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A TEM study of the habit plane structure of intragrainular proeutectoid α precipitates in a Ti–7.26 wt%Cr alloy
Acta Materialia 52 (2004) 2449–2460 www.actamat-journals.com
A TEM study of the habit plane structure of intragrainular proeutectoid a precipitates in a Ti–7.26 wt%Cr alloy F. Ye, W.-Z. Zhang *, D. Qiu Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, PeopleÕs Republic of China Received 11 November 2003; received in revised form 27 January 2004; accepted 27 January 2004
Abstract The crystallography and the structure of the habit plane of proeutectoid a precipitates in b matrix in a Ti–7.26 wt%Cr were studied using TEM. The orientation relationship was found to deviate slightly from the ideal Burgers orientation relationship. The habit plane of precipitates was found to be normal to a particular set of DgÕs. A set of dislocations, about 11 nm apart, were observed on the habit plane of the precipitates. The Burgers vector of the dislocations, as determined by g b contrast analyses, is ½1 1 1b =2ð½2 1 1 3a =6Þ. This Burgers vector is different from those determined by previous investigations. The observations are discussed in terms of an analytical O-line model and tend to support the prediction of the O-line model. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Phase transformations; Crystallography; Interface; Dislocation
1. Introduction Ti based alloys are used widely in aerospace, biomedical and energy applications due to their high strength, high fracture toughness and good corrosion resistance. The most commonly used Ti alloys consists dominantly of hcp a precipitates and bcc b matrix. The nature of the interfaces between the a and b phases is important to understand microstructure development during heat treatment process, as well as the properties of the plastic deformation, fracture and fatigue of Ti based alloy. The crystallography of a precipitates in b matrix has been examined by different investigators. According to these investigations [1–8], the orientation relationship (OR) between the a precipitates and the b matrix can be described by Burgers OR [9], i.e. ð0 0 0 1Þa ==ð0 1 1Þb ; ½1 1 0 0a ==½2 1 1b ;
ð1Þ
Recent studies have noticed that the OR is not the ideal Burgers OR but is slightly deviated from it [6,7]. The a precipitates usually have a shape of plate. The flat regions in the broad faces of the plates are referred as the habit plane. The habit plane was determined as ()13 11 11)b 1 by Furuhara et al. [3]. A single set of parallel dislocations were observed on the habit plane [2–8]. The spacing of these dislocations is about 12 nm and their direction is near [5 3 3]b [3]. Menon and Aaronson [2] first determined that the dislocations are of a-type with the Burgers vector of [)1 2 )1 0]a /3([)1 1 1]b / 2) 2. In a later study, Furuhara et al. [3] determined that the dislocations are of c-type with the Burgers vector of [0 0 0 1]a /2([0 )1 1]b /2). According to the contrast of high resolution transmission electron microscopy (HRTEM) image, Miyano et al. [8] suggested a Burgers vector of [0 1 0]b ([0 1 )1 )1]a /2) which is the same as that observed from a Zr–Nb alloy [10]. HRTEM were used to characterize the structure of the habit plane [3–8]. When viewed from [1 1 1]b or [1 1 )2 0]a direction as the dislo-
½1 1 2 0a ==½1 1 1b :
*
Corresponding author. Fax: +86-10-6277-1160. E-mail address: [email protected] (W.-Z. Zhang).
1 Specific indices are assigned in accordance with the particular variant of OR (Eq. (1)) in the following text. 2 A pair of corresponded vectors of lattices a and b are usually given together for convenience of comparison with other investigations.
1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.01.037
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cations are nearly edge-on, the extra plane of (0 )1 1)b corresponding to (0 0 0 2)a could evidently be observed at the spacing of about tens of layers of (0 )1 1)b or (0 0 0 2)a [4–8]. By a careful examination of the other low indexed planes in the HRTEM images [4,5], one finds difference in the matching condition between the correlated planes (1 )1 0)b /(1 )1 0 1)a and ()1 0 1)b /()1 1 0 1)a , i.e. only one pair of the planes is in mismatching condition. However, if the dislocations are of either a-type or c-type, in the HRTEM images viewed from [1 1 1]b / [1 1 )2 0]a , both of (1 )1 0)b and ()1 0 1)b extra planes should be equivalently observed because g b is not zero 3. Therefore, both a-type and c-type are conflict with the HRTEM observations [4,5]. Moreover, one can also notice that an extra plane of (2 )1 )1)b or (1 )1 0 0)a , which is the terrace plane of the structure ledge [3,11], is associated with the core of the dislocation, clearly indicating the dislocation should not be of c-type. Extensive experimental work has been made in another bcc/hcp alloy system, i.e. Zr–2.5 wt%Nb alloy [10,12–15]. A set of parallel linear dislocations were also observed in the habit plane [10,12–14]. When the OR is near Burgers OR, the habit plane normal and dislocation direction are similar to those observed from Ti based alloys [10]. The Burgers vector of [0 1 0]b ([0 1 )1 )1]a /2) was determined by g b contrast analyses [10]. Banerjee et al. [13] also considered the dislocations to be of hc þ ai type. The ratio of the lattice constants of the bcc and hcp phases in Ti–Cr and Zr–Nb, i.e. ab =aa , are 1.099 and 1.094, respectively, while the ratio ca =aa for the hcp phase in the two alloy systems are 1.587 and 1.589. Because these parameters are very close to each other, the Burgers vectors of the dislocations in the habit plane in the two systems might be similar. Considering the controversial conclusions for the Burgers vector of the dislocations from the previously investigations of the Ti alloys, the present study reexamined the crystallography and the structure of the habit plane in a Ti–7.26 wt%Cr alloy. The experimental results have been interpreted by different approaches. According to the invariant line criterion [16], a deviation from the ideal Burgers OR by a rotation of 0.5° around [0 0 0 1]a /[0 )1 1]b is necessary for producing an invariant line which determines the direction of the structure ledge [11]. Zhang and Purdy [17] proposed an O-line criterion to explain the irrational habit plane structures. When the criterion is met, a single set of dislocations locating alternately with the Olines can accommodate the interfacial misfit completely. They used this model to explain the crystallography of Zr–Nb alloy consistently. Recently, Qiu and Zhang [18] 3 g is a reciprocal vector represent the plane normal and b is the Burgers vector.
developed an analytical method to solve the O-lines and used this method to analyze the crystallography of fcc/ bcc systems systematically. While the invariant line criterion joined with the structure ledge model may also yield the O-lines, the Burgers vector of the model is limited to lie on the terrace plane. In contrast, the O-line criterion is not restricted by such a limitation. In the present work, we have used the analytical O-line model to interpret the crystallography of proeutectoid a precipitates in Ti–Cr alloy.
2. Experimental procedure 2.1. Specimen preparation A Ti–7.26 wt%Cr alloy is selected for the experiment study in this work. The composition is close to that of the alloys used in previous studies [1–4]. The alloy was prepared by arc melting in an argon atmosphere. Then it was forged into a bar of 6 6 mm. After encapsulated in quartz tube under vacuum of less than 103 Pa, the bar was homogenized for three days at 1273 K and then quenched in water after breaking the tube. The bar was cut into specimens of 6 6 6 mm in size for further heat treatment. The specimen for the TEM study was heated at 1273 K for 30 min in the b region, and then isothermally reacted at 973 K for 1 h in the a þ b region followed by water quench. Slices of 0.5 mm thick were electric discharge machined from the specimen. The slices were ground to 20 lm thick and final TEM foils were prepared by ion milling with a Gatan ion-miller. Electrolytic thinning was not used to avoid the precipitation of interphase phases at a=b interface [19,20]. 2.2. TEM analysis TEM observation was performed with a JEOL JEM200CX at 200 kV. The OR between a precipitates and b matrix was determined by analyzing the Kikuchi patterns taken from both phases. Kikuchi patterns are usually produced in single crystal by convergent beam electron diffraction method. Because the width of a plates in the present study were about 0.5 lm, the Kikuchi pattern could also be taken from selected area diffraction (SAD) method when the selected area was smaller than a plates. To minimize the effect of the foil bend, two Kikuchi patterns of different phases have been taken from areas near the interface. In order to measure the angle between two planes of different phases precisely, two Kikuchi bands in different patterns of different phases should both pass through the transmission spot as far as possible. After overlap the two negatives taken from the two phases by alignment of the
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numbering readout shadows, the angle between the Kikuchi bands is the angle between the corresponding planes in different phases and the beam direction gives the rotation axis of the planes. The angle could also be observed from the superimposed SAD pattern in a single negative plate, in which the angle is the deviation between the reflection spots corresponding to the Kikuchi bands. Since the age temperature are lower than xs (x start) temperature, x reflections often present when SAD is performed in b phase due to very fine x particles precipitated from b phase. However, because the x particles are very fine and the density is low, the Kikuchi patterns of b phase could remain distinguishable. The habit plane normal and the direction of the dislocations were determined from the edge-on orientation. The g b contrast analysis was performed on the dislocations in the habit plane. The imaging condition was carefully controlled in a strong two-beam or systematic row diffraction condition. In most cases, one crystal was in the
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above condition and the other crystal in a weakly diffracting condition. Strong b reflections {0 2 0}b and {1 1 0}b were mostly used in the g b contrast analysis. Although x reflections may occur, the g b contrast analysis was effectively unaffected. On the basis of the particular variant of OR given in Eq. (1), planes and zone axes are indexed according to the angles between planes, zone axes in different phases, as well as the angle between traces of the habit plane and the dislocations viewed at different zone axes. Due to the twofold symmetry of the [0 )1 1]b /[0 0 0 1]a axis at the Burgers OR, there remains a twofold ambiguity in indexing the planes and directions. However, because of the existence of small deviation from the ideal Burgers OR, these angles are different even though these planes and zone axes are symmetrically equivalent if the phases are related by the exact Burgers OR. Specific indices could be identified according to a careful study of these angles.
Fig. 1. Kikuchi patterns taken from (a) b near [1 1 1]b , (b) a near [1 1 )2 0]a , (c) b near [)1 )2 0]b and (d) a near [)2 )2 4 3]a .
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3. Results 3.1. Orientation relationship Based on the observation of around 50 plates, all examined a plates have an identical OR with respect to the b phase. Careful examination constantly detect that the OR is deviated slightly from the ideal Burgers OR. As explained in Section 2.2, we used the Kikuchi patterns to determine the rotation angle and the rotation axis of two close-packed planes of different phases. Fig. 1 shows the Kikuchi patterns of two phases near interface. The solid (dashed) lines in the figure are the perpendicular bisectors of the Kikuchi bands of b (a) phase. A pair of lines corresponding to the Kikuchi bands of different phases intersect at the right of Fig. 1(b) or (d) and the deviation can be seen clearly at the left of Fig. 1(a) or (c). It can be seen that the OR is very close to the Burgers OR [9], and it can be described by
recorded. The habit plane normal can be determined from any of the SAD patterns. Among these edge-on positions, we found a set of orientations in which the habit plane is normal to a particular set of DgÕs. A Dg is the difference between two reciprocal vectors ga and gb of two phases. To reach an orientation for precise determination of the Dg in a SAD pattern, the TEM foil was rotated to the position such that Kikuchi bands of ga and gb pass through the transmission spot. The relationship between DgÕs and habit plane was observed in Zr–Nb alloy [10] and explained in terms of a property of the O-line [17]. Figs. 2(a) and (b) show two edge-on positions of one a plate whose habit plane is normal to Dgð0 1 1Þb ¼ gð0 1 1Þb gð0 1 1 0Þa and Dgð1 1 2Þb ¼ gð1 1 2Þb gð1 2 1 1Þa . It can also be seen that another Dgð2 0 0Þb ¼ gð2 0 0Þb gð2 1 1 0Þa is apparently normal to the habit plane in Fig. 2(a). However, Dgð2 0 0Þb actually
ð0 0 0 1Þa 0:6° from ð0 1 1Þb towards ð1 1 0Þb with a rotation axis ½1:3 1 1b ð ==½1:2 0:8 2 0a Þ; ð1 1 0 0Þa 0:7° from ð2 1 1Þb towards ð0 0 2Þb with a rotation axis ½1 2:1 0:1b ð ==½2 2 4 3:1a Þ: ð2Þ Because the angles between these two pairs of closepacked planes are both very small and the contrast of Kikuchi bands corresponding to low indexed planes near the transmission spot are very weak, it is difficult to rotate the foil to the precise rotation axis, i.e. the error of the rotation axis should be large (around several degrees). Therefore, the solid line corresponding to (0 )1 1)b in Fig. 1(a) is deviated slightly from the transmission spot. However, this deviation does not significantly affect the result of the rotation angle. Though the rotation angles are very small and a certain degree of scattering (approximately 0.3°) exists due to either the measurement uncertainty or any true scattering, a deviation from exact parallelism between (0 0 0 1)a and (0 )1 1)b , and between (1 )1 0 0)a and (2 )1 )1)b is evidently detected. 3.2. Orientations of the habit plane and dislocations The broad face of an a plate usually defines the habit plane of the plate. Since the broad face in microscopic scale is not strictly flat in a large scale, the habit plane is usually measured in the flat regions of the broad face. To determine the habit plane normal precisely, we rotated the habit plane to the edge-on orientation. When the TEM foil was rotated around the habit plane normal from one edge-on position, a series of SAD patterns corresponding to different edge-on orientations could be
Fig. 2. The determination of the edge-on habit plane by (a) Dgð0 1 1Þb ¼ gð0 1 1Þb gð0 1 1 0Þa and (b) Dgð1 1 2Þb ¼ gð1 1 2Þb gð1 2 1 1Þa . (c) A near edge-on position of the habit plane when Dgð1 0 1Þb ¼ gð1 0 1Þb gð1 0 1 1Þa .
F. Ye et al. / Acta Materialia 52 (2004) 2449–2460
deviates from the habit plane normal because the Kikuchi bands of gð2 0 0Þb and gð2 1 1 0Þa deviate from the transmission spot. Based on the Kikuchi patterns corresponding to the diffraction pattern in Fig. 2(a), the beam direction is approximately [)0.0224 1 )1]b (// [)0.0081 )0.0033 0.0114 )1]a ) near [0 1 )1]b ([0 0 0 )1]a ). The habit plane normal is ()13 10.2 10.1)b (// ()3.6 4.6 )1 )0.1)a ). The measurements of habit plane were found to scatter within 3° due to either the measurement uncertainty or any true scattering. However, the habit plane normal to a particular set of DgÕs is reproducible. The broad face of different a plates was observed to have a consistent structure that is dominate with a set of regularly spaced dislocations. It is reasonable to con-
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sider the periodic dislocations to be the characteristic feature of the habit plane. The spacing of the dislocations is about 11 nm. The dislocation direction is determined mainly by edge-on method. It is scattered around [5 2.6 2.8]b (//[3.2 1 )4.2 0.1]a ) with uncertainty of approximately 5°. The Burgers vector of the dislocations is analyzed in the following section. 3.3. Determination of Burgers vector of the dislocations The Burgers vector of the regularly spaced dislocations was determined by a g b contrast analysis by mainly used {0 2 0}b and {1 1 0}b reflections. Three sets of dark field images of the habit plane are shown in Figs. 3–5. Each set was taken from a single a plate.
Fig. 3. Dark field micrographs of a habit plane generated by (a) g ¼ 0 )1 1b , (b) g ¼ 0 2 0b , and (c) g ¼ 0 0 )2b near [1 0 0]b zone axis; (d) g ¼ 1 0 1b , and (e) g ¼ 0 1 1b near [1 1 )1]b zone axis.
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Fig. 4. Dark field micrographs of a habit plane generated by (a) g ¼ 1 )1 0b , (b) g ¼ 1 1 0b , (c) g ¼ 2 0 0b , (d) g ¼ 0 2 0b near [0 0 1]b zone axis.
Fig. 5. Dark field micrographs of a habit plane generated by (a) g ¼ )1 0 )1b , (b) g ¼ )1 0 1b , (c) g ¼ )2 0 0b near [0 1 0]b zone axis.
F. Ye et al. / Acta Materialia 52 (2004) 2449–2460
The broad face also contains a set of irregularly spaced linear defects besides the regularly spaced dislocations as indicated by small arrows in Fig. 3(e). Since the regular-spaced dislocations were considered as the habit plane structure, the contrast analysis and the later explanation were focused on these regularly spaced dislocations. The contrast of the dislocations are strong when g ¼ (0 )1 1)b ((0 0 0 2)a ) (Fig. 3(a)) and (1 )1 0)b ((1 )1 0 1)a ) (Fig. 4(a)) were employed. The dislocations were always out of contrast when g ¼ (0 1 1)b ((0 1 )1 0)a ) (Fig. 3(e)) was employed. These results are consistent with the contrast analysis by Furuhara et al. [3] and in conflict with the Burgers vector of [)1 1 1]b /2([)1 2 )1 0]a ) [2]. Therefore, the dislocations should not be of a-type. The contrast of dislocations was also out of contrast or very weak when g ¼ (1 1 0)b ((1 0 )1 )1)a ) (Fig. 4(b)) and ()1 0 1)b (()1 1 0 1)a ) (Fig. 5(b)) were employed. The dislocations were always visible when g ¼ (2 0 0)b ((2 )1 )1 0)a ) (Figs. 4(c) and 5(c)) was employed. These results are conflict with the Burgers vector [0 )1 1]b /2/ ([0 0 0 1]a /2) [3]. Therefore, the dislocations should not be of c-type. An alternative choice of the Burgers vector could be [0 1 0]b ([0 1 )1 )1]a /2) as that defined for dislocations in a Zr–Nb alloy [10]. However, the facets that the dislocations are consistent invisible by using g ¼ (0 1 1)b ((0 1 )1 0)a ) (Fig. 3(e)) and visible by using g ¼ (1 0 1)b ((1 0 )1 1)a ) (Figs. 3(d) and 5(a)) are in conflict with the Burgers vector of [0 1 0]b . Therefore, the Burgers vector of the dislocations is not the same as that in the Zr–Nb alloy. Table 1 summarized the observed results of the g b contrast analysis and they are compared with the g b results for different Burgers vectors. The corresponding ga Õs are also given in the table. Based on the above consideration and invisible results summarized in Table 1, we suggest that the Burgers vector of the dislocations is more properly assigned to be [1 )1 1]b /2([2 )1 )1 3]a / 6). The angle between the Burgers vector and the direction of the dislocations is about 61.5°. It reveals that the dislocations are mixed dislocations.
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4. Discussion 4.1. Comparing with previous experimental results 4.1.1. Comparing with Ti–X system The present experimental results are given in Table 2 and compared with the previous experimental results in [3]. The OR in [3] is not given in the table because the deviation from the ideal Burgers OR was not determined in this reference. According to the careful measurement in this work, the OR is near the ideal Burgers OR, but none of the parallelism relationship described in the Burgers OR is exact. This deviation is also determined in recent investigations in which the deviations of [0 )1 1]b /[0 0 0 1]a and [1 1 1]b /[1 1 )2 0]b are 0.78° and 0.56° [6] or 0.31° and 0.34° [7] with the rotation axes near [1 1 1]b and [0 )1 1]b , respectively [6]. The deviation between [1 1 1]b and [1 1 )2 0]b in the present experiment is 0.5° and the rotation axis is also very close to [0 )1 1]b . The OR determined in the present work is similar to that in previous works [6,7] and the difference between them is possibly due to the experiment error and difference in the compositions. The habit plane normal and the invariant line direction have been measured with great effort for improving precision and they are in good agreement with the previous experiments [1,3,5–8] within experimental uncertainty. The Burgers vector of the dislocations in the habit plane characterized in this work is [1 )1 1]b / 2([2 )1 )1 3]a /6) which is different from the results in previous experiments [2,3,8]. However, this Burgers vector is consistent with the HRTEM observations when viewed from [1 1 1]b /[1 1 )2 0]a [4–8]. In the HRTEM images (0 )1 1)b and (0 0 0 2)a mismatch evidently at the core of the dislocations. Moreover only one pair of planes (1 )1 0)b /(1 )1 0 1)a and ()1 0 1)b / ()1 1 0 1)a is mismatch but the other pair is not [4,5]. In addition, the extra plane (2 )1 )1)b or (1 )1 0 0)a , which is the terrace plane of the structure ledge [3,11], is associated with the core of the dislocations [4,5]. This result is better explained by the Burgers vector determined in the present study because
Table 1 Comparison between observed and theoretical contrast results of the dislocations for different Burgers vectors Fig(s).
gb
ga
Observed contrast
g b for bL ¼ [1 )1 1]b /2
g b for bL ¼ [)1 1 1]b /2
g b for bL ¼ [0 )1 1]b /2
g b for bL ¼ [0 1 0]b
3(a) 3(d) and 5(a) 4(a) 3(e) 4(b) 5(b) 4(c) and 5(c) 3(b) and 4(d) 3(c)
0 )1 1 101 1 )1 0 011 110 )1 0 1 200 020 0 0 )2
0002 1 0 )1 1 1 )1 0 1 0 1 )1 0 1 0 )1 )1 )1 1 0 1 2 )1 )1 0 0 1 )1 )2 0 )1 1 )2
Strong Medium Strong None None None Medium Medium Medium
1 1 1 0 0 0 1 )1 )1
0 0 )1 1 0 1 )1 1 )1
1 1/2 1/2 0 )1/2 1/2 0 )1 )1
)1 0 )1 1 1 0 0 2 0
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Table 2 The experiment results, the O-line model calculation results and the difference between the present experiment results and other results bL
OR
Habit plane
Invariant line
Ddisl (nm)
c ¼ 0:6° d ¼ 0:8° c ¼ 0:8° d ¼ 0:9° c ¼ 0° d ¼ 0:5° c ¼ 0:6° d ¼ 0:7° –
()13 11.5 10.3)b ()2.9 3.9 )1 )0.3)a ()13 9.6 11.9)b ()3.0 4.0 )1 0.5)a ()13 10.9 10.9)b ()3.0 4.0 )1 0)a ()13 10.2 10.1)b ()3.6 4.6 )1 )0.1)a ()13 11 11)b
[5 2.8 3.2]b [2.5 1 )3.5 0.2]a [5 2.7 3.3]b [2.5 1 )3.5 0.3]a [5 3.0 3.0]b [2.5 1 )3.5 0]a [5 2.6 2.8]b [3.2 1 )4.2 0.1]a [5 3 3]b
10.9
O-line with [1 )1 1]b
[1 )1 1]b /2 [2 )1 )1 3]a /6 [0 1 0]b [0 1 )1 )1]a /2 [0 )1 1]b /2 [0 0 0 1]a /2 [1 )1 1]b /2 [2 )1 )1 3]a /6 [0 )1 1]b /2 [0 0 0 1]a /2 Same
3.0°
3.2°
0.1
O-line with[0 1 0]b
Different
5.1°
3.8°
0.8
O-line with [0 )1 1]b /2
Different
2.0°
3.2°
0.8
Exp. [3]
Different
c ¼ 0° d ¼ 0:1° c ¼ 0:2° d ¼ 0:2° c ¼ 0:6° d ¼ 0:2° –
2.3°
3.2°
1
Calculation results
Present experiment Previous experiment [3] Comparision with present experimental results
11.8 11.2 11 12
c – Angle between (0 0 0 1)a and (0 )1 1)b . d – Angle between (1 1 0 0)a and (2 )1 )1)b .
gð0 1 1Þb bL½1 1 1b=2 ¼ 1, gð1 1 0Þb bL½1 1 1b=2 ¼ 1, gð1 0 1Þb bL½1 1 1b=2 ¼ 0 and gð2 1 1Þb bL½1 1 1b=2 ¼ 1. This new Burgers vector is also consistent with the relationship between the habit plane and DgÕs. According to the property of the O-line [17], if the habit plane contains a set of dislocations, this habit plane must be normal to a group of DgÕs whose related gÕs must lie in the zone axis of the Burgers vector of the dislocations. As shown in Fig. 2, two DgÕs, i.e. Dgð0 1 1Þb and Dgð1 1 2Þb , are normal to the habit plane and the gð0 1 1Þb and gð1 1 2Þb lie in the zone axis of the [1 )1 1]b /2 suggesting the Burgers vector of the dislocations in the habit plane is [1 )1 1]b /2. Fig. 2(c) gives another Dgð1 0 1Þb ¼ gð1 0 1Þb gð1 0 1 1Þa . gð1 0 1Þb is normal to [)1 1 1]b /2 and [0 1 0]b . The habit plane at this condition deviates slightly from edge-on orientation. This deviation is observed consistently on different a plates and hence the Burgers vector is unlikely among these vectors. 4.1.2. Comparing with Zr–Nb system Because the ratios of the lattice constants in Ti–Cr and Zr–Nb systems are similar, it may be expected that the crystallography in the two systems are also similar. However, the OR, habit plane and dislocation direction in the systems are similar but the Burgers vector in Zr– Nb system is [0 1 0]b ([0 1 )1 )1]a /2) which is quite different from that in Ti–Cr system. The previous HRTEM experiment [4–8] is also consistent with the Burgers vector of [0 1 0]b ([0 1 )1 )1]a /2), because gð0 1 1Þb bL½0 1 0b ¼ 1, gð1 1 0Þb bL½0 1 0b ¼ 1, gð1 0 1Þb bL½0 1 0b ¼ 0 and gð2 1 1Þb bL½0 1 0b ¼ 1. However, it can be seen from Table 1 that this Burgers vector is in conflict with the results of the g b analysis. It is also inconsistent with the description of the habit plane in terms of DgÕs. It remains to be an open question that
why the Burgers vectors of the dislocations in these two systems are different. 4.2. O-line model calculation When the habit plane of a transformation system contains a set of parallel dislocations, the crystallography of the system should be well explained by the O-line model [17]. We also observed a set of DgÕs normal to the habit plane in agreement with the property of the Olines [17]. In this section, we use the analytical O-line model [18] to explain the crystallography and the habit plane structure between a and b phases in Ti–Cr alloy. The calculation procedure is similar to that for fcc/bcc system [18]. The details of the calculation for the present alloy are summarized in Appendix A The calculation procedure needs to select a Burgers vector as input data. The Burgers vectors of lattice b are possibly one of three h1 0 0ib or one of four h1 1 1ib /2. Among these possible Burgers vectors, [1 )1 1]b / 2([2 )1 )1 3]a /6) and [1 1 )1]b /2([2 )1 )1 )3]a /6), [1 1 1]b / 2([1 1 )2 0]a /3) and [)1 1 1]b /2([)1 2 )1 0]a /3), [0 1 0]b ([0 1 )1 )1]a /2) and [0 0 1]b ([0 1 )1 1]a /2) are symmetrically equivalent according to the lattice correspondence and can give O-line solutions which correspond to the OR near the particular variant of Burgers OR described by Eq. (1), while the O-lines are not solvable for [1 0 0]b ([2 )1 )1 0]a /3). Therefore, we only examine the O-line solutions for the Burgers vectors of [1 )1 1]b / 2([2 )1 )1 3]a /6), [0 1 0]b ([0 1 )1 )1]a /2) and [1 1 1]b / 2([1 1 )2 0]a /3). The maximum spacing of the dislocations given by the O-line solution for bL ¼ [1 1 1]b /2 is 2.4 nm which is too small compared with the observed result. Therefore, we only compared the other two O-line solutions.
F. Ye et al. / Acta Materialia 52 (2004) 2449–2460
The calculation results are given and compared with the experimental observations in Table 2. Only deviation angles of the ORs are given and compared in the table. The rotation axes of the close-packed planes for the O-line solution of bL ¼ [1 )1 1]b /2 are [1.4 1 1]b (// [1.3 0.7 )2 0]a ) for planes (0 0 0 1)a and (0 )1 1)b , and [)1 )2.0 0.03]b (//[)2 )2 4 3.2]a ) for planes (1 )1 0 0)a and (2 )1 )1)b . The difference angles of the rotation axes between the experimental (Eq. (2)) and the O-line solution for bL ¼ [1 )1 1]b /2 are 2.1° and 2.0° corresponding to the two pairs of the close-packed planes, respectively. The OR of the O-line solution for bL ¼ [1 )1 1]b /2 are similar to that for bL ¼ [0 1 0]b . It is interesting that the other results for different Burgers vectors are also quite close to each other in contrast to the fcc/bcc systems, for which the calculated habit plane and the dislocation direction usually change obviously with the Burgers vectors [18]. The differences in the calculated habit planes and the dislocation directions corresponding to different Burgers vectors are within the experiment error. Thus the observed habit plane, the direction and spacing of the dislocations can be explained by the Oline solutions with either Burgers vectors. Based on the present experimental results of the g b contrast analysis and the relationship between DgÕs and the habit plane, we suggest that the Burgers vector of the dislocations should be [1 )1 1]b /2([2 )1 )1 3]a /6). Furuhara et al. [3] determined a Burgers vector of the dislocations as [0 )1 1]b /2([0 0 0 1]a /2). When Burgers vector of [0 )1 1]b /2 is used to define the O-lines, our calculated results confirm the results from the previous study of invariant line model calculation [11]. These results are also close to those of the other O-line solutions, as given in Table 2. However, the OR corresponds to [0 )1 1]b being parallel to [0 0 0 1]a , which is inconsistent with the experimental observation. As mentioned in previous sections, this Burgers vector is also conflict with the HRTEM observation [4,5] and the results of the g b contrast analysis. Therefore, the description of the dislocations with the Burgers vector of [0 )1 1]b /2([0 0 0 1]a /2) is likely incorrect.
5. Conclusion 1. The OR between a and b in a Ti–7.26 wt%Cr alloy is not at the exact Burgers OR. It can be described by (0 0 0 1)a 0.6° from (0 )1 1)b and (1 )1 0 0)a 0.7° from (2 )1 )1)b . The habit plane determined by edge-on method is normal to a particular set of DgÕs. Its orientation is ()13 10.2 10.1)b (//()3.6 4.6 )1 )0.097)a ). 2. The habit plane is characterized by a single set of dislocations at spacing of 11 nm. The direction of the dislocations is approximately [5 2.6 2.8]b (// [3.2 1 )4.2 0.1]a ). The Burgers vector of the dislocations is determined to be [1 )1 1]b /2([2 )1 )1 3]a /6).
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This Burgers vector is different from the previous conclusions [2,3,8]. 3. The experimental observations are consistent with the calculation results from an analytical O-line model [18]. However, because the O-line calculation results for different Burgers vectors are close to each other, it is difficult to distinguish the Burgers vectors according to the comparison of the calculation data.
Acknowledgements The support of National Natural Science Foundation of China Grant Nos. 59871021 and 50271035 is gratefully acknowledged.
Appendix A The calculation procedure of analytical O-line model for bcc/hcp system summarized below is similar to that for fcc/bcc system except the construction of the pure strain. More detailed calculation procedure and theoretical explanation can be found in [18]. (1) Construct the pure strain Any O-line transformation strain A can be decomposed into A ¼ RB:
ðA:1Þ
B is the pure strain, which is expressed as a diagonal matrix. The rotation matrix R defines an OR corresponding to the O-line condition. The first step of the calculation is to construct the pure strain B. The pure strain in fcc/bcc system is the Bain strain [21]. In bcc/ hcp system, it is constructed by different method but it can also be determined from a given OR as explained below. We start to construct the transformation strain at the ideal Burgers OR. Three non-coplanar vectors in lattice b are selected as [1 0 0]b , [1 1 1]b /2 and [0 )1 1]b . The correlated vectors by the transformation are [2 )1 )1 0]a / 3, [1 1 )2 0]a /3 and [0 0 0 1]a according to the nearest neighbor principle [22]. In three indices systems, they can be expressed in SLb and SLa as column vectors, 2 3 2 3 1 1=2 0 1 1 0 SLb ¼ 4 0 1=2 1 5 and SLa ¼ 4 0 1 0 5: 0 1=2 1 0 0 1 ðA:2Þ The superscript L indicates that the vector is expressed on basis of the crystal lattice. Three non-coplanar vectors are chosen as the axis of a new orthogonal coordinates named N1 which is x//[2 )1 )1]b //[1 )1 0 0]a , y//[1 1 1]b //[1 1 )2 0]a and z//[0 )1 1]b //[0 0 0 1]a . We can expressed selected lattice vectors as column vectors in the orthogonal coordinates
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2 qffiffi
2 3
0 pffiffi
6 qffiffi S b ¼ ab 6 4 1
3 2
3
0
0
0
3
7 7 0 5 pffiffiffi 2
2 pffiffi3 and
Sa ¼ aa 4
2 1 2
0
0 1 0
3 0 0 5:
ca aa
ðA:3Þ The lattice constants we used are the same as those used by previous investigators [2,3,11], i.e. ab ¼ 0:325 nm and aa ¼ 0:29564 nm, ca ¼ 0:46928 nm. Then the transformation matrix Ab for the ideal Burgers OR can be determined as 1
Ab ¼ Sa ðSb Þ :
ðA:4Þ
where bð¼ Qb bL ) is Burgers vector of lattice b expressed in N2 and xB is transformed from x due to the pure strain, as defined by 0
xB ¼ ðB1 Þ x :
ðA:9dÞ
(3) Construct an O-line strain A0 According to Eq. (A.1), to construct an O-line strain A0 , lattice a should be rotated after the transformation of the pure strain so that A0 can satisfy with the following invariant line condition 0 ðA1 0 Þx ¼ x :
ðA:10Þ
Following the method developed in the PTMC [23], A0 can be determined from A0 ¼ R1 R02 B;
ðA:11aÞ
Sa , Sb and the coordinates N1 is the same as that used in the O-line calculation in Zr–Nb alloy (but with a different OR variant) [17]. The transformation can be expressed as
R1 and R2 can be defined as R1 ¼ ½x ; bu ; x bu ;
ðA:11bÞ
xa ¼ Ab xb :
R2 ¼ ½xB ; bB ; xB bB ;
ðA:11cÞ
ðA:5Þ
Let xb in Eq. (A.5) to be a unit vector so that all points xa defined an ellipsoid. The directions and lengths of the axes of the ellipsoid can be expressed by the eigenvectors and eigenvalues of a real symmetric matrix F ¼ ðA0b Ab Þ
1=2
¼ PBP1 :
ðA:6Þ
B is a diagonal matrix contains three eigenvalues of F and P contains three eigenvectors of F as column vectors. In the orthogonal coordinates N2 defined by the column vector in P, the pure strain transformation can be expressed by xa ¼ Bxb :
ðA:7Þ
where bu ¼ b=jbj, bB ¼ Bb=jBbj, respectively. (4) Narrow the selection of O-line solution When a is rotated around x , numerous O-line solutions can be generated. To restrict the choice, Zhang and Purdy [17] proposed two possible criterions. One is that the spacing of the dislocations should be large. The other is that deviation from the Burgers OR should be small. In most cases as the present system, these two criterions will give close results. We select the first criterion in the following calculation. To rotate lattice a around x , we select three column vectors in R1 as axis of an orthogonal coordinates N3 . Then the rotation matrix in this coordinates is 2 3 1 0 0 R3 ¼ 4 0 cos h sin h 5: ðA:12Þ 0 sin h cos h
The following calculation is made in this orthogonal coordinates N2 . The coordinate conversion matrix of lattice b in this coordinates, as well as in the latter calculation, is
where a is the rotation angle. The transformation matrix expressed in the coordinates N3 becomes
Qb ¼ P1 Sb SL1 b :
AN3 ¼ R3 ðR01 A0 R1 Þ:
ðA:8Þ
The following calculation steps are the same as that used for the fcc/bcc system [18]. (2) Determine the invariant line in reciprocal space x The O-line condition requires that the Burgers vector should be normal to the invariant line in reciprocal space x [17]. In addition, the length of x is not changed by pure strain. Therefore, the unit vector x can be determined by solve the following equations: x0 B xB ¼ 1
ðA:9aÞ
x0 x ¼ 1
ðA:9bÞ
0
b x ¼ 0;
ðA:9cÞ
ðA:13Þ
If we still express the transformation in coordinates N2 , the transformation matrix is A ¼ R1 AN3 R01 ¼ R1 R3 R02 B:
ðA:14Þ
The coordinate conversion matrix of lattice a is determined by Qa ¼ AQb SLb SL1 a :
ðA:15Þ
The habit plane normal is Dgp ¼ T0 gb ;
ðA:16Þ
where T ¼ I A1 and gb is a reciprocal vector normal to b, i.e. gb b ¼ 0.
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0:2941 0:1214 3:489 10 4 2:964 102 0:2696 3:379 103 5 3 3 2:400 10 2:844 10 0:4693 2 3 0:2941 0:1215 4:772 103 4 2:949 102 0:2695 4:547 103 5 3:277 103 3:847 103 0:4692 2 3 0:2941 0:1213 0 4 2:980 102 0:2696 0 5 0 0 0:4693
A 2 3 3
0 0 0:2298 0:2298 5 0:2298 0:2298 3 0 0 0:2298 0:2298 5 0:2298 0:2298 3 0 0 0:2298 0:2298 5 0:2298 0:2298
Qa 2
2
Oc ¼ T0 b=jbj :
ðA:18Þ
The O-cell walls locate at the poor matching regions which are alternated with the O-lines, the good matching regions. Different A and hence different Ddisl are obtained as angle a is varied in a range of )15° to 15°. The optimum OR is determined when the spacing of dislocations reaches its maximum value. The corresponding dislocation direction, which must be parallel to the invariant line x in direct space, is an eigenvectors of A corresponding to the eigenvalue 1, because Ax ¼ x. A direct vector or reciprocal vector on the lattice basis can be expressed on the orthogonal coordinates by xi ¼ Qi xLi
0
L or gi ¼ ðQ1 i Þ gi ;
ðA:19Þ
where i represents lattice a or b. The conversion matrix is convenient for calculating the angles between vectors in hcp lattice or in different lattices by expressing these vectors in the orthogonal coordinates N2 [25]. Then OR is determined by this method. The calculation results of the OR, the habit plane, and the direction and spacing of the dislocations corresponding to different O-line solutions for different Burgers vectors are given in Table 2. Table 3 gives the coordinate conversion matrix and the transformation strain matrix. Qb Õs for different cases are same since the calculation procedure of Qb are same. Some elements of Qa or A are similar for different cases, because the ORs for different cases are similar.
3
[0 )1 1]b /2
[0 0 0 1]a /2
[0 1 )1 )1]a /2
[0 1 0]b
[1 )1 1]b /2
[2 )1 )1 3]a /6
Qb 2 0:3250 4 0 0 2 0:3250 4 0 0 2 0:3250 4 0 0 bL
Table 3 Coordinates conversion matrixes and transformation strain matrixes
ðA:17Þ
Oc is the reciprocal vector that defines a set of O-cell walls determined by [24] 3 0:1123 0 1:108 0 5 0 1:021
0:1112 1:108 9:611 103
Ddisl ¼ jDgp j=jOc Dgp j:
0:9050 4 9:121 102 7:386 103 2 0:9051 4 9:072 102 1:008 102 2 0:9050 4 9:170 102 0
0:1118 1:108 7:151 103
3 7:591 103 3 5 7:352 10 1:021 3 1:038 102 9:893 103 5 1:021
The spacing of O-lines or dislocations on the habit plane is [24]
References [1] Menon ESK, Aaronson HI. Acta Metall 1986;34:1963. [2] Menon ESK, Aaronson HI. Acta Metall 1986;34:1975. [3] Furuhara T, Howe JM, Aaronson HI. Acta Metall Mater 1991;39:2873. [4] Furuhara T, Ogawa T, Maki T. Philos Mag Lett 1995;72:175. [5] Furuhara T, Makino T, Idei Y, Ishigaki H, Takada A, Maki T. Mater Trans JIM 1998;39:31. [6] Mills MJ, Hou DH, Suri S, Viswanathan GB. In: Pond RC, Clark WAT, King AH, Williams DB, editors. Boundaries and Interfaces in Materials: The David A Smith Symposium. Warrendale, PA: The Minerals, Metals & Materials Society; 1998. p. 295. [7] Miyano N, Ameyama K, Weatherly GC. ISIJ Int 2000;40:S199. [8] Miyano N, Ameyama K, Weatherly GC. Mater Trans 2002;43:1547. [9] Burgers WG. Physica 1934;1:561. [10] Zhang WZ, Purdy GR. Acta Metall Mater 1993;41:543. [11] Furuhara T, Aaronson HI. Acta Metall Mater 1991;39:2857.
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[12] Zhang WZ, Perovic V, Perovic A, Weatherly GC, Purdy GR. Acta Mater 1998;46:3443. [13] Banerjee S, Dey GK, Srivastava D, Ranganathan S. Metall Mater Trans A 1997;28:2201. [14] Perovic V, Weatherly GC. Acta Metall 1989;37:813. [15] Luo CP, Weatherly GC. Metall Trans A 1988;19:1153. [16] Dahmen U. Acta Metall 1982;30:63. [17] Zhang WZ, Purdy GR. Philos Mag 1993;68:291. [18] Qiu D, Zhang WZ. Philos Mag 2003;83:3093. [19] Banerjee D, Wiliams JC. Scr Metall 1983;17:1125.
[20] Banerjee D, Shelton CG, Ralph B, Williams JC. Acta Metall 1988;36:125. [21] Bain EC. Trans. Metall Soc AIME 1924;70:25. [22] Bollmann W. Crystal defects and crystalline interfaces. Berlin: Springer; 1970. [23] Wayman CM. Introduction to the crystallography of martensitic transformation. New York: MacMillan; 1964. [24] Zhang WZ, Purdy GR. Philos Mag 1993;68:279. [25] Zhang WZ, Ye F, Zhang C, Qi Y, Fang HS. Acta Mater 2000;48:2209.
A TEM study of the habit plane structure of intragrainular proeutectoid a precipitates in a Ti–7.26 wt%Cr alloy F. Ye, W.-Z. Zhang *, D. Qiu Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, PeopleÕs Republic of China Received 11 November 2003; received in revised form 27 January 2004; accepted 27 January 2004
Abstract The crystallography and the structure of the habit plane of proeutectoid a precipitates in b matrix in a Ti–7.26 wt%Cr were studied using TEM. The orientation relationship was found to deviate slightly from the ideal Burgers orientation relationship. The habit plane of precipitates was found to be normal to a particular set of DgÕs. A set of dislocations, about 11 nm apart, were observed on the habit plane of the precipitates. The Burgers vector of the dislocations, as determined by g b contrast analyses, is ½1 1 1b =2ð½2 1 1 3a =6Þ. This Burgers vector is different from those determined by previous investigations. The observations are discussed in terms of an analytical O-line model and tend to support the prediction of the O-line model. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Phase transformations; Crystallography; Interface; Dislocation
1. Introduction Ti based alloys are used widely in aerospace, biomedical and energy applications due to their high strength, high fracture toughness and good corrosion resistance. The most commonly used Ti alloys consists dominantly of hcp a precipitates and bcc b matrix. The nature of the interfaces between the a and b phases is important to understand microstructure development during heat treatment process, as well as the properties of the plastic deformation, fracture and fatigue of Ti based alloy. The crystallography of a precipitates in b matrix has been examined by different investigators. According to these investigations [1–8], the orientation relationship (OR) between the a precipitates and the b matrix can be described by Burgers OR [9], i.e. ð0 0 0 1Þa ==ð0 1 1Þb ; ½1 1 0 0a ==½2 1 1b ;
ð1Þ
Recent studies have noticed that the OR is not the ideal Burgers OR but is slightly deviated from it [6,7]. The a precipitates usually have a shape of plate. The flat regions in the broad faces of the plates are referred as the habit plane. The habit plane was determined as ()13 11 11)b 1 by Furuhara et al. [3]. A single set of parallel dislocations were observed on the habit plane [2–8]. The spacing of these dislocations is about 12 nm and their direction is near [5 3 3]b [3]. Menon and Aaronson [2] first determined that the dislocations are of a-type with the Burgers vector of [)1 2 )1 0]a /3([)1 1 1]b / 2) 2. In a later study, Furuhara et al. [3] determined that the dislocations are of c-type with the Burgers vector of [0 0 0 1]a /2([0 )1 1]b /2). According to the contrast of high resolution transmission electron microscopy (HRTEM) image, Miyano et al. [8] suggested a Burgers vector of [0 1 0]b ([0 1 )1 )1]a /2) which is the same as that observed from a Zr–Nb alloy [10]. HRTEM were used to characterize the structure of the habit plane [3–8]. When viewed from [1 1 1]b or [1 1 )2 0]a direction as the dislo-
½1 1 2 0a ==½1 1 1b :
*
Corresponding author. Fax: +86-10-6277-1160. E-mail address: [email protected] (W.-Z. Zhang).
1 Specific indices are assigned in accordance with the particular variant of OR (Eq. (1)) in the following text. 2 A pair of corresponded vectors of lattices a and b are usually given together for convenience of comparison with other investigations.
1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.01.037
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cations are nearly edge-on, the extra plane of (0 )1 1)b corresponding to (0 0 0 2)a could evidently be observed at the spacing of about tens of layers of (0 )1 1)b or (0 0 0 2)a [4–8]. By a careful examination of the other low indexed planes in the HRTEM images [4,5], one finds difference in the matching condition between the correlated planes (1 )1 0)b /(1 )1 0 1)a and ()1 0 1)b /()1 1 0 1)a , i.e. only one pair of the planes is in mismatching condition. However, if the dislocations are of either a-type or c-type, in the HRTEM images viewed from [1 1 1]b / [1 1 )2 0]a , both of (1 )1 0)b and ()1 0 1)b extra planes should be equivalently observed because g b is not zero 3. Therefore, both a-type and c-type are conflict with the HRTEM observations [4,5]. Moreover, one can also notice that an extra plane of (2 )1 )1)b or (1 )1 0 0)a , which is the terrace plane of the structure ledge [3,11], is associated with the core of the dislocation, clearly indicating the dislocation should not be of c-type. Extensive experimental work has been made in another bcc/hcp alloy system, i.e. Zr–2.5 wt%Nb alloy [10,12–15]. A set of parallel linear dislocations were also observed in the habit plane [10,12–14]. When the OR is near Burgers OR, the habit plane normal and dislocation direction are similar to those observed from Ti based alloys [10]. The Burgers vector of [0 1 0]b ([0 1 )1 )1]a /2) was determined by g b contrast analyses [10]. Banerjee et al. [13] also considered the dislocations to be of hc þ ai type. The ratio of the lattice constants of the bcc and hcp phases in Ti–Cr and Zr–Nb, i.e. ab =aa , are 1.099 and 1.094, respectively, while the ratio ca =aa for the hcp phase in the two alloy systems are 1.587 and 1.589. Because these parameters are very close to each other, the Burgers vectors of the dislocations in the habit plane in the two systems might be similar. Considering the controversial conclusions for the Burgers vector of the dislocations from the previously investigations of the Ti alloys, the present study reexamined the crystallography and the structure of the habit plane in a Ti–7.26 wt%Cr alloy. The experimental results have been interpreted by different approaches. According to the invariant line criterion [16], a deviation from the ideal Burgers OR by a rotation of 0.5° around [0 0 0 1]a /[0 )1 1]b is necessary for producing an invariant line which determines the direction of the structure ledge [11]. Zhang and Purdy [17] proposed an O-line criterion to explain the irrational habit plane structures. When the criterion is met, a single set of dislocations locating alternately with the Olines can accommodate the interfacial misfit completely. They used this model to explain the crystallography of Zr–Nb alloy consistently. Recently, Qiu and Zhang [18] 3 g is a reciprocal vector represent the plane normal and b is the Burgers vector.
developed an analytical method to solve the O-lines and used this method to analyze the crystallography of fcc/ bcc systems systematically. While the invariant line criterion joined with the structure ledge model may also yield the O-lines, the Burgers vector of the model is limited to lie on the terrace plane. In contrast, the O-line criterion is not restricted by such a limitation. In the present work, we have used the analytical O-line model to interpret the crystallography of proeutectoid a precipitates in Ti–Cr alloy.
2. Experimental procedure 2.1. Specimen preparation A Ti–7.26 wt%Cr alloy is selected for the experiment study in this work. The composition is close to that of the alloys used in previous studies [1–4]. The alloy was prepared by arc melting in an argon atmosphere. Then it was forged into a bar of 6 6 mm. After encapsulated in quartz tube under vacuum of less than 103 Pa, the bar was homogenized for three days at 1273 K and then quenched in water after breaking the tube. The bar was cut into specimens of 6 6 6 mm in size for further heat treatment. The specimen for the TEM study was heated at 1273 K for 30 min in the b region, and then isothermally reacted at 973 K for 1 h in the a þ b region followed by water quench. Slices of 0.5 mm thick were electric discharge machined from the specimen. The slices were ground to 20 lm thick and final TEM foils were prepared by ion milling with a Gatan ion-miller. Electrolytic thinning was not used to avoid the precipitation of interphase phases at a=b interface [19,20]. 2.2. TEM analysis TEM observation was performed with a JEOL JEM200CX at 200 kV. The OR between a precipitates and b matrix was determined by analyzing the Kikuchi patterns taken from both phases. Kikuchi patterns are usually produced in single crystal by convergent beam electron diffraction method. Because the width of a plates in the present study were about 0.5 lm, the Kikuchi pattern could also be taken from selected area diffraction (SAD) method when the selected area was smaller than a plates. To minimize the effect of the foil bend, two Kikuchi patterns of different phases have been taken from areas near the interface. In order to measure the angle between two planes of different phases precisely, two Kikuchi bands in different patterns of different phases should both pass through the transmission spot as far as possible. After overlap the two negatives taken from the two phases by alignment of the
F. Ye et al. / Acta Materialia 52 (2004) 2449–2460
numbering readout shadows, the angle between the Kikuchi bands is the angle between the corresponding planes in different phases and the beam direction gives the rotation axis of the planes. The angle could also be observed from the superimposed SAD pattern in a single negative plate, in which the angle is the deviation between the reflection spots corresponding to the Kikuchi bands. Since the age temperature are lower than xs (x start) temperature, x reflections often present when SAD is performed in b phase due to very fine x particles precipitated from b phase. However, because the x particles are very fine and the density is low, the Kikuchi patterns of b phase could remain distinguishable. The habit plane normal and the direction of the dislocations were determined from the edge-on orientation. The g b contrast analysis was performed on the dislocations in the habit plane. The imaging condition was carefully controlled in a strong two-beam or systematic row diffraction condition. In most cases, one crystal was in the
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above condition and the other crystal in a weakly diffracting condition. Strong b reflections {0 2 0}b and {1 1 0}b were mostly used in the g b contrast analysis. Although x reflections may occur, the g b contrast analysis was effectively unaffected. On the basis of the particular variant of OR given in Eq. (1), planes and zone axes are indexed according to the angles between planes, zone axes in different phases, as well as the angle between traces of the habit plane and the dislocations viewed at different zone axes. Due to the twofold symmetry of the [0 )1 1]b /[0 0 0 1]a axis at the Burgers OR, there remains a twofold ambiguity in indexing the planes and directions. However, because of the existence of small deviation from the ideal Burgers OR, these angles are different even though these planes and zone axes are symmetrically equivalent if the phases are related by the exact Burgers OR. Specific indices could be identified according to a careful study of these angles.
Fig. 1. Kikuchi patterns taken from (a) b near [1 1 1]b , (b) a near [1 1 )2 0]a , (c) b near [)1 )2 0]b and (d) a near [)2 )2 4 3]a .
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3. Results 3.1. Orientation relationship Based on the observation of around 50 plates, all examined a plates have an identical OR with respect to the b phase. Careful examination constantly detect that the OR is deviated slightly from the ideal Burgers OR. As explained in Section 2.2, we used the Kikuchi patterns to determine the rotation angle and the rotation axis of two close-packed planes of different phases. Fig. 1 shows the Kikuchi patterns of two phases near interface. The solid (dashed) lines in the figure are the perpendicular bisectors of the Kikuchi bands of b (a) phase. A pair of lines corresponding to the Kikuchi bands of different phases intersect at the right of Fig. 1(b) or (d) and the deviation can be seen clearly at the left of Fig. 1(a) or (c). It can be seen that the OR is very close to the Burgers OR [9], and it can be described by
recorded. The habit plane normal can be determined from any of the SAD patterns. Among these edge-on positions, we found a set of orientations in which the habit plane is normal to a particular set of DgÕs. A Dg is the difference between two reciprocal vectors ga and gb of two phases. To reach an orientation for precise determination of the Dg in a SAD pattern, the TEM foil was rotated to the position such that Kikuchi bands of ga and gb pass through the transmission spot. The relationship between DgÕs and habit plane was observed in Zr–Nb alloy [10] and explained in terms of a property of the O-line [17]. Figs. 2(a) and (b) show two edge-on positions of one a plate whose habit plane is normal to Dgð0 1 1Þb ¼ gð0 1 1Þb gð0 1 1 0Þa and Dgð1 1 2Þb ¼ gð1 1 2Þb gð1 2 1 1Þa . It can also be seen that another Dgð2 0 0Þb ¼ gð2 0 0Þb gð2 1 1 0Þa is apparently normal to the habit plane in Fig. 2(a). However, Dgð2 0 0Þb actually
ð0 0 0 1Þa 0:6° from ð0 1 1Þb towards ð1 1 0Þb with a rotation axis ½1:3 1 1b ð ==½1:2 0:8 2 0a Þ; ð1 1 0 0Þa 0:7° from ð2 1 1Þb towards ð0 0 2Þb with a rotation axis ½1 2:1 0:1b ð ==½2 2 4 3:1a Þ: ð2Þ Because the angles between these two pairs of closepacked planes are both very small and the contrast of Kikuchi bands corresponding to low indexed planes near the transmission spot are very weak, it is difficult to rotate the foil to the precise rotation axis, i.e. the error of the rotation axis should be large (around several degrees). Therefore, the solid line corresponding to (0 )1 1)b in Fig. 1(a) is deviated slightly from the transmission spot. However, this deviation does not significantly affect the result of the rotation angle. Though the rotation angles are very small and a certain degree of scattering (approximately 0.3°) exists due to either the measurement uncertainty or any true scattering, a deviation from exact parallelism between (0 0 0 1)a and (0 )1 1)b , and between (1 )1 0 0)a and (2 )1 )1)b is evidently detected. 3.2. Orientations of the habit plane and dislocations The broad face of an a plate usually defines the habit plane of the plate. Since the broad face in microscopic scale is not strictly flat in a large scale, the habit plane is usually measured in the flat regions of the broad face. To determine the habit plane normal precisely, we rotated the habit plane to the edge-on orientation. When the TEM foil was rotated around the habit plane normal from one edge-on position, a series of SAD patterns corresponding to different edge-on orientations could be
Fig. 2. The determination of the edge-on habit plane by (a) Dgð0 1 1Þb ¼ gð0 1 1Þb gð0 1 1 0Þa and (b) Dgð1 1 2Þb ¼ gð1 1 2Þb gð1 2 1 1Þa . (c) A near edge-on position of the habit plane when Dgð1 0 1Þb ¼ gð1 0 1Þb gð1 0 1 1Þa .
F. Ye et al. / Acta Materialia 52 (2004) 2449–2460
deviates from the habit plane normal because the Kikuchi bands of gð2 0 0Þb and gð2 1 1 0Þa deviate from the transmission spot. Based on the Kikuchi patterns corresponding to the diffraction pattern in Fig. 2(a), the beam direction is approximately [)0.0224 1 )1]b (// [)0.0081 )0.0033 0.0114 )1]a ) near [0 1 )1]b ([0 0 0 )1]a ). The habit plane normal is ()13 10.2 10.1)b (// ()3.6 4.6 )1 )0.1)a ). The measurements of habit plane were found to scatter within 3° due to either the measurement uncertainty or any true scattering. However, the habit plane normal to a particular set of DgÕs is reproducible. The broad face of different a plates was observed to have a consistent structure that is dominate with a set of regularly spaced dislocations. It is reasonable to con-
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sider the periodic dislocations to be the characteristic feature of the habit plane. The spacing of the dislocations is about 11 nm. The dislocation direction is determined mainly by edge-on method. It is scattered around [5 2.6 2.8]b (//[3.2 1 )4.2 0.1]a ) with uncertainty of approximately 5°. The Burgers vector of the dislocations is analyzed in the following section. 3.3. Determination of Burgers vector of the dislocations The Burgers vector of the regularly spaced dislocations was determined by a g b contrast analysis by mainly used {0 2 0}b and {1 1 0}b reflections. Three sets of dark field images of the habit plane are shown in Figs. 3–5. Each set was taken from a single a plate.
Fig. 3. Dark field micrographs of a habit plane generated by (a) g ¼ 0 )1 1b , (b) g ¼ 0 2 0b , and (c) g ¼ 0 0 )2b near [1 0 0]b zone axis; (d) g ¼ 1 0 1b , and (e) g ¼ 0 1 1b near [1 1 )1]b zone axis.
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Fig. 4. Dark field micrographs of a habit plane generated by (a) g ¼ 1 )1 0b , (b) g ¼ 1 1 0b , (c) g ¼ 2 0 0b , (d) g ¼ 0 2 0b near [0 0 1]b zone axis.
Fig. 5. Dark field micrographs of a habit plane generated by (a) g ¼ )1 0 )1b , (b) g ¼ )1 0 1b , (c) g ¼ )2 0 0b near [0 1 0]b zone axis.
F. Ye et al. / Acta Materialia 52 (2004) 2449–2460
The broad face also contains a set of irregularly spaced linear defects besides the regularly spaced dislocations as indicated by small arrows in Fig. 3(e). Since the regular-spaced dislocations were considered as the habit plane structure, the contrast analysis and the later explanation were focused on these regularly spaced dislocations. The contrast of the dislocations are strong when g ¼ (0 )1 1)b ((0 0 0 2)a ) (Fig. 3(a)) and (1 )1 0)b ((1 )1 0 1)a ) (Fig. 4(a)) were employed. The dislocations were always out of contrast when g ¼ (0 1 1)b ((0 1 )1 0)a ) (Fig. 3(e)) was employed. These results are consistent with the contrast analysis by Furuhara et al. [3] and in conflict with the Burgers vector of [)1 1 1]b /2([)1 2 )1 0]a ) [2]. Therefore, the dislocations should not be of a-type. The contrast of dislocations was also out of contrast or very weak when g ¼ (1 1 0)b ((1 0 )1 )1)a ) (Fig. 4(b)) and ()1 0 1)b (()1 1 0 1)a ) (Fig. 5(b)) were employed. The dislocations were always visible when g ¼ (2 0 0)b ((2 )1 )1 0)a ) (Figs. 4(c) and 5(c)) was employed. These results are conflict with the Burgers vector [0 )1 1]b /2/ ([0 0 0 1]a /2) [3]. Therefore, the dislocations should not be of c-type. An alternative choice of the Burgers vector could be [0 1 0]b ([0 1 )1 )1]a /2) as that defined for dislocations in a Zr–Nb alloy [10]. However, the facets that the dislocations are consistent invisible by using g ¼ (0 1 1)b ((0 1 )1 0)a ) (Fig. 3(e)) and visible by using g ¼ (1 0 1)b ((1 0 )1 1)a ) (Figs. 3(d) and 5(a)) are in conflict with the Burgers vector of [0 1 0]b . Therefore, the Burgers vector of the dislocations is not the same as that in the Zr–Nb alloy. Table 1 summarized the observed results of the g b contrast analysis and they are compared with the g b results for different Burgers vectors. The corresponding ga Õs are also given in the table. Based on the above consideration and invisible results summarized in Table 1, we suggest that the Burgers vector of the dislocations is more properly assigned to be [1 )1 1]b /2([2 )1 )1 3]a / 6). The angle between the Burgers vector and the direction of the dislocations is about 61.5°. It reveals that the dislocations are mixed dislocations.
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4. Discussion 4.1. Comparing with previous experimental results 4.1.1. Comparing with Ti–X system The present experimental results are given in Table 2 and compared with the previous experimental results in [3]. The OR in [3] is not given in the table because the deviation from the ideal Burgers OR was not determined in this reference. According to the careful measurement in this work, the OR is near the ideal Burgers OR, but none of the parallelism relationship described in the Burgers OR is exact. This deviation is also determined in recent investigations in which the deviations of [0 )1 1]b /[0 0 0 1]a and [1 1 1]b /[1 1 )2 0]b are 0.78° and 0.56° [6] or 0.31° and 0.34° [7] with the rotation axes near [1 1 1]b and [0 )1 1]b , respectively [6]. The deviation between [1 1 1]b and [1 1 )2 0]b in the present experiment is 0.5° and the rotation axis is also very close to [0 )1 1]b . The OR determined in the present work is similar to that in previous works [6,7] and the difference between them is possibly due to the experiment error and difference in the compositions. The habit plane normal and the invariant line direction have been measured with great effort for improving precision and they are in good agreement with the previous experiments [1,3,5–8] within experimental uncertainty. The Burgers vector of the dislocations in the habit plane characterized in this work is [1 )1 1]b / 2([2 )1 )1 3]a /6) which is different from the results in previous experiments [2,3,8]. However, this Burgers vector is consistent with the HRTEM observations when viewed from [1 1 1]b /[1 1 )2 0]a [4–8]. In the HRTEM images (0 )1 1)b and (0 0 0 2)a mismatch evidently at the core of the dislocations. Moreover only one pair of planes (1 )1 0)b /(1 )1 0 1)a and ()1 0 1)b / ()1 1 0 1)a is mismatch but the other pair is not [4,5]. In addition, the extra plane (2 )1 )1)b or (1 )1 0 0)a , which is the terrace plane of the structure ledge [3,11], is associated with the core of the dislocations [4,5]. This result is better explained by the Burgers vector determined in the present study because
Table 1 Comparison between observed and theoretical contrast results of the dislocations for different Burgers vectors Fig(s).
gb
ga
Observed contrast
g b for bL ¼ [1 )1 1]b /2
g b for bL ¼ [)1 1 1]b /2
g b for bL ¼ [0 )1 1]b /2
g b for bL ¼ [0 1 0]b
3(a) 3(d) and 5(a) 4(a) 3(e) 4(b) 5(b) 4(c) and 5(c) 3(b) and 4(d) 3(c)
0 )1 1 101 1 )1 0 011 110 )1 0 1 200 020 0 0 )2
0002 1 0 )1 1 1 )1 0 1 0 1 )1 0 1 0 )1 )1 )1 1 0 1 2 )1 )1 0 0 1 )1 )2 0 )1 1 )2
Strong Medium Strong None None None Medium Medium Medium
1 1 1 0 0 0 1 )1 )1
0 0 )1 1 0 1 )1 1 )1
1 1/2 1/2 0 )1/2 1/2 0 )1 )1
)1 0 )1 1 1 0 0 2 0
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Table 2 The experiment results, the O-line model calculation results and the difference between the present experiment results and other results bL
OR
Habit plane
Invariant line
Ddisl (nm)
c ¼ 0:6° d ¼ 0:8° c ¼ 0:8° d ¼ 0:9° c ¼ 0° d ¼ 0:5° c ¼ 0:6° d ¼ 0:7° –
()13 11.5 10.3)b ()2.9 3.9 )1 )0.3)a ()13 9.6 11.9)b ()3.0 4.0 )1 0.5)a ()13 10.9 10.9)b ()3.0 4.0 )1 0)a ()13 10.2 10.1)b ()3.6 4.6 )1 )0.1)a ()13 11 11)b
[5 2.8 3.2]b [2.5 1 )3.5 0.2]a [5 2.7 3.3]b [2.5 1 )3.5 0.3]a [5 3.0 3.0]b [2.5 1 )3.5 0]a [5 2.6 2.8]b [3.2 1 )4.2 0.1]a [5 3 3]b
10.9
O-line with [1 )1 1]b
[1 )1 1]b /2 [2 )1 )1 3]a /6 [0 1 0]b [0 1 )1 )1]a /2 [0 )1 1]b /2 [0 0 0 1]a /2 [1 )1 1]b /2 [2 )1 )1 3]a /6 [0 )1 1]b /2 [0 0 0 1]a /2 Same
3.0°
3.2°
0.1
O-line with[0 1 0]b
Different
5.1°
3.8°
0.8
O-line with [0 )1 1]b /2
Different
2.0°
3.2°
0.8
Exp. [3]
Different
c ¼ 0° d ¼ 0:1° c ¼ 0:2° d ¼ 0:2° c ¼ 0:6° d ¼ 0:2° –
2.3°
3.2°
1
Calculation results
Present experiment Previous experiment [3] Comparision with present experimental results
11.8 11.2 11 12
c – Angle between (0 0 0 1)a and (0 )1 1)b . d – Angle between (1 1 0 0)a and (2 )1 )1)b .
gð0 1 1Þb bL½1 1 1b=2 ¼ 1, gð1 1 0Þb bL½1 1 1b=2 ¼ 1, gð1 0 1Þb bL½1 1 1b=2 ¼ 0 and gð2 1 1Þb bL½1 1 1b=2 ¼ 1. This new Burgers vector is also consistent with the relationship between the habit plane and DgÕs. According to the property of the O-line [17], if the habit plane contains a set of dislocations, this habit plane must be normal to a group of DgÕs whose related gÕs must lie in the zone axis of the Burgers vector of the dislocations. As shown in Fig. 2, two DgÕs, i.e. Dgð0 1 1Þb and Dgð1 1 2Þb , are normal to the habit plane and the gð0 1 1Þb and gð1 1 2Þb lie in the zone axis of the [1 )1 1]b /2 suggesting the Burgers vector of the dislocations in the habit plane is [1 )1 1]b /2. Fig. 2(c) gives another Dgð1 0 1Þb ¼ gð1 0 1Þb gð1 0 1 1Þa . gð1 0 1Þb is normal to [)1 1 1]b /2 and [0 1 0]b . The habit plane at this condition deviates slightly from edge-on orientation. This deviation is observed consistently on different a plates and hence the Burgers vector is unlikely among these vectors. 4.1.2. Comparing with Zr–Nb system Because the ratios of the lattice constants in Ti–Cr and Zr–Nb systems are similar, it may be expected that the crystallography in the two systems are also similar. However, the OR, habit plane and dislocation direction in the systems are similar but the Burgers vector in Zr– Nb system is [0 1 0]b ([0 1 )1 )1]a /2) which is quite different from that in Ti–Cr system. The previous HRTEM experiment [4–8] is also consistent with the Burgers vector of [0 1 0]b ([0 1 )1 )1]a /2), because gð0 1 1Þb bL½0 1 0b ¼ 1, gð1 1 0Þb bL½0 1 0b ¼ 1, gð1 0 1Þb bL½0 1 0b ¼ 0 and gð2 1 1Þb bL½0 1 0b ¼ 1. However, it can be seen from Table 1 that this Burgers vector is in conflict with the results of the g b analysis. It is also inconsistent with the description of the habit plane in terms of DgÕs. It remains to be an open question that
why the Burgers vectors of the dislocations in these two systems are different. 4.2. O-line model calculation When the habit plane of a transformation system contains a set of parallel dislocations, the crystallography of the system should be well explained by the O-line model [17]. We also observed a set of DgÕs normal to the habit plane in agreement with the property of the Olines [17]. In this section, we use the analytical O-line model [18] to explain the crystallography and the habit plane structure between a and b phases in Ti–Cr alloy. The calculation procedure is similar to that for fcc/bcc system [18]. The details of the calculation for the present alloy are summarized in Appendix A The calculation procedure needs to select a Burgers vector as input data. The Burgers vectors of lattice b are possibly one of three h1 0 0ib or one of four h1 1 1ib /2. Among these possible Burgers vectors, [1 )1 1]b / 2([2 )1 )1 3]a /6) and [1 1 )1]b /2([2 )1 )1 )3]a /6), [1 1 1]b / 2([1 1 )2 0]a /3) and [)1 1 1]b /2([)1 2 )1 0]a /3), [0 1 0]b ([0 1 )1 )1]a /2) and [0 0 1]b ([0 1 )1 1]a /2) are symmetrically equivalent according to the lattice correspondence and can give O-line solutions which correspond to the OR near the particular variant of Burgers OR described by Eq. (1), while the O-lines are not solvable for [1 0 0]b ([2 )1 )1 0]a /3). Therefore, we only examine the O-line solutions for the Burgers vectors of [1 )1 1]b / 2([2 )1 )1 3]a /6), [0 1 0]b ([0 1 )1 )1]a /2) and [1 1 1]b / 2([1 1 )2 0]a /3). The maximum spacing of the dislocations given by the O-line solution for bL ¼ [1 1 1]b /2 is 2.4 nm which is too small compared with the observed result. Therefore, we only compared the other two O-line solutions.
F. Ye et al. / Acta Materialia 52 (2004) 2449–2460
The calculation results are given and compared with the experimental observations in Table 2. Only deviation angles of the ORs are given and compared in the table. The rotation axes of the close-packed planes for the O-line solution of bL ¼ [1 )1 1]b /2 are [1.4 1 1]b (// [1.3 0.7 )2 0]a ) for planes (0 0 0 1)a and (0 )1 1)b , and [)1 )2.0 0.03]b (//[)2 )2 4 3.2]a ) for planes (1 )1 0 0)a and (2 )1 )1)b . The difference angles of the rotation axes between the experimental (Eq. (2)) and the O-line solution for bL ¼ [1 )1 1]b /2 are 2.1° and 2.0° corresponding to the two pairs of the close-packed planes, respectively. The OR of the O-line solution for bL ¼ [1 )1 1]b /2 are similar to that for bL ¼ [0 1 0]b . It is interesting that the other results for different Burgers vectors are also quite close to each other in contrast to the fcc/bcc systems, for which the calculated habit plane and the dislocation direction usually change obviously with the Burgers vectors [18]. The differences in the calculated habit planes and the dislocation directions corresponding to different Burgers vectors are within the experiment error. Thus the observed habit plane, the direction and spacing of the dislocations can be explained by the Oline solutions with either Burgers vectors. Based on the present experimental results of the g b contrast analysis and the relationship between DgÕs and the habit plane, we suggest that the Burgers vector of the dislocations should be [1 )1 1]b /2([2 )1 )1 3]a /6). Furuhara et al. [3] determined a Burgers vector of the dislocations as [0 )1 1]b /2([0 0 0 1]a /2). When Burgers vector of [0 )1 1]b /2 is used to define the O-lines, our calculated results confirm the results from the previous study of invariant line model calculation [11]. These results are also close to those of the other O-line solutions, as given in Table 2. However, the OR corresponds to [0 )1 1]b being parallel to [0 0 0 1]a , which is inconsistent with the experimental observation. As mentioned in previous sections, this Burgers vector is also conflict with the HRTEM observation [4,5] and the results of the g b contrast analysis. Therefore, the description of the dislocations with the Burgers vector of [0 )1 1]b /2([0 0 0 1]a /2) is likely incorrect.
5. Conclusion 1. The OR between a and b in a Ti–7.26 wt%Cr alloy is not at the exact Burgers OR. It can be described by (0 0 0 1)a 0.6° from (0 )1 1)b and (1 )1 0 0)a 0.7° from (2 )1 )1)b . The habit plane determined by edge-on method is normal to a particular set of DgÕs. Its orientation is ()13 10.2 10.1)b (//()3.6 4.6 )1 )0.097)a ). 2. The habit plane is characterized by a single set of dislocations at spacing of 11 nm. The direction of the dislocations is approximately [5 2.6 2.8]b (// [3.2 1 )4.2 0.1]a ). The Burgers vector of the dislocations is determined to be [1 )1 1]b /2([2 )1 )1 3]a /6).
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This Burgers vector is different from the previous conclusions [2,3,8]. 3. The experimental observations are consistent with the calculation results from an analytical O-line model [18]. However, because the O-line calculation results for different Burgers vectors are close to each other, it is difficult to distinguish the Burgers vectors according to the comparison of the calculation data.
Acknowledgements The support of National Natural Science Foundation of China Grant Nos. 59871021 and 50271035 is gratefully acknowledged.
Appendix A The calculation procedure of analytical O-line model for bcc/hcp system summarized below is similar to that for fcc/bcc system except the construction of the pure strain. More detailed calculation procedure and theoretical explanation can be found in [18]. (1) Construct the pure strain Any O-line transformation strain A can be decomposed into A ¼ RB:
ðA:1Þ
B is the pure strain, which is expressed as a diagonal matrix. The rotation matrix R defines an OR corresponding to the O-line condition. The first step of the calculation is to construct the pure strain B. The pure strain in fcc/bcc system is the Bain strain [21]. In bcc/ hcp system, it is constructed by different method but it can also be determined from a given OR as explained below. We start to construct the transformation strain at the ideal Burgers OR. Three non-coplanar vectors in lattice b are selected as [1 0 0]b , [1 1 1]b /2 and [0 )1 1]b . The correlated vectors by the transformation are [2 )1 )1 0]a / 3, [1 1 )2 0]a /3 and [0 0 0 1]a according to the nearest neighbor principle [22]. In three indices systems, they can be expressed in SLb and SLa as column vectors, 2 3 2 3 1 1=2 0 1 1 0 SLb ¼ 4 0 1=2 1 5 and SLa ¼ 4 0 1 0 5: 0 1=2 1 0 0 1 ðA:2Þ The superscript L indicates that the vector is expressed on basis of the crystal lattice. Three non-coplanar vectors are chosen as the axis of a new orthogonal coordinates named N1 which is x//[2 )1 )1]b //[1 )1 0 0]a , y//[1 1 1]b //[1 1 )2 0]a and z//[0 )1 1]b //[0 0 0 1]a . We can expressed selected lattice vectors as column vectors in the orthogonal coordinates
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2 qffiffi
2 3
0 pffiffi
6 qffiffi S b ¼ ab 6 4 1
3 2
3
0
0
0
3
7 7 0 5 pffiffiffi 2
2 pffiffi3 and
Sa ¼ aa 4
2 1 2
0
0 1 0
3 0 0 5:
ca aa
ðA:3Þ The lattice constants we used are the same as those used by previous investigators [2,3,11], i.e. ab ¼ 0:325 nm and aa ¼ 0:29564 nm, ca ¼ 0:46928 nm. Then the transformation matrix Ab for the ideal Burgers OR can be determined as 1
Ab ¼ Sa ðSb Þ :
ðA:4Þ
where bð¼ Qb bL ) is Burgers vector of lattice b expressed in N2 and xB is transformed from x due to the pure strain, as defined by 0
xB ¼ ðB1 Þ x :
ðA:9dÞ
(3) Construct an O-line strain A0 According to Eq. (A.1), to construct an O-line strain A0 , lattice a should be rotated after the transformation of the pure strain so that A0 can satisfy with the following invariant line condition 0 ðA1 0 Þx ¼ x :
ðA:10Þ
Following the method developed in the PTMC [23], A0 can be determined from A0 ¼ R1 R02 B;
ðA:11aÞ
Sa , Sb and the coordinates N1 is the same as that used in the O-line calculation in Zr–Nb alloy (but with a different OR variant) [17]. The transformation can be expressed as
R1 and R2 can be defined as R1 ¼ ½x ; bu ; x bu ;
ðA:11bÞ
xa ¼ Ab xb :
R2 ¼ ½xB ; bB ; xB bB ;
ðA:11cÞ
ðA:5Þ
Let xb in Eq. (A.5) to be a unit vector so that all points xa defined an ellipsoid. The directions and lengths of the axes of the ellipsoid can be expressed by the eigenvectors and eigenvalues of a real symmetric matrix F ¼ ðA0b Ab Þ
1=2
¼ PBP1 :
ðA:6Þ
B is a diagonal matrix contains three eigenvalues of F and P contains three eigenvectors of F as column vectors. In the orthogonal coordinates N2 defined by the column vector in P, the pure strain transformation can be expressed by xa ¼ Bxb :
ðA:7Þ
where bu ¼ b=jbj, bB ¼ Bb=jBbj, respectively. (4) Narrow the selection of O-line solution When a is rotated around x , numerous O-line solutions can be generated. To restrict the choice, Zhang and Purdy [17] proposed two possible criterions. One is that the spacing of the dislocations should be large. The other is that deviation from the Burgers OR should be small. In most cases as the present system, these two criterions will give close results. We select the first criterion in the following calculation. To rotate lattice a around x , we select three column vectors in R1 as axis of an orthogonal coordinates N3 . Then the rotation matrix in this coordinates is 2 3 1 0 0 R3 ¼ 4 0 cos h sin h 5: ðA:12Þ 0 sin h cos h
The following calculation is made in this orthogonal coordinates N2 . The coordinate conversion matrix of lattice b in this coordinates, as well as in the latter calculation, is
where a is the rotation angle. The transformation matrix expressed in the coordinates N3 becomes
Qb ¼ P1 Sb SL1 b :
AN3 ¼ R3 ðR01 A0 R1 Þ:
ðA:8Þ
The following calculation steps are the same as that used for the fcc/bcc system [18]. (2) Determine the invariant line in reciprocal space x The O-line condition requires that the Burgers vector should be normal to the invariant line in reciprocal space x [17]. In addition, the length of x is not changed by pure strain. Therefore, the unit vector x can be determined by solve the following equations: x0 B xB ¼ 1
ðA:9aÞ
x0 x ¼ 1
ðA:9bÞ
0
b x ¼ 0;
ðA:9cÞ
ðA:13Þ
If we still express the transformation in coordinates N2 , the transformation matrix is A ¼ R1 AN3 R01 ¼ R1 R3 R02 B:
ðA:14Þ
The coordinate conversion matrix of lattice a is determined by Qa ¼ AQb SLb SL1 a :
ðA:15Þ
The habit plane normal is Dgp ¼ T0 gb ;
ðA:16Þ
where T ¼ I A1 and gb is a reciprocal vector normal to b, i.e. gb b ¼ 0.
F. Ye et al. / Acta Materialia 52 (2004) 2449–2460
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0:2941 0:1214 3:489 10 4 2:964 102 0:2696 3:379 103 5 3 3 2:400 10 2:844 10 0:4693 2 3 0:2941 0:1215 4:772 103 4 2:949 102 0:2695 4:547 103 5 3:277 103 3:847 103 0:4692 2 3 0:2941 0:1213 0 4 2:980 102 0:2696 0 5 0 0 0:4693
A 2 3 3
0 0 0:2298 0:2298 5 0:2298 0:2298 3 0 0 0:2298 0:2298 5 0:2298 0:2298 3 0 0 0:2298 0:2298 5 0:2298 0:2298
Qa 2
2
Oc ¼ T0 b=jbj :
ðA:18Þ
The O-cell walls locate at the poor matching regions which are alternated with the O-lines, the good matching regions. Different A and hence different Ddisl are obtained as angle a is varied in a range of )15° to 15°. The optimum OR is determined when the spacing of dislocations reaches its maximum value. The corresponding dislocation direction, which must be parallel to the invariant line x in direct space, is an eigenvectors of A corresponding to the eigenvalue 1, because Ax ¼ x. A direct vector or reciprocal vector on the lattice basis can be expressed on the orthogonal coordinates by xi ¼ Qi xLi
0
L or gi ¼ ðQ1 i Þ gi ;
ðA:19Þ
where i represents lattice a or b. The conversion matrix is convenient for calculating the angles between vectors in hcp lattice or in different lattices by expressing these vectors in the orthogonal coordinates N2 [25]. Then OR is determined by this method. The calculation results of the OR, the habit plane, and the direction and spacing of the dislocations corresponding to different O-line solutions for different Burgers vectors are given in Table 2. Table 3 gives the coordinate conversion matrix and the transformation strain matrix. Qb Õs for different cases are same since the calculation procedure of Qb are same. Some elements of Qa or A are similar for different cases, because the ORs for different cases are similar.
3
[0 )1 1]b /2
[0 0 0 1]a /2
[0 1 )1 )1]a /2
[0 1 0]b
[1 )1 1]b /2
[2 )1 )1 3]a /6
Qb 2 0:3250 4 0 0 2 0:3250 4 0 0 2 0:3250 4 0 0 bL
Table 3 Coordinates conversion matrixes and transformation strain matrixes
ðA:17Þ
Oc is the reciprocal vector that defines a set of O-cell walls determined by [24] 3 0:1123 0 1:108 0 5 0 1:021
0:1112 1:108 9:611 103
Ddisl ¼ jDgp j=jOc Dgp j:
0:9050 4 9:121 102 7:386 103 2 0:9051 4 9:072 102 1:008 102 2 0:9050 4 9:170 102 0
0:1118 1:108 7:151 103
3 7:591 103 3 5 7:352 10 1:021 3 1:038 102 9:893 103 5 1:021
The spacing of O-lines or dislocations on the habit plane is [24]
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