predicted, Widmanst\"atten ferrite terminates when $x^\upgamma=x_{Ae3'}$.

\begin{figure}[ht!]
\centering
\subfigure[]{
\includegraphics[width=0.35\textwidth]{widmanstatten.eps}}\hskip1cm
\subfigure[]{
\includegraphics[width=0.35\textwidth]{bainite.eps}}\\
\subfigure[]{
\includegraphics[width=0.5\textwidth]{Caballero_Acta_2009.eps}}
\caption{(a)  Widmanst\"atten ferrite embedded in pearlite. Image courtesy of Rolando M. N\'u\~{n}ez of the Pontifical Catholic University of Peru. (b) Bainite embedded in martensite. Image courtesy of Gebril M. A. M. El-Fallah. (c) Fe-0.29C-1.45Si-1.97Mn-0.95Cr-0.26Mo-0.67Co\,wt\%. Selected data from Caballero et al. \cite{Caballero:2009} where further results like this are available. }
\label{fig:Caballero_2009}\end{figure}



\section{Conclusions}

 I am confident that all now accept that the austenite to bainite transformation is accomplished by a displacive mechanism that results in an invariant-plane strain shape deformation that has a large shear component. And  that solute drag due to segregation at the interface does not occur, that there is no partitioning of substitutional solutes involved on any length scale, eliminating the possibility of a solute drag due to chemical segregation the interface. There is a remarkable lack of atomic mobility at the temperatures where bainite forms. The diffusional-displacive mechanism is not, therefore, a relevant interpretation when there is clear evidence that substitutional atoms are configurationally frozen during the transformation. 
 
 There is overwhelming evidence that when the carbon in austenite remains in solid solution, the bainite transformation is halted when the concentration reaches a value where austenite and ferrite of the same composition have identical free energies. 
 
 It is possible in principle that ferrite can inherit the X/Fe atom ratio of austenite while maintaining local equilibrium at the interface. This requires a tiny amount of partitioning of substitutional solute, that has never been detected experimentally for any transformation in steel. The concept is dubious theoretically because a gradient-energy term becomes important when dealing with sharp concentration gradients, making this mode unsustainable. 
 
 The popular $\mathbb{W}B_{\rm S}$ limitation to the growth of plates does not have a firm foundation because analysis of experimental data using more rigorous theory shows no discrepancy between the calculated and measured lengthening rates. 
 
It is remarkable that modern instrumentation has permitted the measurement of large supersaturations of carbon in solution within bainitic ferrite, together with the tetragonality associated with carbon ordering inherent in the Bain strain.  This in turn has sparked a flurry of publications on the growth of plates that trap carbon, but the recent work has significant omissions and errors. 

It is possible to treat the problem of carbon trapping rigorously \cite{Olson:1990}, but evidence suggests that Hehemann's idea that bainite forms without any diffusion with the carbon partitioning subsequently,  stands. If there is a transition from paraequilibrium growth to partial carbon trapping before diffusionless growth, experimental evidence indicates it is over a narrow temperature range. 

It is now possible to treat Widmanst\"atten ferrite, bainite and martensite transformations in a manner that respects a large variety of observed characteristics, making it possible not only to predict the three transformation temperatures, but to calculated whether all three can form in a given steel. The computer programs for doing such estimates for steel design are available freely on \cite{MAP} and even form the core of complex calculations for microstructural evolution in weld metals \cite{Bhadeshia:1985} and steels in general.

It may even be appropriate to avoid starting publications with the statement that ``bainite is controversial, therefore $\ldots$"!
 
\section*{Acknowledgments}
I much appreciate the discussions I had with Imed-Eddine Benrabah, Bernhard Bloder, and Gregory Olson, although the views presented here are my own.


\section*{Conflict of interest}

The author declares that he has no conflict of interest.


\section{Appendix}

There are some errors and omissions in \cite{Benrabah:2023,Benrabah:2024b} which have been resolved by discussion \cite{Bhadeshia:benrabah}.  In order to estimate the resistance to the motion of $\upalpha/\upgamma$ interface due to solid solution hardening, Benrabah et al. \cite{Benrabah:2023} used an empirical equation for the yield strength $\sigma^\upgamma_{\rm y}$ of austenite \cite{Castellanos:2020},  
\begin{subequations}\begin{align}
\sigma^\upgamma_{\rm y} = &\Bigl(67.8+493w_{\rm N}+354w_{\rm C} +20.8w_{\rm Si}+3.7w_{\rm Cr}+4.5w_{\rm Mo}+1.9w_{\rm Mn}+0.2w_{\rm Ni}\nonumber\\
& ~~~~~+0.1w_{\rm V} +0.4 w_{\rm Al} +0.4w_{\rm Cu} + 0.2w_{\rm Ti}\Bigr) \times \nonumber\\
&\bigl(1-0.26\times10^{-2}T +0.47\times 10^{-5}T^2-0.326\times 10^{-9}T^3\bigr)\label{eqn:Castellanos_1}\\[3pt]
\sigma^\upgamma_{\rm y} = &\Bigl(87.8+354w_{\rm C} +15.1w_{\rm Si}+2.5w_{\rm Cr}+14.5w_{\rm Mo}+4.5w_{\rm W}+8.1w_{\rm Mn}+0.2w_{\rm Ni}\nonumber\\
& ~~~~~+18.5w_{\rm V} +40w_{\rm Nb}+5.4w_{\rm Al} +12.8w_{\rm Cu} + 26.2w_{\rm Ti}\Bigr) \times \nonumber\\
&\bigl(1-0.22\times10^{-3}T +4.2\times 10^{-6}T^2-3.0\times 10^{-9}T^3\bigr).\label{eqn:Castellanos_2}
\end{align}\end{subequations} These two equations reflect different datasets with Equation~\ref{eqn:Castellanos_2} dealing with more solutes.
The equation used in \cite{Benrabah:2023} as a representation of the solid solution hardening of austenite  was
\begin{align}
\sigma^\upgamma_{\rm y} = &\Bigl(87.8+\frac{1}{100}\bigl[354w_{\rm C} +\underline{3.7w_{\rm Cr}}+14.5w_{\rm Mo}+\underline{20.8w_{\rm Si}}+\underline{1.9w_{\rm Mn}}+0.2w_{\rm Ni}\bigr]\Bigr) \times \nonumber\\
&\underline{\bigl(1-0.26\times10^{-2}T +0.47\times 10^{-5}T^2-0.326\times 10^{-9}T^3\bigr)}.\label{eqn:Benrabah}
\end{align}
The underlined terms, for reasons unknown, originate from Equation~\ref{eqn:Castellanos_1} but the remainder from Equation~\ref{eqn:Castellanos_2}. The fraction (1/100) is introduced here because Benrabah et al. substituted weight {\em fractions} rather than weight {\em percentages} in applying the  empirical relations from \cite{Castellanos:2020}. Furthermore, the solid solution strengthening incorrectly includes the strength of iron (87.8\,MPa).

The equation for the resistance to interface motion in \cite{Benrabah:2023} is as follows,
\begin{equation} \Delta G_{\rm mech}=V_{\rm m} (0.15 {\rm M}E_{\rm s}b\sqrt{\rho}+\sigma_{\rm ss})\,b\,m_{\rm Schmid} \Lambda^{-1} \label{eqn:mech2} \end{equation}
with the values assumes for partial molar volume $V_{\rm m}=7.09\times10^{-6}\,{\rm m^3\,mol^{-1}}$, the magnitude of the Burgers vector $b=2\times10^{-10}\,$m, the Taylor factor ${\rm M}=3$, the dislocation density of the austenite $\rho=10^{13}\,{\rm m}^{-2}$, the shear modulus  varies with temperature which according to  \cite{Fukuhara:1993} is $E_{\rm S}\approx 76.171-0.0345(T-273)\,$GPa where $T$ is in Kelvin, and the Schmid factor $m_{\rm Schmid}=0.5$. Using these values with Equation~\ref{eqn:mech2} gives values consistent with the original work, though strain energy has yet to be accounted for, Figure~\ref{fig:mech}.

The original work \cite{Benrabah:2023} contains a typographical error where the Burgers vector $b$ is an order of magnitude larger than the intended $2\times10^{-10}\,$m \cite{Bhadeshia:benrabah}.

\renewcommand{\refname}{REFERENCES}  \footnotesize
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  \providecommand{\doi}{doi:\discretionary{}{}{}\begingroup
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\providecommand{\eprint}[2][]{\url{#2}}
\providecommand{\BIBand}{and}
\providecommand{\bibinfo}[2]{#2}
\ifx\xfnm\undefined \def\xfnm[#1]{\unskip,\space#1}\fi
\makeatletter\def\@biblabel#1{#1.}\makeatother
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%Type = Techreport
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\end{thebibliography}



\end{document}
