Lattice distortion and atomic displacements during the fccbcc martensitic transformation Cyril CAYRON
CEA, LITEN, Minatec, 17 rue des martyrs, 38054 Grenoble,
France. Present address: Ecole Polytechnique Fédérale de Lausanne, Laboratoire de Métallurgie Thermomécanique, Rue de la Maladière 71b, CP 526, 2002 Neuchâtel, Switzerland.
[email protected] &
[email protected]
Synopsis Lattice deformations and continuous displacements of iron atoms during face-
centred-cubic to body-centred-cubic martensitic transformation are calculated with the hardsphere packing rule. This approach is an alternative to the classical theories based on shear mechanisms. Abstract
From our previous models of martensitic transformation, the continuous matrices
of atomic displacements and lattice deformations from face-centred-cubic (fcc) to bodycentred-cubic (bcc) phases are calculated in agreement with different possible final orientation relationships, such as Bain, Pitsch and Kurdjumov-Sachs (KS). The angular distortion introduced in the calculations appears a natural order parameter of transition. The distortion corresponding to KS is the only one that respects the parallelism of a dense direction and of a dense plane of both the fcc and bcc phases. This paper gives an alternative to the classical crystallographic theories and shear concepts associated to martensitic transformations. Keywords: Atomic displacements, Lattice distortion, Martensite 1. Introduction
Martensitic transformations between face centred cubic (fcc) austenite and body centred cubic (bcc) martensite have been studied for more than one century. Among the earlier works, one can cite those of Adolf Martens (1878) and Floris Osmond (1900) linked to the development of metallography. Ninety years ago, Bain (1924) proposed a simple model of fcc-bcc transformation. Few years later, the orientation relationship (OR) between austenite and martensite was measured by X-ray diffraction by Kurdjumov & Sachs (1930), Nishiyama (1934) and Wassermann (1933). The KS and NW OR are separated by 5°, and both are at 10° far from the Bain OR. The discrepancy between the “expected” Bain OR and the experimental ORs made these authors propose separately similar models of lattice distortion by shear and dilatation, now called KSN model. This approach does not seem to have convinced the scientists of that time who continued their efforts to conciliate the theoretical Bain distortion with the experimentally determined ORs and the shapes of the martensite laths
(Greninger & Troiano, 1949). This lead in the 1950s to the phenomenological theory of martensite transformation / crystallography (PTMT or PTMC). Two main beliefs are in the core of this theory: a) martensitic transformations occur according to a shear mechanism, and b) the Bain distortion is the distortion with the lowest deformations (Bowles & Mackenzie, 1954; Nishiyama, 1978; Christian 2002; Bhadeshia, 1987). However, the exact atomic displacements of the iron atoms during the transformation are beyond the scope of PTMC. Actually, neither the Bain distortion with the associated PTMC, the KSN model, nor the γ→ε→α two-step model proposed by Cayron, Barcelo, de Carlan (2010) take into account the atomic movements; they are pure lattice transformation models. To our knowledge, Bogers & Burgers (1964) were the first to consider the iron atoms as hard spheres and they placed them at the core of their theory. Their approach, later refined by Olson & Cohen (1972), is not based on the Bain distortion; however it continues to assume that the transformation results from a (multi)shear mechanism. We recently proposed another hard-sphere model of fcc-bcc martensitic transformations (Cayron, 2013), which is based on the Pitsch distortion (1959). We called it “one-step” in comparison with the “two-step” model published few years earlier (Cayron, Barcelo, de Carlan, 2010). This model qualitatively explains the continuous features observed in the Xray diffraction and Electron BackScatter Diffraction (EBSD)
pole figures between the
classical ORs such as NW, KS and Greninger-Troiano (GT), by taking into account the deformation field imposed by the transformation in the fcc surrounding field. The one-step model proved that it is possible to: a) find at least one matrix other than Bain to model the lattice distortion, b) this matrix is compatible with a hard-sphere representation of the atoms, and c) the transformation results from a homogeneous lattice distortion which is not a shear mechanism, contrarily to the common idea. Many questions could be raised from this approach: a) what is the link between the Bain and Pitsch distortions, b) what are the exact atomic displacements during the distortion, c) is it possible to find other distortion matrices, for example, one that gives directly the KS OR? Indeed, our recent automated treatments of EBSD maps showed that, even if the martensitic grains exhibit spreading between all the classical ORs, Pitsch is generally found in low proportions in comparison with KS, and the central part of the laths along the long lath direction is not in Pitsch but is in KS OR (Cayron, 2014). The present theoretical work aims at showing that it is possible to find such new matrices of continuous lattice distortions compatible with a hard-sphere model. They will be given for Bain, Pitsch and KS ORs. We believe that the distortion matrix corresponding to the KS OR is of fundamental importance to build a general theory of fcc-bcc martensitic transformation.
As in (Cayron, 2013), it is assumed that during the γ →α martensitic transformation the iron atoms are hard spheres of same diameter in both phases, which implies, by considering the γ and α dense directions, that:
2 aγ = 3 aα
(1)
The lattice parameter of pure iron extrapolated at room temperature is aγ = 0.3573 nm, which should give aα = 0.2917 nm, whereas it is actually aα = 0.2866 nm. The difference of 2% can be attributed to the electronic and magnetic properties of iron; it will not be taken into consideration for the sake of simplicity of the crystallographic calculations. For convenience, in the rest of the paper, the unity is attributed to aγ the lattice parameter of the fcc phase, and the reference coordinate basis formed by the vectors [100]γ, [010]γ, [001] γ is called (X0, Y0, Z0) = B0. 2. Model of atomic displacements with Bain OR
The Bain distortion is a contraction of 20% along one γ axis and dilatations along the two γ axis perpendicular to the contraction axis. The final Bain OR is [100]γ // [100]α , [011]γ // [010]α and [0 1 1]γ // [001]α. To our knowledge, the exact link between the continuous contraction and dilatation values has never been established. That is very surprising because the calculations are straightforward when assuming that the iron atoms are hard spheres that “roll” on each other during the distortion. In order to obtain a coherency of the reference frames used with our previous paper (Cayron, 2013), we will take for convention that the contraction occurs along the X0-axis (and not along the Z0-axis as often chosen). Let us consider the basis B1 = (x, y, z) such that the x, y and z axes are orientated along the [110] γ and [ 1 10] γ and [100]γ axes, respectively. The (x, y) basis is (X0,Y0) rotated by π/4 around Z0. During the Bain distortion the x and y directions are rotated in opposite directions, as shown in Fig. 1a. Let us call α/2 the semi-angle of rotation of x and y. By
projection, it is easy to show that the deformation (contraction) along X0 is given by ∆ =
√2 cos 4 + 2 = √2 sin 4 − 2 and the resulting deformation (dilatation) along Y0 is given by ∆ = √2 sin 4 + 2. Since the atoms along the [100]γ directions are shared by the (X0,Y0) = (001)γ and (X0,Z0) = (010)γ planes, the
resulting deformation
(dilatation) along Z0 is the same as the one calculated along Y0, as shown in Fig. 2. Therefore, the Bain distortion matrix in the reference basis B0 is Bain 0
D
0 0 sin(π 4 − α 2) (α ) = 2 0 sin(π 4 + α 2) 0 0 0 sin(π 4 + α 2)
(2)
The transformation starts with the angle α = 0, D 0Bain (0) is the identity matrix; and during the distortion, the [110]γ and [ 1 10]γ axes which make an angle of π/2 are transformed into [ 1 11]α and [1 1 1]α which make an angle of arccos(1/3). The angular difference is π/2arccos(1/3) = arcsin(1/3). Thus, the fcc-bcc transformation is completed when the rotation angle reaches αmax = arcsin(1/3). By using trigonometric relations and simplification of radical forms, it can be shown that α max 2 −1 )= sin( 2 6 α 1 max cos( )= 2 6. ( 2 − 1)
(3)
Therefore, the distortion matrix of the full transformation is
D0Bain
α max α max 0 0 sin( 2 ) − cos( 2 ) α max α max = 0 sin( ) + cos( ) 0 2 2 α max α max 0 0 sin( ) + cos( ) 2 2
2 − 3 = 0 0
0 2 3 0
(4)
0 − 0.81 0 0 0 = 0 1.15 0 2 0 0 1.15 3
This gives the classical values of Bain deformation if one assumes that the atom size remains unchanged during the transformation, i.e. a compression of 18.3% long the X0-axis and a dilatation of 15.5% along the Y0 and Z0 axes.
3. Model of atomic displacements with Pitsch OR
The Pitsch OR is [110]γ // [111]α , [ 1 10]γ // [11 2 ]α and [001]γ // [ 1 10]α. These axes form an orthogonal (but not orthonormal) reference basis B1 = (x,y,z). During the Pitsch distortion, the axis x remains invariant, i.e. undistorted and unrotated; we called it “neutral line” in (Cayron, 2013). The y axis is rotated by an angle α; and the z axis remains parallel to the axis Z0 and is elongated in order to respect the hard-sphere packing of the iron atoms, as shown in Fig. 1b. There are two ways to calculate the distortion. The first one consists in using Fig. 3, and doing the calculations in a Cartesian coordinate system. The distance PM = QM = 1 becomes after deformation PM’ = 1 + sin () and QM’= 1 − sin (). Thus:
∆ = ∆ = 1 + sin ()
(5)
The second way consists in noticing that the elongation along Z0 with Pitsch distortion should be equal to the one obtained with Bain because both models are based on hard-sphere packing, and thus:
∆ = √2 sin + 4 2
(6)
Actually trigonometric relations show that both equations are equal. Hence, the Pitsch distortion matrix in the reference basis B1 = (x,y,z) is: 1 sin(α ) 0 cos(α ) α D BPitsch ( ) = 1 0 0
0 1 + sin(α )
(7)
0
In the reference basis B0, the basis B1 is given by the coordinate transformation matrix: 1 / 2 − 1 / 2 0 1 1 0 1 / 2 0 , for which the inverse is [B1 → B0 ] = − 1 1 0 0 0 0 1 0 1
[B0 → B1 ] = 1 / 2
The Pitsch distortion matrix in the basis B0 is therefore: D0Pitsch(α ) = [B0 → B1 ] D/ B [B1 → B0 ]
(8)
1
which becomes after calculations: 1 − sin(α ) + cos(α ) 2 1 − sin(α ) − cos(α ) Pitsch D0 (α ) = 2 0
1 + sin(α ) − cos(α ) 2 1 + sin(α ) + cos(α ) 2 0
0 1 + sin(α ) 0
(9)
It can be equivalently expressed by: 1 + 2 cos(π 4 + α ) 2 1 − 2 cos(π 4 − α ) Pitsch D0 (α ) = 2 0
1 − 2 cos(π 4 + α ) 2 1 + 2 cos(π 4 − α ) 2 0
0 2 sin(π 4 + α 2) 0
(10)
The transformation starts with the angle α = 0 and D 0Pitsch (0) is the identity matrix. During the distortion, the [ 1 10] γ axis is rotated by an angle of π/2- arccos(1/3) = arcsin(1/3). Thus, the fcc-bcc transformation is completed when the rotation angle reaches αmax = arcsin(1/3).
By using equation (9) with sin(αmax) = 1/3 and cos(αmax) = √8⁄3, the distortion matrix of the complete transformation is:
D0Pitsch
1 + 2 3 1 − 2 = 3 0
2− 2 3 2+ 2 3 0
0 0 2 / 3
(11)
This is the matrix already reported by Cayron (2013). It can be diagonalized and the
eigenvalues are (1, √8⁄3, 2⁄√3) ≈ (1, 0.943, 1.155).
One can also apply a polar decomposition to this matrix and its intermediate state matrices given in equation (9), such that D0Pitsch (α ) = R Pitsch (α ) S Pitsch (α ) where R Pitsch is a (rigid body) rotation matrix and S Pitsch a symmetric matrix given by S Pitsch (α ) =
The symbol
T
T
D0Pitsch (α ) D0Pitsch (α )
(12)
at the left of a matrix or a vector means “transpose”. The equation becomes
after calculations, and comparison with equation (4) S Pitsch (α ) = D0Bain (α )
(13)
Therefore, the matrix takes the form D0Pitsch = R Pitsch D0Bain . The rotation matrix R Pitsch of this decomposition appears naturally in the present approach whereas it is “artificially” created in the PTMC. When the transformation is completed (α = αmax), R Pitsch is a rotation of 9.73° around [00 1 ]γ. 4. Model of atomic displacements with Kurdjumov-Sachs OR
The KS OR is [110]γ // [111]α , [1 1 2]γ // [11 2 ]α and [ 1 11]γ // [ 1 10]α. These γ axes, after normalization, form an orthonormal reference basis B1. The simple KSN shear/dilatation model proposed by Kurdjumov, Sachs (1930) and by Nishiyama (1934) can’t be applied because it does not respect the hard-sphere packing. We propose here a new distortion that a) respects the hard-sphere packing of the iron atoms such that the atoms moves relative to each other exactly as described previously for the final Bain or Pitsch OR, b) imposes that the close packed direction [110]γ remains invariant, and c) imposes that the close packed plane ( 1 11)γ remains globally invariant. The distortion matrix can be calculated by choosing an elementary non-orthogonal frame Be constituted by the normalized axes x = (1/√2).[110]γ, y = (1/√2).[101]γ and z = [100]γ. The [110]γ and [101]γ directions define the ( 1 11)γ plane that is transformed into the ( 1 10)α plane by the distortion. Indeed, by considering the Bain matrix given in equation (2), the angle β between the [110]γ and [101]γ axes appears as a function of the angle α:
cos ( ) = sin 4 + 2
(14)
For the fcc phase α = 0 and β = 60°. When the transformation is complete, α = αmax =
arcsin(1/3) = 19.4°, and β can be calculated using the equation sin 4 + 2 =
!"# $ .
It
follows that β is such that cos(β) = 1/3 , i.e. β = 70.5°, which is the angle between the [111]α and [11 1 ]α directions of the ( 1 10)α plane of the bcc phase. Now, the effect of the distortion on the axis z = [100]γ is calculated such that the hard-sphere packing is continuously respected. The geometrical view of the problem is illustrated in Fig. 4. Initially in the fcc phase the three iron atoms in points O, P, K are in contact and form a triangle belonging to the ( 1 11)γ plane. The iron atom in M, at the head of the arrow PM = [100]γ, is in the upper layer 1/3.( 1 11)γ plane of the classical fcc stacking sequence ABCABC. As the angle β = (OPK) = 60° is progressively increased to 70.5°, the atom in M is moved such that the distance PM is shorten while the distance PO = PK = MO = MK remains
constant and equal to the atom diameter, i.e. √2/2. Thus, the projection H of M on the ( 1 11)γ
plane remains on the line along [211]γ. By considering the Bain matrix given in equation (2),
the angle γ between the [211]γ and [100]γ directions appears as a function of the angle α: cos (&) = and thus
√2 sin 4 − 2 '1 + sin 4 − 2
1 − sin 4 − 2 sin(&) = ( 1 + sin 4 − 2
(15)
(16)
Now, let us associate the orthonormal basis Bs = (xs, ys, zs) to the basis Be = (x, y, z) as usually done for the structural tensor, i.e. xs is parallel to x, ys belongs to the (x, y) plane, and zs is oriented in the same direction than z relatively to the (x, y) plane. At the beginning of the transformation, the basis Bs corresponds to the fixed basis B1; then, during the transformation, difference Bs evolves with the basis Be = (x,y,z). The coordinates of the x, y and z vectors in the basis Bs gives the coordinate transformation matrix from Bs to Be, which is function of the angles β and γ 1 cos( β ) [B s → Be ](β , γ ) = 0 sin( β ) 0 0
z cos(β ) cos(γ ) 2 β z sin( ) cos(γ ) 2 z sin(γ )
The norm of the vector z = [100]γ is given by equation (2): )*) = √2 sin 4 − 2.
(17)
By using equations (14) to (16) and trigonometric rules, it can also be checked that . 1 + sin 4 − 2 β ( cos ( ) = , 2 , 2 0 1 − sin 4 − 2 β , , sin ( 2) = ( 2 +
(18)
Thus, equation (17) can be written 1 cos(β ) [B s → Be ] (β , α ) = 0 sin(β ) 0 0
2 1 − sin (π 4 − α 2) 2 sin 2 (π 4 − α 2) 1 + sin 2 (π 4 − α 2) 1 − sin 2 (π 4 − α 2) 2 sin(π 4 − α 2) 1 + sin 2 (π 4 − α 2) 2 sin 2 (π 4 − α 2)
(19)
By using equation (14) and noting X = cos(β), equation (19) can be written 1 [Bs → Be ]( β ) = 0 0
X
2X
1− X 2
2X
0
2 X
1− X 1+ X 1− X 1+ X
with X = cos(β)
(20)
The initial state of the fcc phase is defined by β = 60°, X = 1/2 and thus 1 [Bs → Be ](60°) = 0 0
1
2
3 0
2
2 1 6 1 3 2
(21)
Its inverse is 1 − 1 − 2 3 3 [Be → Bs ](60°) = 0 2 − 2 3 3 0 3 0
(22)
During the fcc-bcc transformation, X varies from 1/2 (β = 60°) to 1/3 (β = 70.5°). The distortion matrix calculated from the initial fcc phase to an intermediate state parameterized by the angle β is given by the matrix:
DeKS ( β ) = [Be → B s ] (60°) [Bs → Be ] ( β ) 1 = 0 0
1− X 2 3
X− 2
(23)
1− X 1+ X ( 6X − 2 X − 2 X ) 1− X 3 1+ X 2 1− X ( 2X − X ) 3 1+ X 1− X 6 X 1+ X
1
1− X 2 3 0
with X = cos(β) This matrix gives the distortions along the axes x, y, z of Be . At this basis, one can associate the correspondence matrix [B0 → Be ] given by 1 / 2 1 / 2 [B0 → Be ] = 1 / 2 0 0 1/ 2
1 0 0
(24)
Its inverse is 0 [Be → B0 ] = 0 1
2 0 −1
0 2 −1
(25)
The correspondence matrix allows us to calculate the distortion matrix in the basis B0 D0KS ( β ) = [B0 → Be ] DeKS ( β ) [Be → B0 ] = d ijKS ( β )
with the components d ijKS ( β ) in line i and column j given by
(26)
(
)
KS 1 1− X 2X + 2 X + X d 11 ( β ) = 6 1+ X 1 1− X d 21KS ( β ) = − 2X + 2 X + X 6 1+ X d 31KS ( β ) = 2 1 − X 2 X − X 6 1+ X d KS ( β ) = 1 − X − 1 1 − X 2 X + 2 X 12 6 1+ X 1 1− X KS 2X + 2 X d 22 ( β ) = 1 − X + 6 1+ X 2 1− X d 32KS ( β ) = − 2X − X 6 1+ X 2 d 13KS ( β ) = 1 − X − 1 1 − X 2 X + 2 X 3 6 1+ X 2 d KS ( β ) = − 1 − X + 1 1 − X 2 X + 2 X 23 3 6 1+ X 2 d KS ( β ) = 2 1 − X − 2 1 − X 2 X − X 33 3 6 1+ X
(
(27)
)
(
)
(
)
(
)
(
)
(
)
(
(
)
)
with X = cos(β) This matrix gives all the intermediate states resulting from the continuous distortion of the fcc lattice (β = 60°, X = 1/2) toward the final distorted fcc lattice (β = 70.5°, X= 1/3) that corresponds to a bcc lattice in KS OR. The complete distortion is obtained with X=1/3:
D0KS
6 6 3 6 −1 2 1 18 + 3 − 18 + 3 18 3 6 6 − 3 6 1 = − +1 + 18 18 18 3 6 6 + 1 9 − 1 3 − 6 9 + 13 3 3
(28)
This matrix has 1, 1 and 1.088 for eigenvalues. The eigenvector associated to 1.008 is a vector close to [4, 7, -3] γ. The linear subspace associated to 1 (multiplicity =2) is reduced to the line along the eigenvector [110]γ. This fact implies that the matrix D0KS can’t be diagonalized. However, it can be indirectly diagonalized by polar decomposition. D0KS = R KS S KS with R KS a rotation matrix and S KS a symmetric matrix given by S KS =
T
D0KS D0KS
(29)
which becomes after calculations, and comparison with equation (4) S KS = D0Bain
(30)
Therefore, as for Pitsch, the matrix of distortion associated to the KS OR takes the form D0KS = R KS D0Bain . The rotation R KS is a rotation of 11.06° around [21, 25, -60] γ.
5. Comparison of the distortions associated to Bain, Pitsch and KS orientations 5.1. 3D representations
The distortions calculated for Bain, Pitsch and KS orientations are represented in three dimensions in Fig. 5 by using VPython as computing language. The initial fcc crystal (8x8 cells) is coloured in blue, with the X0, Y0 and Z0 directions oriented along the [100]γ, [010]γ and [00 1 ]γ directions (this frame is indirect for graphical reasons). With Bain, the crystal is compressed along X0 and dilated along Y0 and Z0, and these directions are unrotated. With Pitsch, the direction Z0 is unrotated and the direction X0+Y0 = [110]γ (noted N for neutral) is invariant. With KS, the X0, Y0 and Z0 directions are rotated but the direction X0+Y0 is invariant (it is the N direction) and the ( 1 11) γ plane is unrotated (it is the globally invariant plane). The fcc crystals after distortions are coloured in red. It can be checked visually that they are effectively bcc crystals. 5.2. Effect of the distortions on the directions and planes
The matrices D 0OR of equations (4), (11) and (28) give in the initial fcc basis B0γ the images by distortion of the directions u 0γ , for the Bain, Pitsch and KS ORs, respectively. The inverse of their transpose gives the distortion of the planes g 0γ . The distorted direction or plane vectors are written u ' γ0 = D 0OR u 0γ and g ' γ0 =
T
(D ) OR 0
−1
g 0γ , for OR = Bain, Pitsch or KS.
(31)
It is possible to calculate the images in the final bcc reference frame by using the correspondence matrices. These matrices are easily calculated from the orientation relationships. Indeed, the OR can be expressed by a triplet of pairs of direction vectors that are parallel and of proportional norms: xγ = k.xα, yγ = k.yα, zγ = k.zα, where k =
aγ aα
= 3 from equation (1). These vectors define a 2
common base Biγ = ( x γ , y γ , z γ ) = k . Biα = k ( xα , y α , z α ) . Let us call the transformation matrices between the reference frames B0γ and B 0α and this common basis: [B0γ → Biγ ] = [ x γ , y γ , z γ ] and
[B
]
α
→ kBiα = k[ xα , y α , z α ] , respectively. The correspondence matrix for the directions is then
0
given by
[B
α 0
] [
][
→ B 0γ = B 0α → kB iα B 0γ → Biγ
]
−1
[
][
= k B 0α → Biα B 0γ → Biγ
]
−1
= k Rαγ
(32)
where Rαγ = 1 k [B 0α → B 0γ ] is a rotation matrix between two the cubic phases. The correspondence matrix for the planes are given by the inverse of the transpose of [B0α → B0γ ]
which is equal to 1 k Rαγ = 1 2 [B 0α → B 0γ ]. Therefore, the distorted direction vectors and k distorted plane vectors in the final α bcc reference frame B 0α are given by u 'α0 = k Rαγ u ' γ0
α γ 1 and g ' 0 = k Rαγ g ' 0
(33)
It can be checked that the images of the directions and planes noted in the final α bcc reference frame are the same whatever the distortion matrix, Bain, Pitsch or KS of equations (4), (11) and (28). The distortion matrices, the correspondence matrices and the images of some low index directions and planes are reported in Table 1 for Bain, Pitsch and KS. 6. Discussion 6.1. Particularity of the distortion associated to the KS orientation
Whatever the matrix D 0OR calculated with Bain, Pitsch or KS ORs, it can be checked D 0OR that its determinant is det( D 0OR ) =
4 3
2
3
≈ 1.088 , as expected for a transformation between bcc and
fcc phases in relationship by a hard sphere model. This can also be proved from the polar decomposition of the matrices. In the PTMC, the distortion is assumed to result from an invariant plane shear S PTMC = I + d T p , where I is the identity matrix, d is the displacement vector and p is the normal of the invariant plane (Bowles & Mackenzie, 1954). The characteristic roots of S PTMC are 1, 1 and 1+ p T d . The two unit values are correlated to the shear plane which is totally invariant. The volume change is thus fully given by the dilatation component 1 + p T d . In our approach, the volume change results from the lattice dilatation along the eigenvectors of the matrices (4) and (11) associated to Bain and Pitsch ORs; these directions exist because the matrices can be diagonalized. The particularity of the distortion matrix (28) associated to the KS OR is that it can’t be diagonalized. It has 1, 1 and 1.088 as characteristic roots, such as the invariant plane matrix used in PTMC, but the matrix (28) is not of shear type. As S PTMC , the volume change is taken by the dilatation along one unique axis, which is the eigenvector associated to the value 1.088, i.e. here the vector [4,7,-3]γ , but contrarily to the S PTMC for which the volume change occurs perpendicularly to the invariant plane, the vector [4,7,-3]γ vector belongs to the ( 1 11) γ plane, which means that the totality of the volume dilatation is generated by the angular distortion inside this plane (Fig. 6). This can be checked by considering the rectangle on the ( 1 11)γ plane formed by the [110]γ and [101] γ diagonals (with angle of 60°), and the distorted rectangle after transformation, i.e. the rectangle on the ( 1 10)α plane formed by the [111]γ and [11 1 ]γ diagonals (with angle of 70.5°). In the PTMC, the invariant plane is fully invariant, i.e. the directions belonging to this plane
are invariant, whereas in our present approach, the plane of distortion ( 1 11) γ is only globally invariant (it is unrotated); the directions it contains are not dilated or contracted but their angle is changed. All the volume change comes from this distortion. Therefore, the distortion associated to the KS OR is very special and constitutes a very good alternative to the shear matrices used in the PTMC. The concept of disclinations (special pile-ups of dislocations that create rotational discontinuities of lattices) seems very relevant to represent the 60°→70.5° angular distortion on the the ( 1 11) γ plane, as treated in the one-step model (supplementary materials 4 of Cayron, 2013). We also recall that the matrix (28) has by construction the unique property of letting invariant both a close-packed direction and a close-packed plane. Indeed, [110]γ is invariant by D 0KS and is transformed into [111]α by transformed into (110)α by 1
k2
[B
[B
α
0
α 0
]
→ B0γ , and ( 1 11)γ is invariant by
T
(D )
KS −1 0
and is
]
→ B 0γ .
6.2. Natural order parameter
To our knowledge, only complex order parameters could be attributed to fcc-bcc martensitic transformations, most of them associated to the stress field around the martensite (Falk, 1982).
More generally, as frankly said by Clapp (1995) “Certainly one of the
stumbling blocks in defining a martensitic transformation is that there is no obvious order parameter associated with it, such as one has with ferromagnetic transformations (magnetic moment), order-disorder (long range order parameter), etc”. In our approach, whatever the matrix, associated to Bain, Pitsch or KS OR, assuming that the distortion respects the hard sphere packing allows us to introduce a unique natural order parameter of the transformation, which is the angular parameter of the distortion, i.e. α in equations (2) and (10) for Bain and Pitsch, respectively, and β or X = cos(β) in equation (27) for KS. This parameter could probably be used in phase field approaches. 6.3. The one-step model with KS OR
We have shown that it is possible to obtain both the good lattice and correct orientation relationship with a unique matrix that respect the hard sphere packing of the atoms and that can be expressed, by polar decomposition, as a product of a rotation and a Bain distortion, without any ad hoc assumptions such as the ones used in the PTMC. However, our present approach does not explain the martensite shapes such as the plates or the lenticular laths with their habit planes. Some ideas to tackle this issue have been introduced in the initial one-step model built with the distortion associated to the Pitsch OR (Cayron, 2013), and similar approaches seem now possible with the matrix associated to KS OR. The main idea of the
one-step model is that there is a unique OR and that the OR spreading observed in the EBSD or X-ray diffraction pole figures comes from the strain field generated by the transformation itself. It was applied with Pitsch but it can also be applied with KS; in that case, the sequence of events should be slightly modified in comparison to (Cayron, 2013). One has to imagine that the nucleation of martensite in a surrounding austenite field occurs by a distortion leading to KS by equation (28), and that the Pitsch and NW orientations result from the growth of the martensite in the deformed austenite by the continuous rotations noted A and B in our previous works (Cayron, de Carlan & Barcelo, 2010; Cayron, 2013). The modified scheme of this scenario is proposed in Fig. 7. Using KS, instead of Pitsch, as unique distortion matrix seems more reliable because, as mentioned in the introduction, it was shown on the EBSD maps that the central part of the laths along their long direction is in KS and not in Pitsch OR (Cayron, 2014). Moreover, the rotation B with Pitsch is combined with rotation A (see section 4.2 of Cayron, 2013), whereas it is pure with KS because it directly results from the angular ( 1 11)γ → (110)α distortion as explained in section 6.1. It is also probable that the rotation A is not pure, contrarily to what we supposed in the initial one-step model (Fig. 6 of Cayron, 2013). Indeed, there is no {110}γ plane which is transformed into a {111}α plane, as shown in Table 1, even if the close packed direction [110]γ is transformed into the close-packed direction [111]α (neutral line). 6.4. What is the natural fcc-bcc distortion mechanism?
What is the natural martensitic distortion of a small stress-free fcc crystal isolated from any external stress field? Does it follow the distortion matrix (4), (11) or (28) corresponding to Bain, Pitsch or KS, respectively? A pure Bain distortion seems possible, if, as assumed in the PTMC, it is argued that the Bain OR is not observed experimentally because of the strain accommodation of the surrounding austenite matrix. It means that the rigid body rotation would be a consequence of the transformation, but is not intrinsic part of the mechanism. However, we don’t find any crystallographic argument that could support the assumption that the distortion naturally respects a 4-fold γ axis of austenite (the compression axis). Actually, Pitsch OR, and thus equation (11), seems more probable than Bain, because Pitsch OR was experimentally observed in thin TEM lamellas of Fe-N alloys after quenching (Pitsch, 1959), i.e. in samples nearly free of any surrounding austenite. By using distortion matrices such as Pitsch or KS, the rigid body rotation becomes an intrinsic component of the distortion, and the Bain distortion, as expressed by the matrix (4), is just another component of the distortion which characterizes the hard sphere packing rule. Is Pitsch the natural distortion mechanisms for all the martensitic alloys? We think that the distortion matrix (28) associated to the KS OR could be also a very good candidate to build a theory, for
experimental reasons and because it permits to maintain the parallelism of both a close packed plane and a close packed direction. Is this double condition a natural one or is it the consequence of the surrounding austenite matrix? In other words, is it Pitsch the natural path of fcc-bcc deformation and KS the constrained one? We can’t yet answer that question. Experimental observations could be very useful: for example, one could select by EBSD an austenitic grain oriented such that the sample axes X0, Y0, Z0 are along the γ directions, extract a small cube of it by Focus Ion Beam (FIB) and maintain it at the tip of a very thin copper needle on which an absolute reference frame has been carved by FIB. By cooling the sample below Ms, one could observe that the cube is transformed into a parallelepiped, as in Fig. 5. If Bain is respected, one should not observe any rotation of the sample axes X0, Y0, Z0; if Pitsch is respected only Z0 should remain unrotated, and if KS is respected all the three axes should be rotated. Other experiments are also possible but will not be detailed here. 7. Conclusions
The lattice distortions associated to the Bain, Pitsch and KS orientation relationships have been calculated with the simple assumption of a hard-sphere packing of the iron atoms. The distortion matrices of the continuous transformations from fcc to bcc are given in function of a unique parameter, which is an angle of distortion, and which constitute a reasonable order parameter for phase field approaches. The matrix associated to the KS OR is very special; by construction it lets the close packed direction [110]γ invariant -it is transformed into [111]αand the close packed plane ( 1 11)γ globally invariant -it is transformed into (110)α-. The matrix has 1, 1 and 1.088 for eigenvalues, but, importantly, it is not of shear type. Indeed, the whole fcc/bcc volume change is associated to the angular distortion of the globally invariant plane, whereas, it is associated to the dilatation normal to the invariant plane for a shear deformation. Beyond the calculations, the paper proposes a new paradigm for the fcc-bcc martensitic transformations. Up to now, these transformations were considered as “shear” transformations, nearly by definition: “shear transformations are synonymous with martensitic transformation” (Wayman, 1975). In our approach, the distortion matrices leading to the Pitsch or KS OR are not of simple or multiple shear type. More generally, lattice shear seems to be incompatible with the hard sphere packing of the iron atoms, even for classical mechanical twinning. Of course, a shear deformation conveniently describes the transformation from the initial state to the final twinned state, but if the intermediate states are considered, it is clear that a shear would make the atoms interpenetrate, which seems energetically impossible for metals or metallic alloys. Thus, we propose to replace the shear matrices commonly used in twinning and martensitic transformations by atom-compatible lattice distortion matrices, such as the ones presented here for the fcc-bcc transformations.
Table 1
Characteristics of the distortions associated to Bain, Pitsch and Kurdjumov-Sachs
orientation relationships. Bain
Pitsch
KS
Orientation relationships
[B
γ 0
→ B iγ ] =
[ xγ , y γ , z γ ] =
[B
α 0
→ Bi ] = α
[ xα , yα , zα ] =
1 0 0
0 0 1
0 1 0
1 0 0
0
1 2 1 2 0
− 1 2 1 2 0
1 2 1 2
1 3 1 3 1 3
−1 2 1 2 0
0 0 1
1 2 1 2 0
2 1 2 0
1 3 1 3 1 3
−1
1 6 1 6 −2 6
1
3 − 1 3 −1 3 1
6 −1 6 2 6
2 1 2 0
−1
1 6 1 6 −2 6
Rotational part of the correspondence matrix for directions and planes 1 0 0
Rαγ =
0
− 1 2 1 2 0
1 2 1 2
2 −1
2 +1
2 3 2 −1
2 3 2 +1
2 3 2 +2
2 3 2 −2
2 3
2 3
−1 2 1 2 0
1 6 2 3 + 16 1 − 13 6
+1 3 −1 − 1 + 1 6 3 6 1 +1 −2 3 3 6 2 −1 3 6
1
6
Distortion matrix for directions
D 0OR =
2 3 0
0 2 3
0
0
0 0 2 3
1 + 2 3 1 − 2 3 0
2− 2 3 2+ 2 3 0
0 0 2 / 3
6 6 +1 6 −1 2 18 + 3 − 18 3 6 3 6 6 +1 − 6 +1 − 18 18 6 3 6 6 6 1 1 1 + + 9 − 3 − 9 3 3 3
Distortion matrix for planes T
(D )
OR −1 0
=
3 2
0
0
3 2
0
0
0 0 3 2
1 + 2 2 1 − 2 2 0
2− 2 4 2+ 2 4 0
0 0 3 / 2
6 6 −1 +1 − 6 + 1 12 24 4 8 4 6 6 +3 − 6 +1 − 12 24 4 8 4 6 6 6 1 1 1 − − + + 6 2 12 4 4 4
Images of directions 3 γ 6 γ 4 γ 12 γ
1 α , 2 α 4 α, 2 α 4 α 8 α, 4 α Images of planes
3 {100}γ 6 {110}γ 4 {111}γ 12 {112}γ
1 {100}α , 2 {110}α 4 α, 2 α 4 α 8 α, 4 α
Acknowledgements
I would like to thank Prof. Roland Logé for the opportunity he gave
me to continue metallurgy in his laboratory. I am particularly grateful to Prof. Michel Perez for our discussions and his first molecular dynamic calculations. More works are required before publication, but the first results are very interesting. I also thank Gert Nolze for our discussions on EBSD. The exact and detailed features of the pole figures bring a lot of information on the transformation history. References Bain, E.C. (1924) Trans. Amer. Inst. Min. Metall. Eng. 70, 25-35. Bhadeshia, H.K.D.H. (1987) Worked examples in the geometry of crystals, 2d ed, Brookfield: The Institute of Metals. Bogers, A.J. & Burgers, W.G. (1964) Acta Metall. 12, 255-261. Bowles, J.S. & MacKenzie, J.K. (1954) Acta Metall. 2, 129-137. Cayron, C., Barcelo, F. & de Carlan, Y. (2010) Acta Mater. 58, 1395-1402. Cayron, C. (2013) Acta Cryst. A69, 498-509. Cayron, C. (2014) Mater. Charact. 94, 93-110. Christian, J.W. (2002) The theory of transformations in metals and alloys, part II, Oxford: Elsevier Science Ltd; pp 961-1113. Clapp, P.C. (1995) J. Phys. 5, C8, 11-19. Falk, F. (1982) J. Phys. 43, C4, 3-15. Greninger, A.B. & Troiano, A.R. (1949) J. Met. Trans. 185, 590-598. Kurdjumov, G. & Sachs, G. (1930) Z. Phys. 64, 325-343. Martens, A. (1878) Ueber die mikroskopische Untersuchung des Eisens Zeitschrift des Vereines Deutscher Ingenieure 22, 11-18. Nishiyama, Z. (1934) Sci. Rep. Tohoku Imp. Univ. 23, 637-644. Nishiyama, Z. (1978) Martensitic Transformation, Ed. By M.E. Fine, M. Meshii, C.M. Waymann, Materials Science Series, Academic Press, New York. Olson, G.B. & Cohen, M. (1972) Journal of Less-Common Metals 28, 107-118. Osmond, F. (1900) Metallographist, 3, 181–219 Pitsch, W. (1959) Phil. Mag. 4, 577-584. Wassermann, G. (1933) Archiv Eisenhüttenwesen 6, 347–351. Wayman, C. M. (1975) Metallography 8, 105-130 , and republished in Mater. Charact. (1997) 39, 235260.
FIGURES
Fig. 1. Schematic views on the (001)γ plane of the fcc-bcc bcc distortion associated to (a) Bain and (b) Pitsch OR.. The γ fcc phase is in blue and the α bcc phase is in red. red
Fig. 2. Schematic chematic view of the fcc crystal, ystal, (a) in 3D and (b) in a flatten representation of the cube.. The distortion of the X0 and Y0 axes in the (001)γ plane implies a distortion along the Z0 = [001]γ direction due to the hard sphere packing.
Fig. 3. Dilatation and compression of the [100]γ and [010]γ of the (001)γ face during the Pitsch distortion. distortion PQ = [110]γ = [111] α // X0
Fig. 4. 3D scheme of the (x, y, z) basis used to calculate the distortion leading ing to the KS OR. (a) Fcc cube, as represented in (Cayron, 2013) with ( 1 11)γ plane marked by the PQR triangle; x // P0 = ½ [110]γ , y // PK = ½ [101]γ , z = PM = [001]γ . (b) Other 3D representation of the ( 1 11)γ plane. The iron in M moves such that the distance PO = PK = MO =MK remains constant.
Fig. 5. 3D representation with VPython of the fcc-bcc fcc bcc distortion associated to Bain, Pitsch and KS ORs. The initial fcc crystal (8x8 cells) is in blue, with the X0, Y0 and Z0 directions along the [100]γ, [010]γ and [00 1 ]γ directions. With Bain, ain, the crystal is compressed along X0 and extended along Y0 and Z0, and these directions are unrotated. With Pitsch, the direction Z0 is unrotated and the direction X0+Y0 = [110]γ (noted N for neutral) is invariant. With KS, the X0, Y0 and Z0 directions are rotated but the direction N = X0+Y0 is invariant and the ( 1 11) γ plane is unrotated. The fcc crystals after distortions are in red. It can be checked that the atoms inside the fcc distorted crystals form a bcc crystal. crystal
Fig. 6. Distortion tion of the ( 1 11)γ plane into a (1 10)α plane viewed by orienting the 3D crystals along the [ 1 11]γ axis. The planes are delineated by the yellow lines.
Fig. 7. Modified one-step step model. In Fig. 8 of of (Cayron, 2013), the distortion was supposed to lead to Pitsch OR, whereas in the present modified version, the distortion is associated to KS OR. Rotations A and B and associated Pitsch and NW ORs result from the strain field generated by the distortion associated to the KS OR.
