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Interfacial defects in Ti–Nb shape memory alloys
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Acta Materialia 56 (2008) 3088–3097 www.elsevier.com/locate/actamat
Interfacial defects in Ti–Nb shape memory alloys Y.W. Chai a,*, H.Y. Kim a, H. Hosoda b, S. Miyazaki a,* b
a Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305 8573, Japan Precision and Intelligence Laboratory, Tokyo Institute of Technology, Yokohama 226 8503, Japan
Received 31 December 2007; received in revised form 26 February 2008; accepted 28 February 2008 Available online 9 April 2008
Abstract The structure of the martensite (a00 )/martensite (a00 ) and parent (b)/martensite (a00 ) interfaces in a series of binary Ti–Nb alloys with Nb content ranging from 20 to 24 at.% was investigated. Both the a00 /a00 and b/a00 interfaces comprised a series of f2 1 1gb ==f1 1 0ga00 terraces and steps when viewed edge-on (close to ½0 0 1a00 ). Interfacial defects, particularly the transformation disconnections (b, h) superimposed along the terrace–step interface structure, have been identified. They were responsible for accommodating most of the transformation strain along the a00 /a00 and b/a00 interfaces. Using the parameters b and h, the prediction of the a00 habit plane based on the topological model agreed well with the prediction from the phenomenological theory as well with experimental observations. The a00 habit plane in Ti–20Nb alloy is close to f7 5 5gb and moves towards f4 3 3gb in Ti–22Nb and Ti–24Nb alloys. The remaining transformation strain along the b/a00 interface was found to be accommodated by Type 1 twinning on ð1 1 1Þa00 with Burgers vector bLIS close to h 2 1 3ia00 ð h1 1 2ib Þ. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ti–Nb; Shape memory alloys; Transformation disconnections; Twinning dislocations; Strain accommodation
1. Introduction Ti-based alloys consisting of various non-toxic elements, such as Nb, Zr, Ta, Sn, Mo and Pt, which may be suitable candidates for biomedical applications, have been extensively investigated recently [1–7]. More excitingly, some of these alloys were found to exhibit shape memory effect (SME) and superelasticity (SE). In Ti–Nb, Baker [8] first reported that SME was observed in Ti–35 wt.% (or 21.7 at.%) Nb alloy. Recently, a systematic study by Kim et al. [9] also observed SME and SE in Ti–(20–27) at.% Nb binary alloys. SME and SE were also reported in other Ti–Nb-based ternary and even quaternary alloys, such as Ti–Nb–Al [10], Ti–Nb–O [11], Ti–Nb–Ta [12,13], Ti–Nb– Pd [14], Ti–Nb–Zr [15] and Ti–Nb–Zr–Ta [16]. By combining either the SME or SE with other useful properties, such as biocompatibility and low Young’s modulus, these alloys *
Corresponding authors. Tel.:/fax: +81 29 853 5283. E-mail addresses: [email protected] (Y.W. Chai), miyazaki@ ims.tsukuba.ac.jp (S. Miyazaki).
may provide an additional advantage over some conventional biomaterials, e.g. pure Ti, Ti–6Al–4V alloy, stainless steel and Co-based alloys [1,17–19]. The SME and SE in shape memory alloys are closely related to the martensitic transformation (MT) and its reversion between the parent and martensite phases [20]. Transformation disconnections (formerly known as transformation dislocations) have been reported to play an important role in the transformation process [21]. Furthermore, Pond et al. [22,23] also pointed out that these interfacial defects are capable of minimizing the transformation strain arising from the MT. For the case of the b–a00 MT in Ti–Nb-based shape memory alloys, little has been reported on the transformation disconnections and their effects on the shape memory properties of the alloys. Therefore, the aim of this paper is to investigate the nature of the b/a00 interface structures and interfacial defects, particularly transformation disconnections in Ti–Nb-based shape memory alloys. Lattice invariant shear, i.e. twinning or slip, which have been widely reported to be involved in the transformation strain minimization process, will also be discussed briefly. Experimental observations
1359-6454/$34.00 Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2008.02.045
Y.W. Chai et al. / Acta Materialia 56 (2008) 3088–3097
have been carried out using transmission electron microscopy (TEM) and high-resolution TEM (HRTEM). The experimental observations on the martensite (a00 ) crystallography will be discussed in conjunction with predictions from the topological model (TM) [22,23]. 2. Experimental procedure Binary Ti–Nb alloys with Nb contents of 20 at.% (Ms 493 K), 22 at.% (Ms 413 K) and 24 at.% (Ms 338 K), which exhibit the SME, have been chosen for this study. Hereafter, the composition of the alloys is represented in at.% unless stated otherwise. The alloys were prepared by the argon arc melting method. In the melting process, the alloy buttons were melted for 3 min six times, and flipped over each time before melting. In order to minimize segregation, all of the alloy buttons were then homogenized in vacuum at 1273 K for 2 h. After that, they were cold-rolled up to 98.5% of thickness reduction. Specimens for TEM observation were punched out from the cold-rolled strips, chemically cleaned in a solution containing HF, HNO3 and H2O (1:4:5) at 30 °C, sealed in a quartz tube with a 25 Torr partial pressure of high-purity argon gas, and subsequently solution treated at 1173 K for 30 min. The specimens were then water quenched by breaking the quartz tube. The quenching rate has been estimated to about 200–300 K s1. The final specimen thinning was carried out using a twin-jet electropolishing machine (Tenupol-3, Struer) with a solution containing HF, H2SO4 and methanol (2:5:93) at temperatures between 50 and 40 °C. TEM and HRTEM observation was conducted using a JEOL 2010F microscope operated at 200 kV. The lattice parameters of the parent (b, body-centered cubic) and martensite (a00 , orthorhombic) phases in the alloys are shown in Table 1. Hereafter, the parent (b) and martensite (a00 ) phases will be represented by their symbols b and a00 , respectively. 3. Experimental observations 3.1. Martensite (a00 ) interfaces structure TEM micrographs obtained from the Ti–20Nb, Ti–22Nb and Ti–24Nb alloys are shown in Fig. 1a, d and g, respectively. The a00 plates in the figures are viewed edge-on. In Table 1 The lattice parameters of the b (bcc) and a00 (orthorhombic) phases in Ti– 20 at.% Nb, Ti–22 at.% Nb and Ti–24 at.% Nb alloys [9] Alloys (at.%)
Ti–20Nb Ti–22Nb Ti–24Nb
Lattice parameters (nm)
Ms (K)
Parent (b)
Martensite (a00 )
ab
aa00
ba00
ca00
0.328505 0.328537 0.328569
0.31257 0.31518 0.31785
0.48704 0.48431 0.48120
0.46456 0.46427 0.46355
493 413 338
The corresponding martensite start (Ms) temperatures are also indicated.
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Fig. 1a, no retained b phase is observed in the microstructure because the Ms for the Ti–20Nb alloy is too high (493 K). The edge-on a00 plate (labeled a00V2 ) in the figure is shown as being interfaced with an a00 plate of a different variant (labeled a00V1 ). The corresponding orientations of a00V2 and a00V1 plates are viewed parallel to ½0 0 1a00 :V2 and ½0 1 0a00 :V1 , respectively, as indicated by the selected area diffraction (SAD) pattern in Fig. 1b and their indexes in Fig. 1c. For the Ti–22Nb and Ti–24Nb alloys as seen in Fig. 1d and 1g, an edge-on a00 plate and a retained b phase are observed parallel to ½0 0 1a00 and ½0 1 1b , respectively. This is clearly confirmed by the SAD patterns (Fig. 1e and h) and their indexes (Fig. 1f and i). The orientation of the a00 interface, as indicated by the dotted line with plane normal p1 was measured as 10.3 ± 0.5° away from ð1 1 0Þa00 :V2 =ð1 0 1Þa00 :V1 in the Ti–20Nb alloy (see Fig. 1a), 11.0 ± 0.3° away from the ð1 1 0Þa00 =ð2 1 1Þb in the Ti–22Nb alloy (see Fig. 1d), and 11.7 ± 0.3° away from the ð1 1 0Þa00 =ð2 1 1Þb in the Ti–24Nb alloys (see Fig. 1g). By using the HRTEM technique, the interface structure has been further investigated. Fig. 2a and b show the high-resolution image and the inverse fast Fourier transformation (FFT) images, respectively, of the a00V2 =a00V1 interface structure in the Ti–20Nb alloy, whereas Fig. 2c and d reveal the high-resolution images of the b/a00 interfaces in Ti– 22Nb and Ti–24Nb alloys, respectively. Fig. 2a–d indicate that both of the a00V2 =a00V1 and b/a00 interfaces are formed by a series of terraces and steps (indicated by dotted lines) at the atomic scale. The terrace planes of the a00V2 =a00V1 interface is found to be parallel to ð1 1 0Þa00 :V2 =ð1 0 1Þa00 :V1 (see Fig. 2a). They correspond to the ð2 1 1Þb plane when expressed with respect to the b phase. Average length of the terraces was estimated to be 1.6 nm. The steps along the a00V2 =a00V1 interface consist of one or two ð1 1 0Þa00 :V2 plane spacings. It is noticed that a relatively large strain contrast was seen at the interface. The inverse FFT image in Fig. 2b reveals the presence of dislocations. Some of the dislocations with opposite signs will cancel out each other, whereas those remaining in the a00 phase will serve as misfit dislocations to accommodate misfit strain resulting from the difference between the ð0 2 0Þa00 :V2 (0.24 nm) and the ð0 0 2Þa00 :V1 (0.23 nm) planes. Hence, the strain contrast is most probably caused by the mismatch between the ð0 2 0Þa00 :V2 and ð0 0 2Þa00 :V1 plane spacings. The ð0 0 2Þa00 :V1 lattice planes are seen to run continuously across the ð1 1 0Þa00 :V2 =ð1 0 1Þa00 :V1 terraces, suggesting that the a00V2 =a00V1 terraces are semicoherent. For the b/a00 interface in the Ti–22Nb and Ti– 24Nb alloys (see Fig. 2c and d), the terrace planes are parallel to ð2 1 1Þb =ð1 1 0Þa00 . Their average lengths are about 1.5 and 1.3 nm, respectively. It can be seen that the terrace planes are terminated by steps with a height of one or two ð1 1 0Þa00 plane spacings. In Fig. 2c and d, the ð0 2 0Þa00 lattice planes are seen running continuously across the b/a00 terraces, indicating that the ð2 1 1Þb =ð1 1 0Þa00 terrace planes are coherent. No misfit dislocation was observed along the b/a00 interfaces. The strain contrast near the b/a00 interfaces is caused by the coherent particles, namely the
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Fig. 1. TEM micrographs of (a), (d) and (g) revealing edge-on a00 plates that were observed in the Ti–(20,22,24)Nb alloys. The trace of the habit plane is indicated by dotted lines and the habit plane normal is labeled p1. (b) is ½0 0 1a00 :V2 =½0 1 0a00 :V1 SAD pattern taken from the a00V2 =a00V1 interface in (a). (c) is the index of the SAD pattern in (b). (e) and (h) are ½0 1 1b =½0 0 1a00 SAD patterns taken from the b/a00 interfaces in (d) and (g), respectively. Their corresponding indexes are shown in (f) and (i).
athermal omega (x) particles in the b phase. These ellipsoidal x particles (ranging in size from 3 to 8 nm long, and from 1 to 3 nm wide) in Fig. 2c and d, seem to be capable of suppressing the b–a00 MT, by impeding movement of the b/a00 interface. The interaction of these athermal x particles and the a00 phase will be reported in more detail in a future publication and hence will not be discussed any further here.
1 and of about 0.2° was observed between the ½0 1 b ½0 0 1a00 directions. The ð2 1 1Þb and ð1 1 0Þa00 planes in the b and a00 phases also exhibited a small deviation angle, 0.2°. The OR between the b and a00 phases was found to follow the Burgers OR as often observed in hexagonal close-packed (hcp) alloys [24].
3.2. Orientation relationship (OR)
During the TEM observation, it was noticed that the internal microstructure of the a00 phase, particularly the internal twins, changes with respect to the Nb content, as shown in Fig. 3a, c and d, respectively. In the Ti–20Nb alloy, few or no internal twins were observed in many of the a00 plates (see Fig. 3a). Occasionally, some traces of dislocations parallel to ð1 1 1Þa00 were observed within some of the a00 plates. However, as Nb addition increased to 22 at.%, fine internal twins (ranging from a few nm to tens of nm wide), as indicated by the arrows in Fig. 3b, are seen in some of the a00 plates. The ½1 0 1a00 SAD pattern and its index in Fig. 3d indicate that the internal twins are of Type
A set of ORs relating the b, a00 and the athermal x phases were obtained from the TEM and HRTEM observations in Figs. 1 and 2. They are shown as ½0 1 1 ==½0 0 1 00 ==½1 1 20 ; b
a
x
½1 1 1b ==½1 1 0a00 ==½0 0 0 1x ; ð2 1 1Þb ==ð1 1 0Þa00 ==ð1 1 0 0Þx ; where the b, a00 and athermal x phases are represented by the corresponding subscript symbols. A small deviation
3.3. Type 1 f1 1 1ga00 internal twinning
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Fig. 2. (a) indicating the HRTEM image of the a00V2 =a00V1 interface structure in Ti–20Nb alloy. (b) is an inverse FFT image revealing the presence of misfit dislocations along the a00V2 =a00V1 interface. (c) and (d) are the HRTEM images showing the b/a00 interface structure in the Ti–(22,24)Nb alloys, respectively. Athermal omega particles (x) are observed in the b phase.
1 twinning on ð1 1 1Þa00 . These fine ð1 1 1Þa00 internal twins are rather random, short (ranging from 50 to 150 nm long) and often do not extend from one side of the a00 plate to the other. In the Ti–24Nb alloy, the a00 phase with more defined and thicker ð1 1 1Þa00 internal twins (up to 60 nm wide) were observed, as indicated by the arrows in Fig. 3c. 4. Discussion 4.1. Interface structure As seen in the preceding section, the HRTEM micrographs revealed that both a00V2 =a00V1 and b/a00 interfaces exhibit a terraces/steps structure when viewed edge-on along the ½0 0 1a00 :V2 =½0 1 0a00 :V1 direction in the Ti–20Nb alloy (see Fig. 2a) and along the ½0 1 1b =½0 0 1a00 direction in the Ti– 22Nb and Ti–24Nb alloys (see Fig. 2b and c). For the b/a00 interface, the terraces/steps structure is considered as a typical relaxed martensitic interface structure, where the interface (or habit plane) is often irrational and consists a series of rational terrace planes parallel to ð2 1 1Þb =ð1 1 0Þa00 and steps. Such a relaxed interface structure has also been
observed in many other alloys in addition to Ti–Nb alloys [22,25–29]. During the interface relaxation process, two arrays of interfacial defects, i.e. transformation disconnections (b, h) and lattice-invariant shear (LIS) (bLIS, 0), are involved in order to minimize the transformation strain on the terrace plane [22,23]. A schematic diagram illustrating the two interfacial defects at a b/a00 terraces/steps interface is shown in Fig. 4. The transformation disconnection and LIS are represented by their line directions, nDiscon and nLIS, and their Burgers vectors are shown by their components, LIS b = [0, by, bz] and bLIS ¼ ½bLIS x ; 0; bz , expressed with respect to the reference state frame (XYZ). The reference state frame is defined from the experimentally observed b–a00 OR, i.e. X : ½0 1 1b =½0 0 1a00 , Y : ½1 1 1b =½1 1 0a00 and Z : ð2 1 1Þb = ð1 1 0Þa00 . For simplicity, the transformation strain is represented by components exx (parallel to X), and eyy (parallel to Y). Based on the configuration of the interfacial defects as shown in Fig. 4, one can see that their edge Burgers vector components, i.e. by and bLID , are capable of relieving the x strain components, eyy and exx on the ð2 1 1Þb =ð1 1 0Þa00 terrace plane, respectively. exx and eyy can be deduced using the following equations:
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Fig. 3. (a), (b) and (c) are TEM images showing the variation of martensite microstructure including internal twins in the Ti–(20,22,24)Nb alloys. (d) is ½1 0 1a00 SAD pattern and the index that were obtained from the circle in (b), and indicating that the internal twins are of Type 1 twinning on ð1 1 1Þa00 plane.
Fig. 4. A schematic diagram illustrating the transformation strain components, exx and eyy, the transformation disconnections (b = [by, bz], h) LIS 00 and the LIS ðbLIS ¼ ½bLIS terrace interface when x , bz ], 0) in a b/a expressed with respect to the reference state frame, XYZ. The line directions nDiscon and nLIS of the transformation disconnections and the LIS are also indicated in the diagram.
pffiffiffi ca00 2 ab exx ¼ 1 pffiffiffi ; 2 ab þ ca00 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi aa00 þ ba00 3 ab eyy ¼ 1 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 3 ab þ aa00 þ ba00 2
Fig. 5. Variation of the transformation strains, exx and eyy, on the ð2 1 1Þb =ð1 1 0Þa00 terrace interface with respect to the Nb content.
ð1Þ ð2Þ
Using lattice parameters of Ti–(20,22,24)Nb alloys as shown in Table 1, exx and eyy have been calculated as shown in Fig. 5. It should be noted that eyy has a much larger magnitude compared with exx. This suggests that the transformation strain at the b/a00 interface is dominated mostly by strain component eyy. In other words, the trans-
formation disconnections responsible for relieving the strain component eyy are anticipated to play a more significant role in the interfacial strain minimization process compared with the LIS. 4.2. Characterization of the transformation disconnections A dichromatic complex in Fig. 6 illustrates a transformation disconnection (b, h) that is associated with a b/a00 interface in a reference state frame (XYZ). The b and a00
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Fig. 7. Variation of by, bz and h of the transformation disconnections as a function of Nb content.
Fig. 6. (a) Dichromatic complex illustrating a transformation disconnection superimposed on the reference state b/a00 interface. The enlargement on the right indicates the Burgers vector components, by and bz, and the overlap step height (h) of the transformation disconnection. (b) Schematic diagram showing 2 atoms per transforming volume in the b and a00 phases.
atomic sites are represented by white and black squares, respectively. The line direction of the transformation disconnection, nDiscon, is parallel to X : ½0 1 1b =½0 0 1a00 . The Burgers vector, b, of the transformation disconnection, which is responsible for the atomic shearing process during the b–a00 MT, can be determined using the following equation [28]: b ¼ ½0; by ; bz ¼ tðbÞ Ptða00 Þ;
ð3Þ
where P is the matrix of the coordinate transformation that relates the b and a00 crystals (see Appendix A). The vectors t(b) and t(a00 ) are translation vectors in b and a00 crystals, and are equal to ½ 1 0 0b and ½ 1 0 0a00 respectively. The enlargement on the right of Fig. 6 shows b and its two components, by and bz, with respect to the XYZ frame. The overlap step height h is equal to hða00 Þ ¼ d ð110Þa00 or hðbÞ ¼ 2d ð211Þb , with the smaller one being chosen here. For Ti–(20,22,24)Nb alloys, by, bz and h of the transformation disconnection have been deduced and their values are shown as a function of alloy composition in Fig. 7. Clearly, b (=[0, by, bz]) and h are seen to vary as the alloy composition changes, i.e. by and bz decrease as the Nb content changes from 20 to 24 at.%. The former has a much larger magnitude when compared with the latter. On the other hand, h increases with increasing Nb content from 20 to 24 at.%. It is found that the values of b and h are relatively small, and this indicates that the intrinsic mobility of the transformation disconnection in the three alloys is high. Moreover, the transformation disconnection identified above fulfills the criteria for feasible military transforma-
tions [29]. For instance, the number of atoms (same atom species) for a transforming volume is the same between the b and a00 phases, i.e. 2 atoms per each transforming volume (see Fig. 6b). This implies that the transformation disconnection as identified in Fig. 6 and observed in Fig. 2 can move in a conservative and glissile manner on the Z : ð2 1 1Þb =ð1 1 0Þa00 terrace plane along the direction parallel to Y : ½1 1 1b =½1 1 0a00 . Table 2 also includes a rigid body rotation, / (around X : ½0 1 1b =½0 0 1a00 Þ, that is introduced by the bz (see Eq. (B3) in Appendix B). The / in Ti–20Nb is predicted to about 0.20°, and decreases to 0.17° and 0.13° in Ti–(22,24)Nb alloys, respectively. 4.3. Transformation strain accommodation by transformation disconnections At an equilibrium state, an array of the transformation disconnections with an appropriate spacing, k, would relieve eyy effectively and leave no long-range stress field at the b/a00 interface. The average spacing predicted theoretically (see Eq. (B4) in Appendix B) and those observed experimentally (see Fig. 2) in the Ti–(20,22,24)Nb alloys are summarized in Fig. 8. Clearly, they are consistent with each other (deviating by about ±0.1 nm), suggesting that the transformation strain along the b/a00 terrace interface is reduced by an array of transformation disconnections with an average spacing k. The habit plane inclinations (xT), designated as the inclination angle between the habit plane and the ð2 1 1Þb =ð1 1 0Þa00 terrace plane, obtained
Table 2 The transformation disconnection parameters at the b/a00 terrace interface in Ti–(20,22,24)Nb alloys Alloy (at.%)
Topological parameters of disconnection Burgers vector, b
Ti–20Nb Ti–22Nb Ti–24Nb
bx (nm)
by (nm)
bz (nm)
0 0 0
0.0237 0.0204 0.0169
0.0052 0.0041 0.0031
jbj (nm)
h (nm)
/ (°)
0.0243 0.0208 0.0172
0.2631 0.2642 0.2652
0.20 0.17 0.13
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Fig. 8. Nb content dependence of the average spacing of the transformation disconnection, k, and the habit plane inclination, xT, obtained from theoretical prediction (TM model) and experiment.
theoretically (see Eq. (B1) in Appendix B) and observed experimentally for the three alloys, are also summarized in Fig. 8. Close agreement in xT is seen between the predicted and experimentally observed values. xT is clearly dependent on the alloy composition. As the Nb content increases from 20 to 24 at.%, the transformation strain, particularly eyy, along the b/a00 interface decreases (see Fig. 5). Hence, in order to accommodate the strain more effectively, the parameters, by, bz, h and k of the transformation disconnections are anticipated to vary accordingly (see Figs. 7 and 8). Interestingly, it is noticed that the experimentally observed a00V2 =a00V1 interface in Ti–20Nb alloy (see Fig. 1a and 2a), although it is not a b/a00 habit plane, agrees rather well with the predicted b/a00 habit plane orientation (see Fig. 8). This could be due to the fact that the presence of misfit dislocations along the a00V2 =a00V1 interface has almost accommodated the extra misfit strain resulting from the difference between the lattice planes
ð0 0 2Þa00 :V1 and ð0 2 0Þa00 :V2 . Therefore, the parameters of the transformation disconnections, e.g. by, bz, h and k, may not rearrange significantly so that the habit plane orientation will not be altered. Owing to this, the predicted xT using the parameters of the transformation disconnections for the b=a00V2 interface is close to the xT experimentally observed from the a00V2 =a00V1 interface. Based on the values xT in Fig. 8, the habit planes expressed with respect to the b phase for Ti– (20,22,24)Nb alloys, both observed experimentally and predicted using the TM model, have been deduced and plotted in a [0 0 1]b stereogram in Fig. 9. For comparison, the phenomenological theory of martensite crystallography (PTMC) [30–32] was applied to predict the habit plane for the alloys, and these planes are also included in Fig. 9. No significant difference was observed between the a00 habit planes predicted based on the TM and the PTMC (less than the experimental error of ±0.5°). Moreover, the predicted a00 habit planes (from both models) agree well with experimental observation. In summary, Fig. 9 shows that the a00 habit plane is very close to ð7 5 5Þb (deviation less than 0.1°) in Ti–20Nb, and shifts towards ð4 3 3Þb as the Nb content increases from 20 to 24 at.%. 4.4. Characterization of Type 1 ð1 1 1Þa00 twinning dislocation Type 1 ð1 1 0 1Þa0 twins have been considered as the LIS in the b–a0 (hcp) MT [24,33]. In the case of b–a00 MT, it corresponds to Type 1 ð1 1 1Þa00 twins. In Fig. 3, the microstructure of the ð1 1 1Þa00 internal twins is clearly related to the alloy composition, which in turn depends on the transformation strain, exx. Such microstructural variation of ð1 1 1Þa00 ð ð1 1 0 1Þa0 Þ twins with respect to exx (or alloy composition) has been reported in Refs. [24,33]. Recently, Inamura et al. [34] have also reported a similar variation of microstructure in the b–a00 MT in Ti–Nb–Al alloys. The ð1 1 1Þa00 twinning dislocations (bt1) can be characterized based on the dichromatic complex in Fig. 10 [35]. The
Fig. 9. A ½0 0 1b stereogram indicating the a00 habit planes in the Ti–(20,22,24)Nb alloys. They were obtained from the topological model (TM), the phenomenological theory of martensite crystallography (PTMC) and experimental observation.
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Fig. 10. Dichromatic complex for the Type 1 ð1 1 1Þa00 twin. Two crystallographically equivalent Burgers vectors, bt1 and bt2, are indicated. Orthorhombic unit cells are outlined in rectangular dotted lines. The white and black symbols represent the matrix and twin of the a00 crystals, respectively.
Table 3 Type 1 ð1 1 1Þa00 twinning dislocation parameters for Ti–(20,22,24)Nb alloys Alloys (at.%)
Burgers vector, bt1
jbt1 j (nm)
ht ¼ 2dð1 1 1Þa00 ðnmÞ
Ti–20Nb Ti–22Nb Ti–24Nb
½0:4119 0:3600 0:0519a00 ½0:4243 0:3795 0:0449a00 ½0:4459 0:4092 0:0368a00
0.2189 0.2283 0.2432
0.4578 0.4592 0.4604
white and black symbols represent the matrix and twin of the a00 crystals, respectively, and are viewed parallel to h1 1 0ia00 . The Burgers vector bt1 is equal to the difference between vectors tm1 ¼ ½1 0 1a00 and tt1 ¼ 1=2½3 1 0a00 (expressed with respect to the a00 matrix crystal). The vectors correspond to 1=3½2 1 1 3a0 and ½1 0 1 0a0 in the a0 (hcp) crystal, respectively. For Ti–(20,22,24)Nb alloys, bt1 is shown in Table 3. When expresses with respect to the reference state frame (XYZ), it can be seen that bt1 has a strong screw component, i.e. bLIS (parallel to Y : ½ 1 1 1b =½ 1 1 0a00 ), which does not y accommodate the misfit strain, and will lead to a long-range stress field at the b/a00 interface. However, it is postulated that other twinning dislocations with Burgers vectors bt2 (i.e. derived from the difference between vectors tm2 ¼ 1=2½3 1 0a00 and tt2 ¼ ½1 0 1a00 , see Fig. 10) may be also generated simultaneously. The bt2 contains an opposite screw component, bLIS (parallel to Y : ½1 1 1b =½1 1 0a00 ). A y
pair of such twinning dislocations with bt1 and bt2 leads to the elimination of the screw components and results in a Burgers vector (bLIS) with only an edge component bLIS and a x LIS LIS normal component bLIS ð¼ ½bLIS z . The b x ; 0; bz Þ for each alloy is shown in Table 4 and is found to be close to ½2 1 3a00 . This direction corresponds to ½1 1 2b and ½ 1 1 0 2a0 in the b and a0 (hcp) crystals, respectively. By assuming bLIS accommodate exx completely, the average spacing x between pairs of the twinning dislocations, kLIS ð¼ bLIS x =exx Þ, has been estimated for each alloy as shown in Table 4. Note that kLIS decreases rapidly from about 962 nm in Ti–20Nb to about 55 and 14 nm in Ti–(22,24)Nb alloys, respectively. The component bLIS will introduce a rigid body rotation z LIS /LIS ð¼ 2 sinðbLIS =2k ÞÞ of the a00 phase with respect to the z reference state frame XYZ [23]. The /LIS (around Y : ½1 1 1b =½1 1 0a00 Þ for each alloy is shown in Table 4. Note that /LIS increases, while kLIS decreases with Nb content varying from 20 to 24 at.%. The negative sign of /LIS represents anti-clockwise rotation around Y : ½1 1 1b =½ 1 1 0a00 . 4.5. Transformation strain accommodation by LIS: ð1 1 1Þa00 twinning dislocations The edge component bLIS of ð1 1 1Þa00 twins ðbLIS ¼ x is responsible for relieving the transformation strain component exx. In Ti–20Nb, exx is very small LIS ½bLIS x ; 0; bz ])
Table 4 The resultant of twinning dislocation parameters as the LIS for b–a00 MT in Ti–(20,22,24)Nb alloys Alloys (at.%)
bLIS
j bLIS j (nm)
bLIS (nm) x
bLIS (nm) z
kLIS (nm)
/LIS (°)
Ti–20Nb Ti–22Nb Ti–24Nb
½0:0734 0:0303 0:1038a00 ½0:0635 0:0262 0:0898a00 ½0:0520 0:0214 0:0735a00
0.0554 0.0480 0.0392
0.0482 0.0417 0.0341
0.0273 0.0238 0.0194
961.6 55.1 14.2
0.00 0.02 0.08
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(0.01%). A large spacing kLIS of about 962 nm (see Table 4) between pairs of the twinning dislocations is sufficient to accommodate exx. Consequently, the low density of the twinning dislocations may lead to difficulty in forming internal twins in the a00 phase. This explains why internal twins are hardly seen in many of the a00 plates in Ti–20Nb alloy (see Fig. 3a). Instead, slip dislocations with traces parallel to ð1 1 1Þa00 are observed. As the Nb content increases from 20 to 22 at.%, exx increases to 0.08%. This leads to a significant reduction in kLIS to about 55 nm. Such spacing may just be enough for the density of the twinning dislocations to form internal twins. Hence, fine (ranging from 5 to 20 nm wide) but irregular ð1 1 1Þa00 internal twins were observed (see Fig. 3b). Further increase in Nb content to 24 at.% resulted in exx increasing to 0.24%. As a consequence, an increase in the density of twinning dislocations, i.e. kLIS (14 nm), leads to a formation of larger internal twins. Therefore, wider and more defined internal twins (up to 60 nm wide) are observed in the a00 phase in Ti–24Nb (see Fig. 3c). By comparing the / and /LIS (see Tables 2 and 4), it is seen that /LIS in the Ti–(20,22)Nb alloys is significantly smaller. This implies that the OR between the b and a00 phases in the two alloys is mainly due to /. On the other hand, a larger /LIS is predicted in Ti–24Nb alloy. Hence, the OR in Ti–24Nb has contributions from both / (0.13°) and /LIS (0.08°), where an overall rotation of 0:90 0:31 Þ is predicted. 0.16° (clockwise around ½0:30 b Nevertheless, in all three alloys, the predicted and experimentally observed ORs are quite consistent (0.2°). 5. Conclusion 1. The a00V2 =a00V1 and b/a00 interfaces relaxed into a series of ð2 1 1Þb =ð1 1 0Þa00 terraces and steps, when viewed edge-on along the ½0 1 1b =½0 0 1a00 direction. 2. The transformation disconnections (b,h), which could move in a conservative and glissile manner on the ð2 1 1Þb =ð1 1 0Þa00 terrace planes and along the Y: ½1 1 1b =½1 1 0a00 direction during the b–a00 MT, were identified along the a00V2 =a00V1 and b/a00 interfaces in Ti–(20,22,24)Nb alloys. The intrinsic mobility of the transformation disconnections (b,h) in the alloys is higher due to their relatively small magnitude of b and h. Most of the transformation strain at a b/a00 interface in Ti–(20,22,24)Nb alloys was accommodated by transformation disconnections (b,h). 3. No significant deviation between the theoretically predicted a00 habit planes using the TM and PTMC models was observed in Ti–(20,22,24)Nb alloys. The models were quite consistent with experimental observations, in which the a00 habit plane was close to f7 5 5gb in the Ti–20Nb alloy and close to f4 3 3gb for Ti–(22,24)Nb alloys. 4. Type 1 twinning on f1 1 1ga00 was found to be the LIS in the b–a00 MT. Twinning dislocations with Burgers vector, bLIS, close to ½ 2 1 3a00 were responsible for accommodating the strain component exx on the ð2 1 1Þb =ð1 1 0Þa00 terrace interface.
5. The orientation relationships between b and a00 phases in Ti–(20,22,24)Nb alloys predicted by the TM model were consistent with those observed experimentally.
Acknowledgements This work is partially supported by ILC Project from the University of Tsukuba, the Iketani Foundation, the Inamori Foundation and Grants-in-Aid for Fundamental Scientific Research (Kiban A (2002–2004), Wakate B (2006–2007) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. Appendix A (i) P is the coordinate transformation matrix that relates the b (bcc) and a00 (orthorhombic) coordinate frames: 0 1 p11 p21 p31 1 B C @ p12 p22 p32 A; P ¼ ðb Uo Þðo Ao :o Pa00 Þb U1 o ¼ 36ab X p13 p23 p33 ðA1Þ where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ¼ a2a00 þ b2a00 ; U ¼ 4a2a00 þ b2a00 ; p11 ¼ 4½ð3Þ N ab þ 3 U sinðXÞ ab pffiffiffi U 6 cosðXÞ N; p12 ¼ 2½ð6Þ N ab þ 6 U sinðXÞ ab pffiffiffi þ U 6 cosðXÞ N; p13 ¼ 2½ð6Þ N ab þ 6 U sinðXÞ ab pffiffiffi þ U 6 cosðXÞ N;
pffiffiffi p21 ¼ 2½6 N ab þ 3 U sinðXÞ ab U 6 cosðXÞ N; pffiffiffi p22 ¼ ½30 N ab þ 6 U sinðXÞ ab þ U 6 cosðXÞ N; pffiffiffi p23 ¼ ½ð6Þ N ab þ 6 U sinðXÞ ab þ U 6 cosðXÞ N; pffiffiffi p31 ¼ 2½6 N ab þ 3 U sinðXÞ ab U 6 cosðXÞ N; pffiffiffi p32 ¼ ½6 N ab þ 6 U sinðXÞ ab þ U 6 cosðXÞ N; pffiffiffi p33 ¼ ½30 N ab þ 6 U sinðXÞ ab þ U 6 cosðXÞ N:
The symbol X is defined as the angle between ½2 1 0a00 and ð1 1 0Þa00 as shown in Fig. A1, and can be deduced by the following equation: 0 1 X¼
p 2 a2 00 þ b2a00 B C cos1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: 2 4 a2 þ b2 a2 þ b2 a00
a
a00
ðA2Þ
a00
(ii) The matrix oAo describes a deformation that creates a matching ð2 1 1Þb =ð1 1 0Þa00 terrace interface expressed with respect to the reference state coordinate frame(XYZ) as follows:
Y.W. Chai et al. / Acta Materialia 56 (2008) 3088–3097
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(ii) The total rigid body rotation (/) induced by the transformation disconnections and coherency dislocations can be deduced as follows:
bz tanðhÞ: ðB3Þ /¼ h (iii) The average spacing between the transformation disconnections can be determined as follows: Fig. A1. Crystal orientations in the reference state coordinate frame XYZ for the b and a00 crystals.
0 B B B A ¼ o o B @
pffiffi ab 2 ca00
0 0
0
0
ðB4Þ
References
1
C C C: pffiffiffiffiffiffiffiffiffiffiffiffi 0 C a2 00 þb2a00 A a 0 1 pffiffi ab 3
ðA3Þ
(iii) The matrix o Pa00 relates the relaxed a00 and the XYZ frame as follows: 1 0 ca00 pffiffi 0 0 C B ab 2 B pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiC B 2 2 C 2 2 a 00 þb 00 sinðwÞ 4a 00 þb 00 C B a a pffiffi a pffiffi a 0 C: o Pa00 ¼ B ab 3 ab 6 C B B pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi C A @ 2 cosðwÞ 4a2 00 þba00 pffiffia 0 0 a 6 b
ðA4Þ (iv) The matrix bUo is an orthogonal coordinate transformation matrix that relates the XYZ frame and b coordinate frame, 0 1 p1ffiffi p2ffiffi 0 3 6 B C B p1ffiffi p1ffiffi p1ffiffi C : ðA5Þ b Uo ¼ B 3 6C @ 2 A p1ffiffi2 p1ffiffi3 p1ffiffi6
Appendix B The following equations are reproduced from Refs. [22,23]. For more details, the readers are referred to the original references. (i) The inclination angle xT of the a00 habit plane to the ð2 1 1Þb =ð1 1 0Þa00 terrace plane is given by xT ¼ h /=2;
k ¼ h= sinðhÞ:
ðB1Þ
where h is the tan angle of the step height to the average terrace length, and is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 by þ b2y þ 4 ðbz Þðh eyy Þ A: h ¼ tan1 @ ðB2Þ 2 bz
[1] Kuroda D, Niinomi M, Morinaga M, Kato Y, Yashiro T. Mater Sci Eng A 1998;243:244. [2] Hanada S, Matsumoto H, Watanabe S. Intl Congress Series 2005;1284:239. [3] Niinomi M, Kuroda D, Fukunaga K, Morinaga M, Kato Y, Yashiro T, et al. Mater Sci Eng A 1999;263:190. [4] Niinomi M. Sci Technol Adv Mater 2003;4:445. [5] Okazaki Y, Gotoh E. Biomaterials 2005;26:11. [6] Ikeda M, Komatsu S, Nakamura Y. Mater Trans 2004;45:1106. [7] Miyazaki S, Kim HY, Hosoda H. Mater Sci Eng A 2006;438–440:18. [8] Baker C. Metal Sci J 1971;5:92. [9] Kim HY, Ikehara Y, Kim JI, Hosoda H, Miyazaki S. Acta Mater 2006;54:2419. [10] Fukui Y, Inamura T, Hosoda H, Wakashima K, Miyazaki S. Mater Trans 2004;45:1077. [11] Kim JI, Kim HY, Hosoda H, Miyazaki S. Mater Trans 2005;46:852. [12] Kim HY, Sasaki T, Okutsu K, Kim JI, Inamura T, Hosoda H, et al. Acta Mater 2006;54:423. [13] Kim HY, Hashimoto S, Kim JI, Inamura T, Hosoda H, Miyazaki S. Mater Sci Eng 2006;417:120. [14] Ping DH, Mitarai Y, Yin FX. Scripta Mater 2005;52:1287. [15] Kim JI, Kim HY, Inamura T, Hosoda H, Miyazaki S. Mater Sci Eng 2005;403:334. [16] Sakaguchi N, Niinomi M, Akahori T, Takeda J, Toda H. Mater Sci Eng C 2005;25:363. [17] Long Marc, Rack HJ. Biomaterials 1998;19:1621. [18] Wang Kathy. Mater Sci Eng A 1996;213:134. [19] Niinomi M. Mater Sci Eng A 1998;243:231. [20] Otsuka K, Wayman CM. Shape memory materials. Cambridge: Cambridge University Press; 1999. [21] Christian JW. The theory of transformation in metals and alloys: Part I. 2nd ed. Oxford: Pergamon Press; 2002. p. 286. [22] Pond RC, Celotto S, Hirth JP. Acta Mater 2003;51:5385. [23] Pond RC, Ma X, Chai YW, Hirth JP. Dislocations in solids. Amsterdam: Elsevier; 2007. p. 227 [chapter 74]. [24] Hammond C, Kelly PM. Acta Metall 1969;17:869. [25] Mahon GJ, Howe JM, Mahajan S. Philos Mag Lett 1989;59:273. [26] Moritani T, Miyajima N, Furuhara T, Maki T. Scripta Mater 2002;47:193. [27] Chen IW, Chiao YH. Acta Metall 1985;33:1827. [28] Hirth JP, Pond RC. Acta Mater 1996;44:4749. [29] Pond RC, Celotto S. Int Mat Revs 2003;48:225. [30] Bowles JS, Mackenzie JK. Acta Metall 1954;2:129. 138, 224. [31] Wayman CM. Introduction to the crystallography of martensitic transformations. New York: Macmillan; 1964. [32] Bowles JS, Mackenzie JK. Acta Metall 1957;5:134. [33] Bywater KA, Christian JW. Philos Mag 1972;25:1249. [34] Inamura T, Kim JI, Kim HY, Hosoda H, Wakashima K, Miyazaki S. Philos Mag 2007;87:3325. [35] Serra A, Pond RC, Bacon DJ. Acta Metall Mater 1991;39:1469.
Acta Materialia 56 (2008) 3088–3097 www.elsevier.com/locate/actamat
Interfacial defects in Ti–Nb shape memory alloys Y.W. Chai a,*, H.Y. Kim a, H. Hosoda b, S. Miyazaki a,* b
a Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305 8573, Japan Precision and Intelligence Laboratory, Tokyo Institute of Technology, Yokohama 226 8503, Japan
Received 31 December 2007; received in revised form 26 February 2008; accepted 28 February 2008 Available online 9 April 2008
Abstract The structure of the martensite (a00 )/martensite (a00 ) and parent (b)/martensite (a00 ) interfaces in a series of binary Ti–Nb alloys with Nb content ranging from 20 to 24 at.% was investigated. Both the a00 /a00 and b/a00 interfaces comprised a series of f2 1 1gb ==f1 1 0ga00 terraces and steps when viewed edge-on (close to ½0 0 1a00 ). Interfacial defects, particularly the transformation disconnections (b, h) superimposed along the terrace–step interface structure, have been identified. They were responsible for accommodating most of the transformation strain along the a00 /a00 and b/a00 interfaces. Using the parameters b and h, the prediction of the a00 habit plane based on the topological model agreed well with the prediction from the phenomenological theory as well with experimental observations. The a00 habit plane in Ti–20Nb alloy is close to f7 5 5gb and moves towards f4 3 3gb in Ti–22Nb and Ti–24Nb alloys. The remaining transformation strain along the b/a00 interface was found to be accommodated by Type 1 twinning on ð1 1 1Þa00 with Burgers vector bLIS close to h 2 1 3ia00 ð h1 1 2ib Þ. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ti–Nb; Shape memory alloys; Transformation disconnections; Twinning dislocations; Strain accommodation
1. Introduction Ti-based alloys consisting of various non-toxic elements, such as Nb, Zr, Ta, Sn, Mo and Pt, which may be suitable candidates for biomedical applications, have been extensively investigated recently [1–7]. More excitingly, some of these alloys were found to exhibit shape memory effect (SME) and superelasticity (SE). In Ti–Nb, Baker [8] first reported that SME was observed in Ti–35 wt.% (or 21.7 at.%) Nb alloy. Recently, a systematic study by Kim et al. [9] also observed SME and SE in Ti–(20–27) at.% Nb binary alloys. SME and SE were also reported in other Ti–Nb-based ternary and even quaternary alloys, such as Ti–Nb–Al [10], Ti–Nb–O [11], Ti–Nb–Ta [12,13], Ti–Nb– Pd [14], Ti–Nb–Zr [15] and Ti–Nb–Zr–Ta [16]. By combining either the SME or SE with other useful properties, such as biocompatibility and low Young’s modulus, these alloys *
Corresponding authors. Tel.:/fax: +81 29 853 5283. E-mail addresses: [email protected] (Y.W. Chai), miyazaki@ ims.tsukuba.ac.jp (S. Miyazaki).
may provide an additional advantage over some conventional biomaterials, e.g. pure Ti, Ti–6Al–4V alloy, stainless steel and Co-based alloys [1,17–19]. The SME and SE in shape memory alloys are closely related to the martensitic transformation (MT) and its reversion between the parent and martensite phases [20]. Transformation disconnections (formerly known as transformation dislocations) have been reported to play an important role in the transformation process [21]. Furthermore, Pond et al. [22,23] also pointed out that these interfacial defects are capable of minimizing the transformation strain arising from the MT. For the case of the b–a00 MT in Ti–Nb-based shape memory alloys, little has been reported on the transformation disconnections and their effects on the shape memory properties of the alloys. Therefore, the aim of this paper is to investigate the nature of the b/a00 interface structures and interfacial defects, particularly transformation disconnections in Ti–Nb-based shape memory alloys. Lattice invariant shear, i.e. twinning or slip, which have been widely reported to be involved in the transformation strain minimization process, will also be discussed briefly. Experimental observations
1359-6454/$34.00 Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2008.02.045
Y.W. Chai et al. / Acta Materialia 56 (2008) 3088–3097
have been carried out using transmission electron microscopy (TEM) and high-resolution TEM (HRTEM). The experimental observations on the martensite (a00 ) crystallography will be discussed in conjunction with predictions from the topological model (TM) [22,23]. 2. Experimental procedure Binary Ti–Nb alloys with Nb contents of 20 at.% (Ms 493 K), 22 at.% (Ms 413 K) and 24 at.% (Ms 338 K), which exhibit the SME, have been chosen for this study. Hereafter, the composition of the alloys is represented in at.% unless stated otherwise. The alloys were prepared by the argon arc melting method. In the melting process, the alloy buttons were melted for 3 min six times, and flipped over each time before melting. In order to minimize segregation, all of the alloy buttons were then homogenized in vacuum at 1273 K for 2 h. After that, they were cold-rolled up to 98.5% of thickness reduction. Specimens for TEM observation were punched out from the cold-rolled strips, chemically cleaned in a solution containing HF, HNO3 and H2O (1:4:5) at 30 °C, sealed in a quartz tube with a 25 Torr partial pressure of high-purity argon gas, and subsequently solution treated at 1173 K for 30 min. The specimens were then water quenched by breaking the quartz tube. The quenching rate has been estimated to about 200–300 K s1. The final specimen thinning was carried out using a twin-jet electropolishing machine (Tenupol-3, Struer) with a solution containing HF, H2SO4 and methanol (2:5:93) at temperatures between 50 and 40 °C. TEM and HRTEM observation was conducted using a JEOL 2010F microscope operated at 200 kV. The lattice parameters of the parent (b, body-centered cubic) and martensite (a00 , orthorhombic) phases in the alloys are shown in Table 1. Hereafter, the parent (b) and martensite (a00 ) phases will be represented by their symbols b and a00 , respectively. 3. Experimental observations 3.1. Martensite (a00 ) interfaces structure TEM micrographs obtained from the Ti–20Nb, Ti–22Nb and Ti–24Nb alloys are shown in Fig. 1a, d and g, respectively. The a00 plates in the figures are viewed edge-on. In Table 1 The lattice parameters of the b (bcc) and a00 (orthorhombic) phases in Ti– 20 at.% Nb, Ti–22 at.% Nb and Ti–24 at.% Nb alloys [9] Alloys (at.%)
Ti–20Nb Ti–22Nb Ti–24Nb
Lattice parameters (nm)
Ms (K)
Parent (b)
Martensite (a00 )
ab
aa00
ba00
ca00
0.328505 0.328537 0.328569
0.31257 0.31518 0.31785
0.48704 0.48431 0.48120
0.46456 0.46427 0.46355
493 413 338
The corresponding martensite start (Ms) temperatures are also indicated.
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Fig. 1a, no retained b phase is observed in the microstructure because the Ms for the Ti–20Nb alloy is too high (493 K). The edge-on a00 plate (labeled a00V2 ) in the figure is shown as being interfaced with an a00 plate of a different variant (labeled a00V1 ). The corresponding orientations of a00V2 and a00V1 plates are viewed parallel to ½0 0 1a00 :V2 and ½0 1 0a00 :V1 , respectively, as indicated by the selected area diffraction (SAD) pattern in Fig. 1b and their indexes in Fig. 1c. For the Ti–22Nb and Ti–24Nb alloys as seen in Fig. 1d and 1g, an edge-on a00 plate and a retained b phase are observed parallel to ½0 0 1a00 and ½0 1 1b , respectively. This is clearly confirmed by the SAD patterns (Fig. 1e and h) and their indexes (Fig. 1f and i). The orientation of the a00 interface, as indicated by the dotted line with plane normal p1 was measured as 10.3 ± 0.5° away from ð1 1 0Þa00 :V2 =ð1 0 1Þa00 :V1 in the Ti–20Nb alloy (see Fig. 1a), 11.0 ± 0.3° away from the ð1 1 0Þa00 =ð2 1 1Þb in the Ti–22Nb alloy (see Fig. 1d), and 11.7 ± 0.3° away from the ð1 1 0Þa00 =ð2 1 1Þb in the Ti–24Nb alloys (see Fig. 1g). By using the HRTEM technique, the interface structure has been further investigated. Fig. 2a and b show the high-resolution image and the inverse fast Fourier transformation (FFT) images, respectively, of the a00V2 =a00V1 interface structure in the Ti–20Nb alloy, whereas Fig. 2c and d reveal the high-resolution images of the b/a00 interfaces in Ti– 22Nb and Ti–24Nb alloys, respectively. Fig. 2a–d indicate that both of the a00V2 =a00V1 and b/a00 interfaces are formed by a series of terraces and steps (indicated by dotted lines) at the atomic scale. The terrace planes of the a00V2 =a00V1 interface is found to be parallel to ð1 1 0Þa00 :V2 =ð1 0 1Þa00 :V1 (see Fig. 2a). They correspond to the ð2 1 1Þb plane when expressed with respect to the b phase. Average length of the terraces was estimated to be 1.6 nm. The steps along the a00V2 =a00V1 interface consist of one or two ð1 1 0Þa00 :V2 plane spacings. It is noticed that a relatively large strain contrast was seen at the interface. The inverse FFT image in Fig. 2b reveals the presence of dislocations. Some of the dislocations with opposite signs will cancel out each other, whereas those remaining in the a00 phase will serve as misfit dislocations to accommodate misfit strain resulting from the difference between the ð0 2 0Þa00 :V2 (0.24 nm) and the ð0 0 2Þa00 :V1 (0.23 nm) planes. Hence, the strain contrast is most probably caused by the mismatch between the ð0 2 0Þa00 :V2 and ð0 0 2Þa00 :V1 plane spacings. The ð0 0 2Þa00 :V1 lattice planes are seen to run continuously across the ð1 1 0Þa00 :V2 =ð1 0 1Þa00 :V1 terraces, suggesting that the a00V2 =a00V1 terraces are semicoherent. For the b/a00 interface in the Ti–22Nb and Ti– 24Nb alloys (see Fig. 2c and d), the terrace planes are parallel to ð2 1 1Þb =ð1 1 0Þa00 . Their average lengths are about 1.5 and 1.3 nm, respectively. It can be seen that the terrace planes are terminated by steps with a height of one or two ð1 1 0Þa00 plane spacings. In Fig. 2c and d, the ð0 2 0Þa00 lattice planes are seen running continuously across the b/a00 terraces, indicating that the ð2 1 1Þb =ð1 1 0Þa00 terrace planes are coherent. No misfit dislocation was observed along the b/a00 interfaces. The strain contrast near the b/a00 interfaces is caused by the coherent particles, namely the
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Y.W. Chai et al. / Acta Materialia 56 (2008) 3088–3097
Fig. 1. TEM micrographs of (a), (d) and (g) revealing edge-on a00 plates that were observed in the Ti–(20,22,24)Nb alloys. The trace of the habit plane is indicated by dotted lines and the habit plane normal is labeled p1. (b) is ½0 0 1a00 :V2 =½0 1 0a00 :V1 SAD pattern taken from the a00V2 =a00V1 interface in (a). (c) is the index of the SAD pattern in (b). (e) and (h) are ½0 1 1b =½0 0 1a00 SAD patterns taken from the b/a00 interfaces in (d) and (g), respectively. Their corresponding indexes are shown in (f) and (i).
athermal omega (x) particles in the b phase. These ellipsoidal x particles (ranging in size from 3 to 8 nm long, and from 1 to 3 nm wide) in Fig. 2c and d, seem to be capable of suppressing the b–a00 MT, by impeding movement of the b/a00 interface. The interaction of these athermal x particles and the a00 phase will be reported in more detail in a future publication and hence will not be discussed any further here.
1 and of about 0.2° was observed between the ½0 1 b ½0 0 1a00 directions. The ð2 1 1Þb and ð1 1 0Þa00 planes in the b and a00 phases also exhibited a small deviation angle, 0.2°. The OR between the b and a00 phases was found to follow the Burgers OR as often observed in hexagonal close-packed (hcp) alloys [24].
3.2. Orientation relationship (OR)
During the TEM observation, it was noticed that the internal microstructure of the a00 phase, particularly the internal twins, changes with respect to the Nb content, as shown in Fig. 3a, c and d, respectively. In the Ti–20Nb alloy, few or no internal twins were observed in many of the a00 plates (see Fig. 3a). Occasionally, some traces of dislocations parallel to ð1 1 1Þa00 were observed within some of the a00 plates. However, as Nb addition increased to 22 at.%, fine internal twins (ranging from a few nm to tens of nm wide), as indicated by the arrows in Fig. 3b, are seen in some of the a00 plates. The ½1 0 1a00 SAD pattern and its index in Fig. 3d indicate that the internal twins are of Type
A set of ORs relating the b, a00 and the athermal x phases were obtained from the TEM and HRTEM observations in Figs. 1 and 2. They are shown as ½0 1 1 ==½0 0 1 00 ==½1 1 20 ; b
a
x
½1 1 1b ==½1 1 0a00 ==½0 0 0 1x ; ð2 1 1Þb ==ð1 1 0Þa00 ==ð1 1 0 0Þx ; where the b, a00 and athermal x phases are represented by the corresponding subscript symbols. A small deviation
3.3. Type 1 f1 1 1ga00 internal twinning
Y.W. Chai et al. / Acta Materialia 56 (2008) 3088–3097
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Fig. 2. (a) indicating the HRTEM image of the a00V2 =a00V1 interface structure in Ti–20Nb alloy. (b) is an inverse FFT image revealing the presence of misfit dislocations along the a00V2 =a00V1 interface. (c) and (d) are the HRTEM images showing the b/a00 interface structure in the Ti–(22,24)Nb alloys, respectively. Athermal omega particles (x) are observed in the b phase.
1 twinning on ð1 1 1Þa00 . These fine ð1 1 1Þa00 internal twins are rather random, short (ranging from 50 to 150 nm long) and often do not extend from one side of the a00 plate to the other. In the Ti–24Nb alloy, the a00 phase with more defined and thicker ð1 1 1Þa00 internal twins (up to 60 nm wide) were observed, as indicated by the arrows in Fig. 3c. 4. Discussion 4.1. Interface structure As seen in the preceding section, the HRTEM micrographs revealed that both a00V2 =a00V1 and b/a00 interfaces exhibit a terraces/steps structure when viewed edge-on along the ½0 0 1a00 :V2 =½0 1 0a00 :V1 direction in the Ti–20Nb alloy (see Fig. 2a) and along the ½0 1 1b =½0 0 1a00 direction in the Ti– 22Nb and Ti–24Nb alloys (see Fig. 2b and c). For the b/a00 interface, the terraces/steps structure is considered as a typical relaxed martensitic interface structure, where the interface (or habit plane) is often irrational and consists a series of rational terrace planes parallel to ð2 1 1Þb =ð1 1 0Þa00 and steps. Such a relaxed interface structure has also been
observed in many other alloys in addition to Ti–Nb alloys [22,25–29]. During the interface relaxation process, two arrays of interfacial defects, i.e. transformation disconnections (b, h) and lattice-invariant shear (LIS) (bLIS, 0), are involved in order to minimize the transformation strain on the terrace plane [22,23]. A schematic diagram illustrating the two interfacial defects at a b/a00 terraces/steps interface is shown in Fig. 4. The transformation disconnection and LIS are represented by their line directions, nDiscon and nLIS, and their Burgers vectors are shown by their components, LIS b = [0, by, bz] and bLIS ¼ ½bLIS x ; 0; bz , expressed with respect to the reference state frame (XYZ). The reference state frame is defined from the experimentally observed b–a00 OR, i.e. X : ½0 1 1b =½0 0 1a00 , Y : ½1 1 1b =½1 1 0a00 and Z : ð2 1 1Þb = ð1 1 0Þa00 . For simplicity, the transformation strain is represented by components exx (parallel to X), and eyy (parallel to Y). Based on the configuration of the interfacial defects as shown in Fig. 4, one can see that their edge Burgers vector components, i.e. by and bLID , are capable of relieving the x strain components, eyy and exx on the ð2 1 1Þb =ð1 1 0Þa00 terrace plane, respectively. exx and eyy can be deduced using the following equations:
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Fig. 3. (a), (b) and (c) are TEM images showing the variation of martensite microstructure including internal twins in the Ti–(20,22,24)Nb alloys. (d) is ½1 0 1a00 SAD pattern and the index that were obtained from the circle in (b), and indicating that the internal twins are of Type 1 twinning on ð1 1 1Þa00 plane.
Fig. 4. A schematic diagram illustrating the transformation strain components, exx and eyy, the transformation disconnections (b = [by, bz], h) LIS 00 and the LIS ðbLIS ¼ ½bLIS terrace interface when x , bz ], 0) in a b/a expressed with respect to the reference state frame, XYZ. The line directions nDiscon and nLIS of the transformation disconnections and the LIS are also indicated in the diagram.
pffiffiffi ca00 2 ab exx ¼ 1 pffiffiffi ; 2 ab þ ca00 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi aa00 þ ba00 3 ab eyy ¼ 1 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 3 ab þ aa00 þ ba00 2
Fig. 5. Variation of the transformation strains, exx and eyy, on the ð2 1 1Þb =ð1 1 0Þa00 terrace interface with respect to the Nb content.
ð1Þ ð2Þ
Using lattice parameters of Ti–(20,22,24)Nb alloys as shown in Table 1, exx and eyy have been calculated as shown in Fig. 5. It should be noted that eyy has a much larger magnitude compared with exx. This suggests that the transformation strain at the b/a00 interface is dominated mostly by strain component eyy. In other words, the trans-
formation disconnections responsible for relieving the strain component eyy are anticipated to play a more significant role in the interfacial strain minimization process compared with the LIS. 4.2. Characterization of the transformation disconnections A dichromatic complex in Fig. 6 illustrates a transformation disconnection (b, h) that is associated with a b/a00 interface in a reference state frame (XYZ). The b and a00
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Fig. 7. Variation of by, bz and h of the transformation disconnections as a function of Nb content.
Fig. 6. (a) Dichromatic complex illustrating a transformation disconnection superimposed on the reference state b/a00 interface. The enlargement on the right indicates the Burgers vector components, by and bz, and the overlap step height (h) of the transformation disconnection. (b) Schematic diagram showing 2 atoms per transforming volume in the b and a00 phases.
atomic sites are represented by white and black squares, respectively. The line direction of the transformation disconnection, nDiscon, is parallel to X : ½0 1 1b =½0 0 1a00 . The Burgers vector, b, of the transformation disconnection, which is responsible for the atomic shearing process during the b–a00 MT, can be determined using the following equation [28]: b ¼ ½0; by ; bz ¼ tðbÞ Ptða00 Þ;
ð3Þ
where P is the matrix of the coordinate transformation that relates the b and a00 crystals (see Appendix A). The vectors t(b) and t(a00 ) are translation vectors in b and a00 crystals, and are equal to ½ 1 0 0b and ½ 1 0 0a00 respectively. The enlargement on the right of Fig. 6 shows b and its two components, by and bz, with respect to the XYZ frame. The overlap step height h is equal to hða00 Þ ¼ d ð110Þa00 or hðbÞ ¼ 2d ð211Þb , with the smaller one being chosen here. For Ti–(20,22,24)Nb alloys, by, bz and h of the transformation disconnection have been deduced and their values are shown as a function of alloy composition in Fig. 7. Clearly, b (=[0, by, bz]) and h are seen to vary as the alloy composition changes, i.e. by and bz decrease as the Nb content changes from 20 to 24 at.%. The former has a much larger magnitude when compared with the latter. On the other hand, h increases with increasing Nb content from 20 to 24 at.%. It is found that the values of b and h are relatively small, and this indicates that the intrinsic mobility of the transformation disconnection in the three alloys is high. Moreover, the transformation disconnection identified above fulfills the criteria for feasible military transforma-
tions [29]. For instance, the number of atoms (same atom species) for a transforming volume is the same between the b and a00 phases, i.e. 2 atoms per each transforming volume (see Fig. 6b). This implies that the transformation disconnection as identified in Fig. 6 and observed in Fig. 2 can move in a conservative and glissile manner on the Z : ð2 1 1Þb =ð1 1 0Þa00 terrace plane along the direction parallel to Y : ½1 1 1b =½1 1 0a00 . Table 2 also includes a rigid body rotation, / (around X : ½0 1 1b =½0 0 1a00 Þ, that is introduced by the bz (see Eq. (B3) in Appendix B). The / in Ti–20Nb is predicted to about 0.20°, and decreases to 0.17° and 0.13° in Ti–(22,24)Nb alloys, respectively. 4.3. Transformation strain accommodation by transformation disconnections At an equilibrium state, an array of the transformation disconnections with an appropriate spacing, k, would relieve eyy effectively and leave no long-range stress field at the b/a00 interface. The average spacing predicted theoretically (see Eq. (B4) in Appendix B) and those observed experimentally (see Fig. 2) in the Ti–(20,22,24)Nb alloys are summarized in Fig. 8. Clearly, they are consistent with each other (deviating by about ±0.1 nm), suggesting that the transformation strain along the b/a00 terrace interface is reduced by an array of transformation disconnections with an average spacing k. The habit plane inclinations (xT), designated as the inclination angle between the habit plane and the ð2 1 1Þb =ð1 1 0Þa00 terrace plane, obtained
Table 2 The transformation disconnection parameters at the b/a00 terrace interface in Ti–(20,22,24)Nb alloys Alloy (at.%)
Topological parameters of disconnection Burgers vector, b
Ti–20Nb Ti–22Nb Ti–24Nb
bx (nm)
by (nm)
bz (nm)
0 0 0
0.0237 0.0204 0.0169
0.0052 0.0041 0.0031
jbj (nm)
h (nm)
/ (°)
0.0243 0.0208 0.0172
0.2631 0.2642 0.2652
0.20 0.17 0.13
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Fig. 8. Nb content dependence of the average spacing of the transformation disconnection, k, and the habit plane inclination, xT, obtained from theoretical prediction (TM model) and experiment.
theoretically (see Eq. (B1) in Appendix B) and observed experimentally for the three alloys, are also summarized in Fig. 8. Close agreement in xT is seen between the predicted and experimentally observed values. xT is clearly dependent on the alloy composition. As the Nb content increases from 20 to 24 at.%, the transformation strain, particularly eyy, along the b/a00 interface decreases (see Fig. 5). Hence, in order to accommodate the strain more effectively, the parameters, by, bz, h and k of the transformation disconnections are anticipated to vary accordingly (see Figs. 7 and 8). Interestingly, it is noticed that the experimentally observed a00V2 =a00V1 interface in Ti–20Nb alloy (see Fig. 1a and 2a), although it is not a b/a00 habit plane, agrees rather well with the predicted b/a00 habit plane orientation (see Fig. 8). This could be due to the fact that the presence of misfit dislocations along the a00V2 =a00V1 interface has almost accommodated the extra misfit strain resulting from the difference between the lattice planes
ð0 0 2Þa00 :V1 and ð0 2 0Þa00 :V2 . Therefore, the parameters of the transformation disconnections, e.g. by, bz, h and k, may not rearrange significantly so that the habit plane orientation will not be altered. Owing to this, the predicted xT using the parameters of the transformation disconnections for the b=a00V2 interface is close to the xT experimentally observed from the a00V2 =a00V1 interface. Based on the values xT in Fig. 8, the habit planes expressed with respect to the b phase for Ti– (20,22,24)Nb alloys, both observed experimentally and predicted using the TM model, have been deduced and plotted in a [0 0 1]b stereogram in Fig. 9. For comparison, the phenomenological theory of martensite crystallography (PTMC) [30–32] was applied to predict the habit plane for the alloys, and these planes are also included in Fig. 9. No significant difference was observed between the a00 habit planes predicted based on the TM and the PTMC (less than the experimental error of ±0.5°). Moreover, the predicted a00 habit planes (from both models) agree well with experimental observation. In summary, Fig. 9 shows that the a00 habit plane is very close to ð7 5 5Þb (deviation less than 0.1°) in Ti–20Nb, and shifts towards ð4 3 3Þb as the Nb content increases from 20 to 24 at.%. 4.4. Characterization of Type 1 ð1 1 1Þa00 twinning dislocation Type 1 ð1 1 0 1Þa0 twins have been considered as the LIS in the b–a0 (hcp) MT [24,33]. In the case of b–a00 MT, it corresponds to Type 1 ð1 1 1Þa00 twins. In Fig. 3, the microstructure of the ð1 1 1Þa00 internal twins is clearly related to the alloy composition, which in turn depends on the transformation strain, exx. Such microstructural variation of ð1 1 1Þa00 ð ð1 1 0 1Þa0 Þ twins with respect to exx (or alloy composition) has been reported in Refs. [24,33]. Recently, Inamura et al. [34] have also reported a similar variation of microstructure in the b–a00 MT in Ti–Nb–Al alloys. The ð1 1 1Þa00 twinning dislocations (bt1) can be characterized based on the dichromatic complex in Fig. 10 [35]. The
Fig. 9. A ½0 0 1b stereogram indicating the a00 habit planes in the Ti–(20,22,24)Nb alloys. They were obtained from the topological model (TM), the phenomenological theory of martensite crystallography (PTMC) and experimental observation.
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Fig. 10. Dichromatic complex for the Type 1 ð1 1 1Þa00 twin. Two crystallographically equivalent Burgers vectors, bt1 and bt2, are indicated. Orthorhombic unit cells are outlined in rectangular dotted lines. The white and black symbols represent the matrix and twin of the a00 crystals, respectively.
Table 3 Type 1 ð1 1 1Þa00 twinning dislocation parameters for Ti–(20,22,24)Nb alloys Alloys (at.%)
Burgers vector, bt1
jbt1 j (nm)
ht ¼ 2dð1 1 1Þa00 ðnmÞ
Ti–20Nb Ti–22Nb Ti–24Nb
½0:4119 0:3600 0:0519a00 ½0:4243 0:3795 0:0449a00 ½0:4459 0:4092 0:0368a00
0.2189 0.2283 0.2432
0.4578 0.4592 0.4604
white and black symbols represent the matrix and twin of the a00 crystals, respectively, and are viewed parallel to h1 1 0ia00 . The Burgers vector bt1 is equal to the difference between vectors tm1 ¼ ½1 0 1a00 and tt1 ¼ 1=2½3 1 0a00 (expressed with respect to the a00 matrix crystal). The vectors correspond to 1=3½2 1 1 3a0 and ½1 0 1 0a0 in the a0 (hcp) crystal, respectively. For Ti–(20,22,24)Nb alloys, bt1 is shown in Table 3. When expresses with respect to the reference state frame (XYZ), it can be seen that bt1 has a strong screw component, i.e. bLIS (parallel to Y : ½ 1 1 1b =½ 1 1 0a00 ), which does not y accommodate the misfit strain, and will lead to a long-range stress field at the b/a00 interface. However, it is postulated that other twinning dislocations with Burgers vectors bt2 (i.e. derived from the difference between vectors tm2 ¼ 1=2½3 1 0a00 and tt2 ¼ ½1 0 1a00 , see Fig. 10) may be also generated simultaneously. The bt2 contains an opposite screw component, bLIS (parallel to Y : ½1 1 1b =½1 1 0a00 ). A y
pair of such twinning dislocations with bt1 and bt2 leads to the elimination of the screw components and results in a Burgers vector (bLIS) with only an edge component bLIS and a x LIS LIS normal component bLIS ð¼ ½bLIS z . The b x ; 0; bz Þ for each alloy is shown in Table 4 and is found to be close to ½2 1 3a00 . This direction corresponds to ½1 1 2b and ½ 1 1 0 2a0 in the b and a0 (hcp) crystals, respectively. By assuming bLIS accommodate exx completely, the average spacing x between pairs of the twinning dislocations, kLIS ð¼ bLIS x =exx Þ, has been estimated for each alloy as shown in Table 4. Note that kLIS decreases rapidly from about 962 nm in Ti–20Nb to about 55 and 14 nm in Ti–(22,24)Nb alloys, respectively. The component bLIS will introduce a rigid body rotation z LIS /LIS ð¼ 2 sinðbLIS =2k ÞÞ of the a00 phase with respect to the z reference state frame XYZ [23]. The /LIS (around Y : ½1 1 1b =½1 1 0a00 Þ for each alloy is shown in Table 4. Note that /LIS increases, while kLIS decreases with Nb content varying from 20 to 24 at.%. The negative sign of /LIS represents anti-clockwise rotation around Y : ½1 1 1b =½ 1 1 0a00 . 4.5. Transformation strain accommodation by LIS: ð1 1 1Þa00 twinning dislocations The edge component bLIS of ð1 1 1Þa00 twins ðbLIS ¼ x is responsible for relieving the transformation strain component exx. In Ti–20Nb, exx is very small LIS ½bLIS x ; 0; bz ])
Table 4 The resultant of twinning dislocation parameters as the LIS for b–a00 MT in Ti–(20,22,24)Nb alloys Alloys (at.%)
bLIS
j bLIS j (nm)
bLIS (nm) x
bLIS (nm) z
kLIS (nm)
/LIS (°)
Ti–20Nb Ti–22Nb Ti–24Nb
½0:0734 0:0303 0:1038a00 ½0:0635 0:0262 0:0898a00 ½0:0520 0:0214 0:0735a00
0.0554 0.0480 0.0392
0.0482 0.0417 0.0341
0.0273 0.0238 0.0194
961.6 55.1 14.2
0.00 0.02 0.08
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(0.01%). A large spacing kLIS of about 962 nm (see Table 4) between pairs of the twinning dislocations is sufficient to accommodate exx. Consequently, the low density of the twinning dislocations may lead to difficulty in forming internal twins in the a00 phase. This explains why internal twins are hardly seen in many of the a00 plates in Ti–20Nb alloy (see Fig. 3a). Instead, slip dislocations with traces parallel to ð1 1 1Þa00 are observed. As the Nb content increases from 20 to 22 at.%, exx increases to 0.08%. This leads to a significant reduction in kLIS to about 55 nm. Such spacing may just be enough for the density of the twinning dislocations to form internal twins. Hence, fine (ranging from 5 to 20 nm wide) but irregular ð1 1 1Þa00 internal twins were observed (see Fig. 3b). Further increase in Nb content to 24 at.% resulted in exx increasing to 0.24%. As a consequence, an increase in the density of twinning dislocations, i.e. kLIS (14 nm), leads to a formation of larger internal twins. Therefore, wider and more defined internal twins (up to 60 nm wide) are observed in the a00 phase in Ti–24Nb (see Fig. 3c). By comparing the / and /LIS (see Tables 2 and 4), it is seen that /LIS in the Ti–(20,22)Nb alloys is significantly smaller. This implies that the OR between the b and a00 phases in the two alloys is mainly due to /. On the other hand, a larger /LIS is predicted in Ti–24Nb alloy. Hence, the OR in Ti–24Nb has contributions from both / (0.13°) and /LIS (0.08°), where an overall rotation of 0:90 0:31 Þ is predicted. 0.16° (clockwise around ½0:30 b Nevertheless, in all three alloys, the predicted and experimentally observed ORs are quite consistent (0.2°). 5. Conclusion 1. The a00V2 =a00V1 and b/a00 interfaces relaxed into a series of ð2 1 1Þb =ð1 1 0Þa00 terraces and steps, when viewed edge-on along the ½0 1 1b =½0 0 1a00 direction. 2. The transformation disconnections (b,h), which could move in a conservative and glissile manner on the ð2 1 1Þb =ð1 1 0Þa00 terrace planes and along the Y: ½1 1 1b =½1 1 0a00 direction during the b–a00 MT, were identified along the a00V2 =a00V1 and b/a00 interfaces in Ti–(20,22,24)Nb alloys. The intrinsic mobility of the transformation disconnections (b,h) in the alloys is higher due to their relatively small magnitude of b and h. Most of the transformation strain at a b/a00 interface in Ti–(20,22,24)Nb alloys was accommodated by transformation disconnections (b,h). 3. No significant deviation between the theoretically predicted a00 habit planes using the TM and PTMC models was observed in Ti–(20,22,24)Nb alloys. The models were quite consistent with experimental observations, in which the a00 habit plane was close to f7 5 5gb in the Ti–20Nb alloy and close to f4 3 3gb for Ti–(22,24)Nb alloys. 4. Type 1 twinning on f1 1 1ga00 was found to be the LIS in the b–a00 MT. Twinning dislocations with Burgers vector, bLIS, close to ½ 2 1 3a00 were responsible for accommodating the strain component exx on the ð2 1 1Þb =ð1 1 0Þa00 terrace interface.
5. The orientation relationships between b and a00 phases in Ti–(20,22,24)Nb alloys predicted by the TM model were consistent with those observed experimentally.
Acknowledgements This work is partially supported by ILC Project from the University of Tsukuba, the Iketani Foundation, the Inamori Foundation and Grants-in-Aid for Fundamental Scientific Research (Kiban A (2002–2004), Wakate B (2006–2007) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. Appendix A (i) P is the coordinate transformation matrix that relates the b (bcc) and a00 (orthorhombic) coordinate frames: 0 1 p11 p21 p31 1 B C @ p12 p22 p32 A; P ¼ ðb Uo Þðo Ao :o Pa00 Þb U1 o ¼ 36ab X p13 p23 p33 ðA1Þ where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ¼ a2a00 þ b2a00 ; U ¼ 4a2a00 þ b2a00 ; p11 ¼ 4½ð3Þ N ab þ 3 U sinðXÞ ab pffiffiffi U 6 cosðXÞ N; p12 ¼ 2½ð6Þ N ab þ 6 U sinðXÞ ab pffiffiffi þ U 6 cosðXÞ N; p13 ¼ 2½ð6Þ N ab þ 6 U sinðXÞ ab pffiffiffi þ U 6 cosðXÞ N;
pffiffiffi p21 ¼ 2½6 N ab þ 3 U sinðXÞ ab U 6 cosðXÞ N; pffiffiffi p22 ¼ ½30 N ab þ 6 U sinðXÞ ab þ U 6 cosðXÞ N; pffiffiffi p23 ¼ ½ð6Þ N ab þ 6 U sinðXÞ ab þ U 6 cosðXÞ N; pffiffiffi p31 ¼ 2½6 N ab þ 3 U sinðXÞ ab U 6 cosðXÞ N; pffiffiffi p32 ¼ ½6 N ab þ 6 U sinðXÞ ab þ U 6 cosðXÞ N; pffiffiffi p33 ¼ ½30 N ab þ 6 U sinðXÞ ab þ U 6 cosðXÞ N:
The symbol X is defined as the angle between ½2 1 0a00 and ð1 1 0Þa00 as shown in Fig. A1, and can be deduced by the following equation: 0 1 X¼
p 2 a2 00 þ b2a00 B C cos1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: 2 4 a2 þ b2 a2 þ b2 a00
a
a00
ðA2Þ
a00
(ii) The matrix oAo describes a deformation that creates a matching ð2 1 1Þb =ð1 1 0Þa00 terrace interface expressed with respect to the reference state coordinate frame(XYZ) as follows:
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(ii) The total rigid body rotation (/) induced by the transformation disconnections and coherency dislocations can be deduced as follows:
bz tanðhÞ: ðB3Þ /¼ h (iii) The average spacing between the transformation disconnections can be determined as follows: Fig. A1. Crystal orientations in the reference state coordinate frame XYZ for the b and a00 crystals.
0 B B B A ¼ o o B @
pffiffi ab 2 ca00
0 0
0
0
ðB4Þ
References
1
C C C: pffiffiffiffiffiffiffiffiffiffiffiffi 0 C a2 00 þb2a00 A a 0 1 pffiffi ab 3
ðA3Þ
(iii) The matrix o Pa00 relates the relaxed a00 and the XYZ frame as follows: 1 0 ca00 pffiffi 0 0 C B ab 2 B pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiC B 2 2 C 2 2 a 00 þb 00 sinðwÞ 4a 00 þb 00 C B a a pffiffi a pffiffi a 0 C: o Pa00 ¼ B ab 3 ab 6 C B B pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi C A @ 2 cosðwÞ 4a2 00 þba00 pffiffia 0 0 a 6 b
ðA4Þ (iv) The matrix bUo is an orthogonal coordinate transformation matrix that relates the XYZ frame and b coordinate frame, 0 1 p1ffiffi p2ffiffi 0 3 6 B C B p1ffiffi p1ffiffi p1ffiffi C : ðA5Þ b Uo ¼ B 3 6C @ 2 A p1ffiffi2 p1ffiffi3 p1ffiffi6
Appendix B The following equations are reproduced from Refs. [22,23]. For more details, the readers are referred to the original references. (i) The inclination angle xT of the a00 habit plane to the ð2 1 1Þb =ð1 1 0Þa00 terrace plane is given by xT ¼ h /=2;
k ¼ h= sinðhÞ:
ðB1Þ
where h is the tan angle of the step height to the average terrace length, and is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 by þ b2y þ 4 ðbz Þðh eyy Þ A: h ¼ tan1 @ ðB2Þ 2 bz
[1] Kuroda D, Niinomi M, Morinaga M, Kato Y, Yashiro T. Mater Sci Eng A 1998;243:244. [2] Hanada S, Matsumoto H, Watanabe S. Intl Congress Series 2005;1284:239. [3] Niinomi M, Kuroda D, Fukunaga K, Morinaga M, Kato Y, Yashiro T, et al. Mater Sci Eng A 1999;263:190. [4] Niinomi M. Sci Technol Adv Mater 2003;4:445. [5] Okazaki Y, Gotoh E. Biomaterials 2005;26:11. [6] Ikeda M, Komatsu S, Nakamura Y. Mater Trans 2004;45:1106. [7] Miyazaki S, Kim HY, Hosoda H. Mater Sci Eng A 2006;438–440:18. [8] Baker C. Metal Sci J 1971;5:92. [9] Kim HY, Ikehara Y, Kim JI, Hosoda H, Miyazaki S. Acta Mater 2006;54:2419. [10] Fukui Y, Inamura T, Hosoda H, Wakashima K, Miyazaki S. Mater Trans 2004;45:1077. [11] Kim JI, Kim HY, Hosoda H, Miyazaki S. Mater Trans 2005;46:852. [12] Kim HY, Sasaki T, Okutsu K, Kim JI, Inamura T, Hosoda H, et al. Acta Mater 2006;54:423. [13] Kim HY, Hashimoto S, Kim JI, Inamura T, Hosoda H, Miyazaki S. Mater Sci Eng 2006;417:120. [14] Ping DH, Mitarai Y, Yin FX. Scripta Mater 2005;52:1287. [15] Kim JI, Kim HY, Inamura T, Hosoda H, Miyazaki S. Mater Sci Eng 2005;403:334. [16] Sakaguchi N, Niinomi M, Akahori T, Takeda J, Toda H. Mater Sci Eng C 2005;25:363. [17] Long Marc, Rack HJ. Biomaterials 1998;19:1621. [18] Wang Kathy. Mater Sci Eng A 1996;213:134. [19] Niinomi M. Mater Sci Eng A 1998;243:231. [20] Otsuka K, Wayman CM. Shape memory materials. Cambridge: Cambridge University Press; 1999. [21] Christian JW. The theory of transformation in metals and alloys: Part I. 2nd ed. Oxford: Pergamon Press; 2002. p. 286. [22] Pond RC, Celotto S, Hirth JP. Acta Mater 2003;51:5385. [23] Pond RC, Ma X, Chai YW, Hirth JP. Dislocations in solids. Amsterdam: Elsevier; 2007. p. 227 [chapter 74]. [24] Hammond C, Kelly PM. Acta Metall 1969;17:869. [25] Mahon GJ, Howe JM, Mahajan S. Philos Mag Lett 1989;59:273. [26] Moritani T, Miyajima N, Furuhara T, Maki T. Scripta Mater 2002;47:193. [27] Chen IW, Chiao YH. Acta Metall 1985;33:1827. [28] Hirth JP, Pond RC. Acta Mater 1996;44:4749. [29] Pond RC, Celotto S. Int Mat Revs 2003;48:225. [30] Bowles JS, Mackenzie JK. Acta Metall 1954;2:129. 138, 224. [31] Wayman CM. Introduction to the crystallography of martensitic transformations. New York: Macmillan; 1964. [32] Bowles JS, Mackenzie JK. Acta Metall 1957;5:134. [33] Bywater KA, Christian JW. Philos Mag 1972;25:1249. [34] Inamura T, Kim JI, Kim HY, Hosoda H, Wakashima K, Miyazaki S. Philos Mag 2007;87:3325. [35] Serra A, Pond RC, Bacon DJ. Acta Metall Mater 1991;39:1469.