Edge-to-edge matching and its applications

Acta Materialia 53 (2005) 1073–1084 www.actamat-journals.com

Edge-to-edge matching and its applications Part I. Application to the simple HCP/BCC system M.-X. Zhang *, P.M. Kelly Division of Materials, School of Engineering, The University of Queensland, St. Lucia, QLD 4072, Australia Received 22 July 2004; received in revised form 29 October 2004; accepted 2 November 2004 Available online 8 December 2004

Abstract The edge-to-edge matching crystallographic model has been used to predict all the orientation relationships (OR) between crystals that have simple hexagonal close packed (HCP) and body-centered cubic (BCC) structures. Using the critical values for the interatomic spacing misfit along the matching directions and the d-value mismatch between matching planes, the model predicted all the four common ORs, namely the Burgers OR, the Potter OR, the Pitsch–Schrader OR and the Rong–Dunlop OR, together with the corresponding habit planes. Taking the cH/aH and aH/aB ratios as variables, where H and B denote the HCP and BCC structures respectively, the model also predicted the relationship between these variables and the four ORs. These predictions are perfectly consistent with the published experimental results. As was the case in the FCC/BCC system, the edge-to-edge matching model has been shown to be a powerful tool for predicting the crystallographic features of diffusion-controlled phase transformations.
2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Orientation relationship; Habit plane; Interface structure; Phase transformation; Edge-to-edge matching

1. Introduction Body-centered cubic (BCC), face-centered cubic (FCC) and hexagonal close packed (HCP) are three common crystal structures for metals and simple compounds, and in many cases in metallic materials both the matrix and the precipitate will have one of these structures. Due to the strong influence of precipitation on the properties of materials, the crystallographic features between these three structures have been studied for over half a century. In all previous studies, the FCC/BCC system was the one most actively researched and has been analyzed using almost all available crystallographic models, such as the structural ledge model [1– 8], near-coincidence site model [9], invariant line model *

Corresponding author. Tel.: +61 733 65 3669; fax: +61 733 65 3888. E-mail address: [email protected] (M.-X. Zhang).

[10–19], O-lattice theory [20–24] and the early work by van der Merwe [25]. More recently the edge-to-edge matching model [26–29] has been successfully applied to this system [28], and all the experimentally observed ORs and their corresponding habit planes have been predicted. By comparison, research on HCP/BCC and HCP/FCC systems has been less extensive. Ramanujan and co-workers [30,31] proposed a discrete lattice plane model and used it to analyze the composition profile and surface energy between HCP and FCC structures in an Al–Ag alloy. Howe [32] used the atomic site correspondence concept to understand the surface relief in the FCC to HCP transformation and indicated that both the diffusion-controlled and martensitic transformation can produce surface relief in this system. A study of the crystallography of the HCP/BCC system using the structural ledge model was focused on Ti–Cr alloys [33–36] and the interface structures were described using this model when a particular OR was assumed. O-lattice

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2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.11.007

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M.-X. Zhang, P.M. Kelly / Acta Materialia 53 (2005) 1073–1084

theory was also used to explain the observed ORs, habit planes and interfacial boundary structures in Zr–Nb alloy [37–39], Ti–Cr alloy [40,41] and others [42,43]. Duly [44] and Luo and Weatherly [45] also used the invariant line model to explain the ORs and habit planes in this system. DulyÕs work [44] indicated that for the Burgers OR [46] an invariant line could be obtained, but not for the Potter OR [47], nor for the Pistch–Shrader (P– S) OR [48]. Although these models can explain some of the crystallographic features in the HCP/BCC system, none of them can actually predict the ORs and the corresponding habit planes from first principles without assumptions. Therefore, the present work will use the edge-to-edge matching model to predict and understand the ORs and their corresponding habit planes in a simple HCP/BCC system. The HCP/FCC system will not be covered in the current work. Four ORs between HCP and BCC structures have been experimentally observed [44]. The Burgers OR [46]: ð0 0 0 1Þ kð0 1 1Þ ; ½2  1 1 0
k½1  1 1
H

B

H

B

The Potter OR [47]: ð0 0 0 1ÞH 2 from ð0 1 1ÞB ; 1  0
k½1  ½2 1 1 1
H

ð0 1  1 1ÞH kð1 1 0ÞB ;

B

The Pistch–Schrader OR [48]:  0
k½1 0 0
; ð0 0 0 1ÞH kð0 1 1ÞB ; ½1 1 2 H B The Rong–Dunlop (R–D) OR [49]:  0 0Þ kð0  ð0 0 0 1ÞH kð0 2 1ÞB ; ð1 1 1 2ÞB ; H   ½2 1 1 0
k½1 0 0
H

B

The reported habit planes corresponding to these ORs vary with the alloy system. In most cases they are irrational. In addition, Zhang et al. [38,41] in Zr– 2.5wt%Nb and Ti–7.26wt% alloys and Furuhara et al. [34,35] in Ti–7.15wt%Cr alloy found an OR that deviated very slightly from the ideal Burgers OR, i.e. the [0 0 0 1]H is 0.6 away from [1 1 0]B rather than there being exact parallelism between [0 0 0 1]H and [1 1 0]B. Zhang et al. [38,41] reported an irrational habit plane that is perpendicular to a set of Dgs and Furuhara [34,35] reported a habit plane that is close to ð1  1 0 0ÞH kð2  1 1ÞB .

According to the model [50], in order to predict the ORs, the close packed or nearly close packed directions need to be identified. The atom row along these directions can be either straight or zigzag. Normally, it is expected that a straight atom row in one phase will match a straight row in the other phase and the zigzag rows will match with zigzag rows. In the case of zigzag atom rows, the calculation of the interatomic spacing misfit along the matching directions should be based on the effective interatomic spacing, which is the projected spacing in the direction of the atom row. The condition for a zigzag atom row to be a possible matching direction is that the perpendicular distance from the center of the atom that is out of line to the line must be less than the radius of the atom, to ensure that the matching direction line goes through all the atoms that are arranged in the zigzag manner. For the HCP structure there are three possible close packed or nearly close packed directions. They are h1 1 2 0iH , h1 0 1 0iH , and h1 1 2 3iH . The first one is a straight atom row and the other two are zigzag atom rows. Although the Æ0 0 0 1æH directions is a close packed zigzag row, it does not satisfy the condition to be a zigzag matching direction. The interatomic spacing along these potential matching directions can be expressed in terms of the lattice parameters, aH and cH. If fH is used to represent the interatomic spacing in HCP for straight atom rows and fH0 denotes the effective interatomic spacing for p zigzag rows, then fH = aH for h1 1 2 0iH , fH0 ¼ 0:5aH 3 for h1 0 1 0iH and fH0 ¼ 0:5 0:5 2 0:5 ða2H þ c2H Þ ¼ 0:5aH ½1 þ ðcH =aH Þ
for h1 1 2 3iH . For the BCC structure there are also four possible close packed or nearly close packed directions. They are Æ1 1 1æB, Æ1 0 0æB, Æ1 1 0æB and Æ1 1 3æB. The first three are straight atom rows and the last one is zigzag atom row. One may argue that Æ1 1 2æB direction can also be a zigzag atom row. Actually this direction does not satisfy the zigzag row condition. Fig. 1 shows these two directions in a ð1 1 0ÞB plane. The interatomic spacing p along these four directions are fB =p( 3/2)aB for Æ1 1 1æB, fB = aB p for Æ1 0 0æB, and fB = aB 2 for Æ1 1 0æB and fB0 ¼ 0:25aB 11 for Æ1 1 3æB. According to the model, there will be five combinations or direction pairs be[001]B [110]B

2. Prediction of orientation relationships According to the edge-to-edge matching model [26– 29,50], the orientation relationship between HCP and BCC depends on both the ratios aH/aB and cH/aH (aB is the lattice parameter of BCC and cH and aH are the lattice parameters of HCP), because the interatomic spacing along directions and the interplanar spacing (d-value) of the two phases vary with these parameters.

[113]B zigzag row

[112]B

[112]B zigzag row

[113]B

Fig. 1. Schematical illustration of the difference of [1 1 3]B zigzagged atom row from [1 1 2]B zigzagged atom row on ð1 1 0ÞB plane.

M.-X. Zhang, P.M. Kelly / Acta Materialia 53 (2005) 1073–1084

Interatomic spacing misfit (%)

tween HCP and BCC, which can be the potential matching directions. These are h1 1  2 0iH =h1 1 1iB , h1 1 2 0iH =  h1 0 0iB , h1 1 2 0iH =h1 1 0iB , h1 0  1 0iH = h1 1 3iB and h1 1  2 3iH =h1 1 3iB . If it is assumed that the BCC structure is the parent phase and HCP is the product, the variation of interatomic spacing misfit along these direction pairs with the parameter of aH/aB for each direction pair can be calculated. If the HCP is made the parent phase and the BCC is made the product, the procedure will work equally well and same results will be obtained. For the direction pairs involving directions h1 1 2 0iH or h1 0  1 0iH , the interatomic spacing misfit is independent of the cH/aH ratio and the variation of the misfit with the aH/aB is shown in Fig. 2(a). For the rest of the direction pairs, the misfit is related to both aH/aB

30 <1 1 2 0>H/<111>B

20

<1 1 2 0>H/<100>B <1 1 2 0>H/<110>B

10

<1 1 2 0>H/<113>B

0 0.7

0.8

0.9

Interatomic spacing misfit (%)

1

1.1

1.2

aH/aB

(a) 30

cH/aH=1.5 20 cH/aH=1.6 10

cH/aH=1.7

0 0.7

(b)

0.8

0.9

1

1.1

1.2

aH/aB

Fig. 2. The variation of interatomic spacing misfit along direction pairs with aH/aB: (a) the direction pairs involving h1 1 2 0iH and h1 0  1 0iH , which are independent of the cH/aH value; (b) interatomic spacing misfit along h1 1 2 3iH =h1 1 3iB direction pair at various cH/aH values.

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and cH/aH and the variations are shown in Fig. 2(b). In the present work the values of cH/aH used varies from 1.5 to 1.7, and the value range of aH/aB is from 0.7 to 1.2, which cover the majority of real HCP/BCC systems. It can be seen the minimum misfit of each direction pair corresponds to a certain aH/aB value. If 10% is used as the critical value for the interatomic spacing misfit, the potential matching directions for given aH/aB and cH/aH values can be identified from Fig. 2. They are listed in Table 1. The selection of 10% as the critical value is based on van der MerweÕs energy calculation [25] along the close packed directions between FCC and BCC. His results indicate that to obtain the minimum strain energy along these directions, the critical misfit is 9% for the Kudjmov–Sachs OR, and 7% for the Nishiyama–Wassermann OR. In addition, calculation of the interatomic spacing misfit along direction parallelisms in more than 40 experimentally observed ORs [2,18,44] in different systems also show that all the interatomic spacing misfits are less than 10%. Hence, it is reasonable to use 10% as a critical value. The next step to predicting the ORs is to identify the matching planes. In the HCP structure, the close packed or nearly close packed planes include {0 0 0 2}H, f1 0 1 1gH and f1 0 1 0gH . In the BCC structure, there are also three close packed or nearly close packed planes, namely {1 1 0}B, {2 0 0}B and {1 1 1}B. Thus, there are a total of nine possible plane pairs. Similar to the interatomic spacing misfit along matching directions, the edge-to-edge matching model requires a critical d-value mismatch between matching planes to form an OR without large angle rotation of the matching planes relative to each other about the matching directions. As there is no rigorous approach to calculating this critical value, it can be estimated through examination of the reported ORs in known systems. Investigating more than 40 experimental observed ORs in various systems [2,18,44] shows that the d-value mismatch between the plane parallelism in over 95% ORs examined is less than 6%. Thus, it is reasonable to use 6% as an approximate critical value. However, in the case that the smallest d-value mismatch between plane

Table 1 Possible matching directions for certain aH/aB range and given cH/aH value aH/aB

cH/aH Any cH/aH

1.5

1.6

1.7

0.7–0.8 0.8–0.9

h1 1 2 0iH =h1 1 1iB h1 1 2 0iH =h1 1 1iB h1 0 1 0iH =h1 1 3iB h1 1 2 0iH =h1 1 1iB h1 0 1 0iH =h1 1 3iB h1 1 2 0iH =h1 0 0iB h1 1 2 0iH =h1 0 0iB h1 0 1 0iH =h1 1 3iB None

None h1 1 2 3iH =h1 1 3iB

None h1 1 2 3iH =h1 1 3iB

h1 1 2 3iH =h1 1 3iB h1 1 2 3iH =h1 1 3iB

h1 1 2 3iH =h1 1 3iB

h1 1 2 3iH =h1 1 3iB

h1 1 2 3iH =h1 1 3iB

None

None

None

None

None

None

0.9–1.0

1.0–1.1 1.1–1.2

M.-X. Zhang, P.M. Kelly / Acta Materialia 53 (2005) 1073–1084

d-value mismatch (%)

30 24

{1 0 1 0}H/{110}B

18 {1 0 1 0}H/{111}B

12 6 0 0.7

0.8

(a)

0.9

1

1.1

1.2

aH/aB

d-value mismatch (%)

30 {0002}H/{110}B

24 18

{0002}H/{111}B

12

{1 0 1 1}H/{110}B

6

{1 0 1 1}H/{111}B 8

0 0.7

0.8

(b)

0.9

1

1.1

1.2

aH/aB 30

d-value mismatch (%)

pairs in a given system is slightly over 6%, but the interatomic spacing misfit along the matching directions is still less than 10%, an OR still exists but with larger angle rotation of the plane pairs relative to each about the matching direction [50]. This rotation angle can be determined using the Dg theory [20,22,23,37,38] as described in [50]. In edge-to-edge matching, the requirement of small interatomic spacing misfit along matching directions (less than 10%) is essential. Since the interplanar spacing or d-values of planes depend on the aH/aB and cH/aH values, the d-value mismatch between these plane pairs also varies with these parameters, as shown in Fig. 3. None of the plane pairs involving (2 0 0)B have been included in this figure because, in all cases, the mismatch was greater than 6%. Fig. 3(a) shows the mismatch for the two plane pairs involving ð1 0  1 0ÞH , where the mismatch is independent of cH/aH. In Fig. 3(b), because the d-value of {0 0 0 2}H and the d-value of f1 0  1 1gH are the same when cH/ aH = 1.5, the two pairs of planes involving (0 0 0 2)H and f1 0  1 1gH overlap and only two curves are shown, one for (1 1 0)B and one for (1 1 1)B. From Fig. 3 it can be seen that the minimum mismatch of each plane pair corresponds to a certain aH/ aB value. Using 6% as the critical value, the potential matching planes that are parallel to each other or may have a small angle of rotation about the matching directions on these planes can be identified for particular aH/ aB and cH/aH values. They are listed at intervals of 0.1 in Table 2, as aH/aB increases from 0.7 to 1.2 for the three selected cH/aH values. Following the rule that the matching directions must lie in the matching planes, the ORs between HCP and BCC can be predicted from Table 1 and Table 2 by the edge-to-edge matching model. The predicted ORs are listed in Table 3 for different aH/aB ranges. In Table 3, for those plane pairs with a d-value mismatch of less 6%, the ORs are written as parallel pairs. Otherwise, they are listed as plane pairs in italic format. It should also be remembered that even in cases where the d-value mismatch is less than 6%, it still necessary to refine the predicted ORs using the Dg theory [50]. In Table 3, there are a total of 11 ORs listed, which are summarized in Table 4. Because the present research has covered a particularly wide range of aH/aB value and cH/aH value, the number of predicted ORs is greater than the observed. In most real systems [44], the aH/aB value was between 0.9 and 1.05 and the cH/aH value between 1.55 and 1.65. Under these conditions, Table 3 lists six ORs. These include the Burgers OR, the Potter OR, the P–S OR and one unknown OR that is related to the R–D OR by a rotation of 10 about ½1 1 2 0
H ½1 0 0
kB . In the following section it will be shown that the other two ORs are actually very close to the Burgers OR and can be described as near Burgers ORs. Therefore, the predictions from the edge-to-edge matching

24

{0002}H/{110}B

18

{0002}H/{111}B

12

{1011} H/{110}B

6

{1011} H/{111}B

0 0.7

0.8

(c)

0.9

1

1.1

1.2

aH/aB 30

d-value mismatch (%)

1076

24

{0002}H/{110}B

18

{0002}H/{111}B

12

{1011} H/{110}B

6 {1011} H/{111}B

0 0.7

(d)

0.8

0.9

1

1.1

1.2

aH/aB

Fig. 3. The variation of d-value mismatch between plane pairs with aH/aB: (a) the plane pairs involving 1 0 1 0H , which are not affected by the cH/aH value; (b) cH/aH = 1.5; (c) cH/aH = 1.6; and (d) cH/aH = 1.7.

model are completely consistent with published experimental results.

3. Refinement of ORs and prediction of habit planes As required by the edge-to-edge matching model, the matching directions are parallel to each other, but the matching planes are allowed to have a few degree rotations about the matching directions. Hence, the pre-

M.-X. Zhang, P.M. Kelly / Acta Materialia 53 (2005) 1073–1084

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Table 2 Possible matching planes where the d-value mismatch is less than 6% for certain aH/aB range and given cH/aH value aH/aB

cH/aH Any cH/aH

1.5

1.6

1.7

0.7–0.8

f1 0 1 0gH =f1 1 0gB f1 0 1 0gH =f1 1 1gB

{0 0 0 2}H/{1 1 1}B f1 0  1 1gH =f1 1 1gB

{0 0 0 2}H/{1 1 1}B f1 0 1 1gH =f1 1 1gB

{0 0 0 2}H/{1 1 1}B f1 0 1 1gH =f1 1 1gB {0 0 0 2}H/{1 1 0}B

0.8–0.9

f1 0 1 0gH =f1 1 0gB

{0 0 0 2}H/{1 1 1}B f1 0  1 1gH =f1 1 1gB {0 0 0 2}H/{1 1 0}B f1 0  1 1gH =f1 1 0gB

{0 0 0 2}H/{1 1 0}B f1 0 1 1gH =f1 1 0g

{0 0 0 2}H/{1 1 0}B f1 0 1 1gH =f1 1 0g

0.9–1.0

None

{0 0 0 2}H/{1 1 0}B f1 0 1 1gH =f1 1 0gB

{0 0 0 2}H/{1 1 0}B f1 0 1 1gH =f1 1 0g

f1 0 1 1gH =f1 1 0g

1.0–1.1

None

{0 0 0 2}H/{1 1 0}B {1 0 1 1}H ={1 1 0}B

{0 0 0 2}H/{1 1 0}B {1 0 1 1}H ={1 1 0}

{1 0 1 1}H ={1 1 0}

1.1–1.2

None

None

None

None

Italic format indicates the mismatch is slightly over 6%.

dicted ORs listed in Table 4 need to be further refined using the Dg theory. In addition, in this process, the habit plane corresponding to each OR can also be predicted. However, the rotation angles of the matching plane vary with the type of matching direction pairs. As mentioned above, the matching directions can be either straight atom rows or zigzag atom rows. Normally a straight row matches a straight row and a zigzag row matches a zigzag row. In the case of straight-tostraight matching, if the d-value mismatch between the matching plane pair is less than 6%, the matching plane pairs can be either parallel or have a small angular deviation. If the d-value mismatch is slightly bigger than 6%, an OR may still result, but with a bigger rotation angle. In the case of zigzag-to-zigzag matching, it is more complicated. There are two types of zigzag atom rows as shown in Fig. 4(a). If the atom Nos. 1 and 3 are in the matching direction, then atom No. 2 is out of the direction. For a Type I zigzag row all these three atoms lie in the matching plane. For a Type II zigzag row, atom No. 2 is not on the matching plane, so that although these three atoms are on one plane, this plane is not the matching plane. Atom No. 2 can be either slightly above or below the matching plane. In the case of a Type I zigzag row matching another Type I row, the matching planes containing this matching direction will either be parallel or will have a small angle of rotation between them, because a large angle rotation about the matching direction will move the atoms out of coincidence as shown in Fig. 4(b). If a Type I row matches a Types II row, a larger angle of rotation is required in order to make the atoms across the interface have maximum matching, as shown in Fig. 4(c). The matching between Type II zigzagged row and the Type II row is not applicable in this simple HCP/BCC system. Normally it involves larger angle rotation.

Because, the computation using the Dg theory requires the actual lattice parameters of both structures, it is impossible to cover all the above predicted ORs. In the present paper, a number of typical systems, for which experimental crystallographic data is already available, will be selected as examples. The systems to be examined include Zr–Nb alloy, Ti–Cr alloy, Fe/ Mo2C system and Fe/M2C system.

3.1. Zr–Nb alloy Zr–2.5wt%Nb alloy is widely used as a pressure tube material in heavy water moderated nuclear reactors. Previous work [38,45,53,54] shows that after solution treatment at 1000 C, different types of BCC precipitates will form from the HCP Zr solid solution matrix. Ageing at 500 C, only BCC Nb-rich b2 phase is precipitated. Ageing above 550 C, a mixture of BCC Nb-rich b2 and Zr-rich b1 precipitates formed. Luo et al. [45] and Banerjee et al. [53] reported the lattice parameters of these phases are: aB = 0.351 nm for b1, aB = 0.331 nm for b2 and aH = 0.322 nm, cH = 0.5123 nm for Zr matrix. Using these data (aH/aB = 0.917 for b1, aH/aB = 0.973 for b2 and cH/aH = 1.591 for matrix), it can be seen from Table 3 that there are six possible ORs. These are the Burger OR, the Potter OR, the P–S OR and Nos. 5, 6, 8 ORs in Table 4. The interatomic spacing misfit along h1 1 2 0iH =h1 1 1iB , h1 1 2 0iH =h1 0 0iB , h1 1 2 3iH =h1 1 3iB and h1 0 1 0iH =h1 1 3iB directions pairs for both b1 and b2 precipitates is listed in Table 5. Taking 10% as a critical value for forming an OR, it can be seen that the ORs between Zr matrix and b1, will be the No. 6 OR, No. 5 OR, the Potter OR or the Burgers OR and the P–S OR. Normally,

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M.-X. Zhang, P.M. Kelly / Acta Materialia 53 (2005) 1073–1084

Table 3 Possible orientation relationships between simple HCP and BCC structures

aH/aB

0.7–0.8

0.8–0.9

0.9 – 1.0

1.0–1.1

1.1–1.2

(a)

1 3

cH/aH 1.5

1.6

1.7

½1 1  2 0
H k½1 1 1
B ð 1 1 0 0ÞH kð1 1 0ÞB

½1 1 2 0
H k½1 1 1
B ð1 1 0 0ÞH kð1 1 0ÞB

½1 1 2 0
H k½1 1 1
B ð1 1 0 0ÞH kð 1 1 0ÞB ½1 1 2 0
H k½1 1 1
B ð0 0 0 2ÞH kð 1 1 0ÞB ½1 1 2 3
H k½1 1 3
B ð1 1 0 0ÞH kð 1 1 0ÞB

½1 1  2 0
H k½1 1 1
B ð 1 1 0 0ÞH kð1 1 0ÞB ½1 1  2 0
H k½1 1 1
B ð0 0 0 2ÞH kð1 1 0ÞB ½1 1  2 0
H k½1 1 1
B ð 1 1 0 1ÞH kð1 1 0ÞB ½1 0  1 0
H k½1 1 3
B ð0 0 0 2ÞH kð1 1 0ÞB ½1 1  2 3
H k½1 1 3
B ð 1 1 0 0ÞH kð1 1 0ÞB ½1 1  2 3
H k½1 1 3
B ð 1 0 1 1ÞH kð1 1 0ÞB

½1 1 2 0
H k½1 1 1
B ð1 1 0 0ÞH kð1 1 0ÞB ½1 1 2 0
H k½1 1 1
B ð0 0 0 2ÞH kð1 1 0ÞB ½1 1 2 0
H k½1 1 1
B ð1 1 0 1ÞH kð1 1 0ÞB ½1 0 1 0
H k½1 1 3
B ð0 0 0 2ÞH kð1 1 0ÞB ½1 1 2 3
H k½1 1 3
B ð1 1 0 0ÞH kð1 1 0ÞB ½1 1 2 3
H k½1 1 3
B ð1 0 1 1ÞH kð1 1 0ÞB

½1 1 2 0
H k½1 1 1
B ð1 1 0 0ÞH kð 1 1 0ÞB ½1 1 2 0
H k½1 1 1
B ð0 0 0 2ÞH kð 1 1 0ÞB ½1 1 2 0
H k½1 1 1
B ð1 1 0 1ÞH kð 1 1 0ÞB ½1 0 1 0
H k½1 1 3
B ð0 0 0 2ÞH kð 1 1 0ÞB ½1 1 2 3
H k½1 1 3
B ð1 1 0 0ÞH kð 1 1 0ÞB ½1 1 2 3
H k½1 1 3
B ð1 0 1 1ÞH kð 1 1 0ÞB

½1 1  2 0
H k½1 1 1
B ð0 0 0 2ÞH kð1 1 0ÞB ½1 1  2 0
H k½1 1 1
B ð 1 1 0 1ÞH kð1 1 0ÞB ½1 1  2 0
H k½1 0 0
B (0 0 0 2)Hi(0 1 1)B ½1 1  2 0
H k½1 0 0
B ð 1 1 0 1ÞH kð0 1 1ÞB ½1 0  1 0
H k½1 1 3
B ð0 0 0 2ÞH kð1 1 0ÞB ½1 1  2 3
H k½1 1 3
B ð 10 1 1ÞH kð1 1 0ÞB

½1 1 2 0
H k½1 1 1
B ð0 0 0 2ÞH kð1 1 0ÞB ½1 1 2 0
H k½1 1 1
B ð1 1 0 1ÞH kð1 1 0ÞB ½1 1 2 0
H k½1 0 0
B (0 0 0 2)Hi(0 1 1)B ½1 1 2 0
H k½1 0 0
B ð1 1 0 1ÞH kð0 1 1ÞB ½1 0 1 0
H k½1 1 3
B ð0 0 0 2ÞH kð1 1 0ÞB ½1 1 2 3
H k½1 1 3
B ð10 1 1ÞH kð1 1 0ÞB

½1 1 2 0
H k½1 1 1
B ð1 1 0 1ÞH kð 1 1 0ÞB ½1 1 2 0
H k½1 0 0
B ð1 1 0 1ÞH kð0 1 1ÞB ½1 1 2 3
H k½1 1 3
B ð10 1 1ÞH kð 1 1 0ÞB

½1 1  2 0
H k½1 0 0
B (0 0 0 2)H/(0 1 1)B ½1 1  2 0
H k½1 0 0
B ð 1 1 0 1ÞH =ð0 1 1ÞB ½1 0  1 0
H k½1 1 3
B ð0 0 0 2ÞH =ð 1 1 0ÞB

½1 1 2 0
H k½1 0 0
B (0 0 0 2)H/(0 1 1)B ½1 1 2 0
H k½1 0 0
B ð1 1 0 1ÞH =ð0 1 1ÞB ½1 0 1 0
H k½1 1 3
B ð0 0 0 2ÞH =ð1 1 0ÞB

½1 1 2 0
H k½1 0 0
B ð1 1 0 1ÞH =ð0 1 1ÞB

None

None

None

1 2

3

Type I zigzag row

2 Type II zigzag row

(b)

Type I zigzag row

Type I zigzag row

(c)

Type II zigzag row

Type I zigzag row

Fig. 4. Schematic illustration of the two types of matching direction with zigzag atom rows: (a) the two types of zigzag row; (b) Type I matches Type I; (c) Type I matches Type II.

The direction pair h1 1 2 0iH =h1 1 1iB corresponds to both the Burgers OR and the Potter OR. Using the Dg theory, the ORs between Zr matrix and b1 phase can be further refined as: ½1 1 2 0
H k½1 1 1
B ;

ð1 1 0 1ÞH 0:24 from ð0 1 1ÞB ð0 0 0 2ÞH 1:2 from ð1 0 1ÞB 

the first four ORs have almost an equal chance of occurring and the last one should be very rare due to the higher misfit.

ð1Þ

with a habit plane close to ð1 1 0 2ÞH  kð1 2 3ÞB . Fig. 5 shows the simulated diffraction patterns along the ½1 1 2 0
H k½1 1 1
B zone axis with a set of parallel Dgs that are normal to the habit plane. Another solution is a predicted OR that is also close to the Potter OR: ½1 1 2 0
H k½1 1 1
B ; ð0 0 0 2ÞH 1:68 from ð0 1 1ÞB ð1 1 0 1Þ 0:24 from ð1 1 0Þ H

B

ð2Þ

Table 4 Predicted ORs from the edge-to-edge matching model in simple HCP/BCC system OR (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

Expression of OR ½1 1 2 0
H k½1 1 1
B ; ð1 1 0 0ÞH kð1 1 0ÞB ½1 1 2 0
H k½1 1 1
B ; ð0 0 0 2ÞH kð1 1 0ÞB ½1 1 2 3
H k½1 1 3
B ; ð1 1 0 0ÞH kð1 1 0ÞB ½1 1 2 0
H k½1 1 1
B ; ð1 1 0 1ÞH kð1 1 0ÞB ½1 0 1 0
H k½1 1 3
B ; ð0 0 0 2ÞH kð1 1 0ÞB ½1 1 2 3
H k½1 1 3
B ; ð1 0 1 1ÞH kð1 1 0ÞB ½1 1 2 0
H k½1 0 0
B ; ð0 0 0 2ÞH kð0 1 1ÞB ½1 1 2 0
H k½1 0 0
B ; ð1 1 0 1ÞH kð0 1 1ÞB ½1 1 2 0
H k½1 0 0
B ; ð0 0 0 2ÞH deviates from (0 1 1)B ½1 1 2 0
H k½1 0 0
B ; ð1 1 0 1ÞH deviates from (0 1 1)B ½1 0 1 0
H k½1 1 3
B ; ð0 0 0 2ÞH deviates from ð1 1 0ÞB

Comments Dyson-Andrews OR [51] Burgers OR Close to the Crawley OR [52] Potter OR Near Burgers OR Near Burgers OR P–S OR Close to the R–D OR Close to the P–S OR Close to the R–D OR Near Burgers OR

M.-X. Zhang, P.M. Kelly / Acta Materialia 53 (2005) 1073–1084 Table 5 Interatomic spacing along the directions pairs for both b1 and b2 in Zr–Nb alloy (%) h1 1 2 0iH =h1 0 0iB Precipitates h1 1 2 0iH =h1 1 1iB 5.9 12.3

b1 b2

8.3 2.7

011B 1 10 2 H

1 1 01H 10 1 B 000 2 H

110B

HCP Zr matrix BCC β1 precipitate

10 1 1 H

Fig. 5. Simulated diffraction patterns along the zone ½1 1 2 0
H k½1  1 1
B , showing a set of parallel Dgs that are normal to the habit plane for the b1 phase in a Zr–Nb alloy.

with a habit plane close to ð1 0  1 2ÞH  kð 12 3ÞB . A similar superimposed diffraction patterns can also be constructed, but in the interests of brevity will not be shown here. The ORs (1) and (2) are close to the Potter OR and the first one is actually in between the Potter OR and the Burgers OR. Perovic and Weatherly [54] and Zhang and Weatherly [38] experimentally observed that the OR between Zr matrix and the b1 phase deviates from the ideal Burgers OR by 1–1.5 rotation between (0 0 0 1)H and (0 1 1)B about the ½1 1  2 0
H k½1  1 1
B axis. This is actually the OR (1). Luo and Weatherly [45] also indicate that within experimental error the OR is the Burgers OR. Perovic and Weatherly [54] even considered that the determined OR is the Potter OR, which is consistent with the OR (2). The direction pair, h1 1  2 3iH =h1 1 3iB , corresponds to OR No. 6. Because this OR involves matching between a Type I zigzag row and a Type II zigzag row, normally, there will be bigger angle rotation of the matching planes. Using the Dg theory, the rotation angle of the matching planes can be calculated to be 9.6. This gives the following OR matrix: 0:41401454

0:91022350

0:00922683

0:64135940 0:64594898

0:28449891 0:30092116

0:71254368 0:70156709

1079

h1 1 2 3iH =h1 1 3iB

h1 0 1 0iH =h1 1 3iB

4.0 10.2

4.2 1.6

Clearly, it is very close to the Burgers OR. The habit plane associated with this OR is 11.6 from ð1 2  1ÞB , which agrees with Zhang and PurdyÕs experimental results (see Fig. 1(a) in [38]). Fig. 6 is the simulated diffraction patterns along the ½1 1 2 3
H k½1 1 3
B zone axis showing a set of parallel Dgs that are normal to the habit plane. The direction pair, h1 0 1 0iH =h1 1 3iB , results in OR No. 5 in Table 4. This is an example of a Type I zigzag row matching a Type I zigzag row. In order to preserve maximum atom matching across the interface, the matching planes should be parallel or be inclined to each other by a small angle. The Dg theory indicates that the rotation angle is 0.37, which results in the following OR matrix: 0:63502255 0:64415508

0:30151134 0:30151134

0:71122238 0:70296170

0:42639254 0:90453403

0:00275356

It corresponds a habit plane that is 8 from ð1  1 0ÞB . The OR can be expressed as: ½2 1 1 0
H 0:54 from ½1 1 1
B ð0 0 0 2ÞH 0:37 from ð 1 1 0ÞB   ½1 1 2 0
H 4:76 from ½0 0 1
B ð4Þ This OR is also very close to the Burgers OR. Similar superimposed diffraction patterns can be simulated to shown the OR and at least two parallel Dgs. So far, the ‘‘edge-to-edge’’ matching model has predicted four different ORs between Zr matrix and the b1 precipitate. ORs (3) and (4) that involve direction pairs, h1 1 2 3iH =h1 1 3iB and h1 0 1 0iH =h1 1 3iB , are

1 100H 1 011H 1 10B

HCP Zr matrix BCC β1 precipitate

1 2 11H 12 1 B

This OR matrix corresponds to an OR that can be expressed as: ½1 1  2 0
H 0:7 from ½1  1 1
B ð0 0 0 2ÞH 0:68 from ð0 1 1ÞB  ½21 1 0
5:48 from ½ 1 0 0
ð3Þ H

B

Fig. 6. Simulated superimposed diffraction patterns along the zone ½1 1 2 3
H k½1 1 3
B , showing a set of parallel Dgs that is normal to the habit plane for b1 phase in Zr–Nb alloy.

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M.-X. Zhang, P.M. Kelly / Acta Materialia 53 (2005) 1073–1084

very close to the Burgers OR. If experimental error is taken into account, it is difficult to distinguish these ORs from the ideal Burgers OR using conventional electron spot diffraction patterns. Hence, it can be regarded as the Burgers OR. OR (2) is the Potter OR and the OR (1) is in between the Burgers OR and the Potter OR. All these ORs have been experimentally observed [38,45,54]. There are limited experimental data on the habit plane, except for Zhang and Purdy [38], which is consistent with one of the predicted ORs. For the b2 phase, which is a Nb rich precipitate, Table 5 indicates that the most likely ORs should result from h1 1  2 0iH =h1 0 0iB and h1 0  1 0iH =h1 1 3iB direction pairs. Like the b1 case, for the direction pair h1 1  2 0iH =h1 0 0iB , the following ORs are obtained after finding three parallel Dgs from the Dg theory. They are: ½1 1  2 0
k½1 0 0
; ð0 0 0 2Þ 3:4 from ð0 1 1Þ H

B

H

B

½ 1 1 0 0
H 3:4 from ½0  1 1
B

ð5Þ

with habit plane ð2  2 0 5ÞH  kð0 4 1ÞB , and ½1  2 1 0
k½1 0 0
; ð 1 0 1 1Þ  4:1 from ð0 1  1Þ H

B

H

H

ð6Þ

B

with habit plane close to ð0  2 2 1ÞH  kð1 1  3ÞB . The OR (5) is 6 from the Potter OR and 6.26 from the Burgers OR. Following Luo and Weatherly [45], the rotation angle between ½ 1 1 0 0
H and ½0  1 1
B is denoted by c. For the P–S OR c = 0 and for the Burgers OR c = 5.26. This OR results in c = 3.4. Luo and Weatherly [45] reported the experimentally observed ORs between b2 and Zr matrix corresponded to c values between 3.8 and 4.1. Obviously, the predictions agree very well these previously reported results. The OR (6) is 6 from the R–D OR but has not been observed experimentally. For the second direction pair, h1 0  1 0iH =h1 1 3iB , the Dg calculation gives the following OR: 1  0
0:7 from ½1 1 1
ð0 1  1 1Þ 0:8 from ð0 1 1Þ  ½2 1 H

B

7.15wt%Cr alloy [33,34,40]. According to Menon and AaronsonÕs results [40], both a precipitates have the same structure and lattice parameters. These are aH = 0.29564 nm, cH = 0.46928 nm. The lattice parameter for the b matrix is aB = 0.325 nm for the normal a and aB = 0.294 nm for the black plate a. Table 3 suggests that the possible ORs between b and a will be the Burgers OR, the Potter OR, the P–S OR and the ORs Nos. 5, 6, 8 in Table 4. According to the reported lattice parameters, the interatomic spacing misfit is listed in Table 6. From this table it is clear that for normal a, the most likely ORs will result from direction pairs, h1 1 2 0iH =h1 1 1iB ; h1 1 2 3iH =h1 1 3iB and h1 0  1 0iH = h1 1 3iB . The first direction pair corresponds to the Burgers OR and the Potter OR. After refinement using the Dg theory, the following ORs can be obtained: ½1 1 2 0
H k½1 1 1
B ; ð0 0 0 2ÞH 0:66 fromð1 1 0ÞB ð1 1 0 1Þ 1:6 from ð0 1 1Þ ð8Þ

H

with habit plane close to ð1 1 0 3ÞH  kð1 2 1ÞB . Fig. 7 is the simulated superimposed diffraction pattern in ½1 1 2 0
H k½1 1 1
B zone axis showing a set of parallel Dgs that are normal to the habit plane. In addition, two more ORs are also predicted. One is the exact Burgers OR with a habit plane that is 7.1 away from ð1 1 0ÞB . The third is very close to the Potter OR: ½1 1 2 0
k½1 1 1
; ð1 1 0 1Þ 0:18 from ð0 1 1Þ H

B

H

B

ð0 0 0 2ÞH 1:2 from ð1 0 1ÞB

ð9Þ

with a habit plane that is 8.1 away from (0 1 1)B.

B

½1 1  2 0
H 4:8 from ½0 0  1
B

B

1 1 00H 1 1 01H

ð7Þ

1 1 03H 1 2 1B

0 1 1B

with c = 4.8. It is very close to the Burgers OR. As was the case with the b1 phase, there is insufficient experimental data to test the predicted habit planes.

0002H 1 1 0B

HCP α plate BCC β matrix

101B

3.2. Ti–Cr alloy After solution treatment at 1000 C followed by isothermal treatment in the a + b region (600–700 C), two types of HCP structure a precipitates, normal a and black plate a, formed from BCC b matrix in Ti–

Fig. 7. Simulated superimposed diffraction along the zone ½1 1 2 0
H k½1 1 1
B , showing a set of parallel Dgs that is normal to the habit plane for normal a plate in Ti–Cr alloy.

Table 6 Interatomic spacing along the directions pairs for both normal and black a (%) Precipitates h1 1 2 0iH =h1 1 1iB h1 1 2 0iH =h1 0 0iB

h1 12 3iH =h1 1 3iB

h1 0 1 0iH =h1 1 3iB

Normal a Black a

2.9 13.8

5.0 5.0

5.0 16.1

9.0 0.6

M.-X. Zhang, P.M. Kelly / Acta Materialia 53 (2005) 1073–1084

The calculation of the parallel Dgs using the Dg theory has shown that the direction pair h1 1  2 3iH = h1 1 3iB leads to an OR that is: 1 1
B ð0 0 0 2ÞH 0:76 from ð0 1 1ÞB ½1 1 2 0
H 0:8 from½1  ½ 21 1 0
4:9 from½ 1 0 0
ð10Þ H

B

with a corresponding habit plane that is 3.2 away from ð1 2  1ÞB . The direction pair h 1 0 1 0iH =h1 1 3iB leads to an OR as: ½1 1  2 0
0:52 from ½1 1 1
ð0 0 0 2Þ 0:25 from ð1 1 0Þ H

B

½21 1 0
H 4:76 from ½0 0  1
B

H

B

ð11Þ

which corresponds to a habit plane that is 11.9 from ð1  1 0ÞB . ORs (10) and (11) are very close to the Burgers OR. To test the correctness of these predictions, previous experimental results are summarized as follows. Menon and Aaronson [40] reported the Burgers OR between the normal a and the b matrix. While Furuhara and coworkers [33–35] observed that the ORs between normal a and b are slightly different, which included the exact Burgers OR, a near Burgers OR and an OR with (0 0 0 1)H within 1 to 1.5 away from (1 1 0)B that is actually close to the Potter OR. Recently, Ye, Zhang and Qiu [41] used Kikuchi diffraction patterns and found that the OR between the normal a and b matrix deviates from the ideal Burgers OR by 0.6 rotation of (0 0 0 1)H to ð0  1 1ÞB . It can be seen that the predictions are perfectly consistent with all these previously reported experimental results. However, within the range of experimental errors when these ORs were experimentally determined, they could be considered as the Burgers OR or close to the Burgers OR. But, they are actually different and distinguishable, because of the different habit planes. Furuhara and Aaronson [33,34] reported the habit plane for near Burgers OR is close {1 1 2}B, which is consistent with the predicted habit plane for ORs (8) and (10). Menon and Aaroson [40] determined the habit plane for the Burgers OR to be {1 1 0}B, and Williams [55] reported that a plates formed in Ti–Mo–Al alloy had a {1 1 0}B habit plane with a Burgers OR. These results agree with the ORs (9) and (11). For black a plates, Menon [40] considered they had the same lattice parameter as normal a plates, but the lattice parameter of the corresponding b matrix changed to aB = 0.294 nm. It was also reported that the black a plates and the b matrix obeys the Burgers OR, but correspond to a different habit plane because of the lattice parameter change in the matrix. This is probably incorrect. Firstly, the black a is a Cr supersaturated precipitate that has a different chemical composition from the normal a. This will affect its lattice parameter. Secondly, from Fig. 5(a) and (b) in [40] it can be seen that the lat-

1081

tice parameters of both phases are changed, if we assume that both diffraction patterns were taken at the same camera constant. Finally, the diffraction patterns at Fig. 5(a) of [40] actually shows the Potter OR, rather than the Burgers OR. According to the predicted results described above, the Potter OR corresponds to a habit plane that is close to {1 1 0}B. Hence, the different habit plane of the black a compared with the normal a is the result of a different OR, rather than a change in the lattice parameter change of the matrix. Due to the lack of accurate lattice parameters for black a plate, the edgeto-edge matching model cannot be applied with confidence. 3.3. Mo2C/Fe system Another typical example of simple HCP/BCC system is the Mo2C/Fe system (or the M2C/Fe system, where M denotes any carbide-forming element). According to the International Powder Diffraction File, the lattice parameter of BCC Fe is aB = 0.28664 nm and the lattice parameters of HCP Mo2 C are aH = 0.3002 nm, cH = 0.4724 nm. These give aH/aB = 1.05. Thus, the possible ORs will be ½1 1 2 0
H k½1 0 0
B , ð1 1 0 1ÞH =ð0 1 1Þ; ½1 1 2 0
H k½1 0 0
B , (0 0 0 2)H/(0 1 1)B and ½1 0  1 0
H k ½1 1 3
B , (0 0 0 2)H/(0 1 1)B as listed in Table 3. In this system, the d-value mismatch between {1 1 0 1H } and {0 1 1}B and between {0 0 0 2}H and {0 1 1}B is 12.4 and 16.6%, respectively. These are the two smallest d-value mismatch values between the possible matching planes. But the interatomic spacing misfit along the matching directions h1 1 2 0iH kh1 0 0iB and h1 0  1 0iH k h1 1 3iB is 4.5% and 9.4%. Hence, it is still possible for the direction pair, h1 1 2 0iH kh1 0 0iB , to form an OR with a larger rotation angle between the matching planes. But it is unlikely that the direction pair h1 0 1 0iH kh1 1 3iB will form an OR for two reasons. First, this direction pair is a Type I zigzag row matched with a Type I zigzag row, which requires the matching planes to be parallel or have a small angle of rotation between them in order to preserve the maximum atom matching across the interface. Unfortunately, the big d-value mismatch between the possible matching planes associated with this matching direction pair requires a large angle rotation to form an OR. Secondly, the interatomic spacing misfit along this direction pair is rather large. Hence, this direction pair should not be considered. Based on the available information and using the Dg theory, the following ORs can be obtained from the direction pair h1 1 2 0iH kh1 0 0iB : ð12Þ ½1 1 2 0
k½1 0 0
; ð0 0 0 2Þ 4:4 from ð0 1 1Þ H

B

H

B

with habit plane that is 6.9 from (0 2 1)B. Obviously, this predicted OR deviates from the ideal P–S OR by a 4.4 rotation. Although Berry [56] reported an ideal P–S OR between Mo2C and ferrite, DysonÕs results

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M.-X. Zhang, P.M. Kelly / Acta Materialia 53 (2005) 1073–1084

021B

HCP Mo2C 010B

011B

BCC ferrite matrix

0002H

003B

0001H 001B

2 203H

1 100H 0 1 2B 2 201H

22

Fig. 8. Simulated diffraction along the zone ½1 1 2 0
H k½1 0 0
B showing the near P–S OR, and a set of parallel Dgs that is normal to the habit plane in Mo2C/Fe system.

[57] shows a 3.5 deviation of (0 0 0 1)H and (0 1 1)B when the ½ 2 1 1 0
H is exactly parallel to [1 0 0]B. Recently, Shi and Kelly [58] used more accurate techniques to determine the OR and found that there is a systematic deviation of up to 5.5 from the ideal P–S OR. The latter two experimental results prove the predictions. Fig. 8 is a simulated superimposed diffraction patterns showing this predicted OR and the parallel Dgs. Another possible OR from the same matching direction can be refined as:  0
k½1 0 0
; ð1 1  0 1Þ 11:0 from ð0 1 1Þ ½1 1 2 ð13Þ H

B

H

B

with a habit plane that is 8.2 from (0 1 0)B. If this OR is expressed in matrix format as follows: 0:50000000

0:86602540

0:00000000

0:38899148

0:22458433

0:89344698

0:77374778

0:44672349

0:44916867

it can be seen that, this is actually the ideal R–D OR [49,59], which was found between M2C carbide and ferrite in high speed steel during the tempering process. The carbide has chemical composition as (Cr0.4Fe0.24Mo0.25V0.04W0.07)2C0.86 and has the HCP structure with aH = 0.297 nm, cH = 0.475 nm [59]. A similar superimposed diffraction patterns can also be constructed, but in the interests of brevity will not be shown here. Because both the Mo2C and the M2C precipitates are rod-shaped, the data of habit plane are not available yet. Thus, predictions on habit plane can not be experimentally confirmed.

4. Discussion In addition to the above four systems studied, the model can be used in any other systems involving simple HCP and BCC, many of which are listed in [44]. Compared with all other models [1–23], the edge-to-edge matching model does not need to assume an OR and

it does not involve complicated mathematic calculations. Furthermore, the present model can reveal the relationship between the ORs and the aH/aB and cH/aH values in simple HCP/BCC systems. From Fig. 2(a) it can be seen that with the increase in the aH/aB value, the ORs between HCP and BCC phases changes from the Burgers OR or the Potter OR, then an OR that is close to the Burgers OR, and to the P–S OR or R–D OR. Examination of the available crystallographic data as summarized in [44] shows that this trend in the predicted ORs is consistent with the experimentally observed ORs. For some years, the O-lattice theory and the invariant line model have been applied to these systems in an attempt to understand the ORs, habit planes and the interfacial structures. The invariant line model is based on the minimization of strain energy in the interface and it was considered that an invariant line can be produced in numerous ORs [60]. In general the invariant line corresponds to minimum energy [45,54]. Ye and Zhang [41] indicated that a 0.6 deviation from the ideal Burgers OR in a Ti–Cr alloy is required to produce an invariant line. However, throughout the work on invariant lines an OR, or at least a lattice correspondence between the two phases, has to be assumed in order to calculate an invariant line. O-lattice is another theory that was frequently used to explain the observed ORs. Zhang et al. [37,41] applied this theory to both the Ti–Cr and the Zr–Nb alloys and indicated that the misfit in the habit plane can be accommodated by a single set of [0 1 0]B dislocations lying along the invariant line. Similarly Rong et al. [59] calculated the O-lattice based on the observed ORs and habit planes in an attempt to understand the Burgers OR, the P–S OR and the R–D OR. Once more, these complex calculation can only explain an OR when it has been assumed or observed. The structural ledge model [1–9] is another geometrical model for explaining the crystallography of diffusion controlled phase transformations. It works when an OR has already been defined. The major difference between the structure ledge model and the present edge-to-edge matching model can be summarized as following: the former requires that the atomic matching occurs on the close packed or nearly close packed planes. These are, in fact, the planes that are defined as matching planes in the edge-to-edge matching model. The latter is based on the atomic matching occurring along the ‘‘edges’’ of the matching plane, not in the planes themselves. These ‘‘edges’’ are the close packed or nearly close packed directions lying in the matching planes and are termed matching directions. According to the edge-to-edge matching model, the intersection plane of the matching planes forms one interface between the product and the parent phase. This interfacial plane should be a Moire´ plane [61]. It consists of a series of matched atom rows. Hence, in or-

M.-X. Zhang, P.M. Kelly / Acta Materialia 53 (2005) 1073–1084

der to maximize the atom matching at the interface, it requires maximum atom matching along atom rows or the minimization of the interatomic spacing misfit along the matching direction. In most cases, the interatomic spacing misfit is not zero. Thus, dislocations have to be incorporated to accommodate the misfit along the matching directions. Another factor controlling the strain energy at the interface is the d-value mismatch between the matching planes. When the d-value mismatch is small, the matching planes tend to be parallel to each other. If the mismatch increases, the matching planes will deviate from each other by a small angle rotation about the matching direction. The phase transformation always tries to find matching planes that have a small d-value mismatch so that the matching planes can be parallel to each other in an ‘‘edge-to-edge’’ fashion, or else be at a small angle to each other. Except for the compensating dislocations required by the edge-to-edge matching model to accommodate the misfit along the matching directions, ledges are another feature of the model. Because two sets of parallel planes can form a set of parallel Moire´ planes and each Moire´ plane has the potential to be the intersection plane, when the interfacial plane changes from one Moire´ plane to another, a structural ledge will be generated. The height of the ledge should be whole number multiple of the interplanar spacing of the Moire´ planes. This argument is similar to the conclusion reached by Nie [61]. This type of interface can be shown in Fig. 9. It can be seen that this simulated figure is similar to Fig. 11 of [33]. The motion of these ledges results in the growth of the product phases. Assume the solid lines at left of Fig. 9 represent parent phase or matrix and the dashed lines at the right denote product phase or precipitates, the ledges will move downwards as shown by the arrows. This movement of the ledges is strongly affected by the chemical composition, diffusional flux [62,63] and other factors. This is the dynamical problem of solid–solid phase transformations. The present work just elucidates the possibility of formation of ledges at

Interface

Matching planes

1083

the interface between two phases from crystallographic point view using the edge-to-edge matching model.

5. Conclusions 1. As a simple but powerful tool, the edge-to-edge matching model has been used to predict the ORs and the corresponding habit plane in simple HCP/ BCC system over a wide range of aH/aB and cH/aH values. A total of 17 ORs are predicted, which include all the experimentally observed ORs and some that have not been observed because of the lack of suitable phase systems to actually produce these ORs. 2. The predictions indicate that the ORs and habit planes strongly depend on the lattice parameters of both HCP and BCC phases. For the common simple HCP and BCC systems, in which the aH/aB value is between 0.8 and 1.05, and the cH/aH value ranges from 1.5 to 1.65, four ORs – the Burgers OR (including the near Burger OR), the Potter OR, the P–S OR and the R–D OR – are all predicted by the model. These predictions are completely consistent with previous experimentally determined ORs. 3. According to the edge-to-edge matching model, the intersection plane, which may be or may not be the experimentally determined habit plane, consists of matched atom rows. There may be dislocations along the matching directions to accommodate any interatomic spacing misfit. Ledges can also be generated at the interface. 4. One of the key advantages of the edge-to-edge matching model over all other models is that it is capable of predicting ORs rather than merely explaining the experimentally observed ORs, like all other models have done.

Acknowledgement The authors are grateful to the Australian Research Council (ARC) Large Grant and Discovery Project for funding support.

References

Matching planes Matching directions Fig. 9. Schematic diagram showing the formation of an interface with ledges in terms of the edge-to-edge matching model.

[1] Rigsbee JM, Aaronson HI. Acta Metall 1979;27:351–65. [2] Hall MG, Aaronson HI, Kinsman KR. Surf Sci 1972;31:257. [3] Shiflet GJ, van der Merwe JH. Metall Mater Trans A 1994;25:1895. [4] Forwood CT, Clarebrough LM. Philos Mag B 1989;59:637. [5] Russell KC, Hall MG, Kinsman KR, Aaronson HI. Metall Trans A 1974;5:1503. [6] Shiflet GJ, van der Merwe JH. Acta Mater 1994;42:1189.

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