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The role of anisotropic thermal expansion of shape memory alloys in their functional properties
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Acta Materialia 57 (2009) 5605–5612 www.elsevier.com/locate/actamat
The role of anisotropic thermal expansion of shape memory alloys in their functional properties V.A. L’vov a,b,*, N. Glavatska c, I. Aaltio d, O. So¨derberg d, I. Glavatskyy c, S.-P. Hannula d a
Department of Radiophysics, Taras Shevchenko University, Volodymyrska Str. 64, 01601 Kyiv, Ukraine b Institute of Magnetism, Vernadsky Str. 36-b, 03142 Kyiv, Ukraine c Institute for Metal Physics, NAS of Ukraine, Vernadsky Str. 36, 03142, Kiev, Ukraine d Helsinki University of Technology, Department of Materials Science and Engineering, P.O. Box 6200, Espoo, FI-02015 TKK, Finland Received 12 January 2009; received in revised form 29 July 2009; accepted 29 July 2009 Available online 27 August 2009
Abstract The functional properties of a ferromagnetic shape memory alloy, which undergoes martensitic transformation of a cubic-tetragonal type, have been analyzed using the Landau theory of phase transitions. A key role of the temperature dependency of the crystal lattice parameters in the formation of the elastic and magnetic properties of the martensitic phase has been determined. To this end the temperature dependencies of the shear elastic moduli and magnetic anisotropy energy densities (MAED) of two quasi-stoichiometric Ni– Mn–Ga alloys have been computed from the experimental temperature dependencies of their lattice parameters. It turned out that the differences in the temperature dependencies of the lattice parameters of these alloys gave rise to drastic differences in their shear moduli and MAED as a function of temperature. The possibility of the spontaneous reorientation of the magnetization vector on cooling of the alloy and a right-angle realignment of this vector by a low magnetic field have been predicted. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Shape memory alloys; Martensitic phase transformation; Elastic behavior; Magnetic anisotropy
1. Introduction It is common knowledge that shape memory alloys are now widely used in engineering and medicine due to their responses to external stimuli, such as temperature variation or/and mechanical loading [1,2]. These responses are characterized by a several per cent deformation of the crystal lattice on cooling or axial stressing of the alloy [3–5]. The ferromagnetic shape memory alloys form a special class of functional materials, which exhibit high values of the magnetic anisotropy constant and magnetically induced deformation (see, [6,7], and references therein). Extreme *
Corresponding author. Address: Department of Radiophysics, Taras Shevchenko University, Volodymyrska Str. 64, 01601 Kyiv, Ukraine. Tel.: +380 44 452 01 34. E-mail addresses: [email protected] (V.A. L’vov), simo-pekka. hannula@tkk.fi (S.-P. Hannula).
values of deformation and magneto-deformation of shape memory alloys become attainable after a first-order phase transition of martensitic type, which is accompanied by a jump-like deformation of the crystal lattice and pronounced softening of the elastic moduli on cooling of single crystalline specimens. The temperature dependency of the lattice parameters and soft elastic moduli are among the main functional properties of shape memory alloys, which prescribe the peculiarities of their deformational behavior under mechanical load or external magnetic field. The response of these alloys to the application of a magnetic field is related not only to the physical values mentioned above, but also to the magnetic anisotropy constant [6–10]. Experimental studies have shown abnormally low values for the shear elastic moduli [11] and Young’s moduli [12] of some ferromagnetic shape memory alloys belonging to the Ni–Mn–Ga family. The study of abnormally soft ferromagnetic materials thereupon became topical. The
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.07.058
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experimentally observed softening of the elastic moduli was attributed to a cubic-tetragonal martensitic transformation (MT). The functional properties of soft martensitic alloys can be analyzed within the framework of the Landau theory of phase transitions, which correlates the elastic moduli [13] and the magnetic anisotropy energy density (MAED) [14] with the values of the lattice parameters in the tetragonal phase. The temperature dependence of the lattice parameters of martensite may turn out to be the main factor governing both the elastic and magnetic functional characteristics of alloy specimens. The present article seeks to verify this possibility. Moreover, the possibility of a reorientation of the magnetization vector on martensite cooling and a right-angle rotation of this vector by extremely low magnetic fields is verified by the quantitative computations.
is the energy of anisotropic magnetoelastic interactions. The coefficients c2, a4 and b4 in Eq. (4) are the linear combinations of the second-, third- and fourth-order elastic moduli (enumerated according to Gomonaj and L’vov [13]), D in Eq. (5) is the matrix formed by demagnetization coefficients, m = M(T)/M(T) is the unit magnetic vector, M is magnetization vector, H is the external magnetic field and d in Eq. (6) is the dimensionless magnetoelastic constant. A simple regrouping of the terms in Eqs. (3), (4), and (6) results in the expression: ðeff Þ ðeff Þ ðeff Þ ð7Þ G ¼ F m þ F e 3r1 u1 r2 u2 þ r3 u3 =6; where the values Þ rðeff ¼ ra þ rðmeÞ ; a a
2. The Landau theory for ferroelastic phase transition 2.1. Landau expansion of the Gibbs potential of an elastic ferromagnet The Gibbs potential of a single crystal of a ferromagnetic shape memory alloy undergoing cubic-tetragonal MT can be expressed in terms of the basic functions of irreducible representations of the cubic group: u1 ¼ ðexx þ eyy þ ezz Þ=3; pffiffiffi u3 ¼ 2ezz eyy exx ; u2 ¼ 3ðexx eyy Þ; ð1Þ r1 ¼ ðrxx þ ryy þ rzz Þ=3; pffiffiffi r3 ¼ 2rzz ryy rxx ; r2 ¼ 3ðrxx ryy Þ; composed of the strain and stress tensors eik and rik, respectively. The coordinate axes are aligned with the fourfold symmetry axes of the crystal. The functions u1 and r1 describe the isotropic deformation process and, therefore, can be omitted when the anisotropic effects are studied. In this case the expression for Gibbs potential is: G ¼ F ðr2 u2 r3 u3 Þ=6;
ð2Þ
where the Helmholtz free energy (F) is the sum of elastic (Fe), magnetic (Fm) and magnetoelastic (Fme) energies, i.e. F ¼ F e þ F m þ F me ;
ð3Þ
where 1 1 2 1 F e ¼ c2 u22 þ u23 þ a4 u3 u23 3u22 þ b4 u22 þ u23 ; 2 3 4
ð8Þ
can be interpreted as the internal stresses caused by magnetoelastic coupling [15]. 2.2. Ferroelastic phase transition in an axially stressed ferromagnetic single crystal The conditions oG/ou2 = oG/ou3 = 0 result in the equations: ðeff Þ
c2 u2 2a4 u2 u3 þ b4 u2 ðu22 þ u23 Þ r2 c 2 u3 þ
a4 ðu23
u22 Þ
þ
b4 u3 ðu22
þ
u23 Þ
=6 ¼ 0;
ðeff Þ r3 =6
ð9Þ
¼ 0:
ðeff Þ
ðeff Þ
In the case of zero effective stress ðr2 ¼ r3 ¼ 0Þ this equation system has solutions which correspond to the paramagnetic cubic phase and three equivalent variants of the paramagnetic tetragonal phase with fourfold symmetry axes oriented in the x, y or z directions. For certainty, let us consider the phase transition from the cubic phase to the z variant of the tetragonal phase. In this case the equality exx = eyy is valid in both phases. Therefore, u2 = 0 and the equation system reduces to one equation: u3 ðc2 þ a4 u3 þ b4 u23 Þ ¼ 0;
ð4Þ
ð10Þ
which has two solutions: u3 ¼ 0;
is the elastic energy, 1 F m ¼ M 2 ðT Þðm D mÞ mHMðT Þ; 2
pffiffiffi ðmeÞ ¼ dex M 2 ðT Þ=3; r2 ¼ 6 3dM 2 ðT Þ m2x m2y ; ðmeÞ r3 ¼ 6dM 2 ðT Þ 2m2z m2y m2x ðmeÞ
r1
ð5Þ
is the anisotropic part of the magnetic energy of the crystal and hpffiffiffi i F me ¼ dM 2 ðT Þ 3 m2x m2y u2 þ 2m2z m2y m2x u3 ; ð6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u3 ¼ ða4 =2b4 Þ 1 þ 1 c2 ðT Þ=ct u0 ðT Þ
ð11Þ
where ct ¼ a24 =4b4 > 0:
ð12Þ
The first solution in Eq. (11) corresponds to the cubic phase and the second one describes the “spontaneous” deformation of the cubic lattice in the course of a first-
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order phase transition. The tendency to volume conservation is observed for cubic-tetragonal phase transitions in shape memory alloys. In this case: u0 3ezz ¼ 3ðc aÞ=a0 2ðc=a 1Þ;
ð13Þ
where a, c and a0 are the lattice parameters of the tetragonal and cubic phases, respectively. According to the Landau theory the coefficient c2 is an increasing function of temperature. The cubic phase is stable when c2(T) > 0 and the tetragonal one is stable if c2(T) < ct. In the temperature range bounded by the inequalities 0 < c2(T) < ct both the cubic and tetragonal phases can be observed. Therefore, the lability temperatures of the tetragonal and cubic phases (T1and T2, respectively) satisfy the equations c2(T1) = ct and c2(T2) = 0. It should be emphasized that when the high temperature phase is ferromagnetic its symmetry is not exactly cubic, ðmeÞ ðmeÞ because the stresses r2 and=or r3 are not equal to zero in the ferromagnetic state. In the other words, the process of ferromagnetic ordering always reduces the symmetry of the cubic crystal lattice. It is worth explaining this statement in more detail. Let the equilibrium direction of the magnetic vector be parallel to the [001] direction of the crystal lattice. In this case the lattice is axially stressed by magnetoelastic stress: rðmeÞ zz ðT Þ
¼
ðmeÞ r3 ðT Þ=2
2
¼ 6dM ðT Þ:
ð14Þ
This stress causes elastic deformation of the crystal, which is commonly known as spontaneous magnetostriction. Magnetostriction results in deformation of the cubic lattice and a reduction in its symmetry from cubic to tetragonal and, therefore, the ferroelastic phase transition of a ferromagnetic single crystal is, rigorously speaking, an isomorphic (tetragonal–tetragonal) first-order phase transition. For the chosen direction of magnetic vector the spontaneous magnetostriction is of the order of dM2/C0 . For the abnormally soft Ni–Mn–Ga alloys the values C0 1 GPa and dM21MPa were reported [11,12,15]. Therefore, magnetostriction of these alloys may be of the order of 0.1%, even in the high temperature phase. This value of magnetostriction of the high temperature phase was measured by Kokorin and Wuttig [16] on approaching the MT temperature range. The crystal lattice with 1 c/ a 0.1% cannot be considered cubic in precise experiments and rigorous theories. Magnetoelastic stress promotes the appearance of the z variant and retards formation of the x and y variants of the are like-sign vallow temperature phase when u0 and rðmeÞ zz ues. Another factor which assists transition of the parent phase to the z variant of the low temperature one is the mechanical stress rzz, if it is identical in sign to u0. The ðeff Þ effective stress r3 ðT ; rzz Þ ¼ 2½rðmeÞ zz ðT Þ þ rzz is a superimposition of mechanically and magnetically induced stresses. Let us consider the situation rxx = ryy = 0. In this case ðeff Þ r2 ¼ 0 and the first equation in the equation system Eq. (9) are satisfied by the value u2 = 0. Thus, the isomor-
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phic phase transition is described by the second equation, which is reduced to the form: ðeff Þ ð15Þ u3 c2 ðT Þ þ a4 u3 þ b4 u23 r3 ðT ; rzz Þ=6 ¼ 0: This equation explicitly shows absence of solution u3 = 0, which corresponds to the cubic phase. The solution of Eq. (15) can be searched in the form: ð0Þ
u3 ðT ; rzz Þ ¼ u3 ðT Þ þ ~uðT ; rzz Þ;
ð16Þ
where ~uðT ; rzz Þ is the linear combination of elastic strain tensor components: u0 ðT Þ; if T < T 1 ; ð0Þ ð17Þ u3 ðT Þ ¼ 0 otherwise: Under the actual experimental conditions the inequality j~uðT ; rzz Þj ju0 ðT Þj holds true and so substitution of Eq. (16) into Eq. (15) shows that Eq. (15) has the approximate solution: h i1=2
ðeff Þ ~uðT ;rÞ¼A1 ðT Þ C pm ðT Þ C 2pm ðT ÞþAðT Þr3 ðT ;rÞ ; ð18Þ where h i2 ð0Þ ð0Þ C pm ðT Þ ¼ 3c2 ðT Þ þ 6a4 u3 ðT Þ þ 9b4 u3 ðT Þ ; h i ð0Þ AðT Þ ¼ 6 a4 þ 3b4 u3 ðT Þ :
ð19Þ
The function Cpm(T) describes the temperature dependence of the shear modulus in the paramagnetic state. The result, Eq. (18), is valid if ðeff Þ ð20Þ C 2pm ðT Þ > AðT Þ r3 ðT ; rÞ : The ferromagnetic lattice is deformed due to magnetoelastic interaction. For this reason its shear modulus, defined as: 1
C fm ¼ ðdu3 =drÞ jr¼0 ;
ð21Þ
is slightly different from the shear modulus of the paramagnetic lattice. Eqs. (16), (18), and (21) result in the relationship: h i1=2 ð22Þ C fm ðT Þ ¼ C 2pm ðT Þ þ 12dM 2 ðT ÞAðT Þ An elastic strain caused by an external force applied to a single crystalline specimen in the z direction can be determined from the expression: 1 uðT ; rÞ ~uðT ; 0Þ: ð23Þ eðelÞ zz ðT ; rÞ ¼ ½~ 3 The temperature-dependent total (spontaneous and elastic) deformation of a compressed alloy can be expressed as: 1 ð0Þ ð24Þ ezz ðT ; rÞ ¼ u3 ðT Þ þ eðelÞ zz ðT ; rÞ: 3 Eqs. (18), (23), and (24) are applicable to both ferromagnetic and paramagnetic alloys. These equations may be used for computations if the numerical values of the coef-
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ficients of the Landau expansion for Gibbs potential are known. In addition, for ferromagnetic alloys a knowledge of the magnetization function M(T) is needed. 2.3. Evaluation of the Landau expansion coefficients and lability temperatures According to Eq. (11) and the definition of the lability temperatures, the spontaneous tetragonal deformation satisfies the following condition: u0 ðT 2 Þ ¼ 2u0 ðT 1 Þ ¼ a4 =b4 :
ð25Þ
Eqs. (12), (13), and (25) result in the relationships: 2c2 ðT 1 Þ ; 1 cðT 2 Þ=aðT 2 Þ c2 ðT 1 Þ b4 ¼ 4c2 ðT 1 Þ=u20 ðT 2 Þ ¼ : 2 ½1 cðT 2 Þ=aðT 2 Þ a4 ¼ 4c2 ðT 1 Þ=u0 ðT 2 Þ ¼
ð26Þ
These relationships enable estimation of the third-order and fourth-order terms in the Landau expansion for elastic energy from the values of lattice parameters at the lability point of the low temperature phase (T1) and the shear elastic modulus of the parent phase C0 (T1) = 3c2(T1) = 3ct just before the beginning of the phase transformation. The lattice parameters are available from X-ray diffraction data and the shear modulus of the parent phase can be determined by dynamic mechanical analysis (DMA) or ultrasonic methods. After evaluation of parameters the a4 and b4 the temperature-dependent coefficient c2(T) can be found from the expression: 8 T T 2 if T 2 < T < T ; < ct T 1 T 2 h i2 ð27Þ c2 ðT Þ ¼ : a4 uð0Þ b4 uð0Þ if T 6 T 2 ; 3 3 where the temperature T > T1 must be close enough to T1. The first line of Eq. (27) provides for the linear dependence of the coefficient c2 on temperature in the vicinity of the phase transition temperature; this dependence is a feature of orthodox Landau theory. The second line of Eq. (27) results from Eqs. (10) and (17), it enables the use of experimental values of anisotropic thermal expansion for the computation of the temperature dependence of coefficient c2(T) in the low temperature phase. The temperature T2 is the temperature at which the ferroelastic phase transition was complete; in the framework of Landau theory this temperature is considered an empirical value. It should be emphasized now that the temperature T1 is correlated with T2. Indeed, the differentiation of Eq. (27) gives: dc2 ct ¼ : ð28Þ dT T ¼T 1 T 1 T 2 The derivative on the left side of Eq. (28) can be estimated from the Clausius–Clapeyron relationship as:
dc2 2qm ; dT T ¼T 1 T 0 u20 ðT 2 Þ
ð29Þ
where q is the absolute value of the latent heat of the phase transition, m is the mass density and T0 is the phase transition temperature, which belongs to the temperature region T2 < T0 < T1 [17]. Therefore, the approximate relationship: T 1 T 2 ct u20 ðT 2 Þ : T2 2qm
ð30Þ
holds true in the case of (T1 T2)/T2 1, because in this case T0 T2. Possible ways of estimating the magnetoelastic parameter d are described in, for example, by Chernenko et al. [14,15]. 3. Theoretical description of the functional properties of shape memory alloys 3.1. Elastic properties Let Alloy 1 and Alloy 2 be representative ferromagnetic shape memory alloys with MT temperatures of 200 and 300 K, respectively. The temperature dependencies of lattice parameters of two martensitic Ni–Mn–Ga alloys with MT temperatures close to these values were measured by Ma et al. [18] and Glavatska [19], respectively. These measurements showed that the MT of both alloys was accompanied by a negative “spontaneous” deformation ezz = c/a 1. It turned out that the value of lattice parameter c for both alloys decreased greatly on cooling and, therefore, increased on heating. So, an abnormally large thermal expansion of the tetragonal lattice in the z direction took place. Alloys 1 and 2 were ferromagnets with Curie temperatures (376 and 368 K, respectively) exceeding their MT temperatures. It is of importance that for Alloy 1 the difference between the Curie temperature and MT temperature was substantially larger than for Alloy 2 (see Section 3.2). For theoretical modeling of the functional properties of these alloys their experimental spontaneous deformaðmeÞ tions cðT ; rðmeÞ zz Þ=aðT ; rzz Þ 1 and magnetization values [20,21] were approximated by polynomials in the temperature range T < T2. The characteristic temperatures T2 = 201 K and T2 = 278 K were accepted for Alloy 1 and Alloy 2, respectively. The coefficients a4 and b4 were evaluated then from Eq. (26) using the tentative value C0 (T1) = 1 GPa, which approximately corresponds to the abnormally low value of Young modulus E 3 GPa (the latter is close to the minimal values Emin (T) reported for a number of Ni–Mn–Ga alloys by Chernenko et al. [12]). The function c2(T) was determined from Eq. (27) and the function u3 that satisfy the equation c T ; rðmeÞ þ rzz 1 zz 1 u3 T ; rðmeÞ ¼ þ r zz zz ðmeÞ 2 a T ; rzz þ rzz
ð31Þ
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was computed for both alloys from Eqs. (16)–(19). The results of the computations are presented in Fig. 1. The functions computed for zero stress provide for agreement between the theoretical and experimental values of spontaneous tetragonal deformation. This agreement justifies the choice of T2 values mentioned above. The breaks in the dashed and dash-dotted curves corresponding to non-zero mechanical stress are caused by the restriction Eq. (20) of the range of applicability of the approximate result Eq. (18). The temperature dependencies of tetragonal deformations presented in Fig. 1 are semi-empirical, because the functions c2(T) were obtained for Alloys 1 and 2 using polynomial approximations of the experimental temperature dependencies of the lattice parameters. It should be emphasized now that the same values of latent heat q = 4.5 J g–1, mass density q = 8 g cm–3 and C0 (T1) = 1 GPa were applied for both alloys. Therefore, the drastic differences in their deformational behaviors, illustrated by the dashed and dash-dotted lines in Fig. 1, are caused only by the differences in the experimental values of the lattice parameters measured for unloaded specimens. It is seen, in particular, that axial stress deforms Alloy 2 substantially more than Alloy 1. The key role of anisotropic thermal expansion in the deformational characteristics of shape memory alloys was also confirmed by the computations of their shear moduli. The results of these computations are presented in Fig. 2. These results were obtained using the q, q and C0 (T1) values mentioned above. The differences in the temperature dependencies of the shear moduli shown in Fig. 2 were caused by the differences in thermal behavior of the lattice parameters of Alloys 1 and 2. There is a close interrelation between anisotropic thermal expansion (spontaneous deformation) of the crystal lattice and the computed values of shear elastic modulus from Eqs. (19), (22), (26), and (27), which explicitly involve the deformation values assigned by Eqs. (11), (13), and (17). In particular, Eq. (26) gives the following values for the coefficients of Landau expansion: a4 = 26 GPa, b4 = 509 GPa for Alloy 1; a4 = 11 GPa,
Fig. 1. Semi-empirical graphs of anisotropic thermal expansion of axially stressed alloys undergoing ferroelastic phase transition. The stress values 0 MPa (solid), 50 MPa (dashed) and 150 MPa (dash-dotted) were chosen for computations. Experimental values reported by Ma et al. [18] and Glavatska et al. [19] are shown by squares and circles, respectively.
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Fig. 2. The temperature dependencies of the shear elastic moduli computed using the experimental temperature dependencies of spontaneous deformation of Alloy 1 (solid line) and Alloy 2 (dash-dotted line). The lability temperatures are shown by vertical dashed lines. The horizontal dashed line shows the value C0 (T1) chosen for computations.
b4 = 92 GPa for Alloy 2. The values estimated for Alloy 2 were substantially smaller and, therefore, this alloy was softer than Alloy 1. That is why the elastic deformation of Alloy 2 exceeded that of Alloy 1 (see Fig. 1). For Alloy 1 the value C0 (T1) = 1 GPa, which was accepted for the computations, resulted in the following estimation of the sound velocity: st = (C0 /q)1/2 1.6 103 m s–1 when T T2 70 K. The same value was obtained by Shapiro et al. [22] from neutron scattering data for the martensitic phase of a Ni–Mn–Ga alloy with a forward MT temperature equal to 212 K. This is additional confirmation of the reasonability of the value C0 (T1) = 1 GPa accepted for the computations. It should be noted, however, that the results presented in Fig. 2 for Alloys 1 and 2 are illustrative and do not pretend to be quantitative descriptions of the temperature dependencies of the shear elastic moduli of the alloys, which were experimentally studied by Ma et al. [18], Glavatska et al. [19], Webster and Ziebeck [20] and Heczko and Straka [21]. Elevation of the C0 (T1) value results in a dramatic increase in the shear modulus over the whole temperature range of stability of the low temperature phase (see Fig. 3). The temperature dependencies of shear moduli presented in Fig. 3 were computed for different values of
Fig. 3. The temperature dependencies of the shear elastic moduli computed using the experimental temperature dependency of spontaneous deformation of Alloy 2 with different C0 (T1) values.
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C0 (T1) but the same value for lability temperature T2 = 278 K. The lability temperature T1, which corresponds to the abrupt drop in shear modulus on cooling of the alloy, is related to the parameter ct = C0 (T1)/3 by Eq. (30). Due to this, elevation of the value of C0 (T1) resulted in a noticeable increase in T1 and a widening of the temperature interval over which the two phases coexisted (see vertical dashed lines in Fig. 3). 3.2. Magnetic properties In this section the temperature dependencies of the magnetic anisotropy energy of ferromagnetic single crystals undergoing a ferroelastic phase transition will be modeled using the semi-empirical functions of anisotropic thermal expansion of two Ni–Mn–Ga alloys (see Fig. 1) and the appropriate experimental values of magnetization M(T). Substituting the expressions Eqs. (11) and (17) for spontaneous deformation into Eq. (6) and accounting for the condition m2x þ m2y ¼ 1 m2z , one can conclude that the energy density that originates from the phase transition is: F u ¼ K u ðT Þm2z ;
ð32Þ
where the value K u ðT Þ ¼ 6dM 2 ðT Þ½1 cðT Þ=aðT Þ;
ð33Þ
tion of the two factors from the total temperature dependence of MAED. The MAED values computed from Eq. (33) for the cubic-shaped specimens of Alloys 1 and 2 are presented in Fig. 4. The computations were carried out using the experimental values for the lattice parameters [18,19] and magnetization [20,21], the latter presented in Fig. 5. The bilateral arrows in Fig. 5 show that the temperature variation of magnetization in the low temperature phase was rather small for Alloy 1, with a low MT temperature, and very pronounced for Alloy 2, with a high MT temperature. As a consequence, the temperature dependence of the MAED of the alloy with a low MT temperature was mainly due to the temperature variation of the lattice parameters (see Fig. 4a), while the temperature dependence of the MAED of the alloy with a high MT temperature was almost equally due to the two factors mentioned above (see Fig. 4b). It might be expected that the temperature variation of magnetization was the main factor in the temperature dependence of MAED if the MT temperature is close to the Curie temperature of the alloy. The dependence of the MAED value on the shape of the specimen is a subject of practical importance. This dependence is illustrated in Fig. 6 for those cases where competition between the magnetocrystalline and magnetostatic contributions to MAED took place. Fig. 6 illustrates how
is usually called the magnetic anisotropy constant. In view of the sharp temperature dependence of this physical value it will be referred to below as the MAED. In Ni–Mn–Ga crystals with c/a < 1 the constant of magnetoelastic coupling is negative (d 23 [14]). The energy Eq. (33) aligns the magnetic vector with the z-axis of the experimental specimen if the latter has a cubic or spherical shape. The MAED of a thin platelet with phases perpendicular to the [001] direction and a bar (i.e. strongly elongated ellipsoid, prism or cylinder) with the axis perpendicular to this direction are approximately equal to: K pl ¼ K u þ 2pM 2 ðT Þ;
ð34Þ
and K r ¼ K u þ pM 2 ðT Þ;
ð35Þ
respectively, because the non-zero demagnetization coefficients are Dzz 4p for the platelet and Dxx Dyy 2p for the bar. As long as the term Ku is negative the absolute values of MAED for the platelet and bar will be smaller than |Ku| if the c-axis is perpendicular to the plane of the platelet or to the axis of the bar. This means that competition between the magnetocrystalline and magnetostatic contributions to MAED expressed by the first and second summands in the right sides of Eqs. (34) and (35) takes place. Eqs. (33)–(35) show that the magnetic anisotropy energy of the low temperature phase depends on temperature via anisotropic thermal expansion of the crystal lattice and via the temperature dependence of the magnetization value. The equations enable a separation of the contribu-
Fig. 4. The temperature dependencies of the magnetic anisotropy energy densities of cubic-shaped specimens of Alloy 1 (a) and Alloy 2 (b). The dashed lines illustrate the contributions of anisotropic thermal expansion to the total temperature dependencies of the magnetic anisotropy energy densities; the dot-dashed lines show the contributions of the temperature dependencies of the magnetization values.
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4. Summary and conclusions
Fig. 5. The experimental values of magnetization of Alloy 1 (squares) [20] and Alloy 2 (circles) [21] used for computation of the magnetic anisotropy energy densities of these alloys. The bilateral arrows show the total changes d1 and d2 in magnetization values in the low temperature phases of Alloys 1 and 2, respectively.
the magnetocrystalline contribution to MAED dominates for all specimens except the thin platelet cut of Alloy 1. However, the magnetostatic contribution substantially reduced the total MAED of all specimens. In some temperature ranges the MAED values of thin platelets and/or rods can be an order of magnitude smaller then the MAED of cubic-shaped specimens cut from the same alloy. Because of this, a right-angle realignment of the magnetization vector is possible in a low magnetic field. Moreover, spontaneous reorientation of the magnetic vector on cooling of the platelet cut from Alloy 1 is predicted by Fig. 6. On the one hand, this prediction needs additional verification, because the computations were performed using experimental values of the magnetization and lattice parameters reported by Webster and Ziebeck [20] and Ma et al. [18], respectively, for two different alloys with almost the same MT temperatures. On the other hand, the prediction looks realistic, because reorientation of the magnetization vector by a low magnetic field was realized experimentally in Glavatska et al. [19] and Heczko et al. [23].
Fig. 6. The temperature dependencies of the magnetic anisotropy energy densities of thin platelets (triangles), bars (squares) and cubic-shaped specimens (circles) of Alloy 1 (dashed lines) and Alloy 2 (solid lines). Crossing of the dash-dotted line by the straight dashed line illustrates the possibility of a reorientation of the magnetic vector on cooling of the thin platelet cut from Alloy 1 to a temperature of 100 K.
A consistent theoretical analysis has disclosed the key role of the temperature dependency of the lattice parameters in the behavior of the functional properties of alloys on cooling below the MT temperature. To illustrate this role the general mathematical expressions obtained in the framework of the Landau theory of phase transitions were used above for the description of the deformation, shear moduli and magnetic anisotropy energy densities of two representative alloys, referred to as Alloy 1 and Alloy 2. Two different temperature dependencies of tetragonal deformation (reported by Ma et al. [18] and Glavatska et al. [19], respectively) and magnetization value (obtained by Webster and Ziebeck [20] Heczko and Straka [21], respectively) were attributed to these alloys, while all other physical characteristics were considered the same for both of them. To emphasize the abnormal softness of the alloys, a low value of shear elastic modulus of the parent phase C0 (T1) = 1 GPa was attributed to the temperature T1 of stabilization of the martensitic phase. The computations carried out for the representative alloys showed the following. 1. The difference in the temperature dependencies of the lattice parameters of two alloys results in dramatic difference in the temperature variation of their shear elastic moduli – at the low temperature limit the shear modulus of Alloy 1 is 3.5 times as large as the shear modulus of Alloy 2 (see Fig. 2). 2. The difference in the values of the shear moduli results in a related difference in the elastic strains caused by axial stressing of the alloys (see Fig. 1). 3. Elevation of the characteristic value for the shear modulus C0 (T1) results in a proportional increase in this modulus over the whole temperature range of stability of the low temperature phase and in a widening of the twophase temperature range. If 1 GPa 6 C 0 ðT 1 Þ 6 3 GPa, Eq. (32) gives 2:4 K 6 T 1 T 2 6 7:2 K for Alloy 1 and 18:6 K 6 T 1 T 2 6 55:8 K for Alloy 2. For Alloy 1 the theoretical estimation agrees reasonably with the typical temperature ranges of two-phase states in shape memory alloys. For Alloy 2 Eq. (30) results in an obvious overestimation of the two-phase temperature range. Thus, a special study is needed to establish the correspondence between the lability temperatures T1 and T2 involved in phase transition theory and the characteristic temperatures of forward and reverse martensitic transformation measured in real experiments. 4. The difference in temperature dependencies of the lattice parameters of ferromagnetic shape memory alloys results in dramatic differences in the temperature variation of their magnetic anisotropy energy densities: the temperature dependence of the MAED value of the alloy with a low MT temperature is caused mainly by temperature variation of the lattice parameters (see Fig. 4a); the temperature dependence of the MAED value of the alloy whose MT temperature is close to
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room temperature is equally contributed to by temperature variation of the lattice parameters and magnetization value (see Fig. 4b). It may be concluded that temperature variation of the magnetization value is the main factor in the temperature dependence of MAED when the MT temperature is close to the Curie temperature of the alloy. 5. Due to the sharp temperature dependencies of the lattice parameters and magnetization value a right-angle realignment of the magnetization vector in a low magnetic field and spontaneous reorientation of the magnetic vector on cooling of shape memory alloys is possible, when competition between the magnetocrystalline and magnetostatic anisotropy energies takes place. It seems that in Ni–Mn–Ga alloys this situation can be realized at room temperature by proper selection of alloy composition (to realize the martensitic transformation above room temperature and provide for a moderate 1 c/a / Ku value at room temperature) and the shape of the specimen (to minimize the MAED value, expressed by Eqs. (34) and (35)).
Acknowledgements This work was supported by INTAS Project No. 061000024-9396. V.A.L. is grateful to the Academy of Finland for support of his stay at the Helsinki University of Technology.
References [1] Otsuka K, Ren X. Intermetallics 1999;7:511. [2] Hannula S-P, So¨derberg O, Ja¨msa¨ T, Lindroos VK. Adv Sci Technol 2006;49:109. [3] Delaey L. Diffusionless transformations. In: Kostorz G, editor. Phase transformations in materials. Weinheim: Wiley VCH; 1991. p. 339. [4] Wayman CM. Prog Mater Sci 1992;36:203. [5] Sittner P, Novak V, Landa M, Lukas P. Mater Sci Eng A 2007;462:12. [6] Kakeshita T, Fukuda T. Int J Appl Electromagn Mech 2006;23:45. [7] So¨derberg O, Aaltio I, Ge Y, Liu XW, Hannula S-P. Adv Sci Technol 2008;59:1. [8] Murray SJ, Marioni M, Allen SM, O’Handley RC, Lograsso TA. Appl Phys Lett 2000;77:886. [9] Likhachev AA, Sozinov A, Ullakko K. Mech Mater 2006;38:551. [10] Heczko O, Straka L, Hannula S-P. Mater Sci Eng A 2006;438:1003. [11] Dai L, Cullen J, Wuttig M. Appl Phys J 2004;95:6957. [12] Chernenko VA, Pons J, Seguı´ C, Cesari E. Acta Mater 2002;50:53. [13] Gomonaj EV, L’vov VA. Phase Transit 1996;56:43. [14] Chernenko VA, L’vov VA, Cesary E, Pons J, Portier R, Zagorodnyuk S. Mater Trans JIM 2002;43:856. [15] Chernenko VA, L’vov VA, Cesari E. JMMM 1999;196:859. [16] Kokorin VV, Wuttig M. JMMM 2001;234:25. [17] L’vov VA, Chernenko VA. Eur Phys J AP 1999;8:25. [18] Ma Y, Awaji S, Watanabe K, Matsumoto M, Kobayashi N. Solid State Commun 2000;113:671. [19] Glavatska N, Glavatskiy I, Mogilny G, Danilkin S, Hohlwain D, Soderberg O, et al. J Phys IV (France) 2003;112:963. [20] Webster PJ, Ziebeck KRA, Town SL, Peak MS. Phil Mag B 1984;49:295. [21] Heczko O, Straka L. Mater Sci Eng A 2004;378:394. [22] Shapiro SM, Vorderwisch P, Habicht K, Hradil K, Schneider H. EPL 2007;77:56004. [23] Heczko O, Soroka A, Hannula S-P. Appl Phys Lett 2008;93:022503.
Acta Materialia 57 (2009) 5605–5612 www.elsevier.com/locate/actamat
The role of anisotropic thermal expansion of shape memory alloys in their functional properties V.A. L’vov a,b,*, N. Glavatska c, I. Aaltio d, O. So¨derberg d, I. Glavatskyy c, S.-P. Hannula d a
Department of Radiophysics, Taras Shevchenko University, Volodymyrska Str. 64, 01601 Kyiv, Ukraine b Institute of Magnetism, Vernadsky Str. 36-b, 03142 Kyiv, Ukraine c Institute for Metal Physics, NAS of Ukraine, Vernadsky Str. 36, 03142, Kiev, Ukraine d Helsinki University of Technology, Department of Materials Science and Engineering, P.O. Box 6200, Espoo, FI-02015 TKK, Finland Received 12 January 2009; received in revised form 29 July 2009; accepted 29 July 2009 Available online 27 August 2009
Abstract The functional properties of a ferromagnetic shape memory alloy, which undergoes martensitic transformation of a cubic-tetragonal type, have been analyzed using the Landau theory of phase transitions. A key role of the temperature dependency of the crystal lattice parameters in the formation of the elastic and magnetic properties of the martensitic phase has been determined. To this end the temperature dependencies of the shear elastic moduli and magnetic anisotropy energy densities (MAED) of two quasi-stoichiometric Ni– Mn–Ga alloys have been computed from the experimental temperature dependencies of their lattice parameters. It turned out that the differences in the temperature dependencies of the lattice parameters of these alloys gave rise to drastic differences in their shear moduli and MAED as a function of temperature. The possibility of the spontaneous reorientation of the magnetization vector on cooling of the alloy and a right-angle realignment of this vector by a low magnetic field have been predicted. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Shape memory alloys; Martensitic phase transformation; Elastic behavior; Magnetic anisotropy
1. Introduction It is common knowledge that shape memory alloys are now widely used in engineering and medicine due to their responses to external stimuli, such as temperature variation or/and mechanical loading [1,2]. These responses are characterized by a several per cent deformation of the crystal lattice on cooling or axial stressing of the alloy [3–5]. The ferromagnetic shape memory alloys form a special class of functional materials, which exhibit high values of the magnetic anisotropy constant and magnetically induced deformation (see, [6,7], and references therein). Extreme *
Corresponding author. Address: Department of Radiophysics, Taras Shevchenko University, Volodymyrska Str. 64, 01601 Kyiv, Ukraine. Tel.: +380 44 452 01 34. E-mail addresses: [email protected] (V.A. L’vov), simo-pekka. hannula@tkk.fi (S.-P. Hannula).
values of deformation and magneto-deformation of shape memory alloys become attainable after a first-order phase transition of martensitic type, which is accompanied by a jump-like deformation of the crystal lattice and pronounced softening of the elastic moduli on cooling of single crystalline specimens. The temperature dependency of the lattice parameters and soft elastic moduli are among the main functional properties of shape memory alloys, which prescribe the peculiarities of their deformational behavior under mechanical load or external magnetic field. The response of these alloys to the application of a magnetic field is related not only to the physical values mentioned above, but also to the magnetic anisotropy constant [6–10]. Experimental studies have shown abnormally low values for the shear elastic moduli [11] and Young’s moduli [12] of some ferromagnetic shape memory alloys belonging to the Ni–Mn–Ga family. The study of abnormally soft ferromagnetic materials thereupon became topical. The
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.07.058
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experimentally observed softening of the elastic moduli was attributed to a cubic-tetragonal martensitic transformation (MT). The functional properties of soft martensitic alloys can be analyzed within the framework of the Landau theory of phase transitions, which correlates the elastic moduli [13] and the magnetic anisotropy energy density (MAED) [14] with the values of the lattice parameters in the tetragonal phase. The temperature dependence of the lattice parameters of martensite may turn out to be the main factor governing both the elastic and magnetic functional characteristics of alloy specimens. The present article seeks to verify this possibility. Moreover, the possibility of a reorientation of the magnetization vector on martensite cooling and a right-angle rotation of this vector by extremely low magnetic fields is verified by the quantitative computations.
is the energy of anisotropic magnetoelastic interactions. The coefficients c2, a4 and b4 in Eq. (4) are the linear combinations of the second-, third- and fourth-order elastic moduli (enumerated according to Gomonaj and L’vov [13]), D in Eq. (5) is the matrix formed by demagnetization coefficients, m = M(T)/M(T) is the unit magnetic vector, M is magnetization vector, H is the external magnetic field and d in Eq. (6) is the dimensionless magnetoelastic constant. A simple regrouping of the terms in Eqs. (3), (4), and (6) results in the expression: ðeff Þ ðeff Þ ðeff Þ ð7Þ G ¼ F m þ F e 3r1 u1 r2 u2 þ r3 u3 =6; where the values Þ rðeff ¼ ra þ rðmeÞ ; a a
2. The Landau theory for ferroelastic phase transition 2.1. Landau expansion of the Gibbs potential of an elastic ferromagnet The Gibbs potential of a single crystal of a ferromagnetic shape memory alloy undergoing cubic-tetragonal MT can be expressed in terms of the basic functions of irreducible representations of the cubic group: u1 ¼ ðexx þ eyy þ ezz Þ=3; pffiffiffi u3 ¼ 2ezz eyy exx ; u2 ¼ 3ðexx eyy Þ; ð1Þ r1 ¼ ðrxx þ ryy þ rzz Þ=3; pffiffiffi r3 ¼ 2rzz ryy rxx ; r2 ¼ 3ðrxx ryy Þ; composed of the strain and stress tensors eik and rik, respectively. The coordinate axes are aligned with the fourfold symmetry axes of the crystal. The functions u1 and r1 describe the isotropic deformation process and, therefore, can be omitted when the anisotropic effects are studied. In this case the expression for Gibbs potential is: G ¼ F ðr2 u2 r3 u3 Þ=6;
ð2Þ
where the Helmholtz free energy (F) is the sum of elastic (Fe), magnetic (Fm) and magnetoelastic (Fme) energies, i.e. F ¼ F e þ F m þ F me ;
ð3Þ
where 1 1 2 1 F e ¼ c2 u22 þ u23 þ a4 u3 u23 3u22 þ b4 u22 þ u23 ; 2 3 4
ð8Þ
can be interpreted as the internal stresses caused by magnetoelastic coupling [15]. 2.2. Ferroelastic phase transition in an axially stressed ferromagnetic single crystal The conditions oG/ou2 = oG/ou3 = 0 result in the equations: ðeff Þ
c2 u2 2a4 u2 u3 þ b4 u2 ðu22 þ u23 Þ r2 c 2 u3 þ
a4 ðu23
u22 Þ
þ
b4 u3 ðu22
þ
u23 Þ
=6 ¼ 0;
ðeff Þ r3 =6
ð9Þ
¼ 0:
ðeff Þ
ðeff Þ
In the case of zero effective stress ðr2 ¼ r3 ¼ 0Þ this equation system has solutions which correspond to the paramagnetic cubic phase and three equivalent variants of the paramagnetic tetragonal phase with fourfold symmetry axes oriented in the x, y or z directions. For certainty, let us consider the phase transition from the cubic phase to the z variant of the tetragonal phase. In this case the equality exx = eyy is valid in both phases. Therefore, u2 = 0 and the equation system reduces to one equation: u3 ðc2 þ a4 u3 þ b4 u23 Þ ¼ 0;
ð4Þ
ð10Þ
which has two solutions: u3 ¼ 0;
is the elastic energy, 1 F m ¼ M 2 ðT Þðm D mÞ mHMðT Þ; 2
pffiffiffi ðmeÞ ¼ dex M 2 ðT Þ=3; r2 ¼ 6 3dM 2 ðT Þ m2x m2y ; ðmeÞ r3 ¼ 6dM 2 ðT Þ 2m2z m2y m2x ðmeÞ
r1
ð5Þ
is the anisotropic part of the magnetic energy of the crystal and hpffiffiffi i F me ¼ dM 2 ðT Þ 3 m2x m2y u2 þ 2m2z m2y m2x u3 ; ð6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u3 ¼ ða4 =2b4 Þ 1 þ 1 c2 ðT Þ=ct u0 ðT Þ
ð11Þ
where ct ¼ a24 =4b4 > 0:
ð12Þ
The first solution in Eq. (11) corresponds to the cubic phase and the second one describes the “spontaneous” deformation of the cubic lattice in the course of a first-
V.A. L’vov et al. / Acta Materialia 57 (2009) 5605–5612
order phase transition. The tendency to volume conservation is observed for cubic-tetragonal phase transitions in shape memory alloys. In this case: u0 3ezz ¼ 3ðc aÞ=a0 2ðc=a 1Þ;
ð13Þ
where a, c and a0 are the lattice parameters of the tetragonal and cubic phases, respectively. According to the Landau theory the coefficient c2 is an increasing function of temperature. The cubic phase is stable when c2(T) > 0 and the tetragonal one is stable if c2(T) < ct. In the temperature range bounded by the inequalities 0 < c2(T) < ct both the cubic and tetragonal phases can be observed. Therefore, the lability temperatures of the tetragonal and cubic phases (T1and T2, respectively) satisfy the equations c2(T1) = ct and c2(T2) = 0. It should be emphasized that when the high temperature phase is ferromagnetic its symmetry is not exactly cubic, ðmeÞ ðmeÞ because the stresses r2 and=or r3 are not equal to zero in the ferromagnetic state. In the other words, the process of ferromagnetic ordering always reduces the symmetry of the cubic crystal lattice. It is worth explaining this statement in more detail. Let the equilibrium direction of the magnetic vector be parallel to the [001] direction of the crystal lattice. In this case the lattice is axially stressed by magnetoelastic stress: rðmeÞ zz ðT Þ
¼
ðmeÞ r3 ðT Þ=2
2
¼ 6dM ðT Þ:
ð14Þ
This stress causes elastic deformation of the crystal, which is commonly known as spontaneous magnetostriction. Magnetostriction results in deformation of the cubic lattice and a reduction in its symmetry from cubic to tetragonal and, therefore, the ferroelastic phase transition of a ferromagnetic single crystal is, rigorously speaking, an isomorphic (tetragonal–tetragonal) first-order phase transition. For the chosen direction of magnetic vector the spontaneous magnetostriction is of the order of dM2/C0 . For the abnormally soft Ni–Mn–Ga alloys the values C0 1 GPa and dM21MPa were reported [11,12,15]. Therefore, magnetostriction of these alloys may be of the order of 0.1%, even in the high temperature phase. This value of magnetostriction of the high temperature phase was measured by Kokorin and Wuttig [16] on approaching the MT temperature range. The crystal lattice with 1 c/ a 0.1% cannot be considered cubic in precise experiments and rigorous theories. Magnetoelastic stress promotes the appearance of the z variant and retards formation of the x and y variants of the are like-sign vallow temperature phase when u0 and rðmeÞ zz ues. Another factor which assists transition of the parent phase to the z variant of the low temperature one is the mechanical stress rzz, if it is identical in sign to u0. The ðeff Þ effective stress r3 ðT ; rzz Þ ¼ 2½rðmeÞ zz ðT Þ þ rzz is a superimposition of mechanically and magnetically induced stresses. Let us consider the situation rxx = ryy = 0. In this case ðeff Þ r2 ¼ 0 and the first equation in the equation system Eq. (9) are satisfied by the value u2 = 0. Thus, the isomor-
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phic phase transition is described by the second equation, which is reduced to the form: ðeff Þ ð15Þ u3 c2 ðT Þ þ a4 u3 þ b4 u23 r3 ðT ; rzz Þ=6 ¼ 0: This equation explicitly shows absence of solution u3 = 0, which corresponds to the cubic phase. The solution of Eq. (15) can be searched in the form: ð0Þ
u3 ðT ; rzz Þ ¼ u3 ðT Þ þ ~uðT ; rzz Þ;
ð16Þ
where ~uðT ; rzz Þ is the linear combination of elastic strain tensor components: u0 ðT Þ; if T < T 1 ; ð0Þ ð17Þ u3 ðT Þ ¼ 0 otherwise: Under the actual experimental conditions the inequality j~uðT ; rzz Þj ju0 ðT Þj holds true and so substitution of Eq. (16) into Eq. (15) shows that Eq. (15) has the approximate solution: h i1=2
ðeff Þ ~uðT ;rÞ¼A1 ðT Þ C pm ðT Þ C 2pm ðT ÞþAðT Þr3 ðT ;rÞ ; ð18Þ where h i2 ð0Þ ð0Þ C pm ðT Þ ¼ 3c2 ðT Þ þ 6a4 u3 ðT Þ þ 9b4 u3 ðT Þ ; h i ð0Þ AðT Þ ¼ 6 a4 þ 3b4 u3 ðT Þ :
ð19Þ
The function Cpm(T) describes the temperature dependence of the shear modulus in the paramagnetic state. The result, Eq. (18), is valid if ðeff Þ ð20Þ C 2pm ðT Þ > AðT Þ r3 ðT ; rÞ : The ferromagnetic lattice is deformed due to magnetoelastic interaction. For this reason its shear modulus, defined as: 1
C fm ¼ ðdu3 =drÞ jr¼0 ;
ð21Þ
is slightly different from the shear modulus of the paramagnetic lattice. Eqs. (16), (18), and (21) result in the relationship: h i1=2 ð22Þ C fm ðT Þ ¼ C 2pm ðT Þ þ 12dM 2 ðT ÞAðT Þ An elastic strain caused by an external force applied to a single crystalline specimen in the z direction can be determined from the expression: 1 uðT ; rÞ ~uðT ; 0Þ: ð23Þ eðelÞ zz ðT ; rÞ ¼ ½~ 3 The temperature-dependent total (spontaneous and elastic) deformation of a compressed alloy can be expressed as: 1 ð0Þ ð24Þ ezz ðT ; rÞ ¼ u3 ðT Þ þ eðelÞ zz ðT ; rÞ: 3 Eqs. (18), (23), and (24) are applicable to both ferromagnetic and paramagnetic alloys. These equations may be used for computations if the numerical values of the coef-
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ficients of the Landau expansion for Gibbs potential are known. In addition, for ferromagnetic alloys a knowledge of the magnetization function M(T) is needed. 2.3. Evaluation of the Landau expansion coefficients and lability temperatures According to Eq. (11) and the definition of the lability temperatures, the spontaneous tetragonal deformation satisfies the following condition: u0 ðT 2 Þ ¼ 2u0 ðT 1 Þ ¼ a4 =b4 :
ð25Þ
Eqs. (12), (13), and (25) result in the relationships: 2c2 ðT 1 Þ ; 1 cðT 2 Þ=aðT 2 Þ c2 ðT 1 Þ b4 ¼ 4c2 ðT 1 Þ=u20 ðT 2 Þ ¼ : 2 ½1 cðT 2 Þ=aðT 2 Þ a4 ¼ 4c2 ðT 1 Þ=u0 ðT 2 Þ ¼
ð26Þ
These relationships enable estimation of the third-order and fourth-order terms in the Landau expansion for elastic energy from the values of lattice parameters at the lability point of the low temperature phase (T1) and the shear elastic modulus of the parent phase C0 (T1) = 3c2(T1) = 3ct just before the beginning of the phase transformation. The lattice parameters are available from X-ray diffraction data and the shear modulus of the parent phase can be determined by dynamic mechanical analysis (DMA) or ultrasonic methods. After evaluation of parameters the a4 and b4 the temperature-dependent coefficient c2(T) can be found from the expression: 8 T T 2 if T 2 < T < T ; < ct T 1 T 2 h i2 ð27Þ c2 ðT Þ ¼ : a4 uð0Þ b4 uð0Þ if T 6 T 2 ; 3 3 where the temperature T > T1 must be close enough to T1. The first line of Eq. (27) provides for the linear dependence of the coefficient c2 on temperature in the vicinity of the phase transition temperature; this dependence is a feature of orthodox Landau theory. The second line of Eq. (27) results from Eqs. (10) and (17), it enables the use of experimental values of anisotropic thermal expansion for the computation of the temperature dependence of coefficient c2(T) in the low temperature phase. The temperature T2 is the temperature at which the ferroelastic phase transition was complete; in the framework of Landau theory this temperature is considered an empirical value. It should be emphasized now that the temperature T1 is correlated with T2. Indeed, the differentiation of Eq. (27) gives: dc2 ct ¼ : ð28Þ dT T ¼T 1 T 1 T 2 The derivative on the left side of Eq. (28) can be estimated from the Clausius–Clapeyron relationship as:
dc2 2qm ; dT T ¼T 1 T 0 u20 ðT 2 Þ
ð29Þ
where q is the absolute value of the latent heat of the phase transition, m is the mass density and T0 is the phase transition temperature, which belongs to the temperature region T2 < T0 < T1 [17]. Therefore, the approximate relationship: T 1 T 2 ct u20 ðT 2 Þ : T2 2qm
ð30Þ
holds true in the case of (T1 T2)/T2 1, because in this case T0 T2. Possible ways of estimating the magnetoelastic parameter d are described in, for example, by Chernenko et al. [14,15]. 3. Theoretical description of the functional properties of shape memory alloys 3.1. Elastic properties Let Alloy 1 and Alloy 2 be representative ferromagnetic shape memory alloys with MT temperatures of 200 and 300 K, respectively. The temperature dependencies of lattice parameters of two martensitic Ni–Mn–Ga alloys with MT temperatures close to these values were measured by Ma et al. [18] and Glavatska [19], respectively. These measurements showed that the MT of both alloys was accompanied by a negative “spontaneous” deformation ezz = c/a 1. It turned out that the value of lattice parameter c for both alloys decreased greatly on cooling and, therefore, increased on heating. So, an abnormally large thermal expansion of the tetragonal lattice in the z direction took place. Alloys 1 and 2 were ferromagnets with Curie temperatures (376 and 368 K, respectively) exceeding their MT temperatures. It is of importance that for Alloy 1 the difference between the Curie temperature and MT temperature was substantially larger than for Alloy 2 (see Section 3.2). For theoretical modeling of the functional properties of these alloys their experimental spontaneous deformaðmeÞ tions cðT ; rðmeÞ zz Þ=aðT ; rzz Þ 1 and magnetization values [20,21] were approximated by polynomials in the temperature range T < T2. The characteristic temperatures T2 = 201 K and T2 = 278 K were accepted for Alloy 1 and Alloy 2, respectively. The coefficients a4 and b4 were evaluated then from Eq. (26) using the tentative value C0 (T1) = 1 GPa, which approximately corresponds to the abnormally low value of Young modulus E 3 GPa (the latter is close to the minimal values Emin (T) reported for a number of Ni–Mn–Ga alloys by Chernenko et al. [12]). The function c2(T) was determined from Eq. (27) and the function u3 that satisfy the equation c T ; rðmeÞ þ rzz 1 zz 1 u3 T ; rðmeÞ ¼ þ r zz zz ðmeÞ 2 a T ; rzz þ rzz
ð31Þ
V.A. L’vov et al. / Acta Materialia 57 (2009) 5605–5612
was computed for both alloys from Eqs. (16)–(19). The results of the computations are presented in Fig. 1. The functions computed for zero stress provide for agreement between the theoretical and experimental values of spontaneous tetragonal deformation. This agreement justifies the choice of T2 values mentioned above. The breaks in the dashed and dash-dotted curves corresponding to non-zero mechanical stress are caused by the restriction Eq. (20) of the range of applicability of the approximate result Eq. (18). The temperature dependencies of tetragonal deformations presented in Fig. 1 are semi-empirical, because the functions c2(T) were obtained for Alloys 1 and 2 using polynomial approximations of the experimental temperature dependencies of the lattice parameters. It should be emphasized now that the same values of latent heat q = 4.5 J g–1, mass density q = 8 g cm–3 and C0 (T1) = 1 GPa were applied for both alloys. Therefore, the drastic differences in their deformational behaviors, illustrated by the dashed and dash-dotted lines in Fig. 1, are caused only by the differences in the experimental values of the lattice parameters measured for unloaded specimens. It is seen, in particular, that axial stress deforms Alloy 2 substantially more than Alloy 1. The key role of anisotropic thermal expansion in the deformational characteristics of shape memory alloys was also confirmed by the computations of their shear moduli. The results of these computations are presented in Fig. 2. These results were obtained using the q, q and C0 (T1) values mentioned above. The differences in the temperature dependencies of the shear moduli shown in Fig. 2 were caused by the differences in thermal behavior of the lattice parameters of Alloys 1 and 2. There is a close interrelation between anisotropic thermal expansion (spontaneous deformation) of the crystal lattice and the computed values of shear elastic modulus from Eqs. (19), (22), (26), and (27), which explicitly involve the deformation values assigned by Eqs. (11), (13), and (17). In particular, Eq. (26) gives the following values for the coefficients of Landau expansion: a4 = 26 GPa, b4 = 509 GPa for Alloy 1; a4 = 11 GPa,
Fig. 1. Semi-empirical graphs of anisotropic thermal expansion of axially stressed alloys undergoing ferroelastic phase transition. The stress values 0 MPa (solid), 50 MPa (dashed) and 150 MPa (dash-dotted) were chosen for computations. Experimental values reported by Ma et al. [18] and Glavatska et al. [19] are shown by squares and circles, respectively.
5609
Fig. 2. The temperature dependencies of the shear elastic moduli computed using the experimental temperature dependencies of spontaneous deformation of Alloy 1 (solid line) and Alloy 2 (dash-dotted line). The lability temperatures are shown by vertical dashed lines. The horizontal dashed line shows the value C0 (T1) chosen for computations.
b4 = 92 GPa for Alloy 2. The values estimated for Alloy 2 were substantially smaller and, therefore, this alloy was softer than Alloy 1. That is why the elastic deformation of Alloy 2 exceeded that of Alloy 1 (see Fig. 1). For Alloy 1 the value C0 (T1) = 1 GPa, which was accepted for the computations, resulted in the following estimation of the sound velocity: st = (C0 /q)1/2 1.6 103 m s–1 when T T2 70 K. The same value was obtained by Shapiro et al. [22] from neutron scattering data for the martensitic phase of a Ni–Mn–Ga alloy with a forward MT temperature equal to 212 K. This is additional confirmation of the reasonability of the value C0 (T1) = 1 GPa accepted for the computations. It should be noted, however, that the results presented in Fig. 2 for Alloys 1 and 2 are illustrative and do not pretend to be quantitative descriptions of the temperature dependencies of the shear elastic moduli of the alloys, which were experimentally studied by Ma et al. [18], Glavatska et al. [19], Webster and Ziebeck [20] and Heczko and Straka [21]. Elevation of the C0 (T1) value results in a dramatic increase in the shear modulus over the whole temperature range of stability of the low temperature phase (see Fig. 3). The temperature dependencies of shear moduli presented in Fig. 3 were computed for different values of
Fig. 3. The temperature dependencies of the shear elastic moduli computed using the experimental temperature dependency of spontaneous deformation of Alloy 2 with different C0 (T1) values.
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C0 (T1) but the same value for lability temperature T2 = 278 K. The lability temperature T1, which corresponds to the abrupt drop in shear modulus on cooling of the alloy, is related to the parameter ct = C0 (T1)/3 by Eq. (30). Due to this, elevation of the value of C0 (T1) resulted in a noticeable increase in T1 and a widening of the temperature interval over which the two phases coexisted (see vertical dashed lines in Fig. 3). 3.2. Magnetic properties In this section the temperature dependencies of the magnetic anisotropy energy of ferromagnetic single crystals undergoing a ferroelastic phase transition will be modeled using the semi-empirical functions of anisotropic thermal expansion of two Ni–Mn–Ga alloys (see Fig. 1) and the appropriate experimental values of magnetization M(T). Substituting the expressions Eqs. (11) and (17) for spontaneous deformation into Eq. (6) and accounting for the condition m2x þ m2y ¼ 1 m2z , one can conclude that the energy density that originates from the phase transition is: F u ¼ K u ðT Þm2z ;
ð32Þ
where the value K u ðT Þ ¼ 6dM 2 ðT Þ½1 cðT Þ=aðT Þ;
ð33Þ
tion of the two factors from the total temperature dependence of MAED. The MAED values computed from Eq. (33) for the cubic-shaped specimens of Alloys 1 and 2 are presented in Fig. 4. The computations were carried out using the experimental values for the lattice parameters [18,19] and magnetization [20,21], the latter presented in Fig. 5. The bilateral arrows in Fig. 5 show that the temperature variation of magnetization in the low temperature phase was rather small for Alloy 1, with a low MT temperature, and very pronounced for Alloy 2, with a high MT temperature. As a consequence, the temperature dependence of the MAED of the alloy with a low MT temperature was mainly due to the temperature variation of the lattice parameters (see Fig. 4a), while the temperature dependence of the MAED of the alloy with a high MT temperature was almost equally due to the two factors mentioned above (see Fig. 4b). It might be expected that the temperature variation of magnetization was the main factor in the temperature dependence of MAED if the MT temperature is close to the Curie temperature of the alloy. The dependence of the MAED value on the shape of the specimen is a subject of practical importance. This dependence is illustrated in Fig. 6 for those cases where competition between the magnetocrystalline and magnetostatic contributions to MAED took place. Fig. 6 illustrates how
is usually called the magnetic anisotropy constant. In view of the sharp temperature dependence of this physical value it will be referred to below as the MAED. In Ni–Mn–Ga crystals with c/a < 1 the constant of magnetoelastic coupling is negative (d 23 [14]). The energy Eq. (33) aligns the magnetic vector with the z-axis of the experimental specimen if the latter has a cubic or spherical shape. The MAED of a thin platelet with phases perpendicular to the [001] direction and a bar (i.e. strongly elongated ellipsoid, prism or cylinder) with the axis perpendicular to this direction are approximately equal to: K pl ¼ K u þ 2pM 2 ðT Þ;
ð34Þ
and K r ¼ K u þ pM 2 ðT Þ;
ð35Þ
respectively, because the non-zero demagnetization coefficients are Dzz 4p for the platelet and Dxx Dyy 2p for the bar. As long as the term Ku is negative the absolute values of MAED for the platelet and bar will be smaller than |Ku| if the c-axis is perpendicular to the plane of the platelet or to the axis of the bar. This means that competition between the magnetocrystalline and magnetostatic contributions to MAED expressed by the first and second summands in the right sides of Eqs. (34) and (35) takes place. Eqs. (33)–(35) show that the magnetic anisotropy energy of the low temperature phase depends on temperature via anisotropic thermal expansion of the crystal lattice and via the temperature dependence of the magnetization value. The equations enable a separation of the contribu-
Fig. 4. The temperature dependencies of the magnetic anisotropy energy densities of cubic-shaped specimens of Alloy 1 (a) and Alloy 2 (b). The dashed lines illustrate the contributions of anisotropic thermal expansion to the total temperature dependencies of the magnetic anisotropy energy densities; the dot-dashed lines show the contributions of the temperature dependencies of the magnetization values.
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4. Summary and conclusions
Fig. 5. The experimental values of magnetization of Alloy 1 (squares) [20] and Alloy 2 (circles) [21] used for computation of the magnetic anisotropy energy densities of these alloys. The bilateral arrows show the total changes d1 and d2 in magnetization values in the low temperature phases of Alloys 1 and 2, respectively.
the magnetocrystalline contribution to MAED dominates for all specimens except the thin platelet cut of Alloy 1. However, the magnetostatic contribution substantially reduced the total MAED of all specimens. In some temperature ranges the MAED values of thin platelets and/or rods can be an order of magnitude smaller then the MAED of cubic-shaped specimens cut from the same alloy. Because of this, a right-angle realignment of the magnetization vector is possible in a low magnetic field. Moreover, spontaneous reorientation of the magnetic vector on cooling of the platelet cut from Alloy 1 is predicted by Fig. 6. On the one hand, this prediction needs additional verification, because the computations were performed using experimental values of the magnetization and lattice parameters reported by Webster and Ziebeck [20] and Ma et al. [18], respectively, for two different alloys with almost the same MT temperatures. On the other hand, the prediction looks realistic, because reorientation of the magnetization vector by a low magnetic field was realized experimentally in Glavatska et al. [19] and Heczko et al. [23].
Fig. 6. The temperature dependencies of the magnetic anisotropy energy densities of thin platelets (triangles), bars (squares) and cubic-shaped specimens (circles) of Alloy 1 (dashed lines) and Alloy 2 (solid lines). Crossing of the dash-dotted line by the straight dashed line illustrates the possibility of a reorientation of the magnetic vector on cooling of the thin platelet cut from Alloy 1 to a temperature of 100 K.
A consistent theoretical analysis has disclosed the key role of the temperature dependency of the lattice parameters in the behavior of the functional properties of alloys on cooling below the MT temperature. To illustrate this role the general mathematical expressions obtained in the framework of the Landau theory of phase transitions were used above for the description of the deformation, shear moduli and magnetic anisotropy energy densities of two representative alloys, referred to as Alloy 1 and Alloy 2. Two different temperature dependencies of tetragonal deformation (reported by Ma et al. [18] and Glavatska et al. [19], respectively) and magnetization value (obtained by Webster and Ziebeck [20] Heczko and Straka [21], respectively) were attributed to these alloys, while all other physical characteristics were considered the same for both of them. To emphasize the abnormal softness of the alloys, a low value of shear elastic modulus of the parent phase C0 (T1) = 1 GPa was attributed to the temperature T1 of stabilization of the martensitic phase. The computations carried out for the representative alloys showed the following. 1. The difference in the temperature dependencies of the lattice parameters of two alloys results in dramatic difference in the temperature variation of their shear elastic moduli – at the low temperature limit the shear modulus of Alloy 1 is 3.5 times as large as the shear modulus of Alloy 2 (see Fig. 2). 2. The difference in the values of the shear moduli results in a related difference in the elastic strains caused by axial stressing of the alloys (see Fig. 1). 3. Elevation of the characteristic value for the shear modulus C0 (T1) results in a proportional increase in this modulus over the whole temperature range of stability of the low temperature phase and in a widening of the twophase temperature range. If 1 GPa 6 C 0 ðT 1 Þ 6 3 GPa, Eq. (32) gives 2:4 K 6 T 1 T 2 6 7:2 K for Alloy 1 and 18:6 K 6 T 1 T 2 6 55:8 K for Alloy 2. For Alloy 1 the theoretical estimation agrees reasonably with the typical temperature ranges of two-phase states in shape memory alloys. For Alloy 2 Eq. (30) results in an obvious overestimation of the two-phase temperature range. Thus, a special study is needed to establish the correspondence between the lability temperatures T1 and T2 involved in phase transition theory and the characteristic temperatures of forward and reverse martensitic transformation measured in real experiments. 4. The difference in temperature dependencies of the lattice parameters of ferromagnetic shape memory alloys results in dramatic differences in the temperature variation of their magnetic anisotropy energy densities: the temperature dependence of the MAED value of the alloy with a low MT temperature is caused mainly by temperature variation of the lattice parameters (see Fig. 4a); the temperature dependence of the MAED value of the alloy whose MT temperature is close to
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room temperature is equally contributed to by temperature variation of the lattice parameters and magnetization value (see Fig. 4b). It may be concluded that temperature variation of the magnetization value is the main factor in the temperature dependence of MAED when the MT temperature is close to the Curie temperature of the alloy. 5. Due to the sharp temperature dependencies of the lattice parameters and magnetization value a right-angle realignment of the magnetization vector in a low magnetic field and spontaneous reorientation of the magnetic vector on cooling of shape memory alloys is possible, when competition between the magnetocrystalline and magnetostatic anisotropy energies takes place. It seems that in Ni–Mn–Ga alloys this situation can be realized at room temperature by proper selection of alloy composition (to realize the martensitic transformation above room temperature and provide for a moderate 1 c/a / Ku value at room temperature) and the shape of the specimen (to minimize the MAED value, expressed by Eqs. (34) and (35)).
Acknowledgements This work was supported by INTAS Project No. 061000024-9396. V.A.L. is grateful to the Academy of Finland for support of his stay at the Helsinki University of Technology.
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