Magnetoelastic domains and magnetic field-induced strains in ferromagnetic
shape memory alloys by phase-field simulation
L. J. Li,1J. Y. Li,1,a兲 Y. C. Shu,2H. Z. Chen,2and J. H. Yen2 1
Department of Mechanical Engineering, University of Washington Seattle, Washington 98195-2600, USA 2Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, Republic of China 共Received 14 January 2008; accepted 10 April 2008; published online 29 April 2008兲
Magnetoelastic domains in ferromagnetic shape memory alloys evolve through either variant rearrangement or magnetization rotation, resulting in a large or a small magnetic field-induced strain depending on the magnitude of applied compressive stress. These phenomena are simulated in this letter using an unconventional phase-field model motivated by energy-minimizing multirank laminated domain structures. The results agree well with experiments, and confirm the analysis of Ma and Li关Appl. Phys. Lett. 90, 172504 共2007兲兴 based on an energy minimization theory. © 2008
American Institute of Physics. 关DOI:10.1063/1.2918127兴
Ferromagnetic shape memory alloys 共FSMAs兲 possess both ferroelastic and ferromagnetic ordering simultaneously,1 and this magnetoelastic coupling makes it possible to ma-nipulate the ferroelastic domains of FSMA by magnetic field. The resulted variant rearrangement leads to a magnetic field-induced strain as high as 10%,2–5which is very attractive for actuator applications. However, the blocking stress of FSMA is relatively small, and the magnetic field-induced strain dra-matically drops when the external compressive stress increases,5–7 which seriously limits FSMA’s applications as actuators. Using an energy-minimization theory, Ma and Li7 suggested that when the external compressive stress exceeds a critical threshold, the magnetic field-induced variant rear-rangement in FSMA will be frozen, and magnetization rota-tion will take over instead as the dominant mechanism for domain structure evolution, leading to a magnetostrictive strain that is orders of magnitude smaller. Understanding the formation and evolution of FSMA domain structure thus is not only important from the scientific interests but is also critical to the applications of FSMA.
While a few theoretical models have been developed for FSMA,8–19direct numerical simulation of magnetoelastic do-mains has been rarely attempted.20,21Capturing variant rear-rangement and magnetization rotation under a unified theo-retical framework is particularly challenging. In this letter, we report an unconventional phase-field simulation of FSMA to accomplish that. To demonstrate this, we consider a tetrag-onal FSMA. It has three ferroelastic variants, and each of them is distorted by a transformation strain共i兲and is mag-netized by a saturation magnetization Ms that prefers to be
aligned along the easy axis r共i兲 of the respective variants, with
共i兲=␣I −共␣−兲r共i兲丢r共i兲,
r共1兲=共1,0,0兲, r共2兲=共0,1,0兲, r共3兲=共0,0,1兲, 共1兲 where ␣ and  are material parameters, and I is the unit second rank tensor. Notice that the easy axis of magnetiza-tion is intimately coupled with the transformamagnetiza-tion strain. Nevertheless, the direction of magnetization M = Msm in
each variant is not constrained to the easy axis, and can rotate away from the easy axis with an energy penalty, re-sulting in a magnetostrictive strain m共m兲 that is orders of
magnitude smaller than transformation strain.22 As such, both strain and magnetization are needed to describe the do-main structure of FSMA.
It is well known that phase-changing materials such as FSMA form very characteristic domain structures consisting of multiple variants to reduce the overall energy of the sys-tem. While transformation strains are used as order param-eters in the conventional phase-field simulation of ferroelas-tic domain structures,21,23 Shu and Yen24 and Shu et al.25 suggested that the local volume fractions of variants can be used as field variables instead. For a three-variant system such as tetragonal FSMA, the local transformation strain is related to␥i, the volume fraction of variant i, as follows:
*=␥1共1兲+␥2共2兲+␥3共3兲. 共2兲 Obviously, only two of the␥iare independent, and1and2 are introduced to reflect this constraint, with
␥1=1, ␥2=共1 −1兲2, ␥3=共1 −1兲共1 −2兲. 共3兲 This representation is motivated by the multirank laminated domain configuration, which is energy-minimizing when共i兲 are pariwise compatible.12,26,27Through Eqs.共2兲and共3兲, the equivalence of transformation strain * and =共1,2兲 is established.
When is used as field variables instead of *, the potential energy of a FSMA occupying a domain⍀ can be expressed as7,22 I共,m兲 =
冕
⍀兵W int+ W s ani+ W m ani+ Welas−0· −0H0· M其dx + 0 2冕
R3兩ⵜ兩 2dx, 共4兲where Wint= A1兩ⵜ兩2+ A2兩ⵜm兩2 penalizes gradient of internal variables and thus leads to interfacial energy across magnetoelastic domain walls, Ws
ani = Ks兺i=1
2 i2共1−i兲2 is the anisotropy energy that penalizes the deviation of transforma-tion strain from the ground state, which allows us to express energy wells of FSMA explicitly instead of through expan-sion of polynomial of transformation strain, and thus
simpli-a兲Author to whom correspondence should be addressed. Electronic mail: [email protected].
APPLIED PHYSICS LETTERS 92, 172504共2008兲
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fies the model development considerably, Wmani= Ku共1
−关m·r*共兲兴2兲 is the magnetic anisotropy energy that penal-izes the magnetization rotation away from the easy axis, given by r*=␥1r共1兲+␥2r共2兲+␥3r共3兲, which explicitly couples the ferroelastic orderingand ferromagnetic order-ing m. Moreover, Welas=1
2关−*共兲−
m共m兲兴·C关−*共兲
−m共m兲兴 is the elastic energy resulting from the
incompat-ibility of transformation and magnetostrictive strains, which can be determined by solving mechanical equilibrium equation,24 where is the total strain and C is the elastic stiffness. The last two terms in the first integral are potential energies associated with the applied magnetic field H0 and stress0, and the last integral is the demagnetization energy due to the magnetization distribution in FSMA, which can be determined by solving Maxwell’s equation in full space,7 whereis the magnetic potential and0is the permeability of free space.
Under an external magnetic or mechanical loading, both
and m will evolve to minimize the potential energyI in Eq.共4兲. The evolution ofis given by
t = − L
␦I
␦= L关Fint+ Fsani+ Fmani+ Felas兴, 共5兲
which governs the variant rearrangement process. In Eq.共5兲, L is the mobility constant, Fint= 2A1ⵜ2drives the coarsening of ferroelastic domains, Fs
ani
= −/Ws
ani共兲 and Fmani= −/Wmani共, m兲 select the particular set of
variants, with the second one coupling the transformation strain and the magnetization, and Felas= C关−*共兲 −m共m兲兴·*共兲/ drives the refining of ferroelastic
do-mains. Notice that C关−*共兲−m共m兲兴 is actually the stress
in FSAM that is related to applied stress at boundary.22 On the other hand, the evolution of magnetization is given by Landau–Lifschitz–Gilbert equation
m
t = −␥gm⫻ H
eff−␦␥gm⫻ 共m ⫻ Heff兲, 共6兲
which governs the process of magnetization rotation, where
␥g⬇2.21⫻105m/共A s兲 is the gyromagnetic ratio, ␦ is the dimensionless damping coefficient, and the effective mag-netic field is given by
Heff= − 1 0Ms ␦I ␦m= H e + Ha+ Hs+ H0+ Hd, 共7兲 where He= 2A
2/0Msⵜ2m drives the coarsening of
magnetic domains, Ha= −1/0Ms/mWm
ani共, m兲 selects preferred magnetization direction, Hs= 1/0MsC关−*共兲
−m共m兲兴·m共m兲/m couples the magnetization and
trans-formation and magnetostrictive strains, and Hd= −ⵜis the
demagnetization field. Thus, framework is established to study the formation and evolution of magnetoelastic domains in FSMA in terms of and m.
A two-dimensional simulation of FSMA Ni2MnGa is implemented by solving Eqs. 共5兲 and 共6兲 under the periodic boundary condition共BC兲, with the following mate-rial constants:5,7,11,16 Ks= Ku= 1.65⫻105J/m3, A1/共Ksl0
2兲 = A2/共Kul02兲=10−4, with l0being the length of discretization,
␣= 0.021, = −0.034, Young’s modulus is taken to be 154 GPa, and Poisson’s ratio is assumed to be 0.3. The de-magnetization field consists of fields due to shape anisotropy and internal magnetization incompatibility. The shape
aniso-tropy part is evaluated using demagnetization factors N1 = 0.194, N2= N3= 0.403, which are calculated from the di-mensions of 9⫻5⫻5 mm3rod, while the field arising from internal incompatibility is calculated in Fourier space under the periodic boundary condition.28Since variant 2 is favored by neither the applied stress nor magnetic field, it is excluded from consideration, and a two-dimensional simulation will be sufficient to capture the essence of the problem. We first consider the formation of magnetoelastic domain structure in FSMA subjected to clamped BC and random initial condi-tion, and a self-accommodating domain structure emerges, as shown in Fig.1共a兲in a 128⫻128 cell, in which each arrow actually spans multiple cells to make the illustration clear, otherwise the figure will become too crowded. Increasing the computational size further does not change the simulation results. Note that the domain structure consists of two fer-roelastic variants 1 and 3 of equal volume fraction, separated by 90° domain walls along the 共101¯兲 plane. Furthermore, each of the ferroelastic domain is divided into two different kinds of magnetic domains, separated by 180° domain walls along the共100兲 or 共001兲 plane, and the combined magneto-elastic domains are, indeed, a rank-2 laminate as predicted by the constrained theory.12On the other hand, when a com-pressive stress is applied along the关100兴 axis, a single fer-roelastic variant 1 consists of two magnetic domains, sepa-rated by 180° domain walls, emerges as a rank-1 laminate as also predicted by the constrained theory,12 as shown in Fig.1共b兲.
We then consider a Ni2MnGa rod subjected to a fixed compress stress0 along the longer axis of the rod, parallel to the关100兴 axis of the crystal, and a varying magnetic field H0parallel to the关001兴 axis, a typical experimental configu-ration of FSMA.5A rank-1 laminate with equal volume frac-tion of 180° magnetic domains similar to Fig.1共b兲is used as the initial configuration but with a single layer of variant 3 less than 1% of total volume added to facilitate nucleation. As the magnetic field along the 关001兴 axis increases, the domain structure of FSMA will evolve, through either vari-ant rearrangement or magnetization rotation, depending on the magnitude of the applied compress stress, as suggested by Ma and Li.7 This is, indeed, observed in our simulation, as shown in Figs.2and3. For example, when a small com-pressive stress of 0.6 MPa is applied, small magnetization rotation occurs first, but variant rearrangement quickly takes over as the dominant evolution mechanism, leading to a large jump in magnetic field-induced strain at 0.3 T and, corre-spondingly, a larger slope of magnetization curve, as shown by the solid green curve in Fig. 2, which agrees very well FIG. 1. 共Color online兲 Formation of magnetoelastic domains: 共a兲 Rank-2 domain pattern in FSMA under clamped BC and共b兲 rank-1 domain pattern in FSMA under compressive stress. Green and black colors indicate variant 1, while fuchsia and red indicate variant 3; arrow indicates the magnetiza-tion direcmagnetiza-tion.
172504-2 Li et al. Appl. Phys. Lett. 92, 172504共2008兲
with experimental data.5A closer examination of the inter-mediate domain structure before the saturation of magnetiza-tion reveals a very characteristic rank-2 domain structure, as shown in Fig.3共a兲, where variant 3 is observed to grow at the expense of variant 1, which is separated by a 90° domain wall along the 共101兲 plane. Small magnetization also exists in variant 1, which is consistent with the analysis of Ma and Li.7 On the other hand, when a relatively large compress stress of 3 MPa is applied, the variant rearrangement process is completely blocked, leading to magnetostrictive strain that is orders of magnitude smaller than transformation strain, as shown by the broken blue curve in Fig.2共a兲, also in excellent agreement with experiment.5No change in the slope of mag-netization curve is observed before saturation, another indi-cation of magnetization rotation instead of variant rearrange-ment which is again in good agreerearrange-ment with experirearrange-ment.5 Examination of the intermediate domain structure before magnetization saturation reveals that the ferroelastic variant is indeed unchanged throughout the magnetization process even with the presence of prescribed nucleation layer. Only magnetization rotation occurs, as shown in Fig.3共b兲, where a rank-1 laminate is observed with identical ferroelastic variant but different magnetization directions. Simulation on an in-termediate stress of 1.4 MPa has also been carried out in good agreement with experiment. In all these simulations, all the conditions are kept identical except the magnitude of the applied compressive stress.
In summary, we have developed an unconventional phase-field model to simulate the formation and evolution of magnetoelastic domains and magnetic field-induced strain in
FSMA, where both variant rearrangement and magnetization rotation are captured. The simulation agrees well with ex-periments, and confirms the analysis of Ma and Li7that vari-ant rearrangement in FSMA is blocked at large compressive stress, resulting in much reduced magnetic field-induced strain in FSMA.
We acknowledge the financial support from US ARO 共W911NF-07-1-0410兲 and AFOSR 共FA9550-07-1-0175兲. Y.C.S. also acknowledges the support of TW NSC Grant 共NSC-96-2221-E-002-014兲.
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FIG. 2. 共Color online兲 Axial strain 共a兲 and magnetization 共b兲 vs the applied magnetic field under compressive stress of 0.6 and 3 MPa, respectively.
FIG. 3. 共Color online兲 Magnetoelastic domains at intermediate stages of magnetization under different compressive stresses of 0.6 MPa 共a兲 and 3.0 MPa共b兲. Green and blue colors indicate variant 1, while fuchsia indi-cates variant 3; arrow is used to indicate the magnetization direction.
172504-3 Li et al. Appl. Phys. Lett. 92, 172504共2008兲
