Controlling factor for nucleation of martensite at grain boundary in Fe-Ni bicrystals

Acta Materialia 51 (2003) 1007–1017 www.actamat-journals.com

Controlling factor for nucleation of martensite at grain boundary in Fe-Ni bicrystals M. Ueda a, H.Y. Yasuda abc, Y. Umakoshi ac,∗ a

Department of Materials Science and Engineering, Graduate School of Engineering, Osaka University, 2-1, Yamada-oka, Suita, Osaka 565-0871, Japan b Research Centre for Ultra-High Voltage Electron Microscopy, Osaka University 7-1, Mihogaoka, Ibaraki, Osaka 567-0047, Japan c Frontier Research Centre, Osaka University, 2-1, Yamada-oka, Suita, Osaka 565-0871, Japan Received 4 July 2002; received in revised form 23 October 2002; accepted 25 October 2002

Abstract A favourable nucleation of martensites at a grain boundary was investigated using Fe-Ni bicrystals containing a ⬍ 211 ⬎ symmetric tilt boundary with various tilt angles, focusing on the martensite-start temperature (Ms), the morphology of martensites and the variant selection. Near the grain boundaries, some variants with the habit plane almost parallel to the boundaries were preferentially selected and their variants changed depending on the tilt angle. In addition, the equivalent variants were symmetrically adjoined at the boundary to maintain the compatibility of shape strain of martensites across the boundary. Such characteristic nucleation can be regarded as a self-accommodation across the boundary, which is called cooperative nucleation (CN). The CN effectively reduces the strain energy due to the formation of martensites compared with the independent nucleation within a grain resulting in an increase in Ms. The characteristic variant selection of martensites can also be explained from the minimisation of strain energy.  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Grain boundary character; Phase transformation; Martensite; Strain energy; Iron alloys

1. Introduction Heterogeneous nucleation of martensites is known to occur at lattice defects such as dislocations and grain boundaries [1,2]. Many researchers have investigated the effect of grain boundary on martensitic transformation behaviour ∗ Corresponding author. Tel.: +81-6-6879-7494; fax: +81-66879-7495. E-mail address: [email protected] (Y. Umakoshi).

[1–5], but the exact nature of such boundaries could not be identified. We then investigated the effects of grain boundary on martensitic transformation using Fe-Ni bicrystals [6–8]. Limited grain boundaries with a specific character could activate martensitic transformation effectively; 90ⴰ ⬍ 211 ⬎ symmetric tilt boundary acted as a favourable nucleation site for martensites, while 90ⴰ{211} twist boundary did not [6]. In the vicinity of grain boundary, some variants with the habit plane almost parallel to the boundary were preferentially selected among 24 habit plane variants.

1359-6454/03/$30.00  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(02)00503-7

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Furthermore, the equivalent variants were adjoined at the tilt boundary to maintain the compatibility of the transformation strains across the boundary, resulting in an increase in the martensite-start temperature (Ms). We defined this type of nucleation as ‘cooperative nucleation (CN)’. Thus, conditions for a favourable nucleation at the grain boundary were determined, although the reasons such variants are preferentially chosen and why the CN is favourable at the symmetric tilt boundary are not yet clear. According to our previous findings [6], every symmetric tilt boundary can activate the CN from the crystallographic viewpoint. Since the habit plane of specific variants showing CN must be as parallel to the boundary as possible, the variants vary with a tilt angle of the symmetric tilt boundary. We have investigated martensitic transformation behaviour in the vicinity of the 90ⴰ ⬍ 211 ⬎ symmetric tilt boundary in Fe-Ni bicrystals. On the other hand, an annealing twin boundary in f.c.c. metals is also one ⬍ 211⬎ symmetric tilt boundary with a tilt angle of 180ⴰ. Although several researchers [1,5] have examined the effect of the twin boundary on martensitic transformation behaviour, the details are not yet clear. Systematic investigation on the martensitic transformation behaviour in the vicinity of ⬍ 211 ⬎ symmetric tilt boundaries with various tilt angles will allow us to understand the nature of the CN at the boundaries. The purpose of this work was to determine a controlling factor for a favourable nucleation of martensites at the grain boundary using Fe-Ni bicrystals containing ⬍ 211 ⬎ symmetric tilt boundaries with various tilt angles, focusing on the Ms, the morphology of martensites and the variant selection.

Fe-Ni bicrystals containing the ⬍ 211 ⬎ symmetric tilt boundary were obtained, whose tilt angles (c) were selected to be 90ⴰ, 130ⴰ, 150ⴰ or 180ⴰ as shown in Fig. 1. The specimens (4 mml×6 mmw×0.5mmt) were cut from the bicrystals by spark machining, homogenised at 1273 K for 1 h and were subsequently quenched in ice water. Electrical resistivity measurement to determine the Ms was carried out by a four points method at a cooling rate of 1 Kmin⫺1. The surface relief due to the martensitic transformation was removed by mechanical and electrolytic polishing for crystallographic observation. Crystallography and selected variants of the martensites were analysed by a SEM-EBSP method. The 24 habit plane variants are represented as V1–V12 and V1⬘–V12⬘ defined in the previous paper [6], as shown in appendix. The habit plane normals of their variants are plotted in a stereographic projection for component crystals of bicrystals, where an (x, y, z) system was

2. Experimental procedure A Fe-32at.%Ni single crystal was grown by the floating zone method at a rate of 5 mmh⫺1 under high purity Ar gas flow. The chemical composition of this crystal was determined to be Fe33.1at.%Ni-0.0029at.%C-0.0008at.%N. The crystal was homogenised at 1473 K for 48 h. Two plates with controlled crystal orientations were diffusion bonded at 1273 K for 100 h. Four types of

Fig. 1. A schematic illustration of bicrystals with the ⬍ 211⬎ symmetric tilt boundary and definition of the tilt angle (c).

M. Ueda et al. / Acta Materialia 51 (2003) 1007–1017

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set up at each tilt angle c as shown in Fig. 2. The x and y axes were taken to be parallel to the ⬍ 211 ⬎ tilt axis and the grain boundary normal, respectively.

3. Results 3.1. Variation in the Ms with the tilt angle around ⬍ 211 ⬎ axis The Ms of Fe-32at.%Ni bicrystals containing the ⬍ 211 ⬎ tilt boundary at c=90ⴰ, 130ⴰ, 150ⴰ or 180ⴰ was determined by the change in electrical resistivity on cooling. Three specimens were used for each c, all measured Ms are plotted against c in Fig. 3; it varies with c although there is a small scattering of it at each c. The mean Ms at c=90ⴰ is determined to be about 129 K, which is nearly equal to that at c=150ⴰ. In contrast, bicrystals at c=130ⴰ and 180ⴰ show higher Ms than those at c= 90ⴰ and 150ⴰ. Particularly, the average Ms at c= 180ⴰ exhibits the highest value of 143 K among tested specimens. The difference in the Ms of these bicrystals is not due to their chemical composition since all specimens were prepared from the same single crystal ingot. In the previous work [6], we confirmed that bicrystals with a 90ⴰ ⬍ 211 ⬎ tilt

Fig. 2. A stereographic projection of habit plane normal for component crystals of bicrystals with the symmetric tilt boundary at each tilt angle.

Fig. 3. The martensite-start temperature (Ms) of bicrystals with the ⬍ 211 ⬎ tilt boundary plotted against tilt angles (c).

boundary have higher Ms than single crystals by more than 50 K. All tested ⬍ 211 ⬎ symmetric tilt boundaries act as a favourable nucleation site of martensite but the potency of the boundary for the nucleation changes with c because Ms varies with c. 3.2. Morphology and variant selection of martensites in the vicinity of ⬍ 211 ⬎ tilt boundaries Fig. 4 shows typical optical micrographs of martensites in Fe-Ni bicrystals containing a ⬍ 211 ⬎ tilt boundary at c=90ⴰ (a), 130ⴰ (b), 150ⴰ (c) and 180ⴰ (d). Lenticular martensites are symmetrically formed about the boundary in all bicrystals. The habit plane of martensites near the boundary seems almost parallel to the boundary and the angle between the longitudinal direction of martensites and the grain boundary changes depending on c. On the other hand, randomly nucleated martensites are observed apart from the boundary. All martensites are confirmed to satisfy the Nishiyama-Wassermann (N–W) relationship with the parent phase by the SEM-EBSP method. Fig. 5 shows the orientation images of martensites in the vicinity of ⬍ 211 ⬎ tilt boundary at c= 90ⴰ (a), 130ⴰ (b), 150ⴰ (c) and 180ⴰ (d). Although all N-W variants equally nucleated in single crystals [6], some variants are preferentially selected

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Fig. 4. Optical micrographs of thermally transformed martensites in Fe-Ni bicrystals with ⬍ 211 ⬎ tilt boundary at c=90ⴰ (a), 130ⴰ (b), 150ⴰ (c) and 180ⴰ (d).

Fig. 5. Orientation image of EBSP analysis of thermally transformed martensites in Fe-Ni bicrystals with ⬍ 211 ⬎ symmetric tilt boundary at c=90ⴰ (a), 130ⴰ (b), 150ⴰ (c) and 180ⴰ (d).

M. Ueda et al. / Acta Materialia 51 (2003) 1007–1017

among 24 habit plane variants near the tilt boundary. V11, V2⬘ and V3 variants are preferentially formed near the boundary at c=90ⴰ, as shown in Fig. 5(a). The habit planes of V11 and V2⬘ are closest and second closest to the boundary at c= 90ⴰ, respectively, as shown in Fig. 2. In contrast, the habit plane of V3 is not so close to the boundary. At c=130ⴰ and 150ⴰ, V2⬘ and V3 variants are dominant while the variant V11 selected at c=90ⴰ is not seen, as shown in Fig. 5(b) and (c). On the other hand, the specific variants such as V2 and V4’ are formed near the boundary at c=180ⴰ as shown in Fig. 5(d). It should be noted that the equivalent variants are necessarily adjoined at the boundary to maintain the compatibility of transformation strain at the boundary in all bicrystals, that is, the CN always occurs at the ⬍ 211⬎ tilt boundary regardless of c. Furthermore, a peculiar morphology is sometimes observed near the boundary at c=180ⴰ. Fig. 6 shows an optical micrograph (a) of martensites and the corresponding orientation image by EBSP analysis (b) in the same area near the boundary at c=180ⴰ, though the magnification is different. Four types of CN pairs, the V1⬘–V1⬘, V2–V2, V3–V3 and V2⬘–V2⬘ pairs are observed to be close to each other. As a result, the morphology of the variant group looks like a ‘diamond’ or an ‘X’. This strongly suggests that self-accommodation by plural variants occurs near the boundary at c=180ⴰ. Thus, the morphology of

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martensites and the selected variants in the vicinity of grain boundary are strongly influenced by c.

4. Discussion As might have been expected, the CN of martensites could be observed in the vicinity of all tested symmetric tilt boundaries. The Ms, the morphology of martensites and the selected variant depended strongly on the tilt angle c. Such transformation behaviour must be governed by a simple factor. 4.1. The variant selection near the boundary depending on the tilt angle In the vicinity of the ⬍ 211 ⬎ symmetric tilt boundary, a limited set of variants was preferentially chosen among the 24 N-W habit plane variants depending on c. The V2⬘ variant always appeared irrespective of c, while the additional variants changed with an increase in c from c= 90ⴰ to 180ⴰ in the following sequence: V11 at c= 90ⴰ→V3 at c=130ⴰ and 150ⴰ→V1⬘, V2 and V3 at c=180ⴰ. The specific variants having the habit plane almost parallel to the boundary are preferentially chosen near the grain boundary [6]. To understand such variant selection quantitatively, the deviation angle (d) between the habit plane of martensites and the grain boundary was calculated

Fig. 6. Optical micrograph (a) and orientation image of EBSP analysis (b) for characteristic morphology of martensites observed in the vicinity of ⬍ 211 ⬎ symmetric tilt boundary at c=180ⴰ.

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for all 24 habit plane variants, as shown in Fig. 7. As d approaches 0ⴰ, the habit plane becomes parallel to the boundary. From the c–d curves, a suitable variant at each c can be easily predicted; specific variants with the smallest d must be chosen near each boundary. Actually, such variants are preferentially selected as shown in Fig. 5. For instance, the variant with the smallest d at c=90ⴰ is V11, while that at c=130ⴰ and 150ⴰ is V2⬘. Thus the successive change in the selected variants is fairly consistent with the variation in d with c. Interestingly, various variants were confirmed to nucleate in the vicinity of the 180ⴰ ⬍ 211 ⬎ tilt boundary. Although the suitable variant for nucleation near grain boundary is equivocally limited on the basis of d except for the boundary at c=180ⴰ, plural variants of V1, V2, V3, V1⬘, V2⬘ and V3⬘ have the same d of 26ⴰ at c=180ⴰ as shown in Fig. 7. Therefore, it is quite natural that many variants were selected near the boundary at c= 180ⴰ. Thus, the variant selection depending on c can be explained simply using the c–d relation. However, some variants such as V3 at c=90ⴰ and V4⬘ at c=180ⴰ could not be explained based on d. These additional variants of V3 and V4⬘ were always formed adjoining the specific variants of V2’ and V2, respectively. Therefore, such variants may be activated by the shape deformation of neighbouring variants formed near the grain boundary. As reported in our previous paper [6], if the formation of variants is induced by a prior

martensite, the pair of variants should satisfy the following requirements: (i) the stress transmission factor Nij [9] of a variant pair is closed to ⫺1, (ii) the variant pair has close habit planes. It means that the shape strain of the previously formed variant is effectively transmitted to that of the neighbouring one. Since the observed pairs surely satisfies the above-mentioned requirements, the variants of V3 and V4⬘ can be regarded as additive variants induced by the neighbouring variants nucleated in the vicinity of grain boundary. 4.2. Elastic strain energy for the cooperative nucleation near grain boundary Fe-Ni bicrystals with ⬍ 211 ⬎ symmetric tilt boundaries demonstrate not only higher Ms than the single crystals but also the characteristic nucleation called CN. In this section, the reason the CN by specific variants is favoured at the tilt boundary is discussed. In thermoelastic martensitic transformation, several variants are known to nucleate as a group to reduce total strain of martensites by themselves. Generally, this is called self-accommodation which is described by reduction of shape deformation using an average of shape strain matrices [10,11]. This approach has been applied to martensitic transformation inside grains or in single crystals. In the present study, the approach was applied to martensitic transformation in the vicinity of grain boundary. First, the phenomenological theoretical calculation based on Wechsler, Lieberman and Read’s (WLR) theory [12] was carried out, the shape strains were obtained in the bicrystal coordinate (x, y, z). The following shape strains correspond to the variant of V11 observed in the bicrystals at c=90ⴰ,

PV11 ⫽ A

冢 冢

0.9454 ⫺0.2044 ⫺0.0022 0.0195

1.0729 ⫺0.0196

⫺0.0100 ⫺0.0577

Fig. 7. Change in deviation angle (d) between the habit plane of all variants and the grain boundary with the tilt angle c.

0.9454

0.9996

0.2044 ⫺0.0022

冣 冣

PV11 ⫽ ⫺0.0195 B

1.0729

0.0196 ,

⫺0.0100

0.0577

0.9996

(1)

(2)

M. Ueda et al. / Acta Materialia 51 (2003) 1007–1017

where subscripts A and B represent the component crystals A and B, respectively. When the CN of V11 occurs at the 90ⴰ ⬍ 211 ⬎ tilt boundary, the shape strain matrix of the CN (PV11 CN ) is closer to the identity matrix than the original one as follows: 1 V11 ⫹ PV11 PV11 CN ⫽ (PA B ) 2

⫽

冢

(3) ⫺0.0022

0

0

1.0729

0

0

0.9996

⫺0.0100

Pij ⫹ Pji ⫺dij, 2

U0 ⫽ sijdeij.

(5)

0

Since Hook’s law holds in elastically isotropic matrix, the elastic strain energy can be expressed in terms of strain components as follows: nG (e ⫹ eyy ⫹ ezz)2 1⫺2n xx

⫹ G(e2xx ⫹ e2yy ⫹ e2zz) ⫹ 2G(g2xy ⫹ g2yz ⫹ g2xz)

.

(6)

The strain components of gxy, gyx, gyz and gzy become 0 due to the CN, and the reduction of shape strain can be regarded as a peculiar selfaccommodation across the boundary, which is quite different from that within grains. This indicates that the CN at the tilt boundary is more advantageous to form martensites compared with the independent nucleation within grains. In general, the driving force for martensitic transformation is composed of chemical and nonchemical terms. The latter mainly consists of martensite/austenite interface energy, elastic strain energy due to the transformation. The self-accommodation containing the CN can be regarded as a process to minimise the strain energy, which leads to an increase in the Ms. Eshelby [13] proposed an approach to calculate the elastic strain energy corresponding to phase transformation with a constrained coherent interface. Many researchers have succeeded in explaining various phenomena or crystallographic features of phase transformation using Eshelby’s inclusion theory [14–20]. According to the theory [13,19], the elastic strain energy due to martensitic transformation was calculated from the shape strain matrix as follows. The shape strain matrix must be first converted to the symmetric one eTij, which is given by eTij ⫽

eij

U0 ⫽

冣

0.9454

冕

1013

where G is the shear modulus and n is the Poisson’s ratio. The strain energies accompanying the CN were calculated and plotted against d for 24 habit plane variants in bicrystals at all c as shown in Fig. 8. When a martensite independently nucleates in the single crystal, the strain energy is calculated to be about 1500 J/mol; the energy level is drawn by a broken line in the figure. The elastic strain energy for the CN is found to be smaller than that for the independent nucleation regardless of selected variants. Interestingly, the strain energy for the CN depends strongly on d at any c. As d is close to 0ⴰ or 90ⴰ, the strain energy effectively decreases. This suggests that the CN by specific variants with

(4)

where dij is the Kronecker delta. The strain energy generated by the shape change of eTij in elastic matrix U0 is shown as

Fig. 8. Calculated elastic strain energy for cooperative nucleation of 24 habit plane variants against deviation angle (d) between the habit plane and the grain boundary. Solid marks indicate strain energy for actually observed variant at each tilt angle c.

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such d is favourable near the tilt boundary. In fact, the variants with the smallest d are preferentially selected bas shown in Fig. 5. In addition, the CN of variants with the habit plane almost perpendicular to the boundary was sometimes observed near the boundary at c=180ⴰ, although the microstructure is not shown here. However, the variants with the habit plane almost parallel to the boundary are dominant near the boundary, since the boundary hardly constrains the shear deformation of martensite. In addition, the compatibility of shape strains at the boundary is also important. Even if the strain energy of a paired variant is small, the variant which does not maintain the compatibility of shape strains at the boundary cannot be formed. Thus, the characteristic variant selection near the grain boundary is closely related to both the minimisation of the strain energy induced by the formation of martensite and the strain compatibility at the boundary. Fig. 9 shows a comparison among changes in the strain energy (a), the deviation angle d (b) and the Ms (c) with c. The variation in the strain energy with c is in fairly good agreement with that in d as shown in Fig. 9(a) and (b). However, there are some differences: the strain energy of V2⬘ shows a bottom at c=150ⴰ , while the d exhibits a minimum at c=130ⴰ. The Ms must be closely related to the strain energy due to the formation of martensite, and increases with decreasing strain energy. In the present study, however, the bicrystals at c= 130ⴰ show higher Ms than that at c=150ⴰ, although the strain energy reaches a bottom at c=150ⴰ. In contrast, the deviation angle d for the selected variant of V2⬘ is minimal at c=130ⴰ. Furthermore, the Ms and the smallest d at c=90ⴰ are almost the same as those at c=150ⴰ. Although the strain energy and d for each variant are closely related as shown in Fig. 8, the Ms of bicrystals with the tilt angle c has better correlation with d than the calculated strain energy. We earlier showed [6] that the shear direction of martensite affected the variant selection near the boundary. The grain boundary hardly constrains the shear accompanied by the formation of martensites when the habit plane of selected variants is almost parallel to the boundary. Therefore, the constraint effect of the grain boundary as well as the minimisation of strain energy is respon-

Fig. 9. Changes in calculated elastic strain energy due to the cooperative nucleation (a), deviation angle between habit plane and boundary (b) and Ms (c) of Fe-Ni bicrystals with the tilt angles c. The error bars correspond to the scatter in measured values.

sible for change in Ms and the choice of variant near the boundary. The bicrystals at c=180ⴰ showed significantly higher Ms than the others in spite of larger d value of the selected variants. In the vicinity of the boundary at c=180ⴰ there were plural variants with the same d value, as mentioned above. Accordingly, the CN of four variants occurs simultaneously at the boundary as shown in Fig. 6. The average of shape strains for actually observed variants of V2, V1⬘, V3 and V2⬘ is calculated as follows, 1 V2 V3 V2’ (7) (P ⫹ PV1’ CN ⫹ PCN ⫹ PCN ) 4 CN ⫽

冢

1.0223

0

0

0

0.9762

0

0

0

1.0177

冣

.

M. Ueda et al. / Acta Materialia 51 (2003) 1007–1017

The matrix is closer to the identity matrix than that of CN by only two neighbouring martensites. Needless to say, the strain energy also remarkably decreases to 89J/mol by the theoretical calculation. Therefore, 180ⴰ ⬍ 211⬎ symmetric tilt boundary must demonstrate the highest Ms among the bicrystals tested, although the selected variants have relatively large d value.

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nucleation of martensite acts as a trigger for burst transformation and the self-accommodating group within grains must be constructed postnatally. This is one possible explanation for the preferential nucleation at the boundary although more detailed study is necessary.

5. Conclusions 4.3. Comparison between cooperative nucleation near grain boundary and self-accommodation within a grain The morphology and self-accommodation of martensites have been studied in several alloys [21,22], but the relation between the onset of martensitic transformation and the morphology of martensites has not been fully discussed. On the other hand, we demonstrated that the CN necessarily occurs at the tilt boundaries prior to the transformation within grains since the bicrystals containing tilt boundary always demonstrate the characteristic morphologies of martensites and higher Ms than single crystals [6]. This strongly suggests that the sole nucleation of a martensite accompanied by large strain energy is responsible for the low Ms inside a grain. Then the reason the CN occurs at the grain boundary prior to the transgranular martensitic transformation is discussed. A large number of embryos are assumed to be formed homogeneously in a crystal above Ms during cooling, although their size is smaller than the critical size for their nucleation. If the grain boundary can relieve the strain energy associated with the formation of embryos and the boundary hardly constrains the growth of embryos, the embryos near the specific boundary may exceed the critical size even above the Ms of single crystals. Such specific embryos will grow into martensites near the specific boundaries, showing the CN. In contrast, the transgranular martensite is believed to nucleate alone. To cancel out the transformation strain around the prior martensite, additive variants are formed in the neighbourhood, resulting in a self-accommodating morphology. A similar phenomenon was discussed in the previous section of 4.1 and the detail was reported in the previous paper [6]. Consequently, the independent

The controlling factor for a favourable nucleation of martensite at grain boundary was examined using Fe-Ni bicrystals containing ⬍ 211⬎ symmetric tilt boundaries. The results were summarised and the following conclusions were reached. 1. The martensite-start temperature (Ms), the morphology of martensites and the variant selection in Fe-Ni bicrystals containing ⬍ 211 ⬎ symmetric tilt boundaries depend strongly on the tilt angle (c). 2. Specific variants whose habit plane is almost parallel to the boundary are preferentially selected among 24 habit plane variants near the boundary, showing the cooperative nucleation (CN) irrespective of the tilt angles. 3. A grain boundary may reduce the strain energy for the nucleation of martensite. In particular, the symmetric tilt boundary hardly suppresses the growth of embryos into martensite, since the compatibility requirements are maintained at the boundary and result in higher Ms and the CN. In contrast, the martensitic transformation within grains is influenced by the independent nucleation of a martensite variant. Thus, the CN occurs near the grain boundary prior to the independent nucleation apart from the boundary. 4. The CN can be regarded as a kind of selfaccommodation across the boundary; it necessarily decreases the strain energy due to the formation of martensites compared with an independent one within a grain. As the habit plane of a selected variant by the boundary is parallel or perpendicular to the boundary, the strain energy associated with the CN effectively decreases. Such nucleation effectively accelerates the martensitic transformation at the bound-

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ary. Moreover, a grain boundary hardly constrains the shear accompanied by the formation of martensites when the variant with habit plane almost parallel to the boundary is selected, resulting in the peculiar variant selection near the boundary. 5. The Ms of bicrystals is closely related to the angle between the habit plane of selected variants and the grain boundary; as the habit plane is close to the boundary, the Ms of Fe-Ni bicrystals basically increases except for that at c= 180ⴰ. The highest Ms at c=180ⴰ among tested bicrystals results from the CN by four kinds of variants near the grain boundary, which is attributed to the crystallographic symmetry of the boundary. Acknowledgements This work was partly supported by a Research Grant from the Iron and Steel Institute of Japan. Appendix A The notation of 24 habit plane variants in Fe-Ni alloys is represented in Table A1. The habit plane and shear direction of the variants were calculated based on Wechsler, Lieberman and Read theory [12].

References [1] Magee CL. Phase transformations. Metals Park, OH: ASM, 1969 p.115. [2] Kajiwara S. Metal Trans A 1986;17A:1693. [3] Miyazaki S, Kawai T, Otsuka K. Scripta Metall 1982;16:431. [4] Miyazaki S, Kawai T, Otsuka K. J Phy 1982;43:C4–813. [5] Tsuzaki K, Harada N, Maki T. J Physique IV 1995;5(C8):167. [6] Ueda M, Yasuda HY, Umakoshi Y. Acta Mater 2001;49:3421. [7] Ueda M, Yasuda HY, Umakoshi Y. Acta Mater 2001;49:4251. [8] Ueda M, Yasuda HY, Umakoshi Y. Sci Tech Adv Mater 2002;3:171. [9] Livingstone JD, Chalmers B. Acta Metall 1957;5:322. [10] Saburi T, Wayman CN. Acta Metall 1979;27:979. [11] Murakami Y, Otsuka K, Hanada S, Watanabe S. Mater Sci Engng 1994;A189:191. [12] Wechsler MS, Lieberman DS, Read TA. Trans AIME 1953;197:1503. [13] Eshelby J. Proc R Soc 1957;A241:376. [14] Christian J. Acta Metall 1958;6:377. [15] Shibata M, Ono K. Acta Metall 1975;23:587. [16] Kato M, Miyazaki T, Sunaga Y. Scripta Metall 1977;11:915. [17] Hayakawa M, Oka M. Acta Metall 1984;32:1415. [18] Ledbetter H, Dunn ML. Mater Sci Engng 2000;A285:180. [19] Kato M, Onaka S, Fujii T. Sci Tech Adv Mater 2001;2:375. [20] Kato M. Mater Trans, JIM 1992;33:89. [21] Tanaka K, Oshima R. Mater Trans, JIM 1991;32:325. [22] Okamoto H, Oka M, Tamura I. Trans, JIM 1978;19:674.

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Table A1 Variant notations and crystallographic parameters such as orientation relationship, habit plane normal and shear direction vectors based on the phenomenological theoretical calculation Variant notation

Orientation relationship

V1 V1⬘ V2 V2⬘ V3 V3⬘ V4 V4⬘ V5 V5⬘ V6 V6⬘ V7 V7⬘ V8 V8⬘ V9 V9⬘ V10 V10⬘ V11 V11⬘ V12 V12⬘

(111)γ//(011)M

[21¯ 1¯ ]g / / [01¯ 1]M [1¯ 21¯ ]g / / [01¯ 1]M [1¯ 1¯ 2]g / / [01¯ 1]M

(1¯ 1¯ 1)g / / (011)M [2¯ 11¯ ]g / / [01¯ 1]M [12¯ 1¯ ]g / / [01¯ 1]M [112]g / / [01¯ 1]M (11¯ 1)g / / (011)M [211¯ ]g / / [01¯ 1]M [1¯ 2¯ 1¯ ]g / / [01¯ 1]M [1¯ 12]g / / [01¯ 1]M (1¯ 11)g / / (011)M [2¯ 1¯ 1¯ ]g / / [01¯ 1]M [121¯ ]g / / [01¯ 1]M [11¯ 2]g / / [01¯ 1]M

Habit plane

Shear direction

(0.613 0.176 0.770)g (0.613 0.770 0.176)g (0.770 0.613 0.176)g (0.176 0.613 0.770)g (0.176 0.770 0.613)g (0.770 0.176 0.613)g (0.613 0.176 0.770)g (0.613 0.770 0.176)g (0.770 0.613 0.176)g (0.176 0.613 0.770)g (0.176 0.770 0.613)g (0.770 0.176 0.613)g (0.613 0.770 0.176)g (0.613 0.176 0.770)g (0.176 0.613 0.770)g (0.770 0.613 0.176)g (0.770 0.176 0.613)g (0.176 0.770 0.613)g (0.613 0.770 0.176)g (0.613 0.176 0.770)g (0.176 0.613 0.770)g (0.770 0.613 0.176)g (0.770 0.176 0.613)g (0.176 0.770 0.613)g

[0.692 [0.692 [0.694 [0.198 [0.198 [0.694 [0.692 [0.692 [0.694 [0.198 [0.198 [0.694 [0.692 [0.692 [0.198 [0.694 [0.694 [0.198 [0.692 [0.692 [0.198 [0.694 [0.694 [0.198

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