A dislocation model for the magnetic field induced shape memory effect in Ni 2 MnGa

Transcription

1 Scripta Materialia 53 (2005) A dislocation model for the magnetic field induced shape memory effect in Ni 2 MnGa S. Rajasekhara, P.J. Ferreira * The University of Texas at Austin, Materials Science and Engineering Program, Austin, TX 78712, USA Received 17 May 2005; received in revised form 21 May 2005; accepted 6 June 2005 Available online 6 July 2005 Abstract The magnetically driven shape memory effect in Ni 2 MnGa body-centered tetragonal martensite is onset by the motion of h1 11i type twin dislocations on the {1 1 2} type planes. Calculations show that the application of a 400 ka/m magnetic field can induce a shear stress of approximately 2.8 MPa on these twin dislocations, a value exceeding the Peierls and the yield stress for Ni 2 MnGa. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Shape memory alloys; Magnetic field, Dislocations; Twinning 1. Introduction Actuator materials that respond to a controlled external influence to generate large amounts of strain are in demand. Such behaviors can be used to control aerodynamic devices, valves in automotive systems, sensors that can be operated in harsh conditions [1], and ultimately to replace complex mechanical systems in future devices, from motors to space telescopes [1]. For such applications, light actuators that have long lifetimes, require minimal maintenance and have high response frequencies under accurate control are optimal. Among currently available actuator materials, Ni 2 MnGa magnetic shape memory (MSM) alloys show considerable potential to meet requisite demands, showing 50 times (10% for Ni 2 MnGa [2]) greater strains as compared to magnetostrictive materials (0.24% for Terfenol-D) and considerably larger strains than traditional shape memory alloys, such as Nitinol. In addition, MSM * Corresponding author. Tel.: ; fax: address: ferreira@mail.utexas.edu (P.J. Ferreira). alloys can easily respond at over 50 times greater frequencies (250 Hz) than conventional shape memory alloys without a reduction in output strain. Due to the above mentioned alluring properties, interest in the reasons behind the shape memory behavior in Ni 2 MnGa alloys has peaked. Proposed explanations for the MSM effect suggest large amounts of strains are possible due to twin boundary motion as a result of reorientation of certain twin variants in the martensitic phase, during the application of a magnetic field [4 7]. The driving force for this motion is the difference in total magnetic energy density, which consists of the Zeeman energy density difference, DMB, and the magnetic anisotropy energy density difference, across the twin boundaries. As a result, an applied magnetic field induces a shear stress on the twinning plane, leading to the motion of twin dislocations and causing the translation of the twin boundaries [8 11]. However, a fundamental understanding of the type of dislocations that play a role in the MSM effect is currently lacking, which is crucial for enhancing the MSM effect and for tailoring novel, more efficient MSM alloys with lower twinning stresses /$ - see front matter Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi: /j.scriptamat

2 818 S. Rajasekhara, P.J. Ferreira / Scripta Materialia 53 (2005) Since the motion of twin dislocations is required for Ni 2 MnGa to develop a macroscopic strain in the material, the approach described herein attempts to explain the early stages of dislocation motion. Accordingly, we make certain assumptions that are valid during the low magnetic field/magnetostress regime of Ni 2 MnGa, namely, (1) the magnetostresses induced in the material are proportional to the applied magnetic field and (2) the material has an infinite magnetic anisotropy. Additionally, for the sake of simplicity, the unmodulated crystal structure of Ni 2 MnGa is assumed. 2. Twinning mechanism in Ni 2 MnGa To understand the mechanism responsible for the MSM effect in Ni 2 MnGa, it is important to identify the twinning planes and directions in which twin dislocations operate under the influence of an applied magnetic Fig. 1. (a) Undistorted L2 1 (Fm 3m) structure of Ni 2 MnGa; (b) distorted L2 1 (Fm 3m) structure of Ni 2 MnG with c/a = 0.94; and (c) bct (I4/mmm) structure of Ni 2 MnGa. field. To start with we consider the crystal structure of stoichiometric Ni 2 MnGa. Ni 2 MnGa has a L2 1 (Fm 3m) structure at room temperature (Fig. 1(a)) but undergoes a diffusionless phase transformation to a body-centered tetragonal (bct) (I4/mmm) martensite structure if cooled below 45 C. During the transformation, distortion of the initial cubic L2 1 lattice (c/a ratio = 0.94) occurs (Fig. 1(b)). Thus, the relationship between the lattice constants of the distorted L2 1 structure and the bct structure (Fig. 1(c)) are as follows: a bct ¼ adistorted L2 1 ffiffiffi 0.59 nm p ¼ pffiffiffi ¼ nm ð1aþ 2 2 c bct ¼ c distorted L2 1 ¼ nm ð1bþ The bct martensite structure produced exhibits several twin variants which reorient upon application of a magnetic field. The reorientation of the twin variants occurs on {0 1 1} C (the subscript C denominates the cubic parent phase) type planes along the h0 11i C type directions of the parent L2 1 cubic structure of Ni 2 MnGa [3,12]. The corresponding {hkl} planes and the huvwi directions in the bct coordinate system are the {1 1 2} T (the subscript T denotes the tetragonal martensite phase) planes and the h1 11i T directions. To understand the twinning mechanism in the bct martensite structure of Ni 2 MnGa during the application of a magnetic field, it is important to identify the atomic stacking on the (1 12) T twinning planes (Fig. 2(a) and (b)). This allows us to recognize a change in stacking upon motion of twin dislocations, due to a difference in Zeeman energy and a difference in magnetic anisotropy density between the twin variants. Therefore, to understand the MSM effect in Ni 2 MnGa, a determination of twin dislocation types that take part in the twinning mechanism is crucial. On the basis of experimental evidence, twinning in Ni 2 MnGa occurs along the {1 1 2} T planes of the bct Fig. 2. (a) Three-dimensional view of the Ni 2 MnGa bct structure; (b) two-dimensional (1 1 0) projection of the Ni 2 MnGa bct crystal lattice; and (c) shear operation of a twin dislocation along the [ 111] T direction to initiate the shape memory effect.

3 S. Rajasekhara, P.J. Ferreira / Scripta Materialia 53 (2005) structure ({0 1 1} C type planes in the parent L2 1 structure) [3,12]. Considering the stacking along the (1 12) T planes of the bct structure (Fig. 2(b)), the most likely twin dislocation to move, in response to an external magnetic field, is a partial dislocation in the [ 111] T type directions, as these are the close-packed directions along the (1 12) T type planes. Hence, every time a twin dislocation moves on the [ 111] T direction, it carries a twinning shear (Fig. 2(c)). Subsequently, correlated shear operations on successive (1 1 2) planes will produce a reorientation of a microtwin (Fig. 3). The motion of the twin boundary and consequent thickening of a particular twin variant can thus be accomplished by the repeated glide of twin dislocations on successive (1 1 2) planes. To further understand the twinning mechanism operating in Ni 2 MnGa, we are left with the task of finding the Burgers vector of twin dislocations and consequent twinning shear. As described by Mullner and Kriven [13], the Burgers vector of a twin dislocation can be determined if the twinning symmetry elements of the crystal structure subjected to twinning are identified. In this fashion, the Burgers vector of a twin dislocation can be expressed as [13] c 2 2 b t ¼ c2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 2 1 þ c2 2 2c ð2þ 1c 2 cos c where c 2 and c 1 are lattice related parameters and c is the angle between c 1 and c 2. For the Ni 2 MnGa bct martensite crystal structure, the twinning elementspffiffi are 2 represented in Fig. 4, for which c 1 ¼ c bct =2; c 2 ¼ abct ; 2 c ¼ p=2; b t is the Burgers vector of the twin dislocation along the [ 111] T direction and d is the interplanar spacing between the glide (1 1 2) planes. On these assumptions and following Eq. (2), the Burgers vector of the twin dislocation b t is found to be Å. In vector form, the Burgers vector can be written as b t = 0.144h111i. The twinning shear S, which is expressed Fig. 4. Twinning elements in Ni 2 MnGa bct martensite. as S = b t /d [13], can now be calculated. Since d = 2.02 Å for the Ni 2 MnGa bct martensite, then S = Finally, the strain associated with the motion of b t = 0.144h1 11i twin dislocations can be calculated as e = S/2 = (6%) [13], a value which has been observed in previous works [3 5]. These results are exact for an unmodulated martensitic structure. For the modulated five-layered martensitic structure, the atomic positions are shifted from an ideal bct structure, and thus, the values obtained for the magnitude of the Burgers vector, the twinning shear and the strain are an approximation. However, since the shifts in atomic position between the unmodulated and the fivelayer modulated martensitic structures are very small, the model conserves its validity. Following these calculations, we will next determine the stress required to move the twin dislocations as a function of an applied magnetic field. Fig. 3. Successive shear operations of twinning dislocations on the ( 1 1 2) planes results in the reorientation of a microtwin.

4 820 S. Rajasekhara, P.J. Ferreira / Scripta Materialia 53 (2005) Effect of an applied magnetic field As previously discussed, we will continue to assume the low magnetic field/magnetostress regime of Ni 2 Mn- Ga, where magnetostresses induced in the material are proportional to the applied magnetic field. In this regime, the magnetocrystalline anisotropy of the Ni 2 MnGa bct martensitic phase allows the magnetization vector to remain aligned with the easy axis of magnetization, as observed below approximately H/H A = 0.5 [8], where H A = 1 T is the saturation field [8]. Within this linear regime, we assume that the motion of a twin dislocation is only a function of the Zeeman energy difference between the two twin variants (DBH), where DB is the difference in magnetic flux density between the variants and H is the external magnetic field. In two dimensions we may write the Zeeman free energy density difference as [14]: ZE ¼ DBH ¼ l 0 M S Hðcos h cosðh þ /ÞÞ ð3þ where l 0 is the permeability of vacuum, M S is the saturation magnetization, h is the angle between the direction of magnetization of variant 1 and the magnetic field and / is the angle between the two twin variants (Fig. 5). Thus, the shear stress acting on a dislocation as a result of an applied magnetic field can be expressed as [14] r ¼ F Lb ¼ dðdbhþ ð4þ b where d is interplanar spacing and the other symbols retain their earlier meaning. Inserting Eq. (3) into Eq. (4) gives r ¼ dl 0M S Hðcos h cosðh þ /ÞÞ ; ð5þ b which describes the shear stress acting on a twin dislocation as a function of strength and orientation of the Fig. 5. Martensite twin variants in the presence of a magnetic field. M 1 and M 2 are the magnetization vectors in the twin variants, H is the magnetic field direction, h is the angle between the magnetic field direction and the magnetization vector of twin variant 1 and / is the angle between the twin variants. The glide direction of the twin dislocations is [ 111] T. magnetic field, and orientation between the twin variants. Since the entire process of twin reorientation starts with the motion of twin dislocations of type b t = 0.144h1 11i, the minimum stress required to initiate dislocation motion should be at least greater than the Peierls stress, [15], which can be expressed at 0 K as: r P ¼ 2G ð1 mþ exp 2pw ð6þ b where G is the shear modulus, m is PoissonÕs ratio and w = d/(1 m) for an edge dislocation while w = d for a screw dislocation. Assuming the values d = 2.02 Å, b = Å, m = 0.38 and G = 2 GPa [3] for the Ni 2 Mn- Ga bct martensite, the Peierls stress at 0 K ranges from approximately Pa for an edge dislocation to Fig. 6. Resolved shear stress acting on h111i twin dislocations as a function of an applied magnetic field: (a) h =0 and / =30, 60, 90 ; (b) h =45 and / =30, 60, 90.

5 S. Rajasekhara, P.J. Ferreira / Scripta Materialia 53 (2005) Pa for a screw dislocation. We now compare this value with the stress experienced by the twin dislocation as a function of the applied magnetic field. Assuming the saturation magnetization l 0 M S = 0.62T ( A/m) [3], Fig. 6(a) and (b) show the shear stress experienced by twin dislocations as a function of the applied magnetic field, for h =0 and h =45, and three values of /, (30, 60 and 90 ). Fig. 6(a) and (b) show that even very small magnetic fields are sufficient to overcome the calculated Peierls stress at 0 K. 4. Discussion Experiments show that a magnetic field of approximately 400 ka/m (0.50 T), or a compressive stress of 1 MPa produces a twin free state in Ni 2 MnGa [3]. This is attributed to a reorientation of twin variants in the martensite phase of Ni 2 MnGa. The mechanism discussed herein demonstrates that the reorientation of twin variants can occur by successive shear operations of the type 0.144h1 11i on the ( 1 1 2) planes of the bct Ni 2 MnGa martensite phase. For this reorientation to occur, the shear stress induced by the applied magnetic field and acting on the twin dislocations should be greater than the Peierls stress and the yield stress for Ni 2 MnGa at room temperature [3]. Let us first consider the Peierls stress. Based on the calculations aforementioned, the Peierls stress at 0 K is found to be very small. At room temperature, where atomic vibrations are enhanced, the Peierls stress should be further reduced. Assuming the geometric conditions, / = 90 and h = 45, for which the applied magnetic field induces a maximum shear stress [16] on the 0.144h1 11i twin dislocations, a magnetic field of 0.10 T can generate a shear stress of around 0.05 MPa, which is orders of magnitude larger than the Peierls stress. Thus, very small magnetic fields are sufficient to overcome the calculated Peierls stress for twin dislocations in Ni 2 MnGa. Now let us consider the interplay between the applied magnetic field and the yield stress of Ni 2 MnGa. Under the same geometric conditions, / =90 and h =45, as the applied magnetic field is increased to 0.5 T (regime up to where the magnetostresses induced in the material are proportional to the applied magnetic field), a shear stress of approximately 2.8 MPa is generated (Fig. 6b). This value is comparable to the reported yield stress of Ni 2 MnGa of approximately 1 MPa [3]. Furthermore, as the reported compressive stress of 1 MPa along the [0 0 1] direction [3] is an applied stress, the resolved shear stress acting on the ( 1 1 2) plane, along the [ 1 1 1] direction, is approximately 0.80 MPa. This value of stress is thus less than the calculated 2.8 MPa. As a result, for an applied magnetic field of 0.5 T, twin dislocation motion is expected to occur. This motion will induce the translation of the twin boundaries and eventually, a reorientation of the twin variants. However, this behavior can be significantly altered if impurities are present in the material, which may cause twin dislocation pinning, inhibiting their free motion. Thus, the design of magnetically driven shape memory alloys requires a careful control of impurities. The model described herein can therefore be used, as a first approximation, to calculate the shear stress acting on the twin dislocations in Ni 2 MnGa as a function of an applied magnetic field. If the shear stress acting on the twin dislocations, due to the application of a magnetic field, is considerably lower than the Peierls stress and/or the yield stress of Ni 2 MnGa, twin dislocation motion will be inhibited and the MSM effect will not occur. For this reason, newly developed MSM alloys should exhibit lower twinning stresses, and thus respond to lower applied magnetic fields. In tailoring new alloys, material properties such as the shear modulus, the interplanar spacing between the twinning planes, the Burgers vector of the twin dislocations, and the saturation magnetization should be considered. Hence, a material that exhibits a low shear modulus, a larger saturation magnetization, a high interplanar spacing between twinning planes, and a short Burgers vector should exhibit an enhancement of the MSM effect. With this in mind, theoretical calculations can be made for other Heusler alloys (L2 1 structure) to evaluate their potential as MSM alloys. 5. Conclusions The application of a 400 ka/m (0.50 T) magnetic field to the bct martensite phase of Ni 2 MnGa, can induce a shear stress of approximately 2.83 MPa on h1 1 1i type twin dislocations. As the calculated stress value exceeds the Peierls and the yield stress of Ni 2 Mn- Ga, twin dislocation motion is likely to occur. Once in motion, these twin dislocations generate the translation of the twin boundaries, and subsequently the orientation of the twin variants upon the application of a magnetic field. Acknowledgement The authors would like to thank Prof. Peter Mullner for discussions on this paper, in particular the aspects related with the determination of the Burgers vector for the twin dislocations. References [1] C.H. Joshi, B.R. Rent, in: Aerospace Mechanisms Symposium, May 1999.

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