{225}γ habit planes in martensitic steels: from the PTMC to a continuous model

Fine twinned microstructures with {225}γ habit planes are commonly observed in martensitic steels. The present study shows that an equibalanced combination of twin-related variants associated to the Kurdjumov-Sachs orientation relationship is equivalent to the Bowles and Mackenzie’s version of the PTMC for this specific {225}γ case. The distortion associated to the Kurdjumov-Sachs orientation relationship results from a continuous modeling of the FCC-BCC transformation. Thus, for the first time, an atomic path can be associated to the PTMC.

solves therefore, for this particular case, the question of the atomic path that could not be given by the PTMC. The crystallographic features of the {225} γ plate martensite, like the shape strain and the twinning system, are directly derived from the distortion using simple linear algebra concepts and are in total agreement with the Bowles and Mackenzie analysis. Furthermore, an answer is proposed to an old open question raised by Jaswon and Wheeler in the conclusion of their article, namely, "the reason for the choice of {225} γ habit in preference to an octahedral habit, since both have been shown to satisfy the condition of undergoing no directional change" 3 .

Results
Distortion associated with the Kurdjumov-Sachs orientation relationship. Experimental observations show that the orientation relationship between austenite and martensite is often found to be the Kurdjumov-Sachs relationship 13 : Considering atoms as hard spheres, the FCC-BCC martensitic transformation according to this particular orientation relationship can then be described by the total distortion matrix 12 :

Variants of the distortion associated with the Kurdjumov-Sachs orientation relationship. Due
to the orientation relationship and the crystalline symmetries, there exist 24 variants of Kurdjumov-Sachs. The distortion matrix α D i relative to each variant α i is thus found using the symmetry properties of the FCC-BCC transformation 14,15 . In the present paper, we arbitrarily consider that the distortion matrix D presented in equation (2) is the distortion matrix relative to the first variant = α D D 1 . The absolute basis  γ 0 for expressing the transformation matrix is therefore equal to the basis relative to the first variant   = γ γ 0 1 . By convention, if the basis in which the vectors and matrix are expressed is different from the absolute basis γ 0  . This basis appears explicitly in the notation as a right-down index. For example, a matrix M expressed in the α i crystal is written α M / i 0  . When the basis is the absolute one γ 0  , the basis is not specified in the index. The transformation matrix relative to all the 24 variants in the absolute basis γ 0  can be computed by using an appropriate change of basis: 1 and G γ is the point group of the austenite 15 . It is worth mentioning that in the particular case of FCC-BCC transformation, there is no distinction between orientational and distortional variants.
Among these 24 variants, there are 12 different pairs of twin-related variants. The twin-related variants of each pair have the particularity of sharing the same {111} γ plane and the same 〈 110〉 γ direction. This feature is illustrated in Fig. 1 for each particular {111} γ plane. 3D representation of the crystallographic arrangements of the twin-related variants can be found in the supplementary material 3 of reference ref. 16.
In addition, it can be noted that the transformation matrix corresponding to each variant of these pairs leaves the same plane γ {11 6} untilted. The twin-related variants of each pair can be expressed in the same semi-eigenbasis γ p j  for j = 1, ..., 12 defined by the common 〈 110〉 γ invariant line, the normal to the common γ {11 6} untilted plane and the cross product of these two vectors.
The terminology of semi-eigenbasis is used here in opposition to the classical eigenbasis. Indeed as the distortion associated with the Kurdjumov-Sachs orientation relationship has only one invariant line, it is not diagonalizable, and thus cannot be expressed in an eigenbasis 12 . In the rest of the paper, the mathematical development will be performed explicitly only for the pair p 1 of variants α 1 and α 3 , but the same calculation is applicable to all pairs p j . The transformation matrices of the twin-related variants α 1 and α 3 expressed in their common semi-eigenbasis are then:  Untilted planes (111) (11 6 ) (111) ( 6 11) (111) (11 6 ) (111) ( If expressed in their appropriate semi-eigenbasis  γ p j , the distortion matrices of each of the twin-related variants of every pairs p j are equal to the ones presented in equation (4).
The Kurdjumov-Sachs invariant plane strain. By inspection, it exists only one possibility to achieve an invariant plane strain (IPS) from a linear combination of these two matrices, and it is a mixture with 1:1 volume ratio of each of the twin-related variants, This matrix shows that a fine combination of alternating twin-related variants can produce a shape strain which is an IPS, having the γ {11 6} as invariant plane. The invariant plane of the average distortion D IPS is disoriented by 0.5° from the {225} γ . The magnitude of the shape shear is = ≈ . s 0 19 3 9 and the dilatation normal to the habit plane is δ = − ≈ . = + . . It should be noted that for twin-related variants pairs, a linear combination satisfy the volume conservation:

Interface between the twin-related variants.
To complete the crystallographic study of the {225} γ habit planes one needs to verify the geometrical compatibility of the transformation at the interface between the two twin-related variants. In other words, it is necessary that the interface plane between each twin-related variants is transformed in the same way by both variant. Mathematically, it consists in searching two non-collinear vectors v ∈ {v 1, v 2 } such that for each of them 17,18 . : The computation shows that two non-collinear vectors v I p 1 and v II p 1 belong to the Kernel. Together they define the interface plane between the two twin-related variants. The normal n p 1 to this interface plane is simply found by calculating a cross-product, 1 . Such Kernel can be computed for each pair p j of twin-related variants.
It is usual to express the interface between the two variants in their own basis α 0 i  . The distortion of the planes, expressed in the α i basis  α 0 i , is given by the correspondence matrix for each i = 1, 2, ... 24 in the reciprocal lattice The interface plane between the twin-related variant is then: The results of the predicted interface between two twin-related variants are reported in Table 2 for each pair p j of twins. Six different interfaces belonging to the {110} γ planes family have been found. According to equation (7), they correspond to {112} α .
Equivalence with Bowles & Mackenzie model. In the Bowles and Mackenzie's paper The crystallography of martensite transformations III 1 , the shape deformation associated to their prediction is expressed in the basis γ 0  as follows, = + ′ m P I dp (8) where . This transformation matrix can be expressed in its proper semi-eigenbasis in the same manner that has been used previously for the twin-related variant. It takes a form that is exactly the same as the Kurdjumov-Sachs invariant plane strain D IPS produced by the composition of twin-related variants.
Consequently, the Bowles and Mackenzie's version of the PTMC allows the prediction of the same γ {11 6} planes as the ones that are shown to stay invariant by an appropriate combination of twin-related variants.

Discussion
It has been shown that an heterogenous structure of infinitely small twin-related variants in equal proportion leaves a plane completely invariant. This plane is a γ {11 6} plane disoriented from the well-known {225} γ plane by only 0.5°. The macroscopic shape deformation resulting from such a combination of variants, each of these variants undergoing an invariant line strain, is exactly an invariant plane strain.  19 . Each pair p j of twin-related variants is then associated unequivocally with one of theses twelve {225} γ habit planes. The proportion λ of each of the twin-related variants α 1 and α 3 that is needed to produce the invariant plane strain is unique and the ratio is found to be 1:1. This result corresponds to the value experimentally deduced by Kelly and Nutting 20 when they studied the martensitic transformation in carbon steels. It is also the same proportion that is considered in the Bowles and Mackenzie's model for this specific habit plane.
The present study also shows that the interface plane between two Kurdjumov-Sachs twin-related variants is unique, as there are exactly two non-collinar vectors in the Kernel computed in equation (6). The results suggest that the martensite twins are not created from mechanical twinning, but are due to a particular association of variants, these variants being twin-related. We conclude that a local variant selection occurs to accommodate the phase transformation on the habit plane. Away from the habit plane, however, the transformation might need additional mechanisms, like plasticity by dislocations gliding to accommodate the shape strain related to the transformation. Such additional mechanisms can be observed in {225} γ habit planes martensitic steels 21 .
The shape strain D IPS associated with an equibalanced combination of twin-related variants can be clearly identified in equation (5). This shape strain is exactly the same as the one predicted by the PTMC and computed in equation (9). The calculated shear is 0.19 and the dilatation normal to the habit plane is + 8.9%. The shear strain is in good agreement with the magnitude experimentally measured in ferrous martensite which varies between 0.18 22 and 0.22 23 . However, the dilatation normal to the habit plane is overestimated. Indeed, the dilatation reported in the literature is about + 3% 23 . This discrepancy was expected because the atoms were assumed to be hard spheres of constant radius, whereas a slight decrease of the atomic size (few %) is observed during the transformation. This atomic size change is not captured by our model such that we overestimated the volume change associated to the FCC-BCC transformation, and hence the dilatation normal to the habit plane as well. The hard-sphere assumption is done in both the Bowles 24 . More recently, Stanford and Dunne used similar arguments to explain the austenite/-martensite interface in Fe-Mn-Si alloys 25 . The Bowles and Mackenzie's dilatation parameter was controversial and so is also the hard sphere modeling of the atoms. However, this approach has the advantage of allowing the description of the atomic trajectories. In this respect, using the hard-sphere model may allow more significant insights into the effective transformation mechanism than the consideration of artificial double shear systems, mentioned in the introduction.
It is remarkable that both the atomistic and the phenomenological modeling lead to the same results. In fact, even though these approaches are a priori based on opposite starting hypothesis, they share one common assumption: the atomic correspondence between the austenite and the martensite. The Bowles and Mackenzie model is historically based on the observation of the macroscopic shape strain associated with the transformation. An initial guess on the lattice invariant shear is required and the orientation relationship can then be derived. It only deals with the initial and the final states, but allows to cover a broad range of transformation, morphologies and habit planes. On the contrary, in the model proposed by Cayron, one assumes the final orientation relationship and imposes a steric condition on the atomic trajectories, by the mean of the hard sphere assumption. A precise atomic path can thus be defined for the transformation. The shape strain and the twinning system are then directly derived from the model with simple calculations. The predicted twinning system corresponds exactly to the lattice invariant shear assumed in the PTMC and experimentally observed 19 , which confirms the equivalence of the two models in the {225} γ case.
In their original papers, Bowles and Mackenzie emphasize the phenomenological nature of their theory. As reformulated by Dunne 24 , they stressed that the theory provides "a framework that any proposed transformation mechanism must satisfy". The distortion associated to Kurdjumov-Sachs orientation relationship is shown to fit perfectly in this framework, for this particular {225} γ case. Our model is thus a step toward a complete mechanistic representation of the transformation. In this regard, it might be noted that the mathematical approach used in this paper has also been successfully applied for {557} γ habit planes in dislocated martensite 26 .
The present study explains the invariant nature of the {225} γ habit planes thanks to fundamental, but rather abstract, linear algebra concepts. So, in order to visualize the crystallography of {225} γ thin plates, some computer simulations have been performed at the atomic scale. These simulations consist in computing the two final twin-related BCC lattices within their parent austenitic matrix. The computation of the transformation is based on the matrices α D 1 and α D 3 from equation (3). Only the iron atoms are considered and illustrated in Fig. 3. Figure 3(a), analogous to the schematic Fig. 2 (110) . All the crystallographic features presented schematically in Fig. 2 are also illustrated here. Figure 3(b) shows a tridimensional view of the simulated {225} γ plate. The proposed modeling of the {225} γ habit plane martensite is based on an atomic description of the FCC-BCC phase transformation 12 . A movie of the simulated {225} γ thin plate of martensite was computed. It is available in the supplementary material. Snapshots of this film are presented in Fig. 4.
As previously mentioned, the distortion associated with the Kurdjumov-Sachs orientation relationship leaves two families of plane untilted, {111} γ and γ {11 6} . However, it is usually the second family of planes which is experimentally observed. As mentioned in the introduction and already questioned by Jaswon and Wheeler 3 , a major question is then to understand why. The present approach proposes a clear answer, illustrated in Fig. 5. In the distortion, each of the untilted planes is deformed within the plane, as the spacing between the atoms in this plane changes during the transformation 12 . The matrices computed in equation (4) show that for the (225) γ habit plane the atoms are displaced in opposite directions for each twin of a given pair of twin-related variants such that the average displacement on this plane is zero. This average cancellation is illustrated in Fig. 5(a) and (b). They show the atomic positions in a γ (11 6) habit plane for both variants α 1 and α 3 . The γ (11 6) plane being irrational, it cannot form a 2D crystallographic lattice. Therefore, in the simulations, this plane is defined by all the atomic positions u such that: relatively to the initial FCC lattice in the same manner. Such a displacement cannot be cancelled by any combination of these two variants. This result can also be proved mathematically by applying an analogous approach for (111) γ as the one we used to show the invariant nature of (225) γ planes. We notice that all the volume change intrinsic to the FCC-BCC transformation occurs by the distortion of the (111) γ plane. Therefore {111} γ planes cannot be invariant during the transformation.
In conclusion, this study shows that a fine alternate structure of Kurdjumov-Sachs twin-related variants in equal proportion creates a macroscopic invariant plane strain having γ (11 6) as invariant plane, lying at 0.5° from the observed (225) γ plane. The shape strain resulting from this combination of twin-related variants consists  in a shear of magnitude 0.19 parallel to the habit plane and a dilatation normal to the habit plane of + 8.9%. This shape strain corresponds exactly to the result of the Bowles and Mackenzie's version of the PTMC. For this special (225) γ case, the two models are shown to be equivalent. We also demonstrated that because of the geometrical compatibility at the interface, Kurdjumov-Sachs twin-related variants share an interface plane of type {112} α , which corresponds to the twinning system that is assumed in the PTMC.
In this specific case, a continuous atomic displacement can be associated to the original Bowles and Mackenzie's model, offering, for the first time, a mechanistic dimension to a theory which up to now was phenomenological.

Methods
Mathematical notations and conventions. The mathematical notations and conventions used in the present paper are briefly presented here. The vectors are column vectors and are written in small bold letters. The matrices are written in bold capital letters. A vector v is transformed by a matrix M as follows: The coordinate-transformation matrix between two basis 1  and  2 is noted = → T [ ] 1 2   and is defined such that its columns are the vectors of the basis 2  expressed in the basis  1 : The vector v and the matrix M expressed in 1  , noted  v / 1 and  M / 1 are, then, respectively expressed in 2  by:  To express the transformed vector  ′ γ u / 0 in the α 0 i  basis one needs the transformation matrix presented in equation (18) Based on equation (20), the correspondence matrix is then: Considering the distortion associated with the Kurdjumov-Sachs orientation relationship presented in relation (1) and the present coordinate-transformation matrix, the correspondence matrix α γ → C 1 is: 1 And the correspondence matrix in the reciprocal lattice is then: T i i