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Martensitic transformation in constrained films
PII:
Acta mater. Vol. 46, No. 14, pp. 5095±5107, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00165-7 1359-6454/98 $19.00 + 0.00
MARTENSITIC TRANSFORMATION IN CONSTRAINED FILMS A. L. ROYTBURD, T. S. KIM, QUANMIN SU, J. SLUTSKER and M. WUTTIG{ Department of Materials Science and Engineering, University of Maryland, College Park, MD 20742-2115, U.S.A. (Received 29 December 1997; accepted 18 April 1998) AbstractÐA thermodynamic analysis is presented for the martensitic transformation in a constrained ®lm upon cooling and heating. It is shown that this transformation proceeds with a variable self-strain corresponding to a variable polydomain structure of the martensite phase. Even the equilibrium microstructure develops irreversibly with changing temperature, i.e. the microstructure evolution paths for the direct and the reverse transformations are dierent. During the reverse transformation the stressed austenite, incompatible to martensite, is formed. The theory predicts a considerable shift of the temperature interval of transformation and its broadening due to the constraint. Experimental studies of the stress evolution with changing temperature in NiTi polycrystalline ®lms on Si substrates support the principal thermodynamic conclusions. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Phase transformations under constraint take place in all multiphase materials where a transforming phase is in contact with nontransforming phases. Interest in the problem of constrained transformations has increased recently due to growing applications of arti®cial materials and composites with transforming components, in bulk as well as in thin-®lm form. The martensitic transformation in constrained particles embedded in a nontransformed matrix has been widely studied and used as an eective mechanism of toughening of ceramics. Another area of applications of constrained martensitic transformations is smart composites with shape memory alloys as the active components [1±3]. Layer composites containing transforming shape memory material are particularly interesting for studies of constrained martensitic transformations and for applications as well. There are two typical con®gurations of composites with transforming constrained ®lms: symmetrical and asymmetrical. The ®rst con®guration consists of two identical ®lms on both sides of a substrate (``trimorph'') [Fig. 1(a)]; the second one is a ®lm on a substrate (``bimorph'') [Fig. 1(b)]. The trimorph can be considered as an elementary cell of multilayer composites or heterostructures. Bimorphs are often used as actuators or sensors, because the transformation in the ®lm results in considerable bending. For the same reason the bimorphs are usually used for studies of martensitic transformation itself. {To whom correspondence should be addressed.
Several studies of the martensitic transformation in NiTi ®lms on Si substrates have been performed recently [4, 5]. In this paper we present experimental results on the stress evolution in bimorphs and trimorphs consisting of NiTi polycrystalline ®lms on Si substrates with SiO2 buer layers. We discuss the principal results of experimental studies from the point of view of the thermodynamics of a martensitic transformation in a constrained ®lm. The high level of stress in a ®lm due to its mis®t with a substrate suggests that they can compete with internal stresses in the martensite or martensite±austenite microstructure and considerably change the self-accommodation of the martensitic transformation in shape memory alloys. Therefore, thermodynamic eects in constrained ®lms should be much more pronounced than in bulk materials. In Section 2 of this paper the mechanics of a constrained ®lm is considered. The dierent physical factors which determine the ®lm±substrate mis®t are discussed and formulas for the calculation of the average stress in anisotropic and isotropic ®lms are presented. The thermodynamics of the transformation in a single-crystalline constrained ®lm is discussed in Section 3. The analysis of a cubic±tetragonal model transformation predicts special eects of the constraint, namely: 1. a structural irreversibility of the evolution of the equilibrium microstructure; and 2. the appearance of an elastic incompatibility between the austenite and the martensite because the interfaces between these phases are not necessarily invariant planes.
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Fig. 1. Self-strain and deformation of a composite containing a transforming constrained ®lm. Trimorph (a) and bimorph (b) con®gurations of the composite are shown.
Experimental results on the preparation of polycrystalline constrained NiTi ®lms and their transformation upon cooling and heating are presented in Section 3. These results, together with other published data, are discussed in Section 4 where it will be seen that they support the thermodynamical concepts presented in Section 3 of this paper. 2. MECHANICS OF CONSTRAINED TRANSFORMATIONS
A phase transformation in a constrained ®lm is a self-regulated process. The transformation changes
the state of stress of the ®lm and the changing stress, in turn, aects the transformation. We consider both sides of the process from the point of view of the observed stress evolution in the ®lm as a function of temperature. A typical temperature dependence of the average stress in a constrained SMA ®lm is presented in Fig. 2. The two sections of the almost linearly increasing stress with decreasing temperature correspond to the dierence of the thermal expansions of the ®lm and substrate. The section in which the stress decreases with temperature re¯ects the change of the martensite fraction during the transformation. The curve corresponds to the transformation during cooling and the temperature Ms marks the beginning of the transformation of austenite to martensite. The temperature Mf indicates the point where the A 4 M transformation is complete. Stress in a ®lm/substrate composite arises due to a mis®t between the two components along their interface. Since the mis®t is in general the result of volume deformations, it is convenient to introduce a three-dimensional mis®t self-strain for the calculation of the state of stress of the composite [6±8]. This mis®t self-strain can be determined as follows. Compare the state of strain of the composite at a temperature, T, with a reference stress-free state at a temperature, TR. (Fig. 2). If the ®lm and the substrate are separated at temperature T, then the comparison with the reference state determines their
Fig. 2. Stress±temperature curve for a transforming constrained ®lm. The evolution of the ®lm microstructure and mis®t is shown.
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self-strains, i.e. the strains which the ®lm and the substrate would have without mutual constraint. If T < TR the self-strain e^ S of the substrate is a result of the thermal contraction: ÿ e^ S a^ S T ÿ TR 1 where a^ S is the tensor of thermal expansion of the substrate. The self-strain of the ®lm is determined by a similar relation if the ®lm is austenitic at the reference temperature. Naturally, if the ®lm transforms, its self-strain should include the transformation selfstrain, e^ t a: ÿ 2 e^ F a^ F T ÿ TR e^ t a where a^ F is a thermal expansion tensor of the ®lm and a is the fraction of martensite. Here, it is assumed that 1. all strains are small so their linear superposition is valid; 2. the coecients of thermal expansion for the martensite and the austenite are equal (^aM a^ A a^ F ); and 3. the temperature and the martensitic microstructure are uniform across the ®lm. The mis®t self-strain is determined by the dierence between the self-strains of the ®lm and the substrate: ÿ e^ e^ F ÿ e^ S e^ T e^ t a; e^ T y^ T ÿ TR 3 where e^ T is the thermal mis®t, y^ a^ F ÿ a^ S . It is assumed above that there is no stress relaxation due to plastic deformation or the changes of the adhesion while varying the temperature. For epitaxial ®lms it means that there is no movement or/and generation of mis®t dislocations during the evolution of the stress as the temperature changes. However, the ®lm/substrate interface may contain immobile interface dislocations which had been formed at/or near the deposition temperature. In this case the reference temperature TR corresponding to the relaxed state will be close to the deposition temperature. In the general case TR depends on the conditions of deposition, epitaxial mis®t and degree of relaxation. Knowledge of the mis®t self-strain, as well as geometric parameters of the composite and the elastic properties of its components, permits the calculation of the stresses, the elastic strains and the composite displacement. If the thickness of the composite is much smaller than its other dimensions, the calculations reduce to a one-dimensional {The following notations are used in thisÿ paper: for the second rank tensors of strain and stress, e^ eik ; i,k 1,2,3 ^ for the fourth rank tensors of elastic moduli, and s; C (Cikjl) and G; for a vector of normal, 4 n s^ ^ e ) sij ; e^ G^e ) Gijkl eij ekl
5097
problem. For a ®lm on an absolutely rigid substrate the stress is{ n^e s^ F ÿG
4
ÿ1
where G n C ÿ C n nC n nC is a planar elastic modulus, C is the elastic modulus tensor of the ®lm, and n is the crystallographic direction of the ®lm normal [9, 10]. If the principal strains of the mis®t strain e^ in the plane of the ®lm are equal, the ®lm has cubic anisotropy and n 100, then sF11 sF22 sF ÿGe G C11 C12 ÿ
2C 212 C11
5 6
where e is the in-plane ®lm strain. In the isotropic case G
E 1ÿ
7
where the quantities E and n represent Young's modulus and Poisson's ratio, respectively. Hereafter, only thin-®lm composites with h/ H < 0.01 are considered (see Fig. 1). In this case the change of the stress across the ®lm of the bimorph is negligible as well as the dierence between the states of stress of the symmetrical and unsymmetrical con®gurations. In both cases the stress is approximately determined by equation (4). The stressed ®lm of a bimorph causes bending of the substrate. Its mean radius of curvature is determined by the Stoney's equation [11]: R
H2 sF 6hGS
8
In equation (8) the quantities GS is the plane modulus of the substrate. An in-plane anisotropy of the stressed ®lm on the substrate results in the appearance of the two principal curvatures re¯ecting the principal stresses in the bimorph. The radius R as a function of temperature is one of the basic characteristics of a bimorph's thermomechanical performance. Conversely, its measurement allows one to calculate the ®lm stress and to obtain information on the evolution of the transformation in the constrained ®lm. According to the above analysis the ®lm stress in Fig. 2 is s^ F 8 > < ÿG^eÿT s^ T T ÿG e^ T e^ t a s^ T T s^ t a > ÿ : ÿG e^ T e^ t 1 s^ T T s^ t 1
T > MS MS > T > Mf T
i.e. the stress of the ®lm in the austenite±martensite or martensite states is composed of the thermal stress s^ T ÿG^eT and the transformation stress s^ t ÿG^et . For an elastically isotropic ®lm with iso-
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tropic transformation strain and an isotropic coecient of thermal expansion, ÿ sF sT st ÿGy T ÿ TR ÿ Get a 9a where sF, sT= ÿ GeT, eT=y(T ÿ TR) and et(a) are components of the biaxial stress and strain. Hereafter the subscript F is omitted and stress s now represents the components of an average biaxial stress in the ®lm, i.e. s0 s11=s22. The quantity y can be found from the single-phase sections of the s(T) diagram through equation (9). Then the average transformation self-strain can be determined from equation (9a) if the stress in a ®lm, sF, is measured. In the average transformation self-strain e^ t a e^ 0 aa T
10
a(T) is the fraction of the martensite phase and e^ 0 is the speci®c (i.e. per unit of volume of martensite) transformation self-strain. It is worth noting that this important characteristic of the transformation, e^ 0 , strongly depends on the state of stress of the ®lm, its crystallographic orientation and the degree of transformation, a. In the following, the thermodynamic aspects of a constrained transformation will be discussed with the goal of obtaining possible values of the speci®c self-strain and its dependence on the conditions of the transformation. The experimentally documented good reproducibility of the dependence s(T) and its quasistatic character justify this thermodynamic approach. 3. THERMODYNAMICS OF THE MARTENSITIC TRANSFORMATION IN A CONSTRAINED FILM
In this section the thermodynamics of the martensitic transformation in a constrained ®lm will be considered to determine the evolution of the equilibrium austenite±martensite microstructure as the temperature changes. The section will ®rst recall the basic features of martensitic transformation. After that, the accommodation of the austenite±martensite mixture to the ®lm constraint during cooling will be considered. Here, the situation is dierent depending on whether the transformation releases part or all of the ®lm stress. The accommodation of the fully transformed ®lm to the constraint is treated next. Finally, an analysis of the retransformation of the constraint-accommodated martensite upon heating will be presented. Throughout the guiding principle will be that the microstructure corresponds to a minimum of the free energy of the ®lm/substrate composite. In general the free energy of the ®lm/substrate composite per unit ®lm volume is given by f a 1 ÿ a f0A af
0 M
1 e a e^ G^e 2
which for isotropic composites reads
11a
f a f
0 A
ÿ 2 Dfa e a G eT et a
11b
In equations (11a)±(b) the ®rst three terms express the energy of an unconstrained two-phase ®lm. The quantities f 0A and f 0M are the speci®c free energies of the unstressed austenite and martensite, Df f 0M ÿ f 0A . The energy e(a) is the internal energy of the ®lm, i.e. the elastic energy of longrange and short-range internal stresses together with the energy of the interfaces. The last term in equation (11b) is the mis®t energy, i.e. the elastic energy due to the ®lm±substrate mis®t [6, 8]. The mis®t energy, through e^ 0 a, and the internal ®lm energy, e(a), depend on the microstructure. Therefore, the problem of minimization of the free energy must include an evaluation of the con®guration of the equilibrium microstructure together with a determination of the equilibrium fraction, a. The exact solution of the problem, at least for some partial cases, can be obtained for the transformation of a single-crystal ®lm. 3.1. A model martensitic transformation To elucidate the physics of constrained transformation we consider the cubic±tetragonal transformation in a ®lm along (001)c as a model. For simplicity, we assume that the components of the transformation self-strain are small. Hence a linear theory is valid [9] and the model accurately describes transformations in ferroelastics such as InTl alloys, although the results can be applied also to a qualitative analysis of b.c.c. t f.c.c. and other transformations. As is known, the cubic phase can transform into three orientation variants of a tetragonal phase with self-strains (Fig. 3) 0 0 1 1 2e 0 0 ÿe 0 0 B B C C e^ 10 @ 0 ÿe 0 A; e^ 20 @ 0 2e 0 A; 0 0 ÿe 0 0 ÿe 0 1 ÿe 0 0 B C e^ 30 @ 0 ÿe 0 A 12 0
0
2e
The self-strain with a positive strain along the tetragonal axis corresponds to the transformation in InTl alloys or the b.c.c. 4 f.c.c. transformation in Cu-based alloys. To decrease the number of parameters we assume that there is no volume change due to the transformation. Usually, the martensitic transformation proceeds through the formation of polydomain plates composed of alternations of domains of two variants. There is no stress due to contacts between domains, because they are mutual twins and interfaces between them are twinning planes. The self-strain of a polydomain martensite is variable, dependent on the fractions of variants. The polydomain plates
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®lm tension. It is equally clear that a combination ^ 21 of these plates with self-strains e^ 12 0 and e 0 is the most eective way to decrease the mis®t if the fractions of the plates of both self-strains are equal. Thus, the speci®c self-strain of the constrained martensitic transformation is e^ 0 e^ 0
Fig. 3. Three domains and equilibrium polydomain structures for the cubic±tetragonal transformation. The quantity a3 is the fraction of domain 3 which varies between a3=0 in (1,2) and a3=1 in (3).
are compatible with austenite provided their domain fractions reduce the average self-strain to a plane strain, i.e. one of the principal strains is zero [9, 10, 12, 13]. These fractions are equal to 1/3 or 2/3 for the self-strains considered here [equation (12)]. The compatible plate has a special orientation along an invariant plane, {110} for our model. The six possible plane self-strains of polydomain martensite are presented in the diagram shown in Fig. 4. There are four equivalent crystallographic plates corresponding to each self-strain e^ ij0 i,j 1,2,3. For example, the self-strain e^ 12 0 is an attribute of the plate with habit orientation (101) and (10 1), and both of them may have domain interfaces, or twinning planes, along (110) or (1 10). Thus, 24 martensite polydomain plates compatible with austenite are possible which do not create long-range stresses and, therefore, do not contribute the elastic energy to the internal energy of the mixture. If the strain is not small the Wechsler± Lieberman±Read theory determines the general form {hkl} of the invariant habit plane corresponding to the compatible plates [12, 13]. 3.2. The martensitic transformation in tensioned ®lms upon cooling The diagram in Fig. 4 allows one to determine what kind of martensite plate forms the equilibrium microstructure of the constrained ®lm. Consider the ®lm in a state of biaxial tension, i.e. eT=y(T ÿ TR) < 0 (y>0, T < TR) and the thermal stresses s = ÿ GeT>0. The mis®t e = eT+et decreases if plates consisting of domains 1 and 2 are formed. It was shown above that the domain fraction in the martensite plates must be 1/3 and 2/ 3, to accommodate the stresses at the austenite/martensite interface. It is clear that a microstructure consisting of 1/3 of domain 1 and 2/3 of domain 2 will create a biaxial plus a uniaxial tension in the ®lm, i.e. it will only incompletely accommodate the
1 ÿ 12 e^ e^ 21 2
e=2 0 0
0 0 e=2 0 0 ÿe
13
This self-strain is shown in the top section of Fig. 4. The polydomain martensite has an average selfstrain e^ 0 to minimize the mis®t energy of the tensioned ®lm. If the plates form a self-accommodated microstructure, e.g. form an array of sets of parallel plates [9, 14], long-range stresses are absent and the internal energy e(a) in equations (11a)±(b) is reduced to e0a where e0 is the density of the energy of interfaces and their intersections. The energy e0 includes a term proportional to l1/2 where l is the martensite plate length. This proportionality leads to the dependence of the transformation on the thickness of a single crystalline or on the grain size of a polycrystalline ®lm [9, 14]. This dependence will not be discussed here. The general free energy [equation (11a)] of the self-accommodated groups of martensite plates now becomes f a f
0 A
ÿ 1 Df e0 a e^ G^e 2
14a
and for the isotropic case f a f
0 A
ÿ e 2 Df e0 a G eT a 2
14b
It follows from the equilibrium equation df ÿ e e ^ e0 Df e0 2G eT a0 Df e0 s^ da 2 2 0
15
that the equilibrium volume fraction of martensite is ÿ ÿ ÿ Df e0 ÿ s^ T e^ 0 ÿ Df e0 ÿ GeT e a0 16 Ge2 =2 2em where 1 1 em e^ 0 G n^e0 Ge2 2 4
17
is the mis®t energy if the mis®t self-strain, e^ , is equal to e^ 0 . The conditions a0=0 and a0=1 permit an evaluation of the start of the martensitic transformation, Ms, and its end, Mf, if the temperature dependencies of Df and sT are taken into account [Fig. 5(a)]. These two temperatures determine the region of stability of the two-phase martensite±austenite mixture, which is dictated by the constraint [15]. Without constraint the self-accom-
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Fig. 4. Variable self-strain of polydomain martensite. The self-strains of domains are shown inside the inner rectangles, the self-strains of martensite plates are shown in bold-framed rectangles, the average self-strains of a group of the plates are shown in the double-framed rectangles. The lengths of the arrows are inversely proportional to the fractions of the domains.
modating transformation should occur at a single point in the (s,T) plane [16].{ The stress in the two-phase ®lm is determined by equations (15) and (16). The s(T) line crosses the axis s = 0 at T = TM. The temperature TM is determined by the equation Df(TM) + e0=0 and can be considered as the martensite start temperature, Ms, of the unconstrained martensitic transformation. The dierential equation for the {This is generally true for the ®rst-order phase transformation. It has been proven recently for martensitic transformations by computer modeling [16]; also, Khachaturyan, A. G., private communications.
temperature dependence of the ®lm stress follows from equation (15). If the temperature dependencies of e0 and e^ 0 e^ 0 are negligible it reads: e^ 0
ds^ dDf L ds L ÿ 1 or dT dT T0 dT T0 e
18
Here, the quantity L is the latent heat of the transformation [Df = L(T ÿ T0)/T0] and the second equation describes the isotropic case considered in this paper. Because equation (18) describes the equilibrium of the two phases, it has the form of the Clapeyron±Clausius equation. However, unlike the classical Clapeyron±Clausius relation, equation (18) contains the variable internal stress in the ®lm
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5101
reverse transformation diers from those characteristics of the direct transformation. The internal energy of the two-phase ®lm in this case has a minimum if the microstructure consists of a set of specially oriented plane-parallel austenite plates [8, 14]. Then, the free energy of the two-phase ®lm is [15] f a f
0 A
ÿ 1 a Df e0 eI a 1 ÿ a e^ G^e 2
19
where 1 eI e^ 0 G p^e0 2
Fig. 5. Stress±temperature curves for a transforming constrained ®lm. The transformation proceeds as an evolution of the equilibrium polydomain structure at (a) sT+st>0, y>0; (b) sT+st<0, y>0; (c) sT+st<0, y < 0. (1,2) is the equidomain composition of domains 1 and 2 (see Fig. 3); (1,2,3) is the three-domain microstructure (see Fig. 3); and (12,21) + A and (31,32) + A denote the microstructures containing austenite and two sets of martensite plates. In Fig. 5(c) the fraction of domain 3 increases along the solid line Ms, Mf and remains constant along the dotted line Ms', Mf'.
which is a function of the degree of transformation, a. The stress±temperature evolution upon cooling is presented in Fig. 5(a). The ®gure represents the case where the total transformation of the ®lm from austenite to martensite does not result in the complete relaxation of tension stress, i.e. sT(Mf)> ÿ st(1) = Ge . As shown above, the equi0 librium martensite microstructure with equidomain composition of domains 1 and 2 has to form. This microstructure does not change upon further cooling towards temperatures lower than the temperature Mf. As a result of this unchanging microstructure the thermal stress continues to grow as it did in the austenitic ®lm. 3.3. The martensitic transformation in tensioned ®lms upon heating If the composite is heated from a temperature below Mf, the reverse transformation from polydomain martensite to austenite will proceed. Since only one variant of austenite exists, the austenite plates can not be compatible to the martensite with the equidomain composition. Therefore, the thermodynamics and microstructural evolution of the
20
When compared to equations (14a)±(b), equation (19) contains the additional term eIa(1 ÿ a), which accounts for the elastic energy of the long-range stress ®elds due to the incompatibility between martensite and austenite. The quantity eI denotes the energy density of the martensite plate with self-strain, e^ 0 [equation (13)], embedded into the austenite. Equation (20) is the same as equation (17) but now p is the normal of the plate of minimum elastic energy. The equilibrium condition ÿ df ÿ Df e0 eI 1 ÿ 2a0 s^ T e^ 0 2em a da ÿ ÿ e Df e0 eI 1 ÿ 2a0 G eT a0 e0 2 21 determines the fraction of stressed incompatible martensite ÿ ÿ Df e0 sT e ÿ eI ÿ a0 22 2 em ÿ eI which decreases from 1 to 0 upon heating. As before, the conditions a0=1 and a0=0, i.e. ÿ 23 a0 1 ÿ Df e0 sT e eI ÿ 2em 0 and
ÿ a0 0 ÿ Df e0 sT e ÿ eI 0
24
determine the temperatures of the beginning and end of the reverse M 4 A transformation, As and Af. The temperature interval of the reverse transformation (As
ds^ dDf da0 dDf ÿ 2eI ÿ gGey 1 ÿ 1 g dT dT dT dT 25
where g = eI/(emÿeI) and Gey<
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Clausius equation (18) which is valid during cooling in the temperature interval (Ms
1 ÿ 31 e^ e^ 32 2
ÿe=2 0 0 ÿe=2 0 0
0 0 e
26
Therefore, the austenite remaining at TM transforms into martensite with zero average self-strain, ÿ 1ÿ e^ ÿ e^ 0 TM e^ 0 0 2 0
27
Since the stress remains zero the ensuing microstructure minimizes the internal energy of the austenite±martensite ®lm as well as the mis®t. After the transformation is complete the equilibrium threedomain martensite structure adjusts the thermal strain by changing its polydomain structure. During cooling below TM the fraction of domain 3 decreases and disappears at Mf0 [see Fig. 5(b)]. The martensite microstructure becomes a two-domain
structure and does not change under further cooling. If the two-domain structure described above is heated domain 3 reappears at Mf0. During further heating the fraction of domain 3 increases without generating any stress. The temperature interval, DT, in which the polydomain martensite demonstrates this ``invar'' property can be large, DT3 = e0/y. Under the present assumptions of a zero transformation volume change and equal coecients of thermal expansion of both phases it is clear that the increase of the fraction of domain 3, a3, compensates the thermal mis®t up to the temperature TR, at which the self-strain becomes zero (a3=1/3). This ``invar'' eect is the result of the variable polydomain structure. However, the reverse transformation will begin at a lower than TR temperature, As where the driving force of the transformation, Df, overcomes the sum of the ®lm mis®t and elastic A±M incompatibility energies. This temperature can be calculated by minimizing the free energy equation (19) provided the dependence of e^ 0 on the variable domain fraction of domain 3 is taken into account. The solution of this problem is rather complex and is not considered here. It is only noted that As appears to be considerably higher than TM which is Mf for transformation at cooling. The s(T) dependence corresponding to the two-phase evolution of the mixture of three-domain martensite and austenite is strongly nonlinear. It can thus be expected that the adaptive invar property of the three-domain martensite in constrained ®lms renders it metastable up to high temperatures. 3.5. The martensitic transformation in compressed ®lms upon cooling Above, a composite with a tensioned austenite ®lm was considered. If during cooling the austenite ®lm is compressed with respect to the reference unstressed state (i.e. y < 0), the evolution of the equilibrium microstructure is more complex. Martensite plates containing domain 3 have to form during the A 4 M transformation. However, their combination with the average self-strain e^ ÿ 0 is less eective in relaxing the constraint energy than single-domain plates with the self-strain e^ 30 . Therefore, the equilibrium fraction of domain 3 exceeds the one keeping the compatibility of martensite plates with austenite, i.e. a3 becomes larger than 2/3 for the model considered above. Thus, incompatible martensite plates are formed even during the austenite±martensite transformation upon cooling. The domain fraction is changed with the change of the stress. This evolution leads to a nonlinear dependence of the stress upon temperature as schematically illustrated in Fig. 5(c). This microstructural evolution is not discussed in this paper. It has been considered for uniaxial constraint in Refs [17, 18]. It is worth noting that the trans-
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formation between single-domain phases is structurally and thermodynamically reversible. 3.6. General remarks The results of the thermodynamic analysis above should be modi®ed if ®lm orientations other than (001) or other types of transformations are considered. Moreover, the results may be considerably distorted due to deviations from the assumed perfect self-accommodation. However, the following principal conclusions re¯ect the general properties of constrained phase equilibrium and the variable self-strain of polydomain martensite: 1. the temperature interval of two-phase equilibria is widened and shifted towards higher temperatures if the stress in the ®lm does not change its sign during the transformation; 2. the transformation is thermodynamically and structurally irreversible; and 3. martensite phases can be stabilized. These results do not depend on particular models. Since experimental studies of the transformations in single crystalline ®lm are not known, the theoretical conclusions will be compared to the characteristics of the martensitic transformation in textured polycrystalline ®lms. Perfect self-accommodation is impossible in polycrystals. Therefore, even the transformation in unconstrained polycrystals occurs in a broad interval of temperatures and stresses. The theoretical description of the microstructural evolution in polycrystalline ®lms is a complex unsolved problem similar to the one in bulk polycrystals. The incompatibility between transforming grains and the irregularity of martensite can be taken into account phenomenologically by including positive terms proportional to second and higher powers of a into the free energy [equations (11a)±(b)]. In the simplest approximation 1 f a Dfa e0 a e1 a2 e^ G^e 2
28
where e1 0^e20 is the elastic energy of repulsive interaction between randomly arranged plates. In this approximation (16) transforms into a0 and e^ 0
ÿDf ÿ s^ T e^ 0 ÿ e0 ÿ 2 em el
ds^ e1 L e1 e^ 0 Gy^ 1ÿ ÿ dT em el T0 em el
29
30
Due to repulsive interaction between martensite plates, the slope ds/dT decreases and does not conform to a Clapeyron±Clausius equation (18). Yet, this equation determines the onset of the transformation. The speci®c self-strain at the beginning of
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transformation depends on the constraining conditions. For well-textured polycrystalline ®lms it should be the same as for a single crystalline ®lm with the orientation corresponding to the texture. For example, for a cubic austenitic ®lm with a {100} texture, e^ 0 at a 1 0 has to be equal to e^ 0 . The accommodation of the incompatibility between transforming grains becomes important as the transformation develops. Thus, as a increases, e^ 0 decreases and approaches the relative volume change due to the transformation. The result is a nonlinear s(T) dependency in the transformation temperature interval. As has been shown before, incompatible austenite plates have to form in the course of the reverse M 4 A transformation. The attractive interaction between them corresponds to the negative term ÿa2eI in the free energy [equation (28)]. Therefore, as for single crystalline ®lms, the slope ds/dT for the reverse M 4 A transformation in polycrystalline ®lms is larger than that for the direct A 4 M transformation. As in equation (25), the ratio (em+el)/(em+elÿeI) determines the dierence of the two slopes. This ratio can increase dramatically as e^ 0 and consequently em and el decreases, so em+el approaches eI. 4. STRESS EVOLUTION IN CONSTRAINED NITI FILM
4.1. Experimental procedure Well adhering 1 mm thick NiTi ®lms were sputter deposited onto thermally oxidized (100 nm SiO2) Si cantilevers of dierent thicknesses utilizing the previously published technique [19]. Periodic post-deposition X-ray, TEM, SEM and RBS characterization served to assure the desired ®lm properties. The ®lm stresses were determined by measuring the cantilever de¯ection [20]. To study the stress evolution in the Ni50Ti50/SiO2/Si ®lm composites as a function of temperature, a cantilever beam specimen was clamped inside a quartz chamber equipped with temperature and vacuum controls. The de¯ection of the cantilever beams was computer monitored and converted into the average internal ®lm stress using Stoney's relationship [see equation (8)]. The quantity Gs is assumed to be equal to 200 GPa [21]. This stress is reported in the experimental results given below. Measurements at elevated temperatures were conducted in a vacuum better than 10ÿ5 torr and those below room temperature were performed in a high-purity helium atmosphere. All heating and cooling rates were adjusted so that the thermocouple indicating the sample temperature and the sample were in thermal equilibrium. The as-deposited amorphous NiTi ®lms were heat treated between 430 and 5908C. They densi®ed during holding for 30 min at 4308C and started to
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Fig. 6. Evolution of stress (above) and internal friction (below) of an amorphous 1 mm NiTi/90 mm Si sample deposited at room temperature. The heating and cooling rates are approximately 1 K/min.
crystallize at 5508C. Grain growth occurred at 5908C and the crystallization was complete at 6308C. After annealing all ®lms had a h110i texture. 4.2. Experimental results The experimental results consist of two groups, the stress evolution of the Ni50Ti50/SiO2/Si bimorph composite during crystallization/recrystallization and during the martensitic transformation. Typical results from the ®rst group are shown in Fig. 6. The ®gure shows the evolution of the stress of Ni50Ti50/SiO2/Si bimorph as a function of temperature. By correlating the stress evolution with internal friction data taken in an identical bimorph before [22], it becomes clear that the variations of the stress in the temperature range 4008C R TR 5508C signal densi®cation (the stress is
increasing with temperature) and recrystallization (the stress is decreasing with temperature) of the initially amorphous NiTi ®lm. The internal friction peak, which is observed above the temperature range of densi®cation, signals recrystallization. The stress±temperature characteristic of the crystallized ®lm/Si bimorph is linear. It will be shown later that the slope conforms exactly to the dierential thermal expansion of the two components of the bimorph. Figures 7 and 8 display the evolution of the stress as the NiTi portion of the bimorph undergoes the martensitic transformation. Figure 7 displays a typical stress±temperature characteristic of a Ni50Ti50/ SiO2/Si bimorph. It can be seen that the Ms and Af temperatures practically coincide and that for small fractions of the martensitic phase the hysteresis is
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an average self-strain e0=3.6 10ÿ3. The value of the average self-strain scales to biaxial modulus G. The experimentally determined values of G for austenite range from 110 GPa [24] to 189 GPa, resulting in 3.6 10ÿ3 R e0 R 7 10ÿ3. 2. The slopes ds/dT of the sections of decreasing stress with temperature are equal to 3±2.5 MPa/K for austenite and 2.6±2.3 MPa/K for martensite. This value corresponds to y = aNiTiÿaSi 1 1 10ÿ4 in good agreement with the reported thermal expansion coecients [24]. 3. The average slopes ds/dT in the temperature intervals in which the transformation occurs, are equal to 1±1.5 MPa/K in Ms
almost absent. Figure 8 shows that the change of the bimorph stress upon transformation depends on the grain size. Some quantitative characteristics of the stress± temperature curves should be noted. 1. The maximum stress near Ms is exceptionally high, 0.7±0.9 GPa. The total change of the stress due to the transformation, st(a = 1) in equation (9), has a magnitude of the same order. However, the minimum stress remains positive, around 0.1±0.3 GPa. For G = 189 GPa [23] the transformation stress of 0.7 GPa corresponds to
Fig. 8. Stress evolution due to the martensite±austenite transformation of two 1 mm NiTi/90 mm Si bimorphs with dierent grain sizes of the NiTi ®lm, approximately 1 mm and 50 nm.
The thermal stress will lead to nonunique de¯ection and render the application of the bimorph dicult. This diculty can be overcome in trimorphs in which the thermoelastic de¯ection is compensated. The evolution of the de¯ection in (crystalline Ni50Ti50)/SiO2/Si/SiO2/(amorphous Ni50Ti50) trimorphs is shown in Fig. 9. The trimorph consists of a Si plate cantilever and crystalline and amorphous NiTi ®lms of equal thickness on its two faces. Since the coecients of thermal expansion of the crystalline and amorphous NiTi are almost equal to each other, thermoelastic stresses are compensated. 5. DISCUSSION
A typical experimentally determined s(T) curve for a NiTi/(SiO2)/Si bimorph is presented in Fig. 7 together with the equilibrium line corresponding to equations (15), (16) and (18). The ®lm is highly textured [25], and each grain has a (110) plane
Fig. 9. A NiTi/Si composite micro-switch: the stress evolution of NiTi/Si bimorphs (top) and trimorphs (bottom) is indicated.
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parallel to the surface of the ®lm. Therefore, the speci®c self-strain for the B2±B19' martensitic transformation, e^ 0 , should be calculated as an average expansion in the (110) plane of the B2 phase. The analysis of preferable variants of monoclinic martensite in tensioned (110) textured ®lms is a rather complicated problem. However, its solution can be avoided by using the recently published calculation of the maximum contraction along [110] resulting from the B2±B19' transformation in a NiTi 50/50 alloy, e[110]= ÿ 5.2 10ÿ2 [26]. Then, the average expansion in the (110) plane, e0, related to the relative volume change due to the transformation, DV/ V, is 2e0 e110 DV=V
31 ÿ2
Since the dilatation DV/V = 0.5 10 lows that e0 2:85 10ÿ2
[27] it fol 32
Using this value of e0, the slope of the equilibrium transformation line becomes ds L 7 MPa=K dT T0 2e0
33
In the relation given by equation (33) values of the latent heat, L = 0.12 GPa [24], and the equilibrium temperature of the austenite±martensite phase, T0=300 K [28], have been used. The increase of Ms as well as of T0 due to the thermal stress is equal to MsÿTM=100 K. This dierence is in good agreement with the experiment if TM=230 K. The experimentally determined value of Mf is about 160 K (see Fig. 7). Thus MsÿMf=166 K which is a much broader temperature interval than usually observed for unconstrained transformations. Such a broad temperature interval was envisaged in the theoretical section. The slope ds/dT in the transformation temperature interval Ms
Fig. 10. Stress±temperature dependence for transforming 2.8 mm ®lm NiTi/(001)Si (after Ref. [5]) with the equilibrium line (TM=240 K).
(100)Si is presented in Fig. 10. While the parameters of the curve during cooling are comparable to the parameters describing the A 4 M transformation in the present experiments, the slope of the portion corresponding to the M 4 A transformation during heating is an order of magnitude larger than the equilibrium slope. It is suggested that this dierence re¯ects the dierent ®lm thicknesses of this and the Zhang±Grummon experiment. The ®lm thickness in the latter is about three times larger leading to a reduced incompatibility between grains during transformation. In thicker ®lms, the transformation microstructure evolves closer to the equilibrium path rendering the present theory more strictly applicable. Applying it to the reverse transformation it shows that the coecient g in equation (25) is very large. This is so, because if e0 becomes closer to a dilatation, the dierence (emÿeI) approaches zero. Correspondingly, the slope ds/dT becomes very steep. 6. SUMMARY
This paper represents a ®rst attempt to understand the martensitic transformation in constrained ®lms. A cubic to tetragonal transformation of single crystalline material has been used as a model system to simplify the analysis. The analysis leads to the conclusion that the constrained martensitic transformation in stressed ®lms in general should be structural and thermodynamically irreversible even if it develops as the evolution of an equilibrium microstructure. This irreversibility is the result of the dierent equilibrium domain microstructures which accommodate the ®lm/substrate and the austentite/martensite interface constraints. The analysis also determined that the equilibrium stress/temperature characteristic in the transformation region at cooling should follow a modi®ed Clausius± Clapeyron relationship. Contrary to the classic relationship, the stress entering it is not a constant
ROYTBURD et al.:
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externally applied stress but the internal stress which varies as the constrained transformation proceeds. For the transformation in polycrystalline ®lms the modi®ed Clausius±Clapeyron relation describes the onset of transformation. The theory predicts considerable broadening of the temperature interval of transformation due to the constraint. These theoretical conclusions are in complete accord with the experimentally observed facts. Thus, while the present theoretical analysis does not yield a quantitative description of the transformation, particularly in polycrystalline ®lms, it sets the stage toward accomplishing this goal. AcknowledgementsÐThis study was supported by the National Science Foundation, Grant No. DMR-97-06815. It also bene®ted from support by the Oce of Naval Research, contract No. N00014-93-10506. REFERENCES 1. George, E. P., Gotthardt, R., Otsuka, K., TroilerMcKinstry, S. and Wun-Fogle, M. (ed.), in MRS Proc., Materials for Smart System II, Vol. 459, MRS, Pittsburgh, 1997. 2. Simmons, W. C., Aksay, I. A. and Huston, D. R. (ed.), in SPIE Proc., Smart Materials Technologies, Vol. 3040, SPIE, 1997. 3. Bidaux, J.-E., Yu, W. J., Gotthardt, R. and Manson, J.-A. E., J. Physique, 1995, IV, C2±543. 4. Krulevitch, P., Ramsey, P. B., Makowiecki, D. M., Lee, A. P., Northrup, M. A. and Johnson, G. C., Thin Solid Films, 1996, 274, 101. 5. Zhang, J., Grummon, D.S., in MRS Proc., Materials for Smart System II, Vol. 459, ed. E.P. George, R. Gotthardt, K. Otsuka, S. Troiler-McKinstry and M. Wun-Fogle (ed.), MRS, Pittsburgh, 1997, p. 451. 6. Roitburd, A. L., Physica status solidi (a), 1976, 37, 329. 7. Roytburd, A. L., in Thin Film Ferroelectric Materials and Devices, ed. R. Ramesh, Kluwer, Dordrecht, 1997, p. 71. 8. Roytburd, A. L., J. appl. Phys., 1998, 83, 228.
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9. Roitburd, A. L., in Solid State Physics, eds H. Ehrenreich, F. Seitz and D. Turnball, Vol. 33, Academic Press, New York, 1978, p. 317. 10. Khachaturyan, A. G., Theory of Structural Transformation in Solids, John Wiley and Sons, New York, 1983. 11. Stoney, G. G., Proc. R. Soc. A, 1909, 82, 172. 12. Wechsler, M. S., Lieberman, D. S. and Read, T. A., Trans. metall. Soc. A.I.M.E., 1953, 127, 1503. 13. Wayman, C. M., Introduction to Crystallography of Martensitic Transformations, Macmillan, New York, 1964. 14. Roitburd, A. L. and Kurdjumov, G. V., Mater. Sci. Engng, 1979, 39, 141. 15. Roytburd, A. L. and Slutsker, J., Mater. Sci. Eng. A, 1997, 238, 23. 16. Wang, Y. and Khachaturyan, A. G., Acta mater., 1997, 45, 759. 17. Roytburd, A. L. and Slutsker, J., J. appl. Phys., 1995, 77, 2745. 18. Roytburd, A. L., J. Physique, 1996, IV, CI±11. 19. Su, Q., Hua, S. Z. and Wuttig, M., Trans. Mater. Res. Soc. Japan., 1994, 18B, 1057. 20. Kim, T. S., Martensitic transformation in NiTi/Si composites. Ph.D. Thesis, University of Maryland, 1995. 21. Brantley, W. A., J. appl. Phys., 1973, 44, 534. 22. Hua, Z. S., Su, C. M. and Wuttig, M., in Proc. Symp. on Damping Multiphase Inorganic Materials, ed. R. B. Bhagat, AMS, Metals Park, OH, 1993, p.165. 23. Su, Q., Hua, S. Z. and Wuttig, M., J. Adhesion Sci. Technol., 1994, 8, 7. 24. Materials Properties Handbook: Titanium Alloys, ASM International, Pittsburgh, 1994. 25. Su, Q., Kim, T. and Wuttig, M., MRS Symposium Proc., Materials for Smart System, Vol.360, ed. S. Troiler-McKinstry, K. Uchino, M. Wun-Fogle, E.P. George and S. Takahashi, MRS, Pittsburgh, 1995, p. 375. 26. Plietsh, R. and Enrich, K., Acta mater., 1997, 45, 2417. 27. Otsuka, K., Sawamura, K. and Shimizu, K., Physica status soloidi (a), 1971, 5, 457. 28. Chang, L. and Grummon, D. S., Phil. Mag. A, 1997, 76, 191.
Acta mater. Vol. 46, No. 14, pp. 5095±5107, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00165-7 1359-6454/98 $19.00 + 0.00
MARTENSITIC TRANSFORMATION IN CONSTRAINED FILMS A. L. ROYTBURD, T. S. KIM, QUANMIN SU, J. SLUTSKER and M. WUTTIG{ Department of Materials Science and Engineering, University of Maryland, College Park, MD 20742-2115, U.S.A. (Received 29 December 1997; accepted 18 April 1998) AbstractÐA thermodynamic analysis is presented for the martensitic transformation in a constrained ®lm upon cooling and heating. It is shown that this transformation proceeds with a variable self-strain corresponding to a variable polydomain structure of the martensite phase. Even the equilibrium microstructure develops irreversibly with changing temperature, i.e. the microstructure evolution paths for the direct and the reverse transformations are dierent. During the reverse transformation the stressed austenite, incompatible to martensite, is formed. The theory predicts a considerable shift of the temperature interval of transformation and its broadening due to the constraint. Experimental studies of the stress evolution with changing temperature in NiTi polycrystalline ®lms on Si substrates support the principal thermodynamic conclusions. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Phase transformations under constraint take place in all multiphase materials where a transforming phase is in contact with nontransforming phases. Interest in the problem of constrained transformations has increased recently due to growing applications of arti®cial materials and composites with transforming components, in bulk as well as in thin-®lm form. The martensitic transformation in constrained particles embedded in a nontransformed matrix has been widely studied and used as an eective mechanism of toughening of ceramics. Another area of applications of constrained martensitic transformations is smart composites with shape memory alloys as the active components [1±3]. Layer composites containing transforming shape memory material are particularly interesting for studies of constrained martensitic transformations and for applications as well. There are two typical con®gurations of composites with transforming constrained ®lms: symmetrical and asymmetrical. The ®rst con®guration consists of two identical ®lms on both sides of a substrate (``trimorph'') [Fig. 1(a)]; the second one is a ®lm on a substrate (``bimorph'') [Fig. 1(b)]. The trimorph can be considered as an elementary cell of multilayer composites or heterostructures. Bimorphs are often used as actuators or sensors, because the transformation in the ®lm results in considerable bending. For the same reason the bimorphs are usually used for studies of martensitic transformation itself. {To whom correspondence should be addressed.
Several studies of the martensitic transformation in NiTi ®lms on Si substrates have been performed recently [4, 5]. In this paper we present experimental results on the stress evolution in bimorphs and trimorphs consisting of NiTi polycrystalline ®lms on Si substrates with SiO2 buer layers. We discuss the principal results of experimental studies from the point of view of the thermodynamics of a martensitic transformation in a constrained ®lm. The high level of stress in a ®lm due to its mis®t with a substrate suggests that they can compete with internal stresses in the martensite or martensite±austenite microstructure and considerably change the self-accommodation of the martensitic transformation in shape memory alloys. Therefore, thermodynamic eects in constrained ®lms should be much more pronounced than in bulk materials. In Section 2 of this paper the mechanics of a constrained ®lm is considered. The dierent physical factors which determine the ®lm±substrate mis®t are discussed and formulas for the calculation of the average stress in anisotropic and isotropic ®lms are presented. The thermodynamics of the transformation in a single-crystalline constrained ®lm is discussed in Section 3. The analysis of a cubic±tetragonal model transformation predicts special eects of the constraint, namely: 1. a structural irreversibility of the evolution of the equilibrium microstructure; and 2. the appearance of an elastic incompatibility between the austenite and the martensite because the interfaces between these phases are not necessarily invariant planes.
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Fig. 1. Self-strain and deformation of a composite containing a transforming constrained ®lm. Trimorph (a) and bimorph (b) con®gurations of the composite are shown.
Experimental results on the preparation of polycrystalline constrained NiTi ®lms and their transformation upon cooling and heating are presented in Section 3. These results, together with other published data, are discussed in Section 4 where it will be seen that they support the thermodynamical concepts presented in Section 3 of this paper. 2. MECHANICS OF CONSTRAINED TRANSFORMATIONS
A phase transformation in a constrained ®lm is a self-regulated process. The transformation changes
the state of stress of the ®lm and the changing stress, in turn, aects the transformation. We consider both sides of the process from the point of view of the observed stress evolution in the ®lm as a function of temperature. A typical temperature dependence of the average stress in a constrained SMA ®lm is presented in Fig. 2. The two sections of the almost linearly increasing stress with decreasing temperature correspond to the dierence of the thermal expansions of the ®lm and substrate. The section in which the stress decreases with temperature re¯ects the change of the martensite fraction during the transformation. The curve corresponds to the transformation during cooling and the temperature Ms marks the beginning of the transformation of austenite to martensite. The temperature Mf indicates the point where the A 4 M transformation is complete. Stress in a ®lm/substrate composite arises due to a mis®t between the two components along their interface. Since the mis®t is in general the result of volume deformations, it is convenient to introduce a three-dimensional mis®t self-strain for the calculation of the state of stress of the composite [6±8]. This mis®t self-strain can be determined as follows. Compare the state of strain of the composite at a temperature, T, with a reference stress-free state at a temperature, TR. (Fig. 2). If the ®lm and the substrate are separated at temperature T, then the comparison with the reference state determines their
Fig. 2. Stress±temperature curve for a transforming constrained ®lm. The evolution of the ®lm microstructure and mis®t is shown.
ROYTBURD et al.:
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self-strains, i.e. the strains which the ®lm and the substrate would have without mutual constraint. If T < TR the self-strain e^ S of the substrate is a result of the thermal contraction: ÿ e^ S a^ S T ÿ TR 1 where a^ S is the tensor of thermal expansion of the substrate. The self-strain of the ®lm is determined by a similar relation if the ®lm is austenitic at the reference temperature. Naturally, if the ®lm transforms, its self-strain should include the transformation selfstrain, e^ t a: ÿ 2 e^ F a^ F T ÿ TR e^ t a where a^ F is a thermal expansion tensor of the ®lm and a is the fraction of martensite. Here, it is assumed that 1. all strains are small so their linear superposition is valid; 2. the coecients of thermal expansion for the martensite and the austenite are equal (^aM a^ A a^ F ); and 3. the temperature and the martensitic microstructure are uniform across the ®lm. The mis®t self-strain is determined by the dierence between the self-strains of the ®lm and the substrate: ÿ e^ e^ F ÿ e^ S e^ T e^ t a; e^ T y^ T ÿ TR 3 where e^ T is the thermal mis®t, y^ a^ F ÿ a^ S . It is assumed above that there is no stress relaxation due to plastic deformation or the changes of the adhesion while varying the temperature. For epitaxial ®lms it means that there is no movement or/and generation of mis®t dislocations during the evolution of the stress as the temperature changes. However, the ®lm/substrate interface may contain immobile interface dislocations which had been formed at/or near the deposition temperature. In this case the reference temperature TR corresponding to the relaxed state will be close to the deposition temperature. In the general case TR depends on the conditions of deposition, epitaxial mis®t and degree of relaxation. Knowledge of the mis®t self-strain, as well as geometric parameters of the composite and the elastic properties of its components, permits the calculation of the stresses, the elastic strains and the composite displacement. If the thickness of the composite is much smaller than its other dimensions, the calculations reduce to a one-dimensional {The following notations are used in thisÿ paper: for the second rank tensors of strain and stress, e^ eik ; i,k 1,2,3 ^ for the fourth rank tensors of elastic moduli, and s; C (Cikjl) and G; for a vector of normal, 4 n s^ ^ e ) sij ; e^ G^e ) Gijkl eij ekl
5097
problem. For a ®lm on an absolutely rigid substrate the stress is{ n^e s^ F ÿG
4
ÿ1
where G n C ÿ C n nC n nC is a planar elastic modulus, C is the elastic modulus tensor of the ®lm, and n is the crystallographic direction of the ®lm normal [9, 10]. If the principal strains of the mis®t strain e^ in the plane of the ®lm are equal, the ®lm has cubic anisotropy and n 100, then sF11 sF22 sF ÿGe G C11 C12 ÿ
2C 212 C11
5 6
where e is the in-plane ®lm strain. In the isotropic case G
E 1ÿ
7
where the quantities E and n represent Young's modulus and Poisson's ratio, respectively. Hereafter, only thin-®lm composites with h/ H < 0.01 are considered (see Fig. 1). In this case the change of the stress across the ®lm of the bimorph is negligible as well as the dierence between the states of stress of the symmetrical and unsymmetrical con®gurations. In both cases the stress is approximately determined by equation (4). The stressed ®lm of a bimorph causes bending of the substrate. Its mean radius of curvature is determined by the Stoney's equation [11]: R
H2 sF 6hGS
8
In equation (8) the quantities GS is the plane modulus of the substrate. An in-plane anisotropy of the stressed ®lm on the substrate results in the appearance of the two principal curvatures re¯ecting the principal stresses in the bimorph. The radius R as a function of temperature is one of the basic characteristics of a bimorph's thermomechanical performance. Conversely, its measurement allows one to calculate the ®lm stress and to obtain information on the evolution of the transformation in the constrained ®lm. According to the above analysis the ®lm stress in Fig. 2 is s^ F 8 > < ÿG^eÿT s^ T T ÿG e^ T e^ t a s^ T T s^ t a > ÿ : ÿG e^ T e^ t 1 s^ T T s^ t 1
T > MS MS > T > Mf T
i.e. the stress of the ®lm in the austenite±martensite or martensite states is composed of the thermal stress s^ T ÿG^eT and the transformation stress s^ t ÿG^et . For an elastically isotropic ®lm with iso-
5098
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tropic transformation strain and an isotropic coecient of thermal expansion, ÿ sF sT st ÿGy T ÿ TR ÿ Get a 9a where sF, sT= ÿ GeT, eT=y(T ÿ TR) and et(a) are components of the biaxial stress and strain. Hereafter the subscript F is omitted and stress s now represents the components of an average biaxial stress in the ®lm, i.e. s0 s11=s22. The quantity y can be found from the single-phase sections of the s(T) diagram through equation (9). Then the average transformation self-strain can be determined from equation (9a) if the stress in a ®lm, sF, is measured. In the average transformation self-strain e^ t a e^ 0 aa T
10
a(T) is the fraction of the martensite phase and e^ 0 is the speci®c (i.e. per unit of volume of martensite) transformation self-strain. It is worth noting that this important characteristic of the transformation, e^ 0 , strongly depends on the state of stress of the ®lm, its crystallographic orientation and the degree of transformation, a. In the following, the thermodynamic aspects of a constrained transformation will be discussed with the goal of obtaining possible values of the speci®c self-strain and its dependence on the conditions of the transformation. The experimentally documented good reproducibility of the dependence s(T) and its quasistatic character justify this thermodynamic approach. 3. THERMODYNAMICS OF THE MARTENSITIC TRANSFORMATION IN A CONSTRAINED FILM
In this section the thermodynamics of the martensitic transformation in a constrained ®lm will be considered to determine the evolution of the equilibrium austenite±martensite microstructure as the temperature changes. The section will ®rst recall the basic features of martensitic transformation. After that, the accommodation of the austenite±martensite mixture to the ®lm constraint during cooling will be considered. Here, the situation is dierent depending on whether the transformation releases part or all of the ®lm stress. The accommodation of the fully transformed ®lm to the constraint is treated next. Finally, an analysis of the retransformation of the constraint-accommodated martensite upon heating will be presented. Throughout the guiding principle will be that the microstructure corresponds to a minimum of the free energy of the ®lm/substrate composite. In general the free energy of the ®lm/substrate composite per unit ®lm volume is given by f a 1 ÿ a f0A af
0 M
1 e a e^ G^e 2
which for isotropic composites reads
11a
f a f
0 A
ÿ 2 Dfa e a G eT et a
11b
In equations (11a)±(b) the ®rst three terms express the energy of an unconstrained two-phase ®lm. The quantities f 0A and f 0M are the speci®c free energies of the unstressed austenite and martensite, Df f 0M ÿ f 0A . The energy e(a) is the internal energy of the ®lm, i.e. the elastic energy of longrange and short-range internal stresses together with the energy of the interfaces. The last term in equation (11b) is the mis®t energy, i.e. the elastic energy due to the ®lm±substrate mis®t [6, 8]. The mis®t energy, through e^ 0 a, and the internal ®lm energy, e(a), depend on the microstructure. Therefore, the problem of minimization of the free energy must include an evaluation of the con®guration of the equilibrium microstructure together with a determination of the equilibrium fraction, a. The exact solution of the problem, at least for some partial cases, can be obtained for the transformation of a single-crystal ®lm. 3.1. A model martensitic transformation To elucidate the physics of constrained transformation we consider the cubic±tetragonal transformation in a ®lm along (001)c as a model. For simplicity, we assume that the components of the transformation self-strain are small. Hence a linear theory is valid [9] and the model accurately describes transformations in ferroelastics such as InTl alloys, although the results can be applied also to a qualitative analysis of b.c.c. t f.c.c. and other transformations. As is known, the cubic phase can transform into three orientation variants of a tetragonal phase with self-strains (Fig. 3) 0 0 1 1 2e 0 0 ÿe 0 0 B B C C e^ 10 @ 0 ÿe 0 A; e^ 20 @ 0 2e 0 A; 0 0 ÿe 0 0 ÿe 0 1 ÿe 0 0 B C e^ 30 @ 0 ÿe 0 A 12 0
0
2e
The self-strain with a positive strain along the tetragonal axis corresponds to the transformation in InTl alloys or the b.c.c. 4 f.c.c. transformation in Cu-based alloys. To decrease the number of parameters we assume that there is no volume change due to the transformation. Usually, the martensitic transformation proceeds through the formation of polydomain plates composed of alternations of domains of two variants. There is no stress due to contacts between domains, because they are mutual twins and interfaces between them are twinning planes. The self-strain of a polydomain martensite is variable, dependent on the fractions of variants. The polydomain plates
ROYTBURD et al.:
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®lm tension. It is equally clear that a combination ^ 21 of these plates with self-strains e^ 12 0 and e 0 is the most eective way to decrease the mis®t if the fractions of the plates of both self-strains are equal. Thus, the speci®c self-strain of the constrained martensitic transformation is e^ 0 e^ 0
Fig. 3. Three domains and equilibrium polydomain structures for the cubic±tetragonal transformation. The quantity a3 is the fraction of domain 3 which varies between a3=0 in (1,2) and a3=1 in (3).
are compatible with austenite provided their domain fractions reduce the average self-strain to a plane strain, i.e. one of the principal strains is zero [9, 10, 12, 13]. These fractions are equal to 1/3 or 2/3 for the self-strains considered here [equation (12)]. The compatible plate has a special orientation along an invariant plane, {110} for our model. The six possible plane self-strains of polydomain martensite are presented in the diagram shown in Fig. 4. There are four equivalent crystallographic plates corresponding to each self-strain e^ ij0 i,j 1,2,3. For example, the self-strain e^ 12 0 is an attribute of the plate with habit orientation (101) and (10 1), and both of them may have domain interfaces, or twinning planes, along (110) or (1 10). Thus, 24 martensite polydomain plates compatible with austenite are possible which do not create long-range stresses and, therefore, do not contribute the elastic energy to the internal energy of the mixture. If the strain is not small the Wechsler± Lieberman±Read theory determines the general form {hkl} of the invariant habit plane corresponding to the compatible plates [12, 13]. 3.2. The martensitic transformation in tensioned ®lms upon cooling The diagram in Fig. 4 allows one to determine what kind of martensite plate forms the equilibrium microstructure of the constrained ®lm. Consider the ®lm in a state of biaxial tension, i.e. eT=y(T ÿ TR) < 0 (y>0, T < TR) and the thermal stresses s = ÿ GeT>0. The mis®t e = eT+et decreases if plates consisting of domains 1 and 2 are formed. It was shown above that the domain fraction in the martensite plates must be 1/3 and 2/ 3, to accommodate the stresses at the austenite/martensite interface. It is clear that a microstructure consisting of 1/3 of domain 1 and 2/3 of domain 2 will create a biaxial plus a uniaxial tension in the ®lm, i.e. it will only incompletely accommodate the
1 ÿ 12 e^ e^ 21 2
e=2 0 0
0 0 e=2 0 0 ÿe
13
This self-strain is shown in the top section of Fig. 4. The polydomain martensite has an average selfstrain e^ 0 to minimize the mis®t energy of the tensioned ®lm. If the plates form a self-accommodated microstructure, e.g. form an array of sets of parallel plates [9, 14], long-range stresses are absent and the internal energy e(a) in equations (11a)±(b) is reduced to e0a where e0 is the density of the energy of interfaces and their intersections. The energy e0 includes a term proportional to l1/2 where l is the martensite plate length. This proportionality leads to the dependence of the transformation on the thickness of a single crystalline or on the grain size of a polycrystalline ®lm [9, 14]. This dependence will not be discussed here. The general free energy [equation (11a)] of the self-accommodated groups of martensite plates now becomes f a f
0 A
ÿ 1 Df e0 a e^ G^e 2
14a
and for the isotropic case f a f
0 A
ÿ e 2 Df e0 a G eT a 2
14b
It follows from the equilibrium equation df ÿ e e ^ e0 Df e0 2G eT a0 Df e0 s^ da 2 2 0
15
that the equilibrium volume fraction of martensite is ÿ ÿ ÿ Df e0 ÿ s^ T e^ 0 ÿ Df e0 ÿ GeT e a0 16 Ge2 =2 2em where 1 1 em e^ 0 G n^e0 Ge2 2 4
17
is the mis®t energy if the mis®t self-strain, e^ , is equal to e^ 0 . The conditions a0=0 and a0=1 permit an evaluation of the start of the martensitic transformation, Ms, and its end, Mf, if the temperature dependencies of Df and sT are taken into account [Fig. 5(a)]. These two temperatures determine the region of stability of the two-phase martensite±austenite mixture, which is dictated by the constraint [15]. Without constraint the self-accom-
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Fig. 4. Variable self-strain of polydomain martensite. The self-strains of domains are shown inside the inner rectangles, the self-strains of martensite plates are shown in bold-framed rectangles, the average self-strains of a group of the plates are shown in the double-framed rectangles. The lengths of the arrows are inversely proportional to the fractions of the domains.
modating transformation should occur at a single point in the (s,T) plane [16].{ The stress in the two-phase ®lm is determined by equations (15) and (16). The s(T) line crosses the axis s = 0 at T = TM. The temperature TM is determined by the equation Df(TM) + e0=0 and can be considered as the martensite start temperature, Ms, of the unconstrained martensitic transformation. The dierential equation for the {This is generally true for the ®rst-order phase transformation. It has been proven recently for martensitic transformations by computer modeling [16]; also, Khachaturyan, A. G., private communications.
temperature dependence of the ®lm stress follows from equation (15). If the temperature dependencies of e0 and e^ 0 e^ 0 are negligible it reads: e^ 0
ds^ dDf L ds L ÿ 1 or dT dT T0 dT T0 e
18
Here, the quantity L is the latent heat of the transformation [Df = L(T ÿ T0)/T0] and the second equation describes the isotropic case considered in this paper. Because equation (18) describes the equilibrium of the two phases, it has the form of the Clapeyron±Clausius equation. However, unlike the classical Clapeyron±Clausius relation, equation (18) contains the variable internal stress in the ®lm
ROYTBURD et al.:
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reverse transformation diers from those characteristics of the direct transformation. The internal energy of the two-phase ®lm in this case has a minimum if the microstructure consists of a set of specially oriented plane-parallel austenite plates [8, 14]. Then, the free energy of the two-phase ®lm is [15] f a f
0 A
ÿ 1 a Df e0 eI a 1 ÿ a e^ G^e 2
19
where 1 eI e^ 0 G p^e0 2
Fig. 5. Stress±temperature curves for a transforming constrained ®lm. The transformation proceeds as an evolution of the equilibrium polydomain structure at (a) sT+st>0, y>0; (b) sT+st<0, y>0; (c) sT+st<0, y < 0. (1,2) is the equidomain composition of domains 1 and 2 (see Fig. 3); (1,2,3) is the three-domain microstructure (see Fig. 3); and (12,21) + A and (31,32) + A denote the microstructures containing austenite and two sets of martensite plates. In Fig. 5(c) the fraction of domain 3 increases along the solid line Ms, Mf and remains constant along the dotted line Ms', Mf'.
which is a function of the degree of transformation, a. The stress±temperature evolution upon cooling is presented in Fig. 5(a). The ®gure represents the case where the total transformation of the ®lm from austenite to martensite does not result in the complete relaxation of tension stress, i.e. sT(Mf)> ÿ st(1) = Ge . As shown above, the equi0 librium martensite microstructure with equidomain composition of domains 1 and 2 has to form. This microstructure does not change upon further cooling towards temperatures lower than the temperature Mf. As a result of this unchanging microstructure the thermal stress continues to grow as it did in the austenitic ®lm. 3.3. The martensitic transformation in tensioned ®lms upon heating If the composite is heated from a temperature below Mf, the reverse transformation from polydomain martensite to austenite will proceed. Since only one variant of austenite exists, the austenite plates can not be compatible to the martensite with the equidomain composition. Therefore, the thermodynamics and microstructural evolution of the
20
When compared to equations (14a)±(b), equation (19) contains the additional term eIa(1 ÿ a), which accounts for the elastic energy of the long-range stress ®elds due to the incompatibility between martensite and austenite. The quantity eI denotes the energy density of the martensite plate with self-strain, e^ 0 [equation (13)], embedded into the austenite. Equation (20) is the same as equation (17) but now p is the normal of the plate of minimum elastic energy. The equilibrium condition ÿ df ÿ Df e0 eI 1 ÿ 2a0 s^ T e^ 0 2em a da ÿ ÿ e Df e0 eI 1 ÿ 2a0 G eT a0 e0 2 21 determines the fraction of stressed incompatible martensite ÿ ÿ Df e0 sT e ÿ eI ÿ a0 22 2 em ÿ eI which decreases from 1 to 0 upon heating. As before, the conditions a0=1 and a0=0, i.e. ÿ 23 a0 1 ÿ Df e0 sT e eI ÿ 2em 0 and
ÿ a0 0 ÿ Df e0 sT e ÿ eI 0
24
determine the temperatures of the beginning and end of the reverse M 4 A transformation, As and Af. The temperature interval of the reverse transformation (As
ds^ dDf da0 dDf ÿ 2eI ÿ gGey 1 ÿ 1 g dT dT dT dT 25
where g = eI/(emÿeI) and Gey<
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Clausius equation (18) which is valid during cooling in the temperature interval (Ms
1 ÿ 31 e^ e^ 32 2
ÿe=2 0 0 ÿe=2 0 0
0 0 e
26
Therefore, the austenite remaining at TM transforms into martensite with zero average self-strain, ÿ 1ÿ e^ ÿ e^ 0 TM e^ 0 0 2 0
27
Since the stress remains zero the ensuing microstructure minimizes the internal energy of the austenite±martensite ®lm as well as the mis®t. After the transformation is complete the equilibrium threedomain martensite structure adjusts the thermal strain by changing its polydomain structure. During cooling below TM the fraction of domain 3 decreases and disappears at Mf0 [see Fig. 5(b)]. The martensite microstructure becomes a two-domain
structure and does not change under further cooling. If the two-domain structure described above is heated domain 3 reappears at Mf0. During further heating the fraction of domain 3 increases without generating any stress. The temperature interval, DT, in which the polydomain martensite demonstrates this ``invar'' property can be large, DT3 = e0/y. Under the present assumptions of a zero transformation volume change and equal coecients of thermal expansion of both phases it is clear that the increase of the fraction of domain 3, a3, compensates the thermal mis®t up to the temperature TR, at which the self-strain becomes zero (a3=1/3). This ``invar'' eect is the result of the variable polydomain structure. However, the reverse transformation will begin at a lower than TR temperature, As where the driving force of the transformation, Df, overcomes the sum of the ®lm mis®t and elastic A±M incompatibility energies. This temperature can be calculated by minimizing the free energy equation (19) provided the dependence of e^ 0 on the variable domain fraction of domain 3 is taken into account. The solution of this problem is rather complex and is not considered here. It is only noted that As appears to be considerably higher than TM which is Mf for transformation at cooling. The s(T) dependence corresponding to the two-phase evolution of the mixture of three-domain martensite and austenite is strongly nonlinear. It can thus be expected that the adaptive invar property of the three-domain martensite in constrained ®lms renders it metastable up to high temperatures. 3.5. The martensitic transformation in compressed ®lms upon cooling Above, a composite with a tensioned austenite ®lm was considered. If during cooling the austenite ®lm is compressed with respect to the reference unstressed state (i.e. y < 0), the evolution of the equilibrium microstructure is more complex. Martensite plates containing domain 3 have to form during the A 4 M transformation. However, their combination with the average self-strain e^ ÿ 0 is less eective in relaxing the constraint energy than single-domain plates with the self-strain e^ 30 . Therefore, the equilibrium fraction of domain 3 exceeds the one keeping the compatibility of martensite plates with austenite, i.e. a3 becomes larger than 2/3 for the model considered above. Thus, incompatible martensite plates are formed even during the austenite±martensite transformation upon cooling. The domain fraction is changed with the change of the stress. This evolution leads to a nonlinear dependence of the stress upon temperature as schematically illustrated in Fig. 5(c). This microstructural evolution is not discussed in this paper. It has been considered for uniaxial constraint in Refs [17, 18]. It is worth noting that the trans-
ROYTBURD et al.:
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formation between single-domain phases is structurally and thermodynamically reversible. 3.6. General remarks The results of the thermodynamic analysis above should be modi®ed if ®lm orientations other than (001) or other types of transformations are considered. Moreover, the results may be considerably distorted due to deviations from the assumed perfect self-accommodation. However, the following principal conclusions re¯ect the general properties of constrained phase equilibrium and the variable self-strain of polydomain martensite: 1. the temperature interval of two-phase equilibria is widened and shifted towards higher temperatures if the stress in the ®lm does not change its sign during the transformation; 2. the transformation is thermodynamically and structurally irreversible; and 3. martensite phases can be stabilized. These results do not depend on particular models. Since experimental studies of the transformations in single crystalline ®lm are not known, the theoretical conclusions will be compared to the characteristics of the martensitic transformation in textured polycrystalline ®lms. Perfect self-accommodation is impossible in polycrystals. Therefore, even the transformation in unconstrained polycrystals occurs in a broad interval of temperatures and stresses. The theoretical description of the microstructural evolution in polycrystalline ®lms is a complex unsolved problem similar to the one in bulk polycrystals. The incompatibility between transforming grains and the irregularity of martensite can be taken into account phenomenologically by including positive terms proportional to second and higher powers of a into the free energy [equations (11a)±(b)]. In the simplest approximation 1 f a Dfa e0 a e1 a2 e^ G^e 2
28
where e1 0^e20 is the elastic energy of repulsive interaction between randomly arranged plates. In this approximation (16) transforms into a0 and e^ 0
ÿDf ÿ s^ T e^ 0 ÿ e0 ÿ 2 em el
ds^ e1 L e1 e^ 0 Gy^ 1ÿ ÿ dT em el T0 em el
29
30
Due to repulsive interaction between martensite plates, the slope ds/dT decreases and does not conform to a Clapeyron±Clausius equation (18). Yet, this equation determines the onset of the transformation. The speci®c self-strain at the beginning of
5103
transformation depends on the constraining conditions. For well-textured polycrystalline ®lms it should be the same as for a single crystalline ®lm with the orientation corresponding to the texture. For example, for a cubic austenitic ®lm with a {100} texture, e^ 0 at a 1 0 has to be equal to e^ 0 . The accommodation of the incompatibility between transforming grains becomes important as the transformation develops. Thus, as a increases, e^ 0 decreases and approaches the relative volume change due to the transformation. The result is a nonlinear s(T) dependency in the transformation temperature interval. As has been shown before, incompatible austenite plates have to form in the course of the reverse M 4 A transformation. The attractive interaction between them corresponds to the negative term ÿa2eI in the free energy [equation (28)]. Therefore, as for single crystalline ®lms, the slope ds/dT for the reverse M 4 A transformation in polycrystalline ®lms is larger than that for the direct A 4 M transformation. As in equation (25), the ratio (em+el)/(em+elÿeI) determines the dierence of the two slopes. This ratio can increase dramatically as e^ 0 and consequently em and el decreases, so em+el approaches eI. 4. STRESS EVOLUTION IN CONSTRAINED NITI FILM
4.1. Experimental procedure Well adhering 1 mm thick NiTi ®lms were sputter deposited onto thermally oxidized (100 nm SiO2) Si cantilevers of dierent thicknesses utilizing the previously published technique [19]. Periodic post-deposition X-ray, TEM, SEM and RBS characterization served to assure the desired ®lm properties. The ®lm stresses were determined by measuring the cantilever de¯ection [20]. To study the stress evolution in the Ni50Ti50/SiO2/Si ®lm composites as a function of temperature, a cantilever beam specimen was clamped inside a quartz chamber equipped with temperature and vacuum controls. The de¯ection of the cantilever beams was computer monitored and converted into the average internal ®lm stress using Stoney's relationship [see equation (8)]. The quantity Gs is assumed to be equal to 200 GPa [21]. This stress is reported in the experimental results given below. Measurements at elevated temperatures were conducted in a vacuum better than 10ÿ5 torr and those below room temperature were performed in a high-purity helium atmosphere. All heating and cooling rates were adjusted so that the thermocouple indicating the sample temperature and the sample were in thermal equilibrium. The as-deposited amorphous NiTi ®lms were heat treated between 430 and 5908C. They densi®ed during holding for 30 min at 4308C and started to
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Fig. 6. Evolution of stress (above) and internal friction (below) of an amorphous 1 mm NiTi/90 mm Si sample deposited at room temperature. The heating and cooling rates are approximately 1 K/min.
crystallize at 5508C. Grain growth occurred at 5908C and the crystallization was complete at 6308C. After annealing all ®lms had a h110i texture. 4.2. Experimental results The experimental results consist of two groups, the stress evolution of the Ni50Ti50/SiO2/Si bimorph composite during crystallization/recrystallization and during the martensitic transformation. Typical results from the ®rst group are shown in Fig. 6. The ®gure shows the evolution of the stress of Ni50Ti50/SiO2/Si bimorph as a function of temperature. By correlating the stress evolution with internal friction data taken in an identical bimorph before [22], it becomes clear that the variations of the stress in the temperature range 4008C R TR 5508C signal densi®cation (the stress is
increasing with temperature) and recrystallization (the stress is decreasing with temperature) of the initially amorphous NiTi ®lm. The internal friction peak, which is observed above the temperature range of densi®cation, signals recrystallization. The stress±temperature characteristic of the crystallized ®lm/Si bimorph is linear. It will be shown later that the slope conforms exactly to the dierential thermal expansion of the two components of the bimorph. Figures 7 and 8 display the evolution of the stress as the NiTi portion of the bimorph undergoes the martensitic transformation. Figure 7 displays a typical stress±temperature characteristic of a Ni50Ti50/ SiO2/Si bimorph. It can be seen that the Ms and Af temperatures practically coincide and that for small fractions of the martensitic phase the hysteresis is
ROYTBURD et al.:
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5105
an average self-strain e0=3.6 10ÿ3. The value of the average self-strain scales to biaxial modulus G. The experimentally determined values of G for austenite range from 110 GPa [24] to 189 GPa, resulting in 3.6 10ÿ3 R e0 R 7 10ÿ3. 2. The slopes ds/dT of the sections of decreasing stress with temperature are equal to 3±2.5 MPa/K for austenite and 2.6±2.3 MPa/K for martensite. This value corresponds to y = aNiTiÿaSi 1 1 10ÿ4 in good agreement with the reported thermal expansion coecients [24]. 3. The average slopes ds/dT in the temperature intervals in which the transformation occurs, are equal to 1±1.5 MPa/K in Ms
almost absent. Figure 8 shows that the change of the bimorph stress upon transformation depends on the grain size. Some quantitative characteristics of the stress± temperature curves should be noted. 1. The maximum stress near Ms is exceptionally high, 0.7±0.9 GPa. The total change of the stress due to the transformation, st(a = 1) in equation (9), has a magnitude of the same order. However, the minimum stress remains positive, around 0.1±0.3 GPa. For G = 189 GPa [23] the transformation stress of 0.7 GPa corresponds to
Fig. 8. Stress evolution due to the martensite±austenite transformation of two 1 mm NiTi/90 mm Si bimorphs with dierent grain sizes of the NiTi ®lm, approximately 1 mm and 50 nm.
The thermal stress will lead to nonunique de¯ection and render the application of the bimorph dicult. This diculty can be overcome in trimorphs in which the thermoelastic de¯ection is compensated. The evolution of the de¯ection in (crystalline Ni50Ti50)/SiO2/Si/SiO2/(amorphous Ni50Ti50) trimorphs is shown in Fig. 9. The trimorph consists of a Si plate cantilever and crystalline and amorphous NiTi ®lms of equal thickness on its two faces. Since the coecients of thermal expansion of the crystalline and amorphous NiTi are almost equal to each other, thermoelastic stresses are compensated. 5. DISCUSSION
A typical experimentally determined s(T) curve for a NiTi/(SiO2)/Si bimorph is presented in Fig. 7 together with the equilibrium line corresponding to equations (15), (16) and (18). The ®lm is highly textured [25], and each grain has a (110) plane
Fig. 9. A NiTi/Si composite micro-switch: the stress evolution of NiTi/Si bimorphs (top) and trimorphs (bottom) is indicated.
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parallel to the surface of the ®lm. Therefore, the speci®c self-strain for the B2±B19' martensitic transformation, e^ 0 , should be calculated as an average expansion in the (110) plane of the B2 phase. The analysis of preferable variants of monoclinic martensite in tensioned (110) textured ®lms is a rather complicated problem. However, its solution can be avoided by using the recently published calculation of the maximum contraction along [110] resulting from the B2±B19' transformation in a NiTi 50/50 alloy, e[110]= ÿ 5.2 10ÿ2 [26]. Then, the average expansion in the (110) plane, e0, related to the relative volume change due to the transformation, DV/ V, is 2e0 e110 DV=V
31 ÿ2
Since the dilatation DV/V = 0.5 10 lows that e0 2:85 10ÿ2
[27] it fol 32
Using this value of e0, the slope of the equilibrium transformation line becomes ds L 7 MPa=K dT T0 2e0
33
In the relation given by equation (33) values of the latent heat, L = 0.12 GPa [24], and the equilibrium temperature of the austenite±martensite phase, T0=300 K [28], have been used. The increase of Ms as well as of T0 due to the thermal stress is equal to MsÿTM=100 K. This dierence is in good agreement with the experiment if TM=230 K. The experimentally determined value of Mf is about 160 K (see Fig. 7). Thus MsÿMf=166 K which is a much broader temperature interval than usually observed for unconstrained transformations. Such a broad temperature interval was envisaged in the theoretical section. The slope ds/dT in the transformation temperature interval Ms
Fig. 10. Stress±temperature dependence for transforming 2.8 mm ®lm NiTi/(001)Si (after Ref. [5]) with the equilibrium line (TM=240 K).
(100)Si is presented in Fig. 10. While the parameters of the curve during cooling are comparable to the parameters describing the A 4 M transformation in the present experiments, the slope of the portion corresponding to the M 4 A transformation during heating is an order of magnitude larger than the equilibrium slope. It is suggested that this dierence re¯ects the dierent ®lm thicknesses of this and the Zhang±Grummon experiment. The ®lm thickness in the latter is about three times larger leading to a reduced incompatibility between grains during transformation. In thicker ®lms, the transformation microstructure evolves closer to the equilibrium path rendering the present theory more strictly applicable. Applying it to the reverse transformation it shows that the coecient g in equation (25) is very large. This is so, because if e0 becomes closer to a dilatation, the dierence (emÿeI) approaches zero. Correspondingly, the slope ds/dT becomes very steep. 6. SUMMARY
This paper represents a ®rst attempt to understand the martensitic transformation in constrained ®lms. A cubic to tetragonal transformation of single crystalline material has been used as a model system to simplify the analysis. The analysis leads to the conclusion that the constrained martensitic transformation in stressed ®lms in general should be structural and thermodynamically irreversible even if it develops as the evolution of an equilibrium microstructure. This irreversibility is the result of the dierent equilibrium domain microstructures which accommodate the ®lm/substrate and the austentite/martensite interface constraints. The analysis also determined that the equilibrium stress/temperature characteristic in the transformation region at cooling should follow a modi®ed Clausius± Clapeyron relationship. Contrary to the classic relationship, the stress entering it is not a constant
ROYTBURD et al.:
MARTENSITIC TRANSFORMATION
externally applied stress but the internal stress which varies as the constrained transformation proceeds. For the transformation in polycrystalline ®lms the modi®ed Clausius±Clapeyron relation describes the onset of transformation. The theory predicts considerable broadening of the temperature interval of transformation due to the constraint. These theoretical conclusions are in complete accord with the experimentally observed facts. Thus, while the present theoretical analysis does not yield a quantitative description of the transformation, particularly in polycrystalline ®lms, it sets the stage toward accomplishing this goal. AcknowledgementsÐThis study was supported by the National Science Foundation, Grant No. DMR-97-06815. It also bene®ted from support by the Oce of Naval Research, contract No. N00014-93-10506. REFERENCES 1. George, E. P., Gotthardt, R., Otsuka, K., TroilerMcKinstry, S. and Wun-Fogle, M. (ed.), in MRS Proc., Materials for Smart System II, Vol. 459, MRS, Pittsburgh, 1997. 2. Simmons, W. C., Aksay, I. A. and Huston, D. R. (ed.), in SPIE Proc., Smart Materials Technologies, Vol. 3040, SPIE, 1997. 3. Bidaux, J.-E., Yu, W. J., Gotthardt, R. and Manson, J.-A. E., J. Physique, 1995, IV, C2±543. 4. Krulevitch, P., Ramsey, P. B., Makowiecki, D. M., Lee, A. P., Northrup, M. A. and Johnson, G. C., Thin Solid Films, 1996, 274, 101. 5. Zhang, J., Grummon, D.S., in MRS Proc., Materials for Smart System II, Vol. 459, ed. E.P. George, R. Gotthardt, K. Otsuka, S. Troiler-McKinstry and M. Wun-Fogle (ed.), MRS, Pittsburgh, 1997, p. 451. 6. Roitburd, A. L., Physica status solidi (a), 1976, 37, 329. 7. Roytburd, A. L., in Thin Film Ferroelectric Materials and Devices, ed. R. Ramesh, Kluwer, Dordrecht, 1997, p. 71. 8. Roytburd, A. L., J. appl. Phys., 1998, 83, 228.
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9. Roitburd, A. L., in Solid State Physics, eds H. Ehrenreich, F. Seitz and D. Turnball, Vol. 33, Academic Press, New York, 1978, p. 317. 10. Khachaturyan, A. G., Theory of Structural Transformation in Solids, John Wiley and Sons, New York, 1983. 11. Stoney, G. G., Proc. R. Soc. A, 1909, 82, 172. 12. Wechsler, M. S., Lieberman, D. S. and Read, T. A., Trans. metall. Soc. A.I.M.E., 1953, 127, 1503. 13. Wayman, C. M., Introduction to Crystallography of Martensitic Transformations, Macmillan, New York, 1964. 14. Roitburd, A. L. and Kurdjumov, G. V., Mater. Sci. Engng, 1979, 39, 141. 15. Roytburd, A. L. and Slutsker, J., Mater. Sci. Eng. A, 1997, 238, 23. 16. Wang, Y. and Khachaturyan, A. G., Acta mater., 1997, 45, 759. 17. Roytburd, A. L. and Slutsker, J., J. appl. Phys., 1995, 77, 2745. 18. Roytburd, A. L., J. Physique, 1996, IV, CI±11. 19. Su, Q., Hua, S. Z. and Wuttig, M., Trans. Mater. Res. Soc. Japan., 1994, 18B, 1057. 20. Kim, T. S., Martensitic transformation in NiTi/Si composites. Ph.D. Thesis, University of Maryland, 1995. 21. Brantley, W. A., J. appl. Phys., 1973, 44, 534. 22. Hua, Z. S., Su, C. M. and Wuttig, M., in Proc. Symp. on Damping Multiphase Inorganic Materials, ed. R. B. Bhagat, AMS, Metals Park, OH, 1993, p.165. 23. Su, Q., Hua, S. Z. and Wuttig, M., J. Adhesion Sci. Technol., 1994, 8, 7. 24. Materials Properties Handbook: Titanium Alloys, ASM International, Pittsburgh, 1994. 25. Su, Q., Kim, T. and Wuttig, M., MRS Symposium Proc., Materials for Smart System, Vol.360, ed. S. Troiler-McKinstry, K. Uchino, M. Wun-Fogle, E.P. George and S. Takahashi, MRS, Pittsburgh, 1995, p. 375. 26. Plietsh, R. and Enrich, K., Acta mater., 1997, 45, 2417. 27. Otsuka, K., Sawamura, K. and Shimizu, K., Physica status soloidi (a), 1971, 5, 457. 28. Chang, L. and Grummon, D. S., Phil. Mag. A, 1997, 76, 191.