Classification and analysis of trigonal martensite laminate twins in shape memory alloys

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ScienceDirect Acta Materialia 89 (2015) 193–204 www.elsevier.com/locate/actamat

Classification and analysis of trigonal martensite laminate twins in shape memory alloys ⇑

Nien-Ti Tsou,a, Chih-Hsuan Chen,b Chuin-Shan Chenc and Shyi-Kaan Wub a

Department of Materials Science and Engineering, National Chiao Tung University, Ta Hsueh Road, HsinChu 300, Taiwan Department of Materials Science and Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan c Department of Civil Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan

b

Received 14 October 2014; revised 3 February 2015; accepted 4 February 2015 Available online 26 February 2015

Abstract—In the trigonal martensite shape memory alloys (SMAs), typically only the well-known classic herringbone pattern satisfies the nonlinear compatibility theory. However, many other interesting trigonal patterns are still possible. Thus, geometrically linear compatibility theory is first used to obtain the full set of rank-2 laminate twin patterns which can possibly form in the trigonal SMAs. There are only seven families of twin pattern, including many new structures, such as checkerboard, toothbrush and non-classic herringbone patterns. These microstructures are confirmed in the trigonal R-phase Ti50.3Ni48.2Fe1.5 SMA. The interfaces separating the seven families and their parent phase (austenite) are also determined and observed. The results show that minor geometrical incompatibility of microstructures can be tolerated in the trigonal R-phase SMAs. Next, a numerical method based on non-linear compatibility theory is developed to evaluate the parameter for measuring the defect of disclination in these patterns. The greater the value of the parameter is, the less likely the pattern could be observed in the experiments. The likelihood of occurrence of all seven families in the typical trigonal SMAs is revealed in the current work. Moreover, 3-dimensional unit cell alignment of the patterns is illustrated. The loading conditions to avoid the disclination are also discussed. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Shape memory alloys; Compatibility; Laminate twins

1. Introduction Shape memory alloys (SMAs) are one of the most important functional materials which are widely used for industrial applications such as actuators. While most of the studies focus on the microstructures in SMAs with the monoclinic or orthorhombic phase transformation [1–4] because of their high deformation output, the trigonal R-phase transformation is particularly useful for actuators with very small temperature hysteresis and relatively high operation frequency [5–7]. Sittner et al. [8] were motivated by the excellent functional fatigue properties of R-phase and studied its transformation process and the thermomechanical behavior of the crystals. Fan et al. [9] discovered that R-phase in Ti50Ni48Fe2 exhibits a very high internal friction compared to the other crystal systems, and thus it is very suitable for high damping applications. R-phase SMAs have a great potential to be utilized as the applications with special purposes. However, since the trigonal R-phase typically has very small transformation strain and weak surface relief effect for experimental observations [10], the development and design in the applications of the trigonal R-phase are still not reliable [8]. This motivates the

⇑ Corresponding author; e-mail: [email protected]

current study. Our focus is on the theoretical calculation for the formation of the trigonal microstructures and their possible stable states, which are particularly useful for correlating the data from experimental observation without ambiguity. The origin of the significant properties mentioned above is the formation of microstructures. The microstructures typically are laminated twins which minimize the overall energy in the crystal [11]. The twin patterns are well explained by the constrained theory [12] which reveals the importance of compatibility between phases and the formation of fine phase mixtures in SMAs. DeSimone and James [13] developed a constrained theory for laminate twins by using linearized compatibility equations. The orientation of a compatible interface separating distinct phases/variants had also been studied rigorously [11,14–18]. There are two types of compatibility theories: nonlinear theory and geometrically linear theory. The former considers the nonlinearity of phase transformation and the rigid rotations between regions of variant, which prevent the occurrence of the geometrical disclination at every junction of interfaces. The requirements of nonlinear theory are strict, and thus limited sets of microstructures can be found. By contrast, the latter assumes that the unit cell distortion of martensite phases is small, and neglects the rigid rotations

http://dx.doi.org/10.1016/j.actamat.2015.02.006 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

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between variants. Thus, some microstructures that are not feasible in nonlinear compatibility theory can be found [11]. These microstructures may have the local incompatibility or misfits between variants resulting in the elastically stressed configurations. This issue has been addressed in the literature. Seiner et al. [19,20] developed a mathematical model to examine the motion, configuration and compatibility of macrotwin interfaces. Heczko et al. [21] analyzed the crossing morphology and illustrated the energy landscape of different types of interfaces. A variety of numerical methods have been used recently to seek the compatible microstructures in SMAs. Molecular dynamics (MD) simulations calculate the interaction between atoms in the crystals, and can capture the process of phase transformations in detail and obtain the low energy microstructures [10,22,23]. Phase field method is one of the main numerical methods which typically uses the timedependent Ginzburg–Landau (TDGL) equations, and describes the microstructure by the order parameters [24– 28]. Most of the resulting twin patterns generated by phase field method have compatible interfaces between the present variants in the crystal. For example, in the work of Shu and Yen [26], the trigonal R-phase twin patterns in the form of compatible stripes, herringbone and other periodic topologies are predicted in the Ti–Ni film. Although MD and phase field method can give detailed microstructure in SMAs, they are typically computationally demanding. Thus, the study is limited to a relatively small region. By contrast, sharp interface approaches treat an interface separating a pair of variants as a discontinuity, giving a significant saving of computation. In sharp interface approaches, twin structures are usually assumed as laminate patterns represented by a binary tree diagram [18,29,30]. This allows a rapid full search for low energy laminate twin patterns based on the compatibility equations [18]. In the context of stable, low energy laminate twin patterns in the trigonal crystal system, most of the observed microstructures are self-accommodating twins rather than a single variant state [31]. In general, they can be treated as rank-2 laminate twins or the mixture of multiple rank2 laminate twins. Where the rank-2 twin indicates “a twin within a twin”, i.e. the mixture of a pair of rank-1 laminate twins [11]. A typical example of low energy rank-2 laminate twin is the “herringbone” with a prismatic topology along one axis. It is the most commonly observed twin pattern and has been extensively studied both experimentally and numerically [10,26,31,32]. It is shown that the herringbone twin pattern can be achieved by six equivalent phase arrangements based on the crystallographical symmetry [33]. However, given the great number of publications and studies, there is no other type of topology of the trigonal twin patterns being discussed or classified, leaving many possible complex and interesting trigonal patterns unexplained, such as non-symmetric herringbone, toothbrush and checkerboard patterns found in the present work. These trigonal twin patterns are geometrically linear compatible, but do not satisfy the non-linear compatibility theory. This indicates that minor geometrical incompatibility of microstructures can be tolerated in the trigonal SMAs. A similar statement was also made by Balandraud and Zanzotto [34], when they examined the compatibility of the microstructures in Cu-based shape memory alloys. In the present work, we develop a sharp interface approach seeking compatible twin structures in the trigonal

crystal system, and identify/classify the microstructures through examining the orientation of the interfaces. The method based on our previous work for the classification of microstructures in ferroelectrics [35] is extended. Being different from the ferroelectrics, solving compatibility equation of non-magnetic SMAs typically leads to multiple solutions for the interface orientation, and thus, further complexity arises. We propose a method which utilizes the geometrically linear compatibility theory to obtain sets of possible solution. The results are used to classify the trigonal microstructures and applied to identify the patterns and crystal variants in the scanning electron microscope (SEM) images of a Ti50.3Ni48.2Fe1.5 SMA. Then an optimization scheme based on the nonlinear compatibility theory is proposed and applied to determine the best sets of interfaces and the rotations of unit cells with lowest incompatibility in the twin patterns for microstructural analysis. In the following sections, microstructures at rank-2 in the trigonal crystal system are shown in detail. Surprisingly, only seven distinct types of geometrically linear compatible rank-2 twins can form. The set of patterns is classified and the incompatibility of each pattern is discussed. The results are validated by the experimental observations in the current study and the model predictions shown in the literature. The present work provides an overview of the set of possible trigonal twin patterns, giving design guideline for the development of smart devices. The approach is rapid, computationally efficient and sufficiently general to allow further extension to other crystal systems and materials. 2. Classification of the trigonal laminate twins 2.1. Geometrically linear compatibility theory In this section, geometrically linear compatibility theory is used, in order to obtain the full set of microstructures which can possibly form in the trigonal SMAs. There are four variants with the strain state ðiÞ ; ði ¼ 1 . . . 4Þ given below [11] 2 3 2 3 a d d a d d 6 7 6 7 d 5; ð1Þ ¼ 4 d a d 5; ð2Þ ¼ 4 d a d 2

ð3Þ

d

d a 3

a d d 6 7 a d 5; ¼4 d d d a

ð4Þ

d 2

a 6 ¼ 4 d d

a d a d

3 d 7 d 5 a ð1Þ

where parameters a and d are the material properties. Now consider a pair of trigonal martensite variants ði; jÞ with transformation strain states ðiÞ ; ðjÞ . A compatible interface with unit normal vector n then satisfies the well-known compatibility equation [11,13]: 1 ðjÞ  ðiÞ ¼ ða  n þ n  aÞ ð2Þ 2 for an arbitrary vector a. Eq. (2) can be solved by making use of the eigenvalues kk ðk ¼ 1; . . . ; 3Þ and eigenvectors ek of the 3  3 matrix M ¼ ðjÞ  ðiÞ . Generally, two solutions of the compatible interface normal can be found whenever k1 ¼ k3 ; k2 ¼ 0 provided that the two martensite variants

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are mutually related by pure rotations. Multiple compatible interface arrangements between a pair of variants are possible, resulting in complex arrangements of martensite twins. Now consider more than two types of trigonal variants in the crystal. As mentioned above, most of the observed microstructures are rank-2 laminates, or the mixture of several rank-2 laminates. Thus, a compatible rank-2 laminate structure can be treated as the building block of a complex microstructure in the crystal, and can be represented by a hierarchical binary tree diagram [30,35] with three levels, as shown in Fig. 1. It contains seven nodes (i ¼ 1; . . . ; 7); each of which has its volume fraction f i and state of average strain. The nodes in the lowest level of the diagram represent the pure trigonal variants, and the remaining nodes associated with a vector ni give the orientation of the compatible interface between the materials represented by their child nodes. For example, the rank-2 interfaces with normal n1 separating two rank-1 twins represented by nodes 2 and 3, and thus, they are also known as macrotwin planes [36,37]. The root node (i ¼ 1) represents the entire laminate twin structure, and we name the pattern “1234”. Note that not all combinations of the trigonal variants can form compatible laminate twin patterns. Here, we apply the compatibility conditions provided in our previous work [35] to check if the trigonal, rank-2 laminate is compatible: 1. Condition (i) requires that interfaces between distinct variants must have the same spacing, wherever they meet at any higher level interface. For the rank-2 tree diagram (see Fig. 1), this means that the volume fraction ratios of nodes 4, 5 and nodes 6, 7 should be identical, i.e. f4 f6 ¼ f5 f7

ð3Þ

2. Condition (ii) requires the interface normals of any two nodes and their parent node must be coplanar. n1  ðn2  n3 Þ ¼ 0

ð4Þ

This condition ensures that the projection of the low rank interfaces onto the higher level interfaces matches in orientation with that of the corresponding interfaces on the opposite side.

Fig. 1. A binary tree diagram for a rank-2 laminate twin pattern “1234”.

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3. Condition (iii) requires that every interface in the crystal must satisfy Eq. (2). This condition can be expressed explicitly in terms of the nodes in the rank-2 tree diagram. For the interface normal vector ni ; ði ¼ 1 . . . 3Þ of node i in the level r, the pure trigonal variants with node numbers p; q in the lowest level must satisfy compatibility equation, where p ¼ 2r i þ k;

q ¼ p þ 2r1 ;

ðk ¼ 0 . . . 2r1  1Þ

ð5Þ

For example, in Fig. 1, nodes 4 and 5 must be compatible across the interface with normal n2 ; nodes 4 and 6 must be compatible across the interface with normal n1 etc. Laminate structures which satisfy the above three conditions can have a perfectly matched twin pattern, and are geometrically linear compatible. It is worth to mention that, generally, the structures with mismatched pattern require separation of length scales between successive laminate composites, i.e. fine phase mixture [12], in order to minimize the overall energy. Thus, under the requirement of the compatibility conditions (i). . .(iii), the length scale of the interface spacing is irrelevant to the twin patterns considered in the present work. Although changing the length scale of interface spacing can make a significant difference to the appearance of a laminate, the interface alignment and compatibility across all the interfaces remain the same in the crystal. Fig. 2(a) and (b) are the examples of two equivalent twin patterns with different length scales. It appears that the two twin patterns look different, however, the arrangements of interface are in fact equivalent: the interfaces with normal ½0; 1; 1p separating variants 1,2; the interfaces with normal ½0; 1; 1p separating variants 3,4; normal ½0; 0; 1p separating variants 1,3 and 2,4. Where the subscript p indicates that these plane normals are indexed with the coordinate of parent phase. 2.2. The set of geometrically linear compatible trigonal rank2 twins Now consider to find all possible geometrically linear compatible trigonal rank-2 laminate twins. In the trigonal martensite which corresponds to the trigonal crystal system, there are 4 martensite variants in total. Meanwhile, there are 4 nodes in the lowest level of a rank-2 tree diagram. Thus, these give up to 44 ¼ 256 possible variant permutations for the trigonal rank-2 laminate. Fortunately, some of these variant permutations are mutually equivalent as they may be related to each other by pure rotations or symmetry. Some of the variant permutations are equivalent due to the identical relative position of variants in the tree diagram. For example, variant orders “1234”, “2143”, “3412” and “4321” produce an identical tree diagram. In

Fig. 2. Equivalent laminate twin topologies: (a) “1234”; (b) “1234” with different interface spacing; (c) “2413” which is a rigid rotation of structure shown in (b) by 90° about y p -axis.

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addition, symmetry operations that leave the topologies identical, such as rigid rotations, reflections and inversions, can be removed. For example, Fig. 2(b) and (c) show that two twin patterns “1234” and “2413” have equivalent topologies, since the structure of “2413” is a rigid rotation of‘ that of “1234” by 90° about the y p -axis. Although these patterns have distinct variant permutations marked in the notation “ . . .”, they have symmetry/rotation related topologies. Thus, we use notation f. . .g to represent families of the equivalent patterns. As the unit cells of each trigonal martensite variant are mutually related by pure rotations, the symmetry operations resulting in equivalent topologies belong to the symmetry group m3m [35]. After the reduction of symmetry/rotation, there are 9 possible variant permutations, as shown in Table 1. Each variant permutation can lead to 23 ¼ 8 distinct families of patterns. This is because Eq. (2) typically gives two solutions of interface normal n and there are three nodes with interfaces in a rank-2 tree diagram. This results in non-uniqueness of microstructures for a given variant permutation. For example, Fig. 3 shows an alternative pattern for the trigonal variant combination of “1234”. All the interface normals of the pattern are obtained from geometrically linear compatibility equation, Eq. (2); the pattern also satisfies the compatibility condition (i). . .(iii). Interface normals n2 and n3 in nodes 2 and 3 are the different sets of solution shown in Fig. 1. Thus, the choice of the solution of interface normal can make a significant change in the topology of the pattern. There are now 9  8 ¼ 72 possible families of patterns to be checked if the compatibility conditions (i). . .(iii) are satisfied. As shown in Table 1, two subsets of variant permutations, 1123 and 1223, cannot form any compatible structure. Each of subsets 1111, 1112, 1122 and 1212 generates its 8 families of pattern with all interfaces parallel, and thus, reduces to rank less than 2. Surprisingly, only three subsets: 1213, 1221 and 1234, can form families of rank-2 twin pattern and satisfy compatibility conditions (i). . .(iii). The three subsets of variant permutation give seven families of geometrically linear compatible rank-2 pattern in total, as shown in Fig. 4. In the variant permutation 1234, two families with distinct topologies both containing all four variants can be found. The first family f1234g1 is the classic herringbone with a prismatic configuration consisting of two groups of symmetric diagonal interfaces separated by a horizontal higher level interface, as shown in Fig. 4(a). The second family is f1234g2 . The corresponding microstructure has crossing horizontal and vertical interfaces giving a “checkerboard” pattern, as shown in Fig. 4(b). Interestingly, family f1221g1 , which has only two crystal variants, is also possible to form a checkerboard pattern (Fig. 4(c)). Family f1221g2 results in a microstructure consisting of two groups of stripes with different orien-

Table 1. The 9 permutations of the trigonal variants after the reduction of symmetry and rigid rotation. Three subsets can form patterns of rank-2 that satisfy compatibility conditions (i). . .(iii). Not compatible

Rank < 2, compatible,

Rank = 2, compatible

1123 1223

1111 1112 1122 1212

1213 1221 1234

Fig. 3. An alternative solution of rank-2 laminate twin pattern “1234”.

Fig. 4. The seven geometrically linear compatible rank-2 laminate twin patterns.

tations meeting at the higher level interfaces, showing a “toothbrush” pattern on at least one surface (Fig. 4(d)). Variant permutation 1213 has three families, and all of them show herringbone patterns on at least one surface, as shown in Fig. 4(e)–(g). Note that these three herringbone patterns are different from family f1234g1 . The herringbone of family f1213g1 has a prismatic configuration consisting of two groups of asymmetric interfaces respective to their higher level interfaces (the midrib) (Fig. 4(e)). Family f1213g2 has a prismatic, symmetric herringbone pattern, however, the higher level of interfaces are diagonal (Fig. 4(f)). Finally, the microstructure of family f1213g3 is the most 3-dimensional pattern among all the geometrically linear compatible rank-2 twins, showing herringbone patterns in all the surfaces. It is worth noting that the notation used here is based on the arrangement of the variant numbers, and can be correlated directly with the terminology of the twinning system reported in the literature [38,39,36,37]. All the interfaces shown in the Fig. 4 can be classified as compound twin planes. The detail of the twinning system is discussed in Appendix A. The seven families mentioned above are pure trigonal martensite exactly compatible laminate patterns. It is of interest to further study the origin of these patterns, i.e. the transition state between these patterns and their parent

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(austenite) phase. Where the parent phase has the strain state p ¼ 0. Typically, compatible austenite–martensite (A–M) interfaces require the martensite structures with certain volume fraction of martensite variants, giving their average transformation strain m with a zero middle eigenvalue k2 ¼ 0. The A–M interface normal vector, nA , can be obtained by substituting m and p into Eq. (2). Then, the lowest level of the tree diagram of structures takes the form of “martensite twin pattern + austenite phase”, i.e. rank-3 laminate, where the number of nodes for austenite phase is the same as that of the martensite twin pattern [18]. This indicates that the different martensite twin patterns can have the same set of the A–M interfaces whenever their average transformation strain m are identical. For example, family f1234g1 will have the same orientation of A– M interfaces as family f1234g2 provided that they have identical composition of the martensite variants 1–4. Here we illustrate the A–M interfaces between austenite and f1234g1 ; f1221g2 ; f1213g1 with selected volume fraction of martensite variants, in Fig. 5. Where the austenite phase is shown translucently to reveal the martensite pattern on the A–M interface. For the illustration purpose, we choose two parameters related to the volume fraction of the martensite variants l1 ¼ 0:5 and l2 ¼ 0:25 as shown in Fig. 5(a). In terms of the tree diagram, l1 indicates the volume fraction ratios of nodes 4, 5 and nodes 6, 7, i.e. l1 ¼ ff 4 ¼ ff 6 ¼ 0:5; l2 is 5 7 equivalent to the volume fraction of node 2, i.e. l2 ¼ f 2 ¼ 0:25. This setting is to avoid the special case of the infinity sets of solution of A–M interface. For example, in the case of the family f1234g1 with l1 ¼ l2 ¼ 0:5, the average transformation strain m ¼ 0, giving arbitrary solutions in Eq. (2). Fig. 5(a–d) shows the two solutions of compatible A–M interfaces of f1234g1 + Austenite and f1221g2 + Austenite, respectively. The twinning planes are either horizontal or vertical in the both cases. Pattern f1213g1 + Austenite has a vertical interface and an oblique A–M interface as shown in Fig. 5(e, f). The A–M interfaces between the trigonal R-phase families and austenite phase are not only verified theoretically but also observed experimentally. This will be explained in detail in Section 2.3. Note that the volume fraction of the martensite variants present in the structure can affect the average transformation strain m , and thus, can alter the orientation of the A–M interface normal [38]. More detailed informa-

Fig. 5. The A–M interfaces between austenite (the translucent regions) and the twin families f1234g1 ; f1221g2 ; f1213g1 with selected volume fraction of martensite variant.

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tion about the A–M interfaces for the families of the trigonal pattern is provided in Appendix A. 2.3. Experimental method and microstructure identification In order to validate the results generated in the previous section, SEM equipped with electron backscatter diffraction (EBSD) is used to find rank-2 twin structures in a bulk Ti50.3Ni48.2Fe1.5 SMA. The specimen is prepared to investigate the morphology of trigonal R-phase. The composition is so selected to obtain a RS temperature (the starting temperature of R-phase transformation) higher than the room temperature. A vacuum arc remelter (VAR) was employed to prepare the TiNiFe ingot using 99.99 wt.% Ti, 99.99 wt.% Ni and 99.98 wt.% Fe as raw materials. The ingot was remelted six times in the VAR to improve homogeneity, and then hot-rolled at 1173 K to a 2 mm-thick plate. Transformation temperature and latent heat of the TiNiFe alloy were determined using differential scanning calorimetry (DSC) (Q10, TA Instruments, USA) with a heating/cooling rate of 10 K/min. The Rs temperature is determined to be 33° C. Specimen for EBSD observation was electropolished at 12 V with an electrolyte consisting of 10% H2SO4 and 90% CH3OH in volume. The polishing temperature was set at 293 K, which was the same with the EBSD observation temperature. FEI Nova 450 SEM equipped with an EDAXTM Hikari XP EBSD Camera was used to scan the kikuchi patterns of the TiNiFe specimen. The orientation of the grains is processed and analyzed by the commercial software, Orientation Imaging Microscopy (OIMTM). The morphologies of the trigonal R-phase are taken with a forward scatter detector (FSD) attached below the EBSD camera to obtain images showing orientation contrast. When multiple interfaces are present within one grain, the interfaces are commonly in certain crystallographic planes which satisfy the compatibility conditions, and thus, the traces of these planes show up as straight lines in the viewing plane. In general, examining the orientation of the lines provides a unique interpretation of the pattern [40], and the family of the twin pattern can then be identified. The crystallographic orientation, specified by Euler angles, of the viewing plane in the grain provided by the EBSD data is used for post-processing of the model results. The 3-dimensional microstructures generated by the current model are rotated, cut and split accordingly, giving the 2-dimensional view, i.e. the projection of the interfaces, of the twin patterns in the viewing plane. The procedures are detailed in the authors’ previous work [40]. Fig. 6 is the FSD image showing variety of herringbone microstructures in one grain. Two families of twin patterns are identified. For comparison, the corresponding schematic twin patterns generated by the current model, where the interfaces are projected onto the region (marked by bold lines) parallel to the viewing plane of the image, are also shown. The regions consisting of different crystal variants are marked in distinct colors, indicated by the legends in the figures. Patterns “1432” and “1342”, which are the members of family f1234g1 , forming the classic herringbone, are shown in Fig. 6(a) and (c), respectively. Classic herringbone f1234g1 are the most commonly found patterns in the trigonal R-phase SMAs, and they are also reported both in simulation [26] and experimental observation [32] in the literature. Note that, here, the interpretation

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Fig. 6. SEM image showing various rank-2 herringbone twin patterns in the trigonal R-phase Ti50.3Ni48.2Fe1.5, along with the patterns generated by the current model: (a) “1432” (family f1234g1 ), (b) “3132” (family f1213g1 ), (c) “1342” (family f1234g1 ) and (d) “4143” (family f1213g1 ). The results of phase identification by examining the contrast of the image and the orientation of interface are also shown in yellow. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

of microstructures is unique and there are no other rank-2 twin patterns that have similar interface projection. Also, as mentioned above, the interface spacing of the pattern is not considered in the present work. Thus, the volume fractions of the crystal variants present in the patterns generated by the current model are adjusted to fit the observed interface spacing in the image. The other family observed in the same grain is family f1213g1 . The orientation of the interfaces shown in Fig. 6(b) and (d) matches those of the members of family f1213g1 : “3132” and “4143”, respectively. This family has not been reported and discussed previously in the literature. One of the reasons can be that the interface projection of the family appears similar to that of the classic herringbone f1234g1 family, and presents ambiguity in identifying microstructure from surface images. The 3-dimensional analysis in current work allows direct mapping of the interface directions, and to distinguish the types of herringbone observed in the experiments. It is also interesting to note that, under the assumption that the regions consisting of the same crystal variant in the same grain give a unique level of contrast in the FSD image, the phase identification can be achieved in some cases. For example, pattern “3132” shown in Fig. 6(b) has the bright zig-zag bands in the FSD image. By examining the interface orientation and the contrast between the FSD image and the result generated by the current model, we can interpret that the regions of the zig-zag bands represent crystal variant 3; the dark regions in Fig. 6(b) represent crystal variant 1; the remaining bright regions represent crystal variant 2. Thus, the variant in each region in pattern “3132” is determined and numbered accordingly in yellow in Fig. 6(b). A similar interpretation can be made for patterns “1342” and “4143”, and the crystal variants of regions are numbered in yellow in Fig. 6(c) and (d), respectively. However, phase identification for pattern “1432” shown in Fig. 6(a) cannot be done by the direct inspection and

further information provided by experimental measurements, such as atomic force microscopy observation, is needed. Let us go back to microstructure identification. The family f1234g1 of classic herringbone twin pattern is also observed in the other grain of the specimen with the different crystallographic orientation/Euler angles, as shown in Fig. 7(a). The orientation of the interface projection in the viewing plane matches twin pattern “1243”. Toothbrush pattern is also observed in the same grain, as shown in Fig. 7(b). The interfaces in the viewing plane match those of pattern “3443” which is the member of family f1221g2 . Similar pattern was also observed in the literature [32]. It is interesting to note that, in the upper-middle of Fig. 7, the toothbrush pattern shares part of its brushes with the herringbone pattern as the ribs. It can also be observed that the orientation of the midrib (marked with dashed line), in the transition region between the two patterns, is slightly different from that of other midribs of herringbone. A possible explanation is that the midrib interface is distorted by the internal stress due to the incompatibility and the mismatching of the regions of phases between the two patterns underneath the viewing plane. Another toothbrush pattern “2442” is found in a different grain as shown in Fig. 8. Most of the interfaces match those generated by the current model, and satisfy geometrically linear compatibility. However, it can be observed that the interfaces of the brushes in the center of the pattern (marked with dashed line in Fig. 8) are not parallel to those of the other brushes and do not satisfy the compatibility equation. This indicates that the microstructure is not at a stress-free state. This nonparallel

Fig. 7. SEM image showing (a) herringbone “1243” (family f1234g1 ) and (b) toothbrush “3443” (family f1221g2 ) twin patterns along with the patterns generated by the current model.

Fig. 8. SEM image showing toothbrush “2442” (family f1221g2 ) twin patterns with nonparallel interface arrangement.

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toothbrush pattern will be discussed in more detail in Section 3.2. Apart from classic herringbone patterns, microstructures with all four trigonal R-phase crystal variants present can also form the checkerboard pattern, family f1234g2 . Fig. 9 shows “1243” in the checkerboard form, while Fig. 7(a) shows the same variant permutation “1243” in the classic herringbone form. It is worth noting that, in the current work, microstructures of this type are found in a small region of the specimen only, and are with relatively low contrast between different R-phase variants. The probability to find the checkerboard patterns in the crystals is relatively lower than that of the classic herringbone patterns. This issue will also be discussed in more detail later on by examining the geometric disclination of the microstructures. Family f1213g2 herringbone pattern was not found in the specimen in the current work. However, the roof type microstructures, which has close relationship with family f1213g2 , have been observed experimentally in the f02 martensite phase in Au-49.5 at.%Cd alloy (which essentially adopt the same microstructures with trigonal R-phase) [31]. The roof type pattern can be regarded as a rank-4 laminate twins consisting of four members of family f1213g2 . An example of roof type pattern which is the combination of “1412”, “2321”, “3234” and “4143” is shown in Fig. 10, where fictitious interfaces marked with bold lines are included in Fig. 10 to aid visualization the mixture of the patterns. This example was also found experimentally and discussed in Murakami et al. [10]. In addition to the pure trigonal martensite twins, we also found the A–M interfaces separating austenite phase and the classic herringbone twin pattern in the specimen, as shown in Fig. 11. Where the top left region in gray, marked with “A”, is austenite and the martensite twin pattern is “1423” which is a member of family f1234g1 . It can be observed that distinct variants in the twin pattern have a similar spacing, i.e. l1 ¼ 0:5; the spacing between rank-1 twins, however, varies in the specimen, i.e. l2 is not constant. The pattern adopts two orientations of the averagely

Fig. 9. SEM image showing checkerboard “1243” (family f1234g2 ) twin patterns. The corresponding result generated by the current model is also shown.

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Fig. 10. Example of the roof type pattern which is the combination of four members of family f1213g2 : “1412”, “2321”, “3234” and “4143”.

Fig. 11. SEM image showing the classic herringbone “1423” (family f1234g1 ) + Austenite phase with the A–M interface normal vectors (a) ½0; 0; 1p and (b) ½1; 0; 0p . The corresponding result generated by the current model is also shown. (c) An incompatible A–M interface.

compatible A–M interfaces, ½0; 0; 1p and ½1; 0; 0p , shown in Fig. 11(a) and (b), respectively, marked by dashed lines. This indicates that the austenite phase transforms to the trigonal R-phase martensite twins along with the propagation of the A–M interfaces with the two distinct orientations. A similar phase transformation mechanism is also reported as the X-interface propagation in the literature [39]. It can be observed that there are several discontinuities between the A–M interfaces in Fig. 11, and the contrast of these regions is not clear. In particular, the greatest discontinuity shown in Fig. 11(c) can be regarded as the A–M interface that separates the austenite phase and the martensite rank-1 laminate twin “23”. However, according to the compatibility theory, the compatible A–M interfaces of this case are also normal to ½0; 0; 1p and ½1; 0; 0p , which do not agree with the blurry A–M interface in Fig. 11(c). Thus, this A–M interface is incompatible. It is worth noting that the austenite phase appears to be rare in the current specimen. Most parts of the material are in the trigonal martensite phases at room temperature. Thus, there are no other types of A–M interfaces discovered in the current specimen. The transition between austenite and martensite twins reveals many interesting structures in the other crystal systems in SMAs [38,39]. However, the focus of the current study is on the trigonal R-phase martensite twins, and thus, the

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detailed classification of the A–M interfaces will be described elsewhere. Some of the trigonal rank-2 structures described above have been reported in the literature, either as the results of direct observation, or from theoretical computations. However, families f1221g1 and f1213g3 , were not observed in the specimen in the current work and haven’t been addressed in the literature. This indicates that twin patterns generated by the geometrically linear compatibility theory may not be in the minimum energy state. Based on the non-linear compatibility theory, most of the patterns formed from their parent austenite phase result in the presence of disclinations at the junctions of interfaces. Thus, a rotation Q for the unit cell of each crystal variant present in the twin pattern is needed to minimize such disclination. In the following section, we propose a numerical method based on the non-linear compatibility theory to solve the best set of solution of the rotation matrices for each family of twin patterns.

3. Microstructure analysis of the trigonal laminate twins 3.1. Numerical method based on non-linear compatibility theory Now consider a pair of the trigonal martensite variants ði; jÞ meet at a given interface with unit normal vector n. The rigid rotation Q of the variant ðjÞ can be obtained by solving the non-linear compatibility equation: QUðjÞ  UðiÞ ¼ b  n

ð6Þ

where b is an arbitrary vector; UðiÞ and UðjÞ are the transformation matrices of the trigonal variants ði; jÞ. Note that the transformation matrix is related to the strain state, such that   12 ½ðU  IÞ þ ðU  IÞT , where I is the unity matrix. Typically, rank-2 twin patterns considered in the present work can be regarded as the assembly of four regions of variants meeting at a line which is shown in Fig. 12 as a point C in a 2-dimensional view. Regions in the figure are numbered following the node number of the corresponding tree diagram, so that regions 4 and 5 are separated by the interface with normal vector n2 ; regions 4 and 6 are separated by the interface with normal vector n1 and so on so forth. Non-linear compatibility theory requires three regions rotated accordingly, one after another, in order to

Qp ¼ 2 cos hp þ r2p1 ð1  cos hp Þ 6 6 rp1 rp2 ð1  cos hp Þ þ rp3 sin hp 4 rp1 rp3 ð1  cos hp Þ  rp2 sin hp

rp1 rp2 ð1  cos hp Þ  rp3 sin hp cos hp þ r2p2 ð1  cos hp Þ rp2 rp3 ð1  cos hp Þ þ rp1 sin hp

form a close loop around the junction. Thus, a “disclination-free” rank-2 laminate twin pattern satisfies the following conditions [11]:

Q1 U6  U4

¼

b1  n1

Q1 Q2 U7  Q1 U6 Q1 Q2 Q3 U5  Q1 Q2 U7 U4  Q1 Q2 Q3 U5

¼ ¼ ¼

b2  n3 b3  n1 b4  n2

ð7Þ

n1 ; n2 and n3 lie on a plane: These conditions are so strict that the solution exists only if the four regions containing different crystal variants and with a particular arrangement of interfaces [11]. In the case of the trigonal rank-2 twin pattern, this means that only the classic herringbone, i.e. family f1234g1 , can form a disclination-free microstructure and the remaining six families cannot be obtained through the non-linear compatibility theory. However, the toothbrush pattern corresponding to family f1221g2 , the checkerboard pattern corresponding to f1234g2 and the roof type pattern corresponding to family f1213g2 have been observed experimentally either in the present work or in the literature [31,32]. This indicates that twin patterns with the geometrical disclinations are still possible in SMAs. Thus, it is worth to utilize the non-linear compatibility theory to measure the level of disclination of all seven families obtained by geometrically linear compatibility theory. A functional C which describes the mean square error of the estimated solution of rotation matrices Qp ðp ¼ 1 . . . 3Þ and vectors bq ðq ¼ 1 . . . 4Þ from the state of disclinationfree, can be written in the tensor form: ( 3 X 3 X C¼ ½ðb1i n1j  Q1ik U 6kj þ U 4ij Þ2 j¼1 i¼1

þ ðb2i n3j  Q1ik Q2kl U 7lj þ Q1ik U 6kj Þ2 þ ðb3i n1j  Q1ik Q2kl Q3lm U 5mj þ Q1ik Q2kl U 7lj Þ2 )12 þ ðb4i n2j  U 4ij þ Q1ik Q2kl Q3lm U 5mj Þ2 

ð8Þ

Note that the final condition “n1 ; n2 and n3 lie on a plane” in Eq. (7) is equivalent to the compatibility condition (ii), Eq. (4), in Section 2.1. Thus this condition is absent in Eq. (8) as all of the seven families do have coplanar interface normals. Now substitute the transformation matrix U4 . . . U7 and n1 ; n2 ; n3 of a given twin family in Fig. 4 into Eq. (8) accordingly. The unknown Qp is 3  3 rotation matrix corresponding to a unit vector rp along the rotation axis and a rotation angle hp , such that

rp1 rp3 ð1  cos hp Þ þ rp2 sin hp

3

7 rp2 rp3 ð1  cos hp Þ  rp1 sin hp 7 5

ð9Þ

cos hp þ r2p3 ð1  cos hp Þ

Thus, there are 4 unknowns for each rotation matrix Qp , and 3 unknowns for each vector bq . These give 24 unknowns in total for Eq. (8). Here, steepest descent

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Fig. 12. A rank-2 laminate twin pattern where four regions meet at a point.

Table 2. The value of C for the seven twin families in the material of Ti50.3Ni48.2Fe1.5 shape memory alloy. Family

C

f1234g1 f1234g2 f1221g1 f1221g2 f1213g1 f1213g2 f1213g3

0 0.0206 0.0294 0.0147 0.0103 0.0103 0.0181

method is used to seek the solution that minimizes the functional C, i.e. total mean square error of compatibility equations. The resulting Qp can make the four regions in the crystal align at their best fit state. The unit cell alignment of each twin family can then be generated. This provides useful information about the composition and the incompatibilities of the trigonal microstructures. 3.2. Results and microstructure analysis We now apply the numerical method to study the bulk Ti50.3Ni48.2Fe1.5 SMA. The lattice parameter of the parent phase is aP ¼ 0:3019 nm; those of the trigonal martensite R-phase are aH ¼ 0:7355 nm and cH ¼ 0:5283 nm in the hexagonal axes [32,33]. Then, the transformation matrix of each of four variants in Eq. (6) can be determined [33]. Note that the parameters used here are for illustration purposes, the current method is sufficiently general for different compositions of SMAs.

201

The value of C for each twin family is determined and listed in Table 2. Pattern of f1234g1 , as mentioned above, is the only family that satisfies nonlinear compatibility equation, Eq. (7), thus, having C ¼ 0. This indicates the trigonal martensite unit cells in the pattern are perfectly matched with proper rotations Qp ðp ¼ 1 . . . 3Þ that minimize Eq. (8). Fig. 13(a) and (b) show the schematic unit cell alignment of “1234”, which is one of the members of family f1234g1 , in different perspectives. Where the distortion of unit cells from their parent phase due to martensitic transformation is exaggerated. Also, for illustration purposes, the unit cell alignment drawn here is corresponding to a part of the microstructure only. It is worth to note that the unit cells that meet at the horizontal interface n1 are surface-contacted, while those that meet at the oblique n2 and n3 interfaces are contacted along the edges of the unit cells. The twinning mode of the former is similar to the so-called modulation twin reported in the literature of magnetic shape memory alloys [36,37] (see Appendix A for more detail); the latter results in regions which do not belong to variants 1–4, and thus no color is used in the figure. Similar contact phenomena of the unit cells can also be found in the other families of twin patterns. Another possible arrangement with all four variants present “1234” is checkerboard pattern. The C value of the pattern in Ti50.3Ni48.2Fe1.5 is 0.0206, see Table 2. This indicates that the unit cells in pattern “1234” cannot fit together at the stress-free state. Fig. 13(c) shows the corresponding unit cell alignment with crossed gaps at every junction of four regions of variants. Shape memory alloys are known to adopt low energy, compatible microstructures, and thus the crystals tend to avoid such incompatibility of the gaps. Further compressive straining must be applied to achieve the formation of the checkerboard pattern. This may imply that family f1234g2 is likely observed in the crystals subjected to compressive stress or in the confined area of the materials. Family f1221g1 gives the greatest value of C among the seven families, resulting in the microstructure with theoretically overlapping unit cells and 3-dimensional complex gaps in the material. The unit cell alignment is so complex, and thus not shown here. It is expected that the family

Fig. 13. Schematic unit cell alignment of (a, b) “1234” (family f1234g1 ), (c) “1234” (family f1234g2 ), (d, e) “1221” (family f1221g2 ), (f) “1213”’ (family f1213g1 ) and (g) “1213” (family f1213g2 ) at the relaxed, stress-free state, where the distortion of unit cells from their parent phase due to martensitic transformation are exaggerated.

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is at a relatively high energy state and is less likely observed in experiments. By contrast, the alternative arrangement of variants 1 and 2, family f1221g2 showing toothbrush pattern, has C ¼ 0:0147, which is the half of the value of family f1221g1 . The unit cell alignment of “1221” belonging to family f1221g2 at stress-free state is shown in Fig. 13(d) and (e). It is interesting to note that, the unit cells rotated according to the best set of solution of Qp ðp ¼ 1 . . . 3Þ in Eq. (8) result in relatively wider gaps between regions compared to the other patterns. This can be the reason why the observed toothbrush patterns can have nonparallel laminate twins, as shown in Fig. 8. In the top view of the unit cell alignment, Fig. 13(e), when the gap between edges l1 and l2 closed by compressive loading, the projection of the resulting interface may vary between l1 and l2 depending on the loading conditions. Similarly, edges l3 and l4 may meet at an interface along any direction between l3 and l4 . Thus, different orientations of interfaces can occur in family f1221g2 toothbrush pattern under complex compressive loading. The herringbone of families f1213g1 ; f1213g2 give the lowest non-zero value of C ¼ 0:0103. Their unit cell alignments are shown in Fig. 13(f) and (g), respectively. Pattern “1213” belonging to family f1213g1 has gaps between variants 1 and 3 along the horizontal interfaces, and the unit cells are slightly mismatched at the oblique interfaces. The mismatched regions and gaps can be closed simply by applying a compressive stress. Pattern “1213” belonging to family f1213g2 , however, may need a complex loading condition to overcome its geometrical subtleties. Fig. 13(g) shows a stress-free, “1213” (family f1213g2 ) herringbone pattern revealing that there are two types of geometric defect along its midrib interfaces: open gaps and theoretically overlapped unit cells. This suggests that a crack-free, low-energy family f1213g2 microstructure can only be found in experiments if the part with gaps subjected to a compressive load, and the part of overlapped unit cell subjected to a tensile load. The loading condition appears difficult to be achieved. Thus, although family f1213g2 has a similar value of C with commonly found family f1213g1 , it has not been found in the specimens in the current study. It is worth to mention that, in the SMAs, stand alone f1213g2 pattern has not been found. However, roof type patterns, which are the combination of four members of family f1213g2 (see Fig. 10), have been reported in the literature. A possible explanation is that the symmetrical arrangement of the four members somehow makes the internal stress of each of members canceled each other out, and thus a metastable state is achieved. The issue is still under discussion and further studies are needed. Finally, family f1213g3 gives the most 3-dimensional microstructure with relatively higher value of C ¼ 0:0181. The resulting unit cell alignment, similar to family f1221g1 , has 3-dimensional complex gaps and theoretically overlapping regions, and thus, is not shown here. The pattern was also not found in our crystals as mentioned in Section 2.3. In summary, by examining the unit cell alignment, the following statements can be made. Classic herringbone patterns f1234g1 typically have no disclination at the junction of the variants, and thus is highly likely to be found in the trigonal SMAs. Family f1213g1 non-classic herringbone pattern, family f1221g2 toothbrush pattern and family f1234g2 checkerboard pattern have certain levels of disclination in the structures, however, their unit cell alignments

reveal that such disclination can be minimized by simple compressive loads. Thus, they can still be observed in the trigonal SMAs. Finally, the unit cell alignments of families f1221g1 ; f1213g2 and f1213g3 show that the crystals require complex loading conditions to minimize the defects of those patterns, so experimental observations and the related theoretical calculation results appear to be rare.

4. Conclusion In this study, 3-dimensional geometrically linear compatibility theory is used to classify compatible trigonal patterns in shape memory alloys. The seven families of periodic compatible rank-2 twins are defined, giving several well-known structures, such as herringbone patterns, and some new structures which have not been addressed in the literature, such as checkerboard and toothbrush patterns. Examples from experimental observations confirm their existence. The origin of the seven trigonal patterns, i.e. “martensite twin pattern + austenite phase” with compatible austenite–martensite (A–M) interfaces, is also studied and verified by SEM images. A numerical method based on non-linear compatibility theory is then developed to obtain the best fit of the unit cell alignment for all the seven families, that minimizes the defect of disclination in the microstructures. These microstructures are analyzed and the geometrical subtleties of certain special patterns, such as roof type, checkerboard, nonparallel toothbrush patterns, are explained. Major advantages of the current methods are the rapidity and completeness of finding compatible twin patterns. The resulting classification is of direct use in interpreting observed twin patterns and, moreover, in identifying phase/variants in microstructures. The methods provide a basis to search for engineered microstructure configurations with optimized properties in shape memory alloys, and are ready to be extended to other crystal systems or higher rank laminates. Acknowledgments The authors wish to acknowledge the support of Ministry of Science and Technology (MOST) Taiwan, Grant No. MOST 102–2218-E-009–019-MY2 and National Taiwan University (NTU), Grant No. NTU 103R891803 and 103R891805 (The NTU Excellence in Research Program), and thank Tzu-Cheng Liu and Jo-Fan Wu for their assistance on generating the figures of the microstructures.

Appendix A. Twining planes generated in the current study In the current study, the notation of the twin patterns is done based on the chosen numbering of the variants, which is suitable for the computation of the tree diagrams. This appendix correlates the interfaces determined in the current work with the commonly accepted terminologies of twin. By using the compatibility theory introduced in Section 2.1, the possible solutions of the interface separating the trigonal R-phase variants 1, 2 are determined and listed in Table 3. Where the subscript p indicates that the plane normals are indexed with the coordinate of parent phase. Solutions for all the other pairs of variants can be related by the rigid rotations. Thus, the results cover all the twin

N.-T. Tsou et al. / Acta Materialia 89 (2015) 193–204 Table 3. The twinning system of the compatible rank-1 laminates (a pair of the trigonal martensite variant).

203

Table 5. The averagely compatible A–M interfaces for the microstructures Austenite + “1234”, “1221” and “1213”.

Variants

n

Twinning type

Microstructure

Volume fraction

nA

“1”:“2”

½1; 0; 0p p1ffiffi ½0; 1; 1 p

Modulation Compound

“A”:“1234”

l1 ¼ 0:5; l2 2 ½0; 1; ml2 – 0:5 l2 ¼ 0:5; l1 2 ½0; 1; l1 – 0:5 l1 ¼ 0:5; l2 ¼ 0:5

½0; 1; 0p ; ½0; 0; 1p

Arbitrary

“A”:“1221”

l1 ¼ 0:5; l2 2 ½0; 1 l2 ¼ 0:5; l1 2 ½0; 1

½0; 1; 0p ; ½0; 0; 1p ½0; 1; 0p ; ½0; 0; 1p

“A”:“1213”

l1 l1 l1 l1 l1

¼ 0:5; ¼ 0:5; ¼ 0:5; ¼ 0:5; ¼ 0:5;

l2 l2 l2 l2 l2

¼0 ¼ 0:125 ¼ 0:25 ¼ 0:375 ¼ 0:5

½0; 1; 0p ; ½1; 0; 0p ½0; 1; 0p ; ½0:99; 0; 0:14p ½0; 1; 0p ; ½0:95; 0; 0:32p ½0; 1; 0p ; ½0:86; 0; 0:51p ½0; 1; 0p ; p1ffiffi2 ½1; 0; 1p

l1 l1 l1 l1

¼ 0:5; ¼ 0:5; ¼ 0:5; ¼ 0:5;

l2 l2 l2 l2

¼ 0:625 ¼ 0:75 ¼ 0:875 ¼1

½0; 1; 0p ; ½0:51; 0; 0:86p ½0; 1; 0p ; ½0:32; 0; 0:95p ½0; 1; 0p ; ½0:14; 0; 0:99p ½0; 1; 0p ; ½0; 0; 1p

2

planes found in the seven families reported in Section 2.2. Note that all of them can be classified as compound twin planes. The unit cell alignment corresponding to each twinning type is illustrated in Fig. 14. Moreover, by examining the geometric relations between the twinned unit cells, one can further identify the “modulation twin” which is reported in the literature of magnetic shape memory alloys [20,36,37]. In this type of twin, the normal direction of the twinning plane is coplanar with the modulation direction of the two martensite variants. Where the modulation direction is the direction pointing along the shortest diagonal of the unit cell of the martensite variant. This results in an arrangement of two surface-contacted unit cells, i.e. the variants are different only in the modulation directions across the interface [20], as shown in Fig. 14(a). Thus, to emphasize such twinning type, we use the term of modulation and distinguish them clearly from the compound twin in Table 3. Next consider the twinning system of the rank-2 interfaces (i.e. macrotwin planes). The results for the pairs of twin “12”:“34”, “12”:“21” and “12”:“13”, which are found in the seven families, are shown in Table 4. Again, other rank-2 configurations can be obtained by the symmetry or rotation operations. The rank-2 interfaces are determined under the compatibility conditions (i)–(iii) provided in Section 2.1, which ensure no local incompatibility across the interfaces. In the case of “12”:“34”, the rank-2 interface ½0; 0; 1p between “12”:“34” is equivalent to the rank-1 interface “1”:“2”, since “1”:“3” and “2”:“4” are both relat-

Fig. 14. Schematic unit cell alignment of variants 1, 2 with (a) modulation twinning and (b) compound twinning. Where the distortion of unit cells from their parent phase due to martensitic transformation are exaggerated. Table 4. The twinning system of the exactly compatible rank-2 laminates (macrotwins). Variants

n

Twinning type

“12”:“34” “12”:“21”

½0; 0; 1p ½1; 0; 0p p1ffiffi ½0; 1; 1 p

Modulation Modulation Compound

“12”:“13”

2 p1ffiffi ½1; 2

0; 1p

½0; 1; 0p

Fictitious + Compound Fictitious + Modulation

½0; 1; 0p ; ½1; 0; 0p

ed to “1”:“2” by the pure rotations. Thus, it can be classified as the modulation twin plane. It is of interest to note that there are typically two possible averagely compatible rank-2 interfaces ½0; 0; 1p ; ½0; 1; 0p for “12”:“34”, if we treat the sub-laminate (rank-1 laminate twins) as a homogeneous medium. However, variants 1,3 and variants 2,4 are not compatible across ½0; 1; 0p . Thus, ½0; 1; 0p is eliminated by the exact compatibility conditions and not shown in the table. Next, the rank-2 interfaces of “12”:“21” are classified as the modulation and compound twins. The rank-2 interfaces of “12”:“13” consist of two types of rank-1 interfaces separating “1”:“1” and “2”:“3”. Where “1”:“1” gives infinity sets of solutions for the compatibility equation, forming the fictitious interfaces, i.e. arbitrary interfaces, while “2”:“3” forming the modulation and compound twin planes. The unit cell alignment of all the macrotwins listed in Table 4 can be found in the Fig. 13. In particular, the modulation macrotwins can be seen in Fig. 13(a, d, f). These trigonal rank-2 laminate twins (macrotwins) can also form averagely compatible A–M interfaces with their parent (austenite) phase, provided that the trigonal twins have their average transformation strain m with a zero middle eigenvalue k2 ¼ 0. Thus, the volume fraction of the martensite variants can alter the orientation of the A– M interface normal [38]. Here we summarize the A–M interface normal nA corresponding to the volume fraction ratio of variants l1 ; l2 for the microstructures “1234”, “1221” and “1213” in Table 5. Other A–M configurations can be obtained by the symmetry or rotation operations. Microstructure “A”:“1234” has vertical or horizontal A– M interfaces when either l1 or l2 equals 0.5. In the special case of l1 ¼ l2 ¼ 0:5, the average transformation strain m ¼ 0, giving arbitrary solutions for nA . Apart from the sets of volume fraction ratio mentioned above, other sets typically give no solution. Microstructure “A”:“1221” also requires one of the volume fraction ratio equals to 0.5. Finally, microstructure “A”:“1213” requires l1 ¼ 0:5 to have compatible solutions for nA . With this condition, one of the solutions nA is ½0; 1; 0p , and the orientation of the other solution rotates about y p -axis from ½1; 0; 0p to ½0; 0; 1p as l2 changes from 0 to 1. It is worth noting that, in the case of l1 ¼ 0:5; l2 ¼ 1 , the solutions nA ¼ ½0; 1; 0p

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or ½0; 0; 1p are identical to those of “A”:“1221”. This is because “A”:“1213” is degenerated into “A”:“12”, and its composition of the variants in the microstructures is the same as that of “A”:“1221”.

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