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Dislocation description of martensite interfaces based on misfit analysis
Materials Science and Engineering A 438–440 (2006) 118–121
Dislocation description of martensite interfaces based on misfit analysis W.-Z. Zhang ∗ , J. Wu Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Received 12 April 2003; received in revised form 21 November 2005; accepted 20 January 2006
Abstract It is shown how the methodology and theoretical relationships developed in the framework of the O-lattice theory can be applied for study of martensite interfaces containing a single set of dislocations. The observed orientation relationship, the habit plane orientation, matching or mismatching of certain sets of planes in the habit plane of lath martensite in an Fe–Ni–Mn alloy can be explained consistently by the present approach. Attention is drawn to the use of measurable g vectors for characterizing the habit plane and for interpreting plane matching in the habit plane. © 2006 Elsevier B.V. All rights reserved. Keywords: Orientation relationship; O-lattice; g approach; Habit plane; Misfit dislocations; Plane matching
1. Introduction Crystallography of martensite in many systems has been successfully predicted by the phenomenological theory of martensite crystallography (PTMC) proposed by Bowles and Mackenzie [1,2] and Wechsler et al. [3], as reviewed by Wayman [4] and Christian [5]. Two conditions were hypothesized in the formal PTMC, i.e. existence of a macroscopic invariant plane as the habit plane and a glissile habit plane. If simultaneous satisfaction of both conditions is imposed, the overall crystallography of the lath martensite often cannot be explained by the theory in a self-consistent manner. Various modifications have been employed, but usually only one of the above conditions can be maintained. One type of modifications allows a small longrange strain, e.g. by introducing a dilatation parameter [2,6] or assuming forced matching of a pair of low index vectors [7,8]. These modifications slightly violate the condition of existence of a macroscopic invariant plane. Another type of modifications may not be in accord with the existence of a glissile interface. The double shear model [9] or the descriptions of the habit plane with two sets of dislocations [10,11] belong to this category. However, the habit plane of lath martensite typically contains a set of regular dislocations visible by using conventional transmission electron microscopy (TEM) [12,13] and by high resolution TEM (HRTEM) [10,14]. ∗
Corresponding author. Tel.: +86 10 62773795; fax: +86 10 62771160. E-mail address: [email protected] (W.-Z. Zhang).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.01.100
The present work is a preliminary attempt to explore an alternative approach based on a study of interfacial structure. The vital role of interfacial structure in the crystallography and growth mechanism of martensite is well recognized, as emphasized by Christian since the 1950s [5,15]. Our approach is based on the O-lattice theory developed by Bollmann [16,17]. It has been explained by Christian [18] why the O-lattice theory is applicable to martensite in the O-line condition. The observations of a single set of discrete dislocations in the habit plane of martensite [13,10] suggest the possibility that the habit plane contains periodic O-lines between the dislocations. This paper shows how to apply the O-line model to the study of martensite crystallography, with a methodology developed mainly for precipitation crystallography [19]. Only the requirement for a macroscopic invariant plane is maintained strictly in this analysis. Lath martensite in an Fe–23.0Ni–3.8Mn alloy is taken as an example for the application.
2. O-line description and application The O-lattice describes a periodic pattern of fit/misfit between two lattices [16]. The O-lattice elements are locations of good matching. Given a transformation between lattices ␣ and , x = Ax␣ , a principal O-lattice vector, xO , can be determined from [16]: TxO = bL ,
(1)
W.-Z. Zhang, J. Wu / Materials Science and Engineering A 438–440 (2006) 118–121
119
Table 1 Comparison of results from an O-line calculation and results in ref. [14] Results in ref. [14]
Calculation
Discrepancy
OR
K–S (experiment)
[−1 0 1]f //[−1 −1 1]b (1 1 1)f ˆ(0 1 1)b = 0.37◦
0. 37◦
Habit plane bO-L ␣
(1 2 1)f (experiment) [−1 0 1]f /2 (assumption)
(1 1.997 1)f [−1 0 1]f /2 ([−1 −1 1]b /2)
0.04◦ No
Invariant line Dislocation spacing
//bLa (assumption) 0.925 nm (calculation)
[−1.12 0.06 1]f 0.92(8) nm
3.9◦ <0.01 nm
Extra plane spacing in unit of (0 −1 1)b planes
8.2 (calculation) 8.6 (experiment)
∼7.3
∼1
where T = I − A−1 defining the displacement field of the lattice deformation, I is a unit matrix and bL is a Burgers vector in lattice ␣. In our calculation, the fcc lattice is selected arbitrarily as the reference lattice ␣. Usually, a measured or nominate OR does not permit A to describe one or more sets of periodic O-lines, but a slight adjustment of the OR may render an O-line strain. Systematic solutions of O-lines for a given system can be determined by either numerical method [20] or analytic method [21]. In the formal PTMC, the OR is fixed by two conditions: (1) the Burgers vector of the dislocations, bO-L ␣ (the superscript stands for O-line), must lie in the plane with the invariant normal (i.e. invariant line in reciprocal space) and (2) the invariant line must lie in a given (rational) slip plane [22]. The O-line strain is subjected only to condition (1) [20]. The remaining one degree of freedom in the OR can be constrained by the criterion of either maximum dislocation spacing or minimum angular deviation from a rational OR (e.g. either N–W or K–S OR for an fcc/bcc system) [20,21,23]. The observed OR for the Fe–23.0Ni–3.8Mn alloy is the K–S OR [14]. According to the previous investigation on the fcc/bcc systems [21,23], to obtain a close OR we should apply the minimum angular deviation criterion and select 1 1 0f /2 corresponding to 1 1 1b /2 to be bO-L ␣ (where and hereafter subscripts f and b indicate the fcc and bcc lattice, respectively). Under these conditions, we can determine A for the O-lines by following the steps of the analytical method, explained elsewhere [21]. A simple parameter to describe the habit plane is a reciprocal vector g, defined by [24] g = T g␣ = g␣ − g ,
(2)
where g␣ and g are reciprocal vectors from lattices ␣ and , and the symbol “ ” denotes a transposition operation. The habit plane containing the O-lines must be normal to a group of parallel g’s that are associated with g␣ ’s in the zone axis of bO-L ␣ [20]. Therefore, the habit plane can be determined by the g associated with any g␣ normal to bO-L ␣ . This relationship is equivalent to the formula, p1 //p2 (S−1 − I), for determining the habit plane, p1 (unit vector), in the PTMC [1,22]. Here S = A, and p2 is the unit normal to the slip plane, which must contain bO-L ␣ . The expression with g is advantageous over the expression with unit vector, because g vectors are measurable in diffraction patterns. This expression facilitates a direct comparison between the theoretical and experimental results of the habit plane. The evidence
confirming the relationship between g vectors and martensite habit planes can be found in the literature, for example, in experimental results provided by Ogawa and Kajiwara (Fig. 3d in ref. [14]) and by Condo and Lovey (Fig. 2b in ref. [25]), though this relationship was not notified by the authors. Strictly, parallelism of g’s can be identified in one overlapped pattern only if the electron beam is along a pair of parallel Burgers vectors [23]. The dislocation configuration can be determined based on an O-cell construction [16]. The following formula is applicable to an interphase interface containing one or two sets of periodic dislocations [26]. Provided with the unit normal of the habit plane p (=g/|g|) and bO-L ␣ of the single set of dislocations in the plane, one can determine the dislocation line direction, i , from [24]: i = T b∗␣ × p,
(3)
where b∗␣ = bL␣ /|bL␣ |2 is a reciprocal Burgers vector. The dislocation spacing is simply di-dis = 1/|i |. Vector i necessarily lies parallel to the invariant line, because both T b∗␣ and g must be normal to the invariant line [20]. The calculation results for the Fe–23.0Ni–3.8Mn alloy (af = 0.3580 nm, ab = 0.2870 nm [14]) are given in Table 1. The OR and the habit plane determined under the O-line condition are in good agreement with the experimental results. The experimental result of the dislocation structure is not available. A rigid model of interfacial structure in the habit plane, consisting of a layer of atoms in both phases, is plotted in Fig. 1 according to the calculated OR. A unit height of step is assumed in the figure for simplicity. As seen from Fig. 1a, the habit plane contains periodic good matching bands separated by poor matching regions where misfit dislocations may form. The diffraction contrast of these narrow spaced dislocations is probably too weak for them to be detected. The proper electron beam direction for one to observe interfacial dislocations with HRTEM is along the edgeon dislocations. The determined dislocation direction in Table 1 O-L is near bO-L ␣ . Fig. 1b is a plot normal to b␣ . The interfacial misfit cannot be shown in this plot, because the misfit is along the viewing direction, as can also be elucidated from a side view of Fig. 1a. This result explains why the habit plane is free of any dislocation when the beam is parallel to bO-L [14]. However, ␣ when the beam is approximately along [1 0 0]b and [0 1 −1]f , an extra half-plane of (0 −1 1)b per 8.6 layers of (0 −1 1)b have been observed [14]. However, the authors who reported this observa-
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Fig. 1. A rigid model of atomic structure of the habit plane in: (a) plane view and (b) near edge-on view along the direction of bO-L . In the plots (䊉, bcc atoms and , fcc atoms), and a possible kinked structure of the dislocations is represented by a zigzag solid line.
tion did not associate the extra (0 −1 1)b planes with the array of (screw) dislocations that they believed to exist in the habit plane based on the PTMC. We will apply the Moir´e plane concept below to demonstrate the association. Any reciprocal vector g represents a set of Moir´e planes [17,24]. The planes defined by g␣ and g , connected by a g, should be in full registry in the interface parallel to the Moir´e plane normal to the g [27]. If the Moir´e planes are inclined to an interface, the related planes will mismatch in the interface at the locations between the Moir´e plane traces. The Moir´e planes in a system containing the O-lines possess special properties [28]: (1) All Moir´e planes must lie in the zone axis of the invariant line; (2) the Moir´e planes are parallel to the habit plane if g␣ · bO-L = 0; a (3) the Moir´e planes will intersect the habit plane at the positions of the O-lines if g␣ · bO-L e planes that are ␣ = ±1. Selected Moir´
Fig. 2. An illustration of intersections of different Moir´e planes in the habit plane.
calculated at the OR in Table 1 are plotted in Fig. 2 in a plane normal to the invariant line. The property (2) further explains the observation of apparent coherent habit plane when the beam is parallel to bO-L ␣ [14], since all related planes in this zone axis must match each other in the habit plane. Consider next the matching status between the related planes (−1 −1 1)f and (0 −1 1)b . Because this (−1 −1 1)f (g␣ ) obeys the property (3), the disregistry between these planes should coincide with the dislocation cores. Fig. 3a shows the traces of the related planes in the habit plane. It can be proved that the spacing of the Moir´e fringes formed by the related plane traces is equal to the spacing of the traces of Moir´e planes associated with these planes, and in turn it is identical to the dislocation spacing d (Fig. 3). Because these planes cannot be viewed at a (nearly) edge-on orientation when dislocations are edge-on, the disregistry periodicity measured from an orientation that permits nearly edge-on planes cannot represent the dislocation spacing. However, the dislocation spacing and plane disregistry geometry are relevant. This situation is depicted in Fig. 3b. The plane of this figure is normal to [1 −0.94 −0.05]f //[1 −0.03 0.03]b , which is in the middle of [1 −1 0]f and [1 0 0]b at the OR in Table 1. This viewing direction is similar to the beam direction in Fig. 14 in ref. [14] for the TEM image of (−1 −1 1)f and (0 −1 1)b planes. All lines in the figure represent traces of different planes. Only (1 1 1)f (and (0 1 1)b ) are almost edge-on, but (−1 −1 1)f and (0 −1 1)b are slightly inclined. If the two phases are separated by the habit plane trace in Fig. 3b, there is one extra (0 −1 1)b plane trace per ∼7 (0 −1 1)b plane traces (∼1.51 nm) in the interface, as listed in Table 1. This particular habit plane trace is also
Fig. 3. (a) The traces of (0 −1 1)b and (−1 −1 1)f planes in the habit plane, plotted with calculated data. (b) The traces of (0 −1 1)b and (1 −1 −1)f planes viewing from an axis in the middle of [1 0 0]b and [1 −1 0]f . The projection of the habit plane is represented by a shadowed area, in which the projected dislocations are indicated. The viewing direction in (b) projected in the habit plane is indicated in (a).
W.-Z. Zhang, J. Wu / Materials Science and Engineering A 438–440 (2006) 118–121
plotted in Fig. 3a to show the difference between the dislocation spacing and the spacing of extra (0 −1 1)b planes seen from the viewing direction. The above extra plane spacing is smaller than the measured value of 8.6 planes. As seen from Fig. 3b, the extra plane spacing decreases with an increase of the angle θ between (1 1 1)f trace and the habit plane trace, as also indicated by Ogawa and Kajiwara [14]. According to their report [14], the measured θ for the observed habit plane is approximately 6◦ , but the calculated θ for the average (1 2 1)f habit plane is 11◦ . Though the present OR and calculation method are slightly different from theirs, the tendency of result should be similar. Namely, it is reasonable that our result of the extra plane spacing corresponding to θ of 10.8◦ is smaller than the measured value corresponding to θ of 6◦ [14]. Since the discrepancy in the extra plane spacing is due to the scattering in the habit plane trace measurement, we may conclude that our dislocation description of the habit plane (Table 1) is consistent with the observation of extra (0 −1 1)b planes, as depicted in Fig. 3b. A construction of an O-lattice usually starts from a longrange strain free state. In this construction, any interface is a macroscopic invariant plane. However, a long-range strain can be introduced by a relative shift between the martensite and parent lattices as the habit plane displaces its position [29]. The results in Table 1 are not affected by the long-range strain introduced in this way. The long-range strain can be expressed in terms of a shear field decomposed from the total displacement field (T) [17,30]. The constraint between the vector elements in different shear fields has been applied to rationalize convergence of numerical solutions of the habit plane of lath martensite [31]. If bO-L ␣ is inclined to the habit plane and if its direction defines the O-line (or dislocation) path, the T decomposition is equivalent to the PTMC [29,30]. However, bO-L in Table 1 lies in ␣ the habit plane. Therefore, the theoretical slip plane, containing the dislocations and bO-L ␣ , is now parallel to the habit plane. This apparent dilemma is not conflict to the O-lattice approach, since the O-line path is more flexible [29]. As seen in Fig. 1, each dislocation consists of large portions of screw dislocations connected by kinks in the theoretical slip plane. The screw portions may glide easily on a slip plane inclined to the habit plane, but the kinks now act as jogs to retard the dislocation motion. Because the constraint to the O-line solution is adjustable, this approach leaves us freedom to explore the mechanism of interface migration with guidance of further systematic experimental studies. 3. Summary The O-line condition has been applied in a quantitative study of crystallography of lath martensite in an Fe–Ni–Mn alloy. The
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habit plane of the martensite is fully accommodated by a single set of near screw dislocations. The calculated results, including the OR, the habit plane orientation and dislocation structure (in terms of matching or mismatching of certain sets of planes) in the habit plane agree with observations in a better self-consistent manner than the previous explanation. Acknowledgements The support of National Natural Science foundation of China, with Project No. 50471012, is gratefully acknowledged. Helpful comments from Min Zhang and Xinfu Gu are appreciated. References [1] J.S. Bowles, J.K. Mackenzie, Acta Metall. 2 (1954) 129–137. [2] J.K. Mackenzie, J.S. Bowles, Acta Metall. 2 (1954) 138–147. [3] M.S. Wechsler, D.S. Lieberman, T.A. Read, Trans. AIME 197 (1953) 1503–1515. [4] C.M. Wayman, Metall. Mater. Trans. 25A (1994) 1787–1795. [5] J.W. Christian, The Theory of Transformation in Metals and Alloys, 3rd ed., Pergamon Press, Oxford, UK, 2002. [6] J.K. Mackenzie, J.S. Bowles, Acta Metall. 7 (1957) 137–149. [7] F.C. Frank, Acta Metall. 1 (1953) 15–21. [8] R.C. Pond, Y.W. Chai, S. Celotto, Mater. Sci. Eng. A378 (2004) 47– 51. [9] N.D.H. Ross, A.G. Crocker, Acta Metall. 18 (1970) 405–418. [10] T. Moritani, N. Miyajima, T. Furuhara, T. Maki, Scripta Mater. 47 (2002) 193–199. [11] B.P.J. Sandvik, C.M. Wayman, Metall. Trans. 14A (1983) 835–844. [12] B.P.J. Sandvik, C.M. Wayman, Metall. Trans. 14A (1983) 809–822. [13] B.P.J. Sandvik, C.M. Wayman, Metall. Trans. 14A (1983) 823–834. [14] K. Ogawa, S. Kajiwara, Philos. Mag. 84 (2004) 2919–2947. [15] B.A. Bilby, J.W. Christain, Institute of Metals Monography and Report Series, No. 18 (1955) 121–72. [16] W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer, Berlin, 1970. [17] W. Bollmann, Crystal Lattices, Interfaces and Matrices, Bollmann, Geneva, 1982. [18] J.W. Christian, Trans. Jpn. Inst. Met. Suppl. 17 (1976) 21–33. [19] W.-Z. Zhang, G.C. Weatherly, Prog. Mater. Sci. 50 (2005) 181–292. [20] W.-Z. Zhang, G.R. Purdy, Philos. Mag. 68A (1993) 291–303. [21] D. Qiu, W.-Z. Zhang, Philos. Mag. 83 (2003) 3093–3116. [22] C.M. Wayman, Introduction to the Crystallography of Martensitic Transformations, MacMillan, New York, 1964. [23] G.C. Weatherly, W.-Z. Zhang, Metall. Mater. Trans. 25A (1994) 1865–1874. [24] W.-Z. Zhang, G.R. Purdy, Philos. Mag. 68A (1993) 279–290. [25] A.M. Condo, F. Lovey, Scripta Mater. 45 (2001) 669–675. [26] W.-Z. Zhang, Appl. Phys. Lett. 86 (2005), 121919-(1-3). [27] W.-Z. Zhang, G.R. Purdy, Mater. Sci. Forum 126–128 (1993) 563–566. [28] W.-Z. Zhang, D. Qiu, X.-P. Yang, F. Ye, Metall. Mater. Trans. 37A (2006) 911–927. [29] W.-Z. Zhang, G.C. Weatherly, Acta Mater. 46 (1998) 1837–1847. [30] W.-Z. Zhang, Philos. Mag. 78A (1998) 913–933. [31] W.-Z. Zhang, G.C. Weatherly, Scripta Mater. 37 (1997) 1569–1574.
Dislocation description of martensite interfaces based on misfit analysis W.-Z. Zhang ∗ , J. Wu Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Received 12 April 2003; received in revised form 21 November 2005; accepted 20 January 2006
Abstract It is shown how the methodology and theoretical relationships developed in the framework of the O-lattice theory can be applied for study of martensite interfaces containing a single set of dislocations. The observed orientation relationship, the habit plane orientation, matching or mismatching of certain sets of planes in the habit plane of lath martensite in an Fe–Ni–Mn alloy can be explained consistently by the present approach. Attention is drawn to the use of measurable g vectors for characterizing the habit plane and for interpreting plane matching in the habit plane. © 2006 Elsevier B.V. All rights reserved. Keywords: Orientation relationship; O-lattice; g approach; Habit plane; Misfit dislocations; Plane matching
1. Introduction Crystallography of martensite in many systems has been successfully predicted by the phenomenological theory of martensite crystallography (PTMC) proposed by Bowles and Mackenzie [1,2] and Wechsler et al. [3], as reviewed by Wayman [4] and Christian [5]. Two conditions were hypothesized in the formal PTMC, i.e. existence of a macroscopic invariant plane as the habit plane and a glissile habit plane. If simultaneous satisfaction of both conditions is imposed, the overall crystallography of the lath martensite often cannot be explained by the theory in a self-consistent manner. Various modifications have been employed, but usually only one of the above conditions can be maintained. One type of modifications allows a small longrange strain, e.g. by introducing a dilatation parameter [2,6] or assuming forced matching of a pair of low index vectors [7,8]. These modifications slightly violate the condition of existence of a macroscopic invariant plane. Another type of modifications may not be in accord with the existence of a glissile interface. The double shear model [9] or the descriptions of the habit plane with two sets of dislocations [10,11] belong to this category. However, the habit plane of lath martensite typically contains a set of regular dislocations visible by using conventional transmission electron microscopy (TEM) [12,13] and by high resolution TEM (HRTEM) [10,14]. ∗
Corresponding author. Tel.: +86 10 62773795; fax: +86 10 62771160. E-mail address: [email protected] (W.-Z. Zhang).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.01.100
The present work is a preliminary attempt to explore an alternative approach based on a study of interfacial structure. The vital role of interfacial structure in the crystallography and growth mechanism of martensite is well recognized, as emphasized by Christian since the 1950s [5,15]. Our approach is based on the O-lattice theory developed by Bollmann [16,17]. It has been explained by Christian [18] why the O-lattice theory is applicable to martensite in the O-line condition. The observations of a single set of discrete dislocations in the habit plane of martensite [13,10] suggest the possibility that the habit plane contains periodic O-lines between the dislocations. This paper shows how to apply the O-line model to the study of martensite crystallography, with a methodology developed mainly for precipitation crystallography [19]. Only the requirement for a macroscopic invariant plane is maintained strictly in this analysis. Lath martensite in an Fe–23.0Ni–3.8Mn alloy is taken as an example for the application.
2. O-line description and application The O-lattice describes a periodic pattern of fit/misfit between two lattices [16]. The O-lattice elements are locations of good matching. Given a transformation between lattices ␣ and , x = Ax␣ , a principal O-lattice vector, xO , can be determined from [16]: TxO = bL ,
(1)
W.-Z. Zhang, J. Wu / Materials Science and Engineering A 438–440 (2006) 118–121
119
Table 1 Comparison of results from an O-line calculation and results in ref. [14] Results in ref. [14]
Calculation
Discrepancy
OR
K–S (experiment)
[−1 0 1]f //[−1 −1 1]b (1 1 1)f ˆ(0 1 1)b = 0.37◦
0. 37◦
Habit plane bO-L ␣
(1 2 1)f (experiment) [−1 0 1]f /2 (assumption)
(1 1.997 1)f [−1 0 1]f /2 ([−1 −1 1]b /2)
0.04◦ No
Invariant line Dislocation spacing
//bLa (assumption) 0.925 nm (calculation)
[−1.12 0.06 1]f 0.92(8) nm
3.9◦ <0.01 nm
Extra plane spacing in unit of (0 −1 1)b planes
8.2 (calculation) 8.6 (experiment)
∼7.3
∼1
where T = I − A−1 defining the displacement field of the lattice deformation, I is a unit matrix and bL is a Burgers vector in lattice ␣. In our calculation, the fcc lattice is selected arbitrarily as the reference lattice ␣. Usually, a measured or nominate OR does not permit A to describe one or more sets of periodic O-lines, but a slight adjustment of the OR may render an O-line strain. Systematic solutions of O-lines for a given system can be determined by either numerical method [20] or analytic method [21]. In the formal PTMC, the OR is fixed by two conditions: (1) the Burgers vector of the dislocations, bO-L ␣ (the superscript stands for O-line), must lie in the plane with the invariant normal (i.e. invariant line in reciprocal space) and (2) the invariant line must lie in a given (rational) slip plane [22]. The O-line strain is subjected only to condition (1) [20]. The remaining one degree of freedom in the OR can be constrained by the criterion of either maximum dislocation spacing or minimum angular deviation from a rational OR (e.g. either N–W or K–S OR for an fcc/bcc system) [20,21,23]. The observed OR for the Fe–23.0Ni–3.8Mn alloy is the K–S OR [14]. According to the previous investigation on the fcc/bcc systems [21,23], to obtain a close OR we should apply the minimum angular deviation criterion and select 1 1 0f /2 corresponding to 1 1 1b /2 to be bO-L ␣ (where and hereafter subscripts f and b indicate the fcc and bcc lattice, respectively). Under these conditions, we can determine A for the O-lines by following the steps of the analytical method, explained elsewhere [21]. A simple parameter to describe the habit plane is a reciprocal vector g, defined by [24] g = T g␣ = g␣ − g ,
(2)
where g␣ and g are reciprocal vectors from lattices ␣ and , and the symbol “ ” denotes a transposition operation. The habit plane containing the O-lines must be normal to a group of parallel g’s that are associated with g␣ ’s in the zone axis of bO-L ␣ [20]. Therefore, the habit plane can be determined by the g associated with any g␣ normal to bO-L ␣ . This relationship is equivalent to the formula, p1 //p2 (S−1 − I), for determining the habit plane, p1 (unit vector), in the PTMC [1,22]. Here S = A, and p2 is the unit normal to the slip plane, which must contain bO-L ␣ . The expression with g is advantageous over the expression with unit vector, because g vectors are measurable in diffraction patterns. This expression facilitates a direct comparison between the theoretical and experimental results of the habit plane. The evidence
confirming the relationship between g vectors and martensite habit planes can be found in the literature, for example, in experimental results provided by Ogawa and Kajiwara (Fig. 3d in ref. [14]) and by Condo and Lovey (Fig. 2b in ref. [25]), though this relationship was not notified by the authors. Strictly, parallelism of g’s can be identified in one overlapped pattern only if the electron beam is along a pair of parallel Burgers vectors [23]. The dislocation configuration can be determined based on an O-cell construction [16]. The following formula is applicable to an interphase interface containing one or two sets of periodic dislocations [26]. Provided with the unit normal of the habit plane p (=g/|g|) and bO-L ␣ of the single set of dislocations in the plane, one can determine the dislocation line direction, i , from [24]: i = T b∗␣ × p,
(3)
where b∗␣ = bL␣ /|bL␣ |2 is a reciprocal Burgers vector. The dislocation spacing is simply di-dis = 1/|i |. Vector i necessarily lies parallel to the invariant line, because both T b∗␣ and g must be normal to the invariant line [20]. The calculation results for the Fe–23.0Ni–3.8Mn alloy (af = 0.3580 nm, ab = 0.2870 nm [14]) are given in Table 1. The OR and the habit plane determined under the O-line condition are in good agreement with the experimental results. The experimental result of the dislocation structure is not available. A rigid model of interfacial structure in the habit plane, consisting of a layer of atoms in both phases, is plotted in Fig. 1 according to the calculated OR. A unit height of step is assumed in the figure for simplicity. As seen from Fig. 1a, the habit plane contains periodic good matching bands separated by poor matching regions where misfit dislocations may form. The diffraction contrast of these narrow spaced dislocations is probably too weak for them to be detected. The proper electron beam direction for one to observe interfacial dislocations with HRTEM is along the edgeon dislocations. The determined dislocation direction in Table 1 O-L is near bO-L ␣ . Fig. 1b is a plot normal to b␣ . The interfacial misfit cannot be shown in this plot, because the misfit is along the viewing direction, as can also be elucidated from a side view of Fig. 1a. This result explains why the habit plane is free of any dislocation when the beam is parallel to bO-L [14]. However, ␣ when the beam is approximately along [1 0 0]b and [0 1 −1]f , an extra half-plane of (0 −1 1)b per 8.6 layers of (0 −1 1)b have been observed [14]. However, the authors who reported this observa-
120
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Fig. 1. A rigid model of atomic structure of the habit plane in: (a) plane view and (b) near edge-on view along the direction of bO-L . In the plots (䊉, bcc atoms and , fcc atoms), and a possible kinked structure of the dislocations is represented by a zigzag solid line.
tion did not associate the extra (0 −1 1)b planes with the array of (screw) dislocations that they believed to exist in the habit plane based on the PTMC. We will apply the Moir´e plane concept below to demonstrate the association. Any reciprocal vector g represents a set of Moir´e planes [17,24]. The planes defined by g␣ and g , connected by a g, should be in full registry in the interface parallel to the Moir´e plane normal to the g [27]. If the Moir´e planes are inclined to an interface, the related planes will mismatch in the interface at the locations between the Moir´e plane traces. The Moir´e planes in a system containing the O-lines possess special properties [28]: (1) All Moir´e planes must lie in the zone axis of the invariant line; (2) the Moir´e planes are parallel to the habit plane if g␣ · bO-L = 0; a (3) the Moir´e planes will intersect the habit plane at the positions of the O-lines if g␣ · bO-L e planes that are ␣ = ±1. Selected Moir´
Fig. 2. An illustration of intersections of different Moir´e planes in the habit plane.
calculated at the OR in Table 1 are plotted in Fig. 2 in a plane normal to the invariant line. The property (2) further explains the observation of apparent coherent habit plane when the beam is parallel to bO-L ␣ [14], since all related planes in this zone axis must match each other in the habit plane. Consider next the matching status between the related planes (−1 −1 1)f and (0 −1 1)b . Because this (−1 −1 1)f (g␣ ) obeys the property (3), the disregistry between these planes should coincide with the dislocation cores. Fig. 3a shows the traces of the related planes in the habit plane. It can be proved that the spacing of the Moir´e fringes formed by the related plane traces is equal to the spacing of the traces of Moir´e planes associated with these planes, and in turn it is identical to the dislocation spacing d (Fig. 3). Because these planes cannot be viewed at a (nearly) edge-on orientation when dislocations are edge-on, the disregistry periodicity measured from an orientation that permits nearly edge-on planes cannot represent the dislocation spacing. However, the dislocation spacing and plane disregistry geometry are relevant. This situation is depicted in Fig. 3b. The plane of this figure is normal to [1 −0.94 −0.05]f //[1 −0.03 0.03]b , which is in the middle of [1 −1 0]f and [1 0 0]b at the OR in Table 1. This viewing direction is similar to the beam direction in Fig. 14 in ref. [14] for the TEM image of (−1 −1 1)f and (0 −1 1)b planes. All lines in the figure represent traces of different planes. Only (1 1 1)f (and (0 1 1)b ) are almost edge-on, but (−1 −1 1)f and (0 −1 1)b are slightly inclined. If the two phases are separated by the habit plane trace in Fig. 3b, there is one extra (0 −1 1)b plane trace per ∼7 (0 −1 1)b plane traces (∼1.51 nm) in the interface, as listed in Table 1. This particular habit plane trace is also
Fig. 3. (a) The traces of (0 −1 1)b and (−1 −1 1)f planes in the habit plane, plotted with calculated data. (b) The traces of (0 −1 1)b and (1 −1 −1)f planes viewing from an axis in the middle of [1 0 0]b and [1 −1 0]f . The projection of the habit plane is represented by a shadowed area, in which the projected dislocations are indicated. The viewing direction in (b) projected in the habit plane is indicated in (a).
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plotted in Fig. 3a to show the difference between the dislocation spacing and the spacing of extra (0 −1 1)b planes seen from the viewing direction. The above extra plane spacing is smaller than the measured value of 8.6 planes. As seen from Fig. 3b, the extra plane spacing decreases with an increase of the angle θ between (1 1 1)f trace and the habit plane trace, as also indicated by Ogawa and Kajiwara [14]. According to their report [14], the measured θ for the observed habit plane is approximately 6◦ , but the calculated θ for the average (1 2 1)f habit plane is 11◦ . Though the present OR and calculation method are slightly different from theirs, the tendency of result should be similar. Namely, it is reasonable that our result of the extra plane spacing corresponding to θ of 10.8◦ is smaller than the measured value corresponding to θ of 6◦ [14]. Since the discrepancy in the extra plane spacing is due to the scattering in the habit plane trace measurement, we may conclude that our dislocation description of the habit plane (Table 1) is consistent with the observation of extra (0 −1 1)b planes, as depicted in Fig. 3b. A construction of an O-lattice usually starts from a longrange strain free state. In this construction, any interface is a macroscopic invariant plane. However, a long-range strain can be introduced by a relative shift between the martensite and parent lattices as the habit plane displaces its position [29]. The results in Table 1 are not affected by the long-range strain introduced in this way. The long-range strain can be expressed in terms of a shear field decomposed from the total displacement field (T) [17,30]. The constraint between the vector elements in different shear fields has been applied to rationalize convergence of numerical solutions of the habit plane of lath martensite [31]. If bO-L ␣ is inclined to the habit plane and if its direction defines the O-line (or dislocation) path, the T decomposition is equivalent to the PTMC [29,30]. However, bO-L in Table 1 lies in ␣ the habit plane. Therefore, the theoretical slip plane, containing the dislocations and bO-L ␣ , is now parallel to the habit plane. This apparent dilemma is not conflict to the O-lattice approach, since the O-line path is more flexible [29]. As seen in Fig. 1, each dislocation consists of large portions of screw dislocations connected by kinks in the theoretical slip plane. The screw portions may glide easily on a slip plane inclined to the habit plane, but the kinks now act as jogs to retard the dislocation motion. Because the constraint to the O-line solution is adjustable, this approach leaves us freedom to explore the mechanism of interface migration with guidance of further systematic experimental studies. 3. Summary The O-line condition has been applied in a quantitative study of crystallography of lath martensite in an Fe–Ni–Mn alloy. The
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habit plane of the martensite is fully accommodated by a single set of near screw dislocations. The calculated results, including the OR, the habit plane orientation and dislocation structure (in terms of matching or mismatching of certain sets of planes) in the habit plane agree with observations in a better self-consistent manner than the previous explanation. Acknowledgements The support of National Natural Science foundation of China, with Project No. 50471012, is gratefully acknowledged. Helpful comments from Min Zhang and Xinfu Gu are appreciated. References [1] J.S. Bowles, J.K. Mackenzie, Acta Metall. 2 (1954) 129–137. [2] J.K. Mackenzie, J.S. Bowles, Acta Metall. 2 (1954) 138–147. [3] M.S. Wechsler, D.S. Lieberman, T.A. Read, Trans. AIME 197 (1953) 1503–1515. [4] C.M. Wayman, Metall. Mater. Trans. 25A (1994) 1787–1795. [5] J.W. Christian, The Theory of Transformation in Metals and Alloys, 3rd ed., Pergamon Press, Oxford, UK, 2002. [6] J.K. Mackenzie, J.S. Bowles, Acta Metall. 7 (1957) 137–149. [7] F.C. Frank, Acta Metall. 1 (1953) 15–21. [8] R.C. Pond, Y.W. Chai, S. Celotto, Mater. Sci. Eng. A378 (2004) 47– 51. [9] N.D.H. Ross, A.G. Crocker, Acta Metall. 18 (1970) 405–418. [10] T. Moritani, N. Miyajima, T. Furuhara, T. Maki, Scripta Mater. 47 (2002) 193–199. [11] B.P.J. Sandvik, C.M. Wayman, Metall. Trans. 14A (1983) 835–844. [12] B.P.J. Sandvik, C.M. Wayman, Metall. Trans. 14A (1983) 809–822. [13] B.P.J. Sandvik, C.M. Wayman, Metall. Trans. 14A (1983) 823–834. [14] K. Ogawa, S. Kajiwara, Philos. Mag. 84 (2004) 2919–2947. [15] B.A. Bilby, J.W. Christain, Institute of Metals Monography and Report Series, No. 18 (1955) 121–72. [16] W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer, Berlin, 1970. [17] W. Bollmann, Crystal Lattices, Interfaces and Matrices, Bollmann, Geneva, 1982. [18] J.W. Christian, Trans. Jpn. Inst. Met. Suppl. 17 (1976) 21–33. [19] W.-Z. Zhang, G.C. Weatherly, Prog. Mater. Sci. 50 (2005) 181–292. [20] W.-Z. Zhang, G.R. Purdy, Philos. Mag. 68A (1993) 291–303. [21] D. Qiu, W.-Z. Zhang, Philos. Mag. 83 (2003) 3093–3116. [22] C.M. Wayman, Introduction to the Crystallography of Martensitic Transformations, MacMillan, New York, 1964. [23] G.C. Weatherly, W.-Z. Zhang, Metall. Mater. Trans. 25A (1994) 1865–1874. [24] W.-Z. Zhang, G.R. Purdy, Philos. Mag. 68A (1993) 279–290. [25] A.M. Condo, F. Lovey, Scripta Mater. 45 (2001) 669–675. [26] W.-Z. Zhang, Appl. Phys. Lett. 86 (2005), 121919-(1-3). [27] W.-Z. Zhang, G.R. Purdy, Mater. Sci. Forum 126–128 (1993) 563–566. [28] W.-Z. Zhang, D. Qiu, X.-P. Yang, F. Ye, Metall. Mater. Trans. 37A (2006) 911–927. [29] W.-Z. Zhang, G.C. Weatherly, Acta Mater. 46 (1998) 1837–1847. [30] W.-Z. Zhang, Philos. Mag. 78A (1998) 913–933. [31] W.-Z. Zhang, G.C. Weatherly, Scripta Mater. 37 (1997) 1569–1574.