Compatibility at Junction Planes between Habit Plane Variants with Internal Twin in Ti Ni Pd Shape Memory Alloy

Compatibility at Junction Planes between Habit Plane Variants

with Internal Twin in Ti-Ni-Pd Shape Memory Alloy

Takeshi Teramoto

, Masaki Tahara

, Hideki Hosoda

and Tomonari Inamura

1_{Department of Innovative and Engineered Materials, Tokyo Institute of Technology, Yokohama 226-8503, Japan} 2_{Precision and Intelligence Laboratory, Tokyo Institute of Technology, Yokohama 226-8503, Japan}

The kinematic compatibility (KC) between habit plane variants (HPVs) was numerically analyzed and classified in Ti-25Ni-25Pd alloy. There are six distinct types of HPV pair that can satisfy the KC condition at averaged junction plane (JP) between twinned HPVs. By analyzing the KC condition between lattice correspondence variants (CVs) in the HPVs, it was found that there are three types of HPV pair in which all the CVs can be compatible due to the existence of the common rotation to keep the KC at all JPs simultaneously. On the other hand, it was found that the incompatibility inevitably remains in the morphologies known as spear (wedge), folk and herring-bone.

[doi:10.2320/matertrans.MB201515]

(Received August 27, 2015; Accepted October 6, 2015; Published November 13, 2015)

Keywords: self-accommodation microstructure, kinematic compatibility, shape memory alloy, titanium-nickel-palladium

1. Introduction

As described in the phenomenological theory of martensite

crystallography (PTMC),1­6) the crystallography of the

isolated martensite plate is determined by the invariant plane (IP) condition at the habit plane that is the interface between the parent and the martensite phase. The martensite plate with

the invariant plane character is called “habit plane variant

(HPV)”. A HPV in shape memory alloys, in general, consists of two kinds of lattice correspondence variants (CVs). The

CV whose volume fraction is larger than 0.5 is termed“main

CV”, and the other is termed“sub CV”in this study. The sub

CV is introduced as the lattice invariant deformation (LID) to satisfy the IP condition at the habit plane.

In shape memory alloys, HPVs connect each other to cancel out the macroscopic change in shape and then

self-accommodation microstructure (SAM) is formed;7) the

connecting HPVs are termed“HPV cluster”, hereafter. Only

limited kinds of morphology appear in experiment, even though there are numerous kinds of HPV cluster in combination logic, depending on the number of HPVs forming the cluster. In cubic-orthorhombic martensitic trans-formation, the fork type microstructure and the spear type microstructure that are constructed by two HPVs are

observed in Ag-Cd.8)_{The triangular cluster and the diamond}

cluster that are constructed by the three and four HPVs

respectively are observed in Ti-Ni-Cu9) _{and Cu-Al-Ni,}10)

respectively.

The arithmetic average of the shape change matrix of HPVs in a cluster has been a guiding principle to evaluate the degree of self-accommodation and to predict the preferential

morphology.8,11­16) The formation frequency of the HPV

clusters in experiment is, however, not always explained by the principle.17­19)

The condition of connection between HPV is essential to consider the preferential morphology and geometry of the interfaces. The evaluation of the misfit at the junction plane

(JP) between HPVs is possible by using the kinematic

compatibility (KC) condition.20)Consider two domains each

with homogeneous deformation gradient (Ui and Uj). The

deformation becomes continuous across the interface be-tween the two domains only if the following equation has realistic solution forQij,aandn,21)whereas the deformation

gradient is discontinuous across the interface;

QijUjUi¼an ð1Þ

Qij is the additional rotation of Uj required for the KC

condition withUiat the interface.aandnare the vectors that

describe the difference between the shape change directions of the domains and the normal direction of the interface,

respectively. The rotation Qijis a measure of the

incompat-ibility at the interface.22,23)The effect of the incompatibility on HPV clusters is essential to understand the mechanism

of the formation of SAM,17­19) except for a special alloy

where formation of a fully compatible microstructure is possible.24­26)The HPV clusters with smallerQijare preferred

in the actual transformation.17­19,22,23)

The cubic-orthorhombic martensitic transformation is one

of a major transformation in shape memory alloys.27­31)

Saburi et al.9) _{proposed that the triangular morphology is}

the preferential microstructure in Ti-Ni-Cu. In our previous

study on a¢-titanium shape memory alloy with¢(bcc)¹¡AA

(orthorhombic) martensitic transformation, two kinds of V-shape morphology and the triangle morphology have been

identified as the preferential HPV cluster by transmission

electron microscopy (TEM) observation and a theoretical evaluation of Qij.32,33) In these previous studies, the alloys

investigated had no LID and only twelve HPVs exist in theory. In general,{111}otype I twin or©211ªotype II exist as LID and there are twenty-four HPVs for a given mode of

LID in the cubic-orthorhombic martensite.4,6,34) The

classi-fication of HPV clusters byQijin the previous study32)does

not represent the general geometrical character of the HPV clusters in the cubic-orthorhombic transformation.

Figure 1 schematically shows HPV, averaged JP between HPVs and local JPs between CVs. The HPV that consists of the main CV ofiand the sub CV ofjis represented by{i,j} +1_{Graduate Student, Tokyo Institute of Technology}

+2_{Corresponding author, E-mail: inamura.t.aa@m.titech.ac.jp}

and is shown in Fig. 1(a). The HPV cluster consists of two HPVs ({i,j} and {k,l}) that are connected at JP. There are two kinds of characteristic interfaces. One is the averaged JP between HPVs (HPVJP) as shown in Fig. 1(b); we assume twinned HPVs as if they are homogeneous domains as treated in most of the previous studies.20,21)_{The other is the local JP}

between CVs (CVJP) that is defined by considering the

inhomogeneous structure of HPV as shown in Fig. 1(c).

CVJP are classified into four groups as CVJP (main«main),

CVJP (sub«sub), CVJP (main«sub) and CVJP (sub«main) as

depicted in Fig. 1(c). Nishidaet al.17,18)_{and Inamuraet al.}19) showed that the KC condition at the CVJP affects the formation frequency of HPV cluster in Ti-Ni. It is, therefore, essential to evaluate the KC condition not only at HPVJP but also at CVJP in an alloy that has LID to understand the underlying principle of the formation of SAM in the general cubic-orthorhombic martensite. Thorough analysis of the HPV clusters that are possible in combination logic among two HPVs is necessary. The analysis of CVJP has not been made, though the KC conditions at HPVJP in cubic-orthorhombic martensitic transformation has been theoret-ically analyzed by K.F. Haneet al.35)

The objective of this study is to reveal and classify the HPV clusters by the KC conditions at HPVJP and/or CVJPs with cubic-orthorhombic martensitic transformation with LID. The analysis is made using the lattice parameters of Ti-25Ni-25Pd alloy as a model alloy because analytical

solution is not obtained in general.21) An emphasis is on

finding the distinct pairs of HPVs in which HPVJP and

CVJPs are bring into compatible by a common rotation, because incompatibility is minimized in such clusters.19)

2. Experimental and Analysis Procedure 2.1 The notations and the lattice parameters

A Ti-25 mol%Ni-25 mol%Pd alloy was fabricated by Ar

arc-melting method (Ar+1%H2). The ingot was

homogen-ized at 1373 K for 21.6 ks. The homogenhomogen-ized ingot was

cold-rolled to be 20% reduction in thickness. The cold-rolled

specimen was wrapped in Ta foil, sealed in evacuated quartz tubes, annealed at 1373 K for 3.6 ks and then quenched by breaking the quartz tube in water. The surface of specimen

was chemical etched in a solution of 25% H2O+25%

HF+25% H2SO4+25% HNO3 at 363 K. ª-2ª XRD

analysis was performed at room temperature (RT, 298 K) and 523 K using an PANalytical X’pert Pro Galaxy system

equipped with an X’celerator module. CuK¡ radiation was

used and Si was the reference material utilized to correct systematic error in the diffraction profiles. Lattice parameters were obtained by a least-squares method on a PC

(CellCalc).36) The calculation was done in the bcc lattice

coordinate; the vectors without subscript is referred to the bcc (parent phase) lattice coordinate. The subscript“o”indicates the orthorhombic (martensite) lattice coordinate. In the Ti-Ni-Pd, the parent phase has the B2 and the martensite has the B19, and there are six CVs following the Au-Cd type lattice correspondence.28)_{The notation and the definition of CVs are} given in Table 1.

The{111}otype I twin is introduced as the major LID in

each HPV and the habit plane is close to the{hkk}-type in Ti-25Ni-25Pd alloy as similar to other alloys that undergo the

cubic-orthorhombic martensitic transformation.27­31) The

lattice parameters used for the analysis are a=306.3 pm,

ao=278.4 pm, bo=466.9 pm and co=444.6 pm,

respec-tively. The diagonalized deformation gradient of the lattice

deformation is diag{0.909, 1.078, 1.026}; the third

compo-nent is larger than unity and the IP condition is satisfied by

the LID of the{111}otype I twin in PTMC.34) The internal

twin as the LID has been experimentally revealed to be the

{111}o type I twin in the present alloy.37) _{The volume}

fraction of the sub CV is 0.303 in PTMC.

2.2 The deformation gradient of HPV

The deformation gradient of the isolated HPV that satisfies the IP condition at the habit plane, namely the KC condition between a HPV and the parent phase, was calculated by the

geometrically non-linear theory of martensite (GNLTM).21)

The deformation gradient of{i,j},U{i,j}is calculated by the following equations;

QBjBi¼aInI ð2Þ

Ufi;jg¼Q0ð­QBjþ ð1­ÞBiÞ ¼Iþbhmh; ð3Þ

where Bi is the deformation gradient of the lattice

deformation of the i-th CV. aI and nI are the vectors that

describe the shear direction and the twinning-plane normal

of the relevant {111}o type I twin as LID. Q and QA are

additional rotations for the KC conditions. ­ is the volume

Averaged HPV

HPVJP

i(Main CV)

j(Sub CV)

Averaged habit plane

(c) Local JPs between CVs (CVJP)

(b) Averaged JP between HPVs (HPVJP) (a) HPV

Main|Main-JP Sub|Sub-JP

Main|Sub-JP

{i,j} {k,l}

{i,j} {k,l}

{i,j}

Sub|Main-JP

Fig. 1 Schematic diagram of HPV (a) and HPV cluster ((b) HPVJP (c) CVJP).

Table 1 Lattice correspondence.

CV [1 0 0]o [0 1 0]o [0 0 1]o

1 [1 0 0] [0 1 1] ½0 1 1

2 ½1 0 0 ½0 1 1 [0 1 1]

3 [0 1 0] [1 0 1] ½1 0 1

4 ½0 1 0 ½1 0 1 [1 0 1]

5 [0 0 1] [1 1 0] ½1 1 0

fraction of thei-th CV (main CV) andIis the identity matrix.

bh and mh are the vectors that describe the shape change

direction and the habit plane normal, respectively. U{i,j} is

identical to the shape change matrix obtained by PTMC.21)

Notice that U{i,j} is an averaged deformation gradient with neglecting the twinned microstructure of HPV. The notation of HPV and the corresponding habit plane are summarized in Table 2.

2.3 KC condition at HPVJP

The KC condition at HPVJP was evaluated by solving the following equation.

Qfi;jgfk;lgUfk;lgUfi;jg¼bHVmHV ð4Þ

Q{i,j}{k,l} is a rotation with rotation angle of ªHPV to make {k,l}to be compatible to{i,j}and,«ªHPV«is the indicator of the misfit at HPVJP.bHPVandmHPVare vectors that represent the difference of the shape change directions of the HPVs and the normal of the HPVJP, respectively.

2.4 KC condition at CVJP

The deformation gradients of CVs that form a HPV are no longer the pure lattice deformation because a small rotation

have been made to form the HPV with the IP character.1­6)

The deformation gradients of CVs (V) that form a HPV are

calculated by the product of the lattice deformation and the additional rotations in eqs. (2) and (3) as following.

Vi¼Q0Bi ð5Þ

Vj¼Q0QBj ð6Þ

Here,ViandVjare the deformation gradients of the main CV

and the sub CV that form the HPV. Then, the following equation is solved forbandnto evaluate the KC condition at CVJP.

Jfi;jgfk;lg

pq VqVp¼apqfi;jgfk;lgnfpqi;jgfk;lg ð7Þ

Jfi;jgfk;lg

pq is a rotation with the rotation angle of ºfpqi;jgfk;lg to make theq-th CV of{k,l}(q=korl) to be compatible to the p-th CV of{i,j}(p=iorj).jºf_pqi;jgfk;lgjis an indicator of the misfit at the CVJP. a_pqfi;jgfk;lg and nf_pqi;jgfk;lg are vectors that represents the twining direction and the normal of the CVJP (the twinning plane).

3. Results and Discussion

3.1 Overview of KC condition at HPVJP and CVJP

There are 24 identical HPVs and the total number of HPV cluster is 480 in the combination logic. Figure 2 shows the

classification of all the possible HPV clusters. The HPV

clusters that are shaded in Fig. 2 can satisfy the KC condition; eq. (4) has 288 realistic solutions that can be categorized into only six types as represented by the symbols I,II,III,IV,VandVI. In contrast to that the KC condition can

be satisfied between any HPVs and the HPV is doubly

degenerated when LID is absent,32) only the limited

combinations of HPVs can satisfy the KC condition at HPVJP when the LID exists. This is due to the existence of the sub CV as the LID and is the general situation in the cubic-orthorhombic martensite. The solutions of eq. (4), Q{i,j}{k,l},ªHPVand mHPV, for the pair-type ofI,II,III,IV,V

and VI in Fig. 2 are summarized for {1, 3} and {k,l} in

Table 3. There are two sets of solution for a given pair of HPVs. Roman numbers in the right side column is the solution number for the convenience. Following analysis for CVJP is given for these six pair-types.

For the six pair-types shown in Table 3, the KC conditions

at CVJPs are summarized in Table 4. In the column ‘No.’,

solution number is newly given for the convenience. There

are specific twin relationships among the CVs.21) The

solutions whose nf_pqi;jgfk;lg is rational and corresponding to

the twinning plane of {111}o type I twin are indicated as

‘{111}I’in the column named‘orientation relationship (OR)’ (No. 5, 7, 9, 11, 16, 18, 22, 24, 29, 31, 33, 35, 40 and 42). Similarly, the solutions corresponding to©211ªotype II twin are indicated as‘©211ªII’(No. 6, 8, 10, 12, 17, 19, 23, 25, 30, 32, 34, 36, 41 and 43). For the pairs of CVs that have rational

nfi;jgfk;lg

pq and afpqi;jgfk;lgare indicated by ‘{011}c’; the relevant twin relationship is{011}ocompound twin (No. 1­4, 14­15, 27­28 and 38­39). OR of‘E’indicates that the CVs are the same across the JP and are brought into coincidence by the rotation (No. 13, 20, 21, 26 and 37).

There are six special solutions in which KC conditions

at HPVJP and/or CVJPs are simultaneously satisfied by a

common rotation to the advantage of reducing the incom-patibility and is discussed in detail in the following sections.

3.2 HPV cluster with fully compatible junction plane

Firstly, three solutions in which fully compatible con-nection of HPVs is possible are shown as depicted in Fig. 3.

For the solutions iii in Table 3, the geometry of the pair

Table 2 Notation of HPV for Ti-25Ni-25Pd h=0.604, k=0.285, l= 0.744.

HV Main CV Sub CV nh

{1, 3} 1 3 (¹h k l)

{1, 4} 1 4 (h k l)

{1, 5} 1 5 (¹h l k)

{1, 6} 1 6 (h l k)

{2, 3} 2 3 (h k¹l)

{2, 4} 2 4 (h¹k l)

{2, 5} 2 5 (h¹l k)

{2, 6} 2 6 (h l¹k)

{3, 1} 3 1 (k¹h l)

{3, 2} 3 2 (k h l)

{3, 5} 3 5 (l¹h k)

{3, 6} 3 6 (l h k)

{4, 1} 4 1 (k h¹l)

{4, 2} 4 2 (¹k h l)

{4, 5} 4 5 (¹l h k)

{4, 6} 4 6 (l h¹k)

{5, 1} 5 1 (k l¹h)

{5, 2} 5 2 (k l h)

{5, 3} 5 3 (l k¹h)

{5, 4} 5 4 (l k h)

{6, 1} 6 1 (k¹l h)

{6, 2} 6 2 (¹k l h)

{6, 3} 6 3 (¹l k h)

typeII viewed from [001] is schematically drawn in Fig. 3(a). In the solutions 3 and 14 in Table 4, the rotation

for the KC condition at CVJP (main«main) and CVJP

(sub«sub) are the same and is equal to the rotation for the

KC at HPVJP;Jf₁₂1;3gf2;4g¼J₃₄f1;3gf2;4g¼Qf1;3gf2;4g. This means

that HPVJP, CVJP (main«main) and CVJP (sub«sub) can

be compatible simultaneously by the common rotation in

exchange for the KC at CVJP (main«sub) and CVJP

(sub«main) are incompatible. The configuration of CVs in

Fig. 3(a), therefore, achieves fully compatible junction plane. This structure is accomplished by the one-to-one connection between main CVs (CV1 and CV2) and the one-to-one connection between sub CVs (CV3 and CV4). LID twin

planes of {1, 3} and {2, 4}, CVJP (main«main) and CVJP

(sub«sub) are edge-on in [001] zone. The width of each CVs

can match completely and the geometry in the microstructure shown in Fig. 3(a) is constructible. Enlarged view of the encircled region in Fig. 3(a) is also shown. The HPVJP

becomes zig-zag morphology that is formed by the {011}

compound twins.

The solutions No. 5, 16, 26 and 37 in Table 4 correspond

to the solution v for the pair-type III in Table 3. The

geometry of this HPV cluster viewed from [111] is depicted

in Fig. 3(b). LID twin planes of{1, 3},{3, 1}and all CVJP

are edge on in Fig. 3(b). As seen in the Tables 3 and 4, the

rotations for the KC at CVJP (main«main), CVJP (sub«sub),

CVJP (main«sub) and CVJP (sub«main) have the relationship

Jf₁₃1;3gf3;1g¼Jf₃₁1;3gf3;1g¼Jf₁₁1;3gf3;1g¼Jf₃₃1;3gf3;1g. This means that all the junction planes can be compatible simultaneously by

the common rotation. In these solutions, CVJP (main«main)

and CVJP (sub«sub) are parallel each other because

n₁₃f1;3gf3;1g¼n₁₁f1;3gf3;1g. On the other hand, CVJP (main«sub)

and CVJP (sub«main) cannot be defined because the main CV

and the sub CV are the same CV in this case;nf₁1_l;3gfk;lg and

nf1;3gfk;lg

3k have no solution in the solution No. 26 and 37

Table 4. The twin planes of the LID, the CVJPs and HPVJP

㻌 {1,3} {1,4} {1,5} {1,6} {2,3} {2,4} {2,5} {2,6} {3,1} {3,2} {3,5} {3,6} {4,1} {4,2} {4,5} {4,6} {5,1} {5,2} {5,3} {5,4} {6,1} {6,2} {6,3} {6,4}

{1,3} I II III IV V VI

{1,4} II I IV III VI V

{1,5} I II V VI III IV

{1,6} II I VI V IV III

{2,3} I II III IV VI V

{2,4} II I IV III V VI

{2,5} I II VI V III IV

{2,6} II I V VI IV III

{3,1} III IV I II V VI

{3,2} IV III II I VI V

{3,5} V VI I II III IV

{3,6} VI V II I IV III

{4,1} III IV I II VI V

{4,2} IV III II I V VI

{4,5} VI V I II III IV

{4,6} V VI II I IV III

{5,1} III IV V VI I II

{5,2} IV III VI V II I

{5,3} V VI III IV I II

{5,4} VI V IV III II I

{6,1} III IV VI V I II

{6,2} IV III V VI II I

{6,3} VI V III IV I II

{6,4} V VI IV III II I

{i,j}

{

k,l

}

Fig. 2 Classification of all possible HPV cluster, shaded pair can satisfy the KC condition at HPVJP.

Table 3 Solutions for KC condition at HPVJP.

{i,j} {k,l} Pair

type

Q{1,3}{k,l}

mHV No.

Rotation axis ªHV

{1, 3}

{2, 3} I [¹0.777, 0,¹0.630] 2.34° (0, 1, 0) i

[¹0.626, 0, 0.780] ¹3.51° (0.234, 0, 0.972) ii

{2, 4} II [0.245,¹0.970, 0] ¹7.06° (0.399, 0.917, 0) iii

[0.426, 0.905, 0] 5.42° (0, 0, 1) iv

{3, 1} III [0.677, 0.677, 0.290] 3.85° (0.707,¹0.707, 0) v

[¹0.475,¹0.475, 0.741] ¹7.46° (¹0.682,¹0.682,¹0.263) vi

{4, 2} IV [0.540,¹0.540, 0.645] ¹8.15° (0.707, 0.707, 0) vii

[¹0.591, 0.591, 0.550] 2.13° (¹0.581, 0.581, 0.569) viii

{5, 3} V [0.668,¹0.329, 0.668] ¹0.400° (0.707, 0,¹0.707) ix

[0.137,¹0.981, 0.137] ¹12.1° (¹0.694,¹0.194,¹0.694) x

{6, 3} VI [¹0.202,¹0.958, 0.202] ¹12.3° (0.707, 0, 0.707) xi

are (10). The KC condition at the junction planes can be kept completely in this HPV cluster.

The solutions No. 8 and 19 for CVJPs in Table 4 correspond to the solutionviiifor pair-typeIVin Table 3. In this pair, Qf1;3gf4;2g¼J14f1;3gf4;2g¼Jf321;3gf4;2g. This means that

the CVJP (main«main) and CVJP (sub«sub) can be compatible

simultaneously by the common rotation and then the HPVJP

between {1, 3} and {2, 4} also becomes compatible. The

geometry viewed from ½111 is schematically shown in

Fig. 3(c). This structure is formed by the one-to-one connections of main CVs (CV1 and CV4) and sub CVs

(CV3 and CV2) to eliminate the CVJP (main«sub) and CVJP

(sub«main). The twinning plane of LID, CVJP (main«main)

and CVJP (sub«sub) are edge-on. An enlarged image of the

encircled region of HPVJP is also shown. The CVs are

connected by ©211ªtype II twin orientation at the CVJPs in

this HPV cluster. The CVJPs form the zig-zag morphology as shown in Fig. 3(c) and the averaged interface normal

corresponds to the HPVJP of the solutionviiiof Table 3.

These three HPV clusters in Fig. 3 have fully compatible junction planes. Relevant twin orientations are held at CVJPs and the HPVJPs are simultaneously compatible. However, it should be noted that the KC condition between martensite and parent phase (invariant plane condition of habit plane) is not kept on the compatible connection of the HPVs because HPV(s) are rotated to make compatible junction plane; the

KC conditions at JPs and the KC conditions at habit planes are competitive. Incompatibility is always placed on some interfaces in general.

3.3 HPV cluster with incompatibility

There are another set of HPV clusters with much smallerQ

for the KC condition at HPVJP, even though some CVJPs cannot be compatible unlike the HPV clusters in the previous section. As shown below, these HPV clusters require very small rotation for the KC at HPVJP, whereas a higher incompatibility remains at CVJPs. As discussed later, it is

interesting that the common morphologies named

“dia-mond”,“wedge”, folk and“spear”8)_{in literatures are formed}

by these HPV clusters.

The schematic image of the solution i for pair-typeI

viewed from [001] is shown in Fig. 4(a). LID twin planes of

{1, 3},{2, 3} and HPVJP are edge on in this image. In the

solutions No. 13, 22 and 33, the rotation for KC condition at

CVJP (sub«sub), CVJP (main«sub) and CVJP (sub«main) are

the same whereas it is different from Q{1,3}{2,3};J33f1;3gf2;3g¼

J₁₃f1;3gf2;3g¼J₃₂f1;3gf2;3g6¼Qf1;3gf2;3g. This means that that CVJP

(sub«sub), CVJP (main«sub) and CVJP (sub«main) can be

compatible simultaneously by the same rotation, in exchange

for the compatibility at CVJP (main«main) and HPVJP, and

vice versa. Figure 4(a) is drawn for the case where the

incompatibility is placed on the CVJP (main«main) and

HPVJP. The orientation relationship between the main CVs

(a)

(b)

(c)

V

VI

I

CV3CV1

CV2

CV3 HPVJP: (0,1,0)

{1,3}

{2,3}

(01-1)ocompound twin (CV1 coordinate)

[001]

{1,3} {5,3}

HPVJP: (0.71,0,-0.71)

(-0.

60,

0.

29,

0.

74)

CV1

CV3

CV5

CV3

(-111)otype I twin (CV1 coordinate)

[111]

{1,3} {6,3}

(-0.

74,

0.29,

0.

60)

CV₃ CV1

CV₃ CV6

(-0.

60,

0.

29,

0.

74)

HPVJP: (-0.67,0.31,0.67)

[211]otype II twin (CV1 coordinate)

[111]

Fig. 4 Schematic images of HPV cluster (I,VandVI), (image of CVJP represent dashed line circled area of HPV cluster).

III

{1,3}

{3,1}

CV1

CV3

CV₃CV1

(0.29,-0.60,0.74)

IV

HPVJP : (-0.58,0.58,0.57)

{1,3} {4,2}

(-0.

60,

0.

29,0.

74)

(-0

.2

9,0

.6

0,0.7

4)

CV4

CV2

CV1

CV3

CVJP

Zig-zag interface

II CVJP

Zig-zag interface

{2,4}

{1,3}

HPVJP : (0.40,0.92,0)

CV4

CV2

CV1

CV3

(a)

(b)

(c)

(01-1)ocompound twin

(CV1 coordinate) (011)ocompound twin

(CV3 coordinate)

(1-11)otype I twin

(CV1 coordinate)

[21-1]otype II twin

(CV1 coordinate)

[211]otype II twin

(CV3 coordinate)

[111]

[111]

[001]

has a deviation from the {011} compound twin due to the incompatibility.

The solution ix in pair-typeV and solution xii in

pair-typeVI in Table 3 has ªHV of only 0.400° and 0.274°,

respectively. This means that the HPVJP becomes almost compatible in average when individually growing relevant HPVs that are satisfying the invariant plane condition of habit plane collide by chance. The schematic image of the HPV cluster for the solutionixfor pair-typeVviewed from [111] is

shown in Fig. 4(b). LID twin planes of {1, 3}, {5, 3} and

HPVJP are edge on in this image. The rotations for the KC

at CVJP (sub«sub), CVJP (main«sub) and CVJP (sub«main)

in the solutions No. 20, 29 and 40 for are common;

Jf1;3gf5;3g

33 ¼Jf131;3gf5;3g¼Jf351;3gf5;3g and are not equal to the

rotations for the KC at HPVJP and CVJP (main«main). These

results mean that CVJP (sub«sub), CVJP (main«sub) and

CVJP (sub«main) can be compatible simultaneously by the

common rotation and, then the CVJP (main«main) is

incompatible, or vice versa. Figure 4(b) is drawn as CVJP

(sub«sub), CVJP (main«sub) and CVJP (sub«main) being

compatible and CVJP (main«main) is incompatible. In this

case, there is a deviation of 3.28° from the{111}type-I twin orientation between main CVs across the HPVJP.

The morphology of HPV cluster corresponding to the solutionxiin the pair-typeVIviewed from½111is depicted in Fig. 4(c). LID twin planes of{1, 3}and{6, 3}are edge on

in this image. The rotations for KC at CVJP (sub«sub), CVJP

(main«sub) and CVJP (sub«main) in the solutions No. 21, 31 and 42 are the same;Jf₃₃1;3gf6;3g¼J₁₃f1;3gf6;3g¼J₃₆f1;3gf6;3gand are not equal to the rotations for KC at HPVJP and CVJP

(main«main). Therefore, CVJP (sub«sub), CVJP (main«sub)

and CVJP (sub«main) can be compatible simultaneously by

the common rotation in exchange of the incompatibility at

CVJP (main«main) and HPVJP, or vice versa. Figure 4(c) is

drawn for the case where the incompatibility is placed on CVJP (main«main). There is a deviation of 8.71° from©211ª type-II twin orientation between main CVs across the HPVJP.

3.4 Comparison with actual microstructure

A comparison between the present results and the actual

microstructure reported in literatures is briefly mentioned

here. Details of actual microstructure are also discussed elsewhere.38)

The six HPV clusters or relevant JPs found in this study are in good agreement with the observed microstructures in the actual transformations as mentioned in following.

As for the three HPV clusters with fully compatible

junction planes in Fig. 3, the “crossing twin”21,27) _is

equivalent to the junction plane of pair-typeII in Fig. 3(a). The HPV cluster that is similar in morphology to the

pair-typeIVin Fig. 3(c) was observed in Ti-Nb-Al alloy.32) The

JP in pair-typeIIIin Fig. 3(b) is the{111}type-I twin and is the same twinning of the LID. These fully compatible HPV cluster or relevant JPs exist in the actual microstructure.

Well known morphology of HPV cluster in

cubic-orthorhombic martensitic transformation are the “spear

(wedge)”, “folk”,“herring-bone”and “diamond”.8,27,37) _The folk, spear and herring-bone are equivalent to the HPV clusters in Fig. 4(a), Fig. 4(b) and Fig. 4(c), respectively.

In our latest study,38) _{spear, folk and herring-bone}

micro-structure were observed in Ti-25 mol%Ni-25 mol%Pd.

Figure 5(a) is a brightfield image of the spear microstructure that is typical microstructure in cubic-orthorhombic

marten-sitic transformation. HPVJP was formed by{6, 1}and{4, 1}.

This HPVJP belongs to pair-typeV, according to Fig. 2.

A selected area electron diffraction (SAED) pattern and

diagrams are summarized in Figs. 5(b)­(e). LID twin in

{6, 1} and {4, 1} was {111}o type I twin as indexed in

Figs. 5(c) and (d). Main«Main-JP was almost {111}o type I

twin as shown in Fig. 5(e). This microstructure is in good agreement with Fig. 4(b). The herring-bone in Fig. 4(c) has been reported in Ti-Au.27)

The diamond is formed by the coalescence of two spears and then the folk and herring-bone inevitably appears as demonstrated in our latest study.39)_{It should be pointed out} that some of the CVJPs are not compatible, even though these are frequently observed in several alloys with

cubic-orthorhombic martensitic transformation such as Ti-Ni-Pd,37)

Ti-Pt,40)_Ti-Pd,41) _Ti-Au.27)_{The JPs on which the remained} incompatibility is placed in the HPV clusters in Fig. 4 are not simply determined by the present analysis. The GNLTM analysis predicts the existence and the geometry of the incompatibility. In the actual microstructure, the incompat-ibility is placed at junction planes and/or habit plane and the interfacial energies that are not explicitly considered in the analysis play an important role. Precise orientation analysis at the junction planes is, therefore, necessary to reveal the place of the incompatibility in the actual microstructure.32)

4. Conclusions

The HPV clusters that are possible in combination logic

were classified in terms of the KC conditions at HPVJP and/

or CVJPs in Ti-25Ni-25Pd alloy and the results are summa-rized as follows.

There are six distinct types of HPV pair that can satisfy the KC condition at HPVJP (pair typeI,II,III,IV,Vand VI).

There are three types of HPV cluster that can form fully

compatible HPVJP and CVJPs (pair-typeII,IIIandIV). This

is originated to the existence of the common rotation that achieves the KC condition at all JPs simultaneously. However, the KC at habit plane is broken on the compatible connection

of HPVs. The CVs can be connected by{111}type-I,{011}

compound or©211ªtype-II twin orientation at CVJPs.

There are another set of three HPV clusters with much

smaller rotation for the KC condition at HPVJP (pair-typeI,

Vand VI), even though some CVJPs cannot be compatible.

These three HPV clusters are equivalent to the spear (wedge), folk and herring-bone morphologies. The twin orientation between CVs is not always held in these HPV cluster.

Acknowledgments

This work was partially supported by the Advanced Low Carbon Technology Research and Development Program (JY240121) from the Japan Science and Technology Agency,

Grant-in-Aid of Scientific Research (S: 26220907 and B:

REFERENCES

1) J. S. Bowles and J. K. Mackenzie:Acta Metall.2(1954) 129­137. 2) J. K. Mackenzie and J. S. Bowles:Acta Metall.2(1954) 138­147. 3) J. S. Bowles and J. K. Mackenzie:Acta Metall.2(1954) 224­234. 4) J. K. Mackenzie and J. S. Bowles:Acta Metall.5(1957) 137­149. 5) M. S. Wechsler, D. S. Lieberman and T. A. Read: Trans. AIME197

(1953) 1503­1515.

6) D. S. Lieberman, M. S. Wechsler and T. A. Read:J. Appl. Phys.26 (1955) 473­484.

7) K. Otuska and C. M. Wayman (ed.): Shape Memory Materials, (Cambridge University Press, Cambridge, 1998).

8) T. Saburi and C. M. Wayman:Acta Metall.27(1979) 979­995. 9) T. Saburi, Y. Watanabe and S. Nenno:ISIJ Int.29(1989) 405­411. 10) S. Tan and H. Xu:Continuum Mech. Thermodyn.2(1990) 241­244. 11) K. Otsuka and K. Shimizu:Trans. Jpn. Inst. Metals15(1974) 103­108. 12) S. Miyazaki, K. Otsuka and C. M. Wayman:Acta Metall.37(1989)

1873­1884.

13) S. Miyazaki, K. Otsuka and C. M. Wayman:Acta Metall.37(1989) 1885­1890.

14) K. Madangopal:Acta Mater.45(1997) 5347­5365. 15) K. Otuska and X. Ren:Prog. Mater. Sci.50(2005) 511­678. 16) M. Nishida, H. Ohgi, I. Itai, A. Chiba and K. Yamauchi:Acta Metall.

Mater.43(1995) 1229­1234.

17) M. Nishida, T. Nishiura, H. Kawano and T. Inamura:Philos. Mag.92 (2012) 2215­2233.

18) M. Nishida, E. Okunishi, T. Nishiura, H. Kawano, T. Inamura, S. Ii and T. Hara:Philos. Mag.92(2012) 2234­2246.

19) T. Inamura, T. Nishiura, H. Kawano, H. Hosoda and M. Nishida: Philos. Mag.92(2012) 2247­2263.

20) J. M. Ball and R. D. James:Arch. Rat. Mech. Anal.100(1987) 13­52. 21) K. Bhattacharya: Microstructure of Martensite, (Oxford University

Press, Oxford, 2003).

22) J. M. Ball and D. Schryvers:J. Phys. IV112(2003) 159­162.

23) X. Balandraud, D. Delpueyo, M. Grediac and G. Zanzotto:Acta Mater. 58(2010) 4559­4577.

24) Y. Song, X. Chen, V. Dabade, T. W. Shield and R. D. James:Nature 502(2013) 85­88.

25) X. Chen, V. Srivastava, V. Dabade and R. D. James:J. Mech. Phys. Solids61(2013) 2566­2587.

26) A. Planes, L. Manosa and A. Saxena: Magnetism and Structure in Functional Materials, (Springer, 2005).

27) T. Inamura and H. Hosoda:Metall. Mater. Trans. A42(2011) 111­120. 28) K. Morii, T. Ohba, K. Otsuka, H. Skamoto and K. Shimizu: Acta

Metall.39(1991) 2719­2725.

29) K. Otsuka, K. Oda, Y. Ueno, M. Piao, T. Ueki and H. Horikawa:Scr. Metall. Mater.29(1993) 1355­1358.

30) T. Biggs, M. J. Witcomb and L. A. Cornish:Mater. Sci. Eng. A273­ 275(1999) 204­207.

31) A. R. G. Brown, D. Clark, J. Eastabrook and K. S. Jepson:Nature201 (1964) 914­915.

32) T. Inamura, H. Hosoda and S. Miyazaki:Philos. Mag.93(2013) 618­634. 33) T. Inamura, H. Y. Kim, H. Hosoda and S. Miyazaki:J. Alloy. Compd.

577S(2013) 592­595.

34) T. Inamura, J. I. Kim, H. Y. Kim, H. Hosoda, K. Wakashima and S. Miyazaki:Philos. Mag.87(2007) 3325­3350.

35) K. F. Hane and T. W. Shield:J. Elasticity59(2000) 267­318. 36) H. Miura: J. Crystallogr. Soc. Japan45(2003) 145­147.

37) R. Delville, S. Kasinathan, Z. Zhang, J. V. Humbeeck, R. D. James and D. Schryvers:Philos. Mag.90(2010) 177­195.

38) T. Teramoto, M. Tahara, H. Hosoda and T. Inamura: Mater. Today, in press.

39) T. Teramoto, M. Tahara, H. Hosoda and T. Inamura: MRS online proceedings library, in press.

40) M. Nishida, M. Matsuda, Y. Yasumoto, S. Yano, Y. Yamabe-Mitarai and T. Hara:Mater. Sci. Technol.24(2008) 884­889.

41) M. Nishida, T. Hara, Y. Morizono, A. Ikeya, H. Kijima and A. Chiba: Acta Mater.45(1997) 4847­4853.

ZA : <1 1 0>o 500nm

{4,1} {6,1}

:CV6 :CV1{6,1} :CV4 :CV1’{4,1}

(e) Main|Main-JP (CV4-CV6) : Almost (111)6type I twin

000 0014

0016

1114

1106

1116

1104 (c) LID twin in {6,1}

: (111)6type I twin

000

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1106

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1111

(d) LID twin in {4,1} : (111)4type I twin

000 0014

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(a) (b)