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Acta Materialia 55 (2007) 5169–5176

Structure of twin boundaries in Mn-based shape memoryalloy: A HRTEM study and the strain energy driving force

Xiaoyu Wang a,b,*, Jihua Zhang a

a School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200030, Chinab School of Materials, Arizona State University, Tempe, AZ 85287, USA

Received 19 December 2006; received in revised form 24 May 2007; accepted 27 May 2007Available online 23 July 2007

Abstract

The paramagnetic–antiferromagnetic transition and the cubic-tetragonal martensitic transformation in Mn-based alloys occur at thesame temperature. The shape memory effect depends on the morphology of twins and the motion of twin boundaries. High-resolutionelectron microscopy was carried out across the twin boundary. Abnormal lattice distortion was observed and determined to be a twin-ning dislocation. A model for the defect was established based on the elasticity theory of Eshelby and Khachaturyan. The twinning wasfound effectively to reduce the strain energy and the introduction of twinning dislocations reduces it further. The strain energy related tothese defects was only 2/7 relative to the value in their absence, providing a thermodynamic mechanism for the formation of these twin-ning dislocations.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: High-resolution electron microscopy (HREM); Martensitic phase transformation; Shape memory alloys (SMA); Twinning; Dislocation structure

1. Introduction

The magnetic shape memory effect (MSME) of magneticshape memory alloys (MSMA) has been widely studied sinceits discovery [1,2]. The mechanism of the effect is attributedto the motion of twin boundaries (TBs) under an externalfield. These twins are the product of the order–disorder ormartensitic phase transformation of these alloys. Duringthe transformation, the symmetry of the cubic high-temper-ature phase is reduced to the tetragonal or orthogonal phase.However, detailed analysis of the mechanical performanceof Ni2MnGa suggests that the motion of TBs is in factachieved by the slipping of twinning dislocations (TDs)[3,4]. It appears that the TDs play an important role in thefield-induced-strain in these alloys, especially under complexstress conditions. Therefore, a thorough study of the TDsshould help to predict the microstructure–property relation-ship and improve the performance of MSMAs.

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.05.050

* Corresponding author. Address: School of Materials, Arizona StateUniversity, Tempe, AZ 85287, USA.

E-mail address: xiaoyu.wang.1@asu.edu (X. Wang).

The TD concept has long been proposed as an explana-tion for the curved boundaries observed between lenticulartwins [5–7]. Each TD produces a ledge on the perfect twinboundary. If a Burgers circuit is drawn around the TD, theBurgers vector is found to be

b ¼ de ð1Þ

where d is the height of the ledge and e is the shear strainvector between the twins. The TD shares many propertieswith ordinary dislocations except that the TD can only slipalong the TB. Since the boundary of the lenticular twin canbe regarded as consisting of the loops of TDs, the shape ofthe twins is closely related to the distribution of TDs andcan be calculated from the theory of dislocation piling.

The relationship between the density of TDs, D, and theangle of deviation of the TB from the twinning plane, h,can be written as

D ¼ ad

tan h

where a refers to the unit-cell spacing.

rights reserved.

5170 X. Wang, J. Zhang / Acta Materialia 55 (2007) 5169–5176

Apparently, such dislocations are a kind of coherent dis-location, as shown by the schematic drawing of an edge-type TD in a face-centered tetragonal (fct) structure inFig. 1. Zhu [8], using high-resolution transmission electronmicroscopy (HRTEM), observed TDs directly on the (11 0)twin boundary in the high-TC superconductor YBCO. InYBCO and most alloys with fct structure, the tetragonality,j1�c/aj, is close to or less than 0.05. Hence, the length ofthe shear strain vector e is also small. According to Eq.(1), the TD Burgers vector is much less than the dimensionof the unit-cell. Thus it is believed that TDs do not producediffraction contrast and HRTEM is the only possiblemethod to investigate these defects.

In this paper, HRTEM is used to study TDs in c-Mn85Fe10Cu5 alloy, which is a typical antiferromagnetic(AF) shape memory alloy with fct crystallographic struc-ture at room temperature [9–11]. Previous study hasfound its twin boundary to be {101} and there mustbe TDs on the TBs since the TBs are not parallel [12].From the softening of the shear elastic constant,C 0 = (C11 � C12)/2 [13–15], and the similarity of micro-structure morphology between the alloy and InTl alloy[16,17], it can be supposed that the tetragonal distortionis mainly caused by a martensitic transformation (MT),which is also called the ‘‘double-shear mechanism’’ [18].However, this conjecture does not contradict the assump-tion that the cubic-tetragonal crystal lattice rearrangementis a result of antiferromagnetic ordering, as will be dis-cussed in detail in the following section. Since the YBCOtwinning is caused by the ordering of oxygen atoms, thedifference in twinning mechanisms must lead to a differ-ence between the alloy and YBCO in the atomic-scale

Fig. 1. Atomic model of a twinning dislocation in a tetragonal lattice. Itseffect is to produce a step in the twin boundary.

configuration near the TD. In fact, most MSMAs, suchas Ni2MnGa and FePd, undergo softening of C 0 andMT to produce the tetragonal twin structure. Therefore,it should be possible to extend the results and conclusionsof the present study to all alloys of this type.

Conversely, since TDs exist in so many different kinds ofmaterials, there must be some intrinsic origin that producessuch defects. In some previous studies, the existence of TDsis assumed in advance and the problem is not quantita-tively discussed. The elastic strain energy of the MT mainlydetermines the morphology. Thus, the curved TBs pro-duced by TDs could be the result of minimization of strainenergy. To confirm this possibility, the elastic energy oftwin structures with and without TDs is calculated andcompared for both elastic isotropic and anisotropic media.Thus, it is determined how TDs can assist in reducing theelastic energy associated with the MT.

2. Experimental

The Mn85.5Fe9.5Cu5 (atom percent) alloy was preparedby medium-frequency induction melting of pure manga-nese and iron under an argon atmosphere. Copper(5 at.%) was added to stabilize the c phase, hindering thec–a transformation upon quenching [19]. The ingot wasannealed at 1233 K for 12 h and quenched in water. Theannealing treatment enables a complete martensiticstructure at room temperature. The experimental TEMinvestigations of the microstructure were performed usinga JEM-2100F(FEG) operated at 200 kV. Thin foils wereprepared by mechanically grinding foil cut from the ingotto �0.1 mm in thickness, followed by electropolishingusing a twin-jet method, at around �30 �C in an electrolyteof 3% perchloric acid ethanol solution. The specimen wasion-milled just before being placed into the electron micro-scope to minimize oxidation.

3. Results

3.1. Effect of oxidation on the HRTEM image

Despite the specimen being cleaned by ion milling, it wasinevitably oxidized when observed under the electronmicroscope. Direct evidence for the existence and type ofoxide was provided by the selected-area electron diffraction(SAED) pattern of different zone axes from the same spec-imen region, as shown in Fig. 2. The oxide was easilyindexed as MnO, which has an NaCl-type structure withoxygen atoms embedded in the octahedral interspacebetween fct Mn cells.

From the SAED patterns, it was found that the scatter-ing intensity of the oxide film was fairly strong and theinteractions of the oxide could thus have a negative effecton the HRTEM image. The moire fringes observed intwo perpendicular directions in the micrograph (seeFig. 3) are due to overlap of the sample matrix with theoxide film, because the modulation period is a common

Fig. 2. The diffraction pattern and indexing with zone axis [100] (a) and[110] (b). The satellite spots are caused by double diffraction.

Fig. 3. HRTEM image of Mn-based alloy with oxidation. Moire fringesare very clear.

X. Wang, J. Zhang / Acta Materialia 55 (2007) 5169–5176 5171

multiple of the lattice constants of c-Mn and MnO. Thesame structural modulation has also been observed inMn80Cu20 alloy [20] and was interpreted as being relatedto spinodal decomposition. However, this hypothesis mightbe not applicable here since both the composition and heattreatment in the present alloy do not allow the occurrenceof spinodal decomposition.

In order to obtain a suitable HRTEM image to enablerecognition of the atomic structure of the TB and TD,

the region of specimen used for imaging must be carefullychosen where the oxidation is not very serious. In addition,the images have to be processed with FFT filtering, so thatonly the Bragg peaks from c-Mn matrix are used for imag-ing. After IFFT transformation, the moire fringes can beremoved. A study of 60� dislocations in Si has pointedout that diffuse scattering from the defects is concentratednear the main reciprocal lattice, meaning that the FFT fil-tering does not cause too much loss of structural informa-tion about the dislocation [21,22].

3.2. HRTEM observation of the {101} twin boundary

Fig. 4 shows the HRTEM image taken across the TBwith [100] zone axis, together with its correspondingSAED pattern and the IFFT image. It is difficult to locatethe TB boundary from direct observation due to the smalltetragonality (c/a = 0.97). By drawing reference linesalong certain directions, the location of the TB can bedetermined from where the lines begin to deviate fromthe lattice. The TD location must be on the TB. It canbe seen that the translation periodicity in the circled areain Fig. 4c is nearly degenerated from two-dimensional(2D) to one-dimensional (1D), not as good as in the sur-rounding area, which indicates the presence of some kindof defect. The loss of translation periodicity in the sameregion is also visible in other HRTEM images for a seriesof different defocus conditions (Fig. 5). On the otherhand, the dimensions of the area are relatively small,�2 · 1 nm. Thus, it can be considered that it is the latticedefect in the specimen rather than an imaging problemthat gives rise to the lattice defect visible in the micro-graph. To determine whether the regions contains a TD,Fig. 4c is transformed into a glancing-angle view inFig. 6, where the periodicity perpendicular to the glancingdirection is exaggerated and the crystallographic orienta-tion change of the lattice fringes near the TB can beobserved directly. In the positions indicated by arrows,there are two lattice fringe misfits of opposite direction.Thus, the number of atomic planes on both sides of theTB is equal for a relatively large area, which indicatesthat the defect is coherent. By comparing Fig. 6 withFig. 1, it can be seen that a similar characteristic is alsopresent in Fig. 1, so that the defect can be confirmed asthe TD illustrated in Fig. 1. However, the area containingthe TD in Fig. 4c is elongated compared with Fig. 1. Thiselongation occurs, because the ledge is inclined to the inci-dent beam, i.e., it is the projection of the inclined ledgethat is visible in the image. This phenomenon has beenreported in a variety of ledges [23–25] and is highly likelyto occur in TDs since the TD is a dislocation loop aroundthe twin lamellae.

3.3. Structural characteristics of TB

YBCO and c-Mn alloy show different atomic structurenear the TB. The TD in YBCO appears like the lattice of

Fig. 4. HRTEM image across twin boundary (a) and corresponding diffraction pattern (b); (c) is the IFFT image of the encircled area in (a). The locationof the TD is circled and the space between the white arrows is the transition zone between the two variants.

5172 X. Wang, J. Zhang / Acta Materialia 55 (2007) 5169–5176

the TB shifting to a neighboring (110) plane. In c-Mn-based alloy, the periodicity perpendicular to the TB isnot disturbed at the TD and the lattice distortion happensonly on lattice planes along the TB (Fig. 4c). This charac-teristic can also be considered as evidence that the cubiclattice undergoes a shear strain in the [011] direction forthe fcc–fct structural transformation instead of normalstrain in the three axes of the cell.

Furthermore, a clear TB can be found in the YBCOlattice images, whereas the strain in the present alloyvaries smoothly from one variant to another, with thetransition taking place over about 10 lattice planes(Fig. 4c). Hence, it is difficult to determine which atomicplanes are in the TB. The lattice curvature of the TB inSMAs has been studied using the Ginzburg–Landau(GL) model [26–28] and the static solution of the mar-tensitic–martensitic domain wall in the model agrees wellwith the experimental result. On the other hand, thecoefficient of the gradient term in the GL model, whichis very important, but difficult to determine, can also beestimated from the width of the interface in HRTEMimages.

4. Discussion

4.1. Property of martensitic transformation in Mn-based

alloys

One of the central issues regarding the MT in Mn-based alloys is whether the transformation is proper orimproper. Measurements of the elastic constant[13,29,30] affirm the existence of the soft mode, whichis evidence of proper MT. Furthermore, the fact thatthe direction of spin is not parallel to the c-axis [31](the angle is 5–10�) also implies the existence of someadditional shear strain other than the normal strain.However, the co-existence of the AF transition and MTand the stress-dependence of the AF transition tempera-ture [14] indicate that the transformation is improper,in order to be consistent with the symmetry of AFtransition.

In view of the microstructure, both kinds of transforma-tions can lead to the same tetragonal lattice then the samemorphology since the twin structure with {110} TB servesas a strain energy minimizer. Thus, it is impossible to deter-

Fig. 5. Defocus series of HRTEM images with defocus 32 nm (a), 64 nm(b), 128 nm (c), 256 nm (d) and their IFFT images, 32 nm (e), 64 nm (f),128 nm (g), 256 nm (h). The contrast characteristics do not change nearthe location of the TD.

Fig. 6. Distorted glancing view of Fig. 4c, the orientation differencebetween the two variants on both sides of the line is exaggerated. Thearrow indicates the core of the TD where there is lattice mismatch, but thenumber of atomic planes is constant.

X. Wang, J. Zhang / Acta Materialia 55 (2007) 5169–5176 5173

mine whether the MT is proper or improper from the mor-phology of twin structure.

On the other hand, when we study the core of TD at theatomic-scale, where the translation symmetry along the TBis broken, the difference between them will emerge. InYBCO, which undergoes typical improper MT, thereshould be the displacement of atoms perpendicular to the

TB and this has been observed [8]. However, accordingto the present HRTEM images, the displacement of atomsis parallel to the TB, namely in the habit plane. This indi-cates that the strain may be pure shear. This fact reinforcesthe opinion of Shimizu et al. [11] and Hocke et al. [32] thatboth proper and improper MT exist in the alloys; theproper MT is induced by the strain from AF ordering(improper MT) so that they can occur together, but the for-mer constitutes most of the strain of the product.

4.2. Elastic energy driving force as the TD origin

The morphology of the product of MT is mainly deter-mined by the elastic energy resulting from the transforma-tion. The energy is related not only to the crystal elasticconstants, but also to the shape and distribution of allthe phases or variants in the system studied. The phase-field method has been successfully developed to simulatethe microstructure morphology of materials based on theprinciple that the stable morphology minimizes the elasticenergy [33–36]. Thus, the appearance of TDs indicates thatthey may help to reduce the elastic energy.

The strain energies of two kinds of microstructure mor-phology, one without TDs and the other with TDs, werecalculated and compared as shown in Fig. 7. The inclusionwhere the shear transformation takes place is in the shape ofa cylinder with length l and radius a, where l� a. The onlynon-zero component of the transformation strain tensor eij

is e13 = e31 = eT. The actual microstructure of the alloyrequires that the same transformation also occurs in thematrix surrounding the inclusion; however, the directionof shear strain is along another Æ101æ direction, which is

Fig. 7. Schematic diagrams showing variant configurations under cubic-tetragonal transformation (a) with and (b) without steps on the interface,and the profile of their strain fields.

5174 X. Wang, J. Zhang / Acta Materialia 55 (2007) 5169–5176

at nearly 60� with respect to that in the inclusion. Therefore,there is little accommodation of strain between matrix andinclusion and transformation of the matrix can be neglectedin the model. The elastic medium is assumed to be isotropicso that Eshelby theory [37] can be applied and the calcula-tion of elastic energy can then be greatly simplified.

According to Eshelby theory, if the MT happens in theneedle-like ellipsoidal inclusion with a = b� c, its totalstrain energy is given by

E ¼ lðeTÞ2V ð2Þ

where l is shear modulus and V is the volume of the inclu-sion. If the ellipsoidal inclusion is oblate (a = b� c), thetotal strain energy is

E ¼ plð2� rÞ2ð1� rÞ

caðeTÞ2V ð3Þ

where r is Poisson’s ratio. For thinner inclusions (c/a), thestrain energy density will obviously be lower.

A more general method in which inclusions of arbitraryshape and elastic anisotropic media can be treated was pro-posed by Khachaturyan [36]. The strain and total strainenergy in the model are divided into

Erelax ¼ Ehomo þ EhetereijðrÞ ¼ �eij þ deijðrÞ ð4Þin which

Ehomo ¼ �X

i

vðiÞCijkleTklðiÞ�eij þ VCijkl�eij�ekl ð5Þ

Eheter ¼ �X

i

ZðvÞ

CijkleTkldeij dvþ

ZðV Þ

Cijkldeijdekl dV ð6Þ

The energy Ehomo, related to homogeneous strain, isonly determined by the volume fraction of all kinds ofphases and variants and is independent of microscopic con-figuration such as shape and distribution. In general cases;

Eheter has no analytical expression, but the results drawn byEshelby, as mentioned above, are applicable when referringto ellipsoidal inclusions in an isotropic medium.

There are two kinds of variants where eTijð1Þ ¼ �eT

ijð2Þand with the same volume fraction shown in Fig. 7a. Theinclusion is divided into n pieces of twins of thickness c,where c� a� l. Evidently, and the strain energy of everypiece of twin is equal to Eheter. From Eq. (3), this is

Ehetertwin ¼

p2lð2� rÞ2ð1� rÞ c2aðeTÞ2V ð7Þ

from which it can be found that nEhetertwin ¼ p

n Enontwin. Twin-ning can thus help reduce the strain energy.

Considering the configuration in Fig. 7b, in which TDsdistribute homogeneously on the TBs, then the slope ofthe TB is uniform. If the two variants also have the samevolume fraction, Ehomo = 0. The inclusion is separatedinto two kinds of regions: the direction of strain doesnot intersect the TB in region I, while they do intersectin region II. The strain energy of region I is the same asthat given by Eq. (5). The strain distribution in Fig. 7dfor region II in the width of 2t, taking the center of theregion as the origin, is given by:

eðxÞ ¼ eTx=tð�t < x < tÞ ð8ÞThe theories of Eshelby and Khachaturyan cannot deal

with the strain energy of the inclusion where the strain var-ies continuously. However, the property of odd functione(x) can be utilized: the strain energies of both sides ofthe origin are equal and thus only the energy of 0 < x < t

needs to be considered. Applying the transformation asEq. (4), then

�e1 ¼1

2eT ð9Þ

de1ðxÞ ¼eT

2x� t

2

� �=tð0 < x < tÞ ð10Þ

and as Eq. (3):

Ehomo1 ¼ p2lð2� rÞ

2ð1� rÞ t2að�e1Þ2 ð11Þ

Taking x1 ¼ xþt2; eT

1 ¼ eT

2; t1 ¼ t

2, then de1ðx1Þ ¼ eT

1 x1=t1

ð�t1 < x < t1Þ, which is of the same form as Eq. (8) andwe return to the same problem.

Repeating the process n times, region II is divided into2n pieces and the recurrence relationship xnþ1 ¼ xnþt

2;

eTnþ1 ¼

eTn2; tnþ1 ¼ tn

2is met at the far right piece. Therefore:

Ehomonþ1 ¼

1

8Ehomo

n ð12Þ

Ehetern ¼ Ehomo

nþ1 þ Eheternþ1 ð13Þ

Substituting the two equations into Eq. (4), the strainenergy in region II is

EII ¼ 2ðEhomo1 þ Eheter

1 Þ ¼ 2ðEhomo1 þ Ehomo

2 þ Eheter2 Þ ¼ � � �

¼ 2X1n¼1

Ehomon ¼ 16

7Ehomo

1 ¼ 2p2lð2� rÞ7ð1� rÞ t2aðeTÞ2

X. Wang, J. Zhang / Acta Materialia 55 (2007) 5169–5176 5175

which is only 2/7 of that without any TDs present. The fi-nal morphology should be the equilibrium state of propertwin separation and TD distributions that minimize the to-tal elastic strain energy.

4.3. Elastic anisotropic media

In order to produce an analytical solution of the elasticfield problem, Eshelby assumed the media to be isotropic.However, most real materials are not isotropic, especiallyMSMAs, where strong anisotropy (A = (C11 � C12)/C44 < < 1) is one of their characteristics. Therefore, it isnecessary to determine whether the conclusion of Section4.1 can be used for anisotropic materials.

It is very clear that there is no analytical solution to theanisotropic problem. We follow the derivation of Khacha-turyan [33] to obtain a solution by a numerical method.According to the derivation in Section 4.1, it is only neces-sary to confirm that the elastic energy of a twin strip is pro-portional to its thickness, if the thickness is much less thanits width.

The result of the calculation is shown in Fig. 8. It isstraightforward to see that the proportional relationshipstill holds in a relatively thin twin, even if the material isan elastic anisotropic alloy. Thus, in SMAs such as the c-Mn-based alloy, the existence of TDs has the same effectas in isotropic media in reducing the strain energy.

4.4. Role of TD in TB motion

According to the discussion above, the TD can beregarded as a kink on the TB. A dislocation in a crystal,as a kink in the lattice, can greatly reduce the critical stressof deformation. By analogy, the TD has the same effect onTB motion.

Under external fields (stress field or magnetic field), theTB can move so that more twins are in the favored orien-tation. However, some Pierls–Nabarro (P–N) force needs

Fig. 8. Thickness dependence of strain energy of different anisotropiccoefficients for thin twin lamella (0 < c/a < 0.1).

to be overcome to start the motion, because of the differ-ent lattice orientations on both sides of the TB. It can beimagined that the force could be considerable if the TB isa perfect plane, when the entire TB has to be moved as awhole. However, with a TD on the boundary, the TBmotion can be realized through the motion of the TD,since the P–N force should be much lower than that ofTB. This property may help SMAs achieve good mechan-ical performance.

5. Conclusion

In this work, HRTEM has been used to characterize thestructure of the (110) TB of the tetragonal phase in anMn-based alloy at the atomic scale. Simulations based onthe Eshelby and Khachaturyan model have been used toexplain the experimental results. We find the following:

(a) The motion of all atoms during the cubic-tetragonaltransition in the Mn-based alloy is along the direc-tion parallel to (110), which is the TB of the low-temperature phase. This phenomenon indicates thatboth soft mode and AF ordering contribute to thetransformation from the cubic to the tetragonalphase.

(b) There are TDs lying on the TB. The TD makes a stepin the TB and the height of the step corresponds to thespacing of the (110) plane. On a larger scale, the TDsmake the curved interface of the twin structure.

(c) The TDs can help to reduce the strain energy inducedduring phase transition. The amount of energydecrease is up to 5/7 of the morphology withoutTDs if TDs are uniformly distributed on the TB. Thisensures the existence of TDs from the viewpoint ofthermodynamics. This conclusion holds in both elas-tic isotropic and anisotropic materials.

Acknowledgement

The present work was financially supported by the Na-tional Natural Science Foundation of China (Grant No.50571065).

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