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Metals 2013, 3(3), 237-282; doi:10.3390/met3030237
Abstract: It is generally accepted that the martensitic transformations (MTs) in the shape memory alloys (SMAs) are mainly characterized by the shear deformation of the crystal lattice that arises in the course of MT, while a comparatively small volume change during MT is considered as the secondary effect, which can be disregarded when the basic characteristics of MTs and functional properties of SMAs are analyzed. This point of view is a subject to change nowadays due to the new experimental and theoretical findings. The present article elucidates (i) the newly observed physical phenomena in different SMAs in their relation to the volume effect of MT; (ii) the theoretical analysis of the aforementioned volume-related phenomena.
The widely used functional properties of the shape memory alloys (SMAs) stem from the martensitic transformation (MT) they exhibit during cooling or under the action of external fields of a different nature. The most important feature of martensitic transformation is the appearance of uniform shear strain of crystal lattice below the MT temperature resulting in the formation of martensitic phase [1,2,3]. Furthermore, MT is accompanied by the local shears within an invariant crystal lattice of the martensitic phase. These shears proceed in different directions and cause formation of the specific microstructure, which consists of the physically equivalent domains of the low temperature martensitic phase with the differently oriented crystallographic axes. Because of their physical equivalence, the domains of martensitic phase (martensite variants) are separated by the coherent, and therefore mobile, interfaces. The shear character of MT strains and appearance/presence of mobile interfaces give rise to a complicate thermoelastic behavior of the shape memory alloys and their engineering and medical applications [3,4,5,6,7]. The most of these applications are based on the shape memory effect and accessibility of large strain values under the moderate mechanical loading.
The ferromagnetic and metamagnetic shape memory alloys (FSMAs and MetaMSMAs) form a special class of the multifunctional materials (see [8,9,10,11,12,13,14] and references therein). These materials undergo both the ferromagnetic ordering and martensitic transformation (MT) on cooling. The considerable change in the martensitic structure of some FSMAs under the magnetic field occurs due to the extremely high mobility of coherent interfaces between the martensite variants. These changes cause the giant (up to 12%) magnetically induced deformation under the moderate (about 0.1 T or even less) magnetic field [15,16,17].
The martensitic transformations are accompanied by the small volume changes. The value of volume change (that is the volume effect of MT) was recognized since early development of SMAs as an important factor influencing the thermoelastic character of MT [1,2,18]. Particularly, a reduced volume effect during MT was observed in Fe-Ni-Co-Ti alloys ; it appeared as one of the crucial factors ensuring thermoelastic character of MT in these unique alloys [20,21,22,23].
The early experiments with the SMAs under hydrostatic pressure P have readily shown the interdependence between volume of SMA and characteristic MT temperature T0 [24,25,26,27]. Depending on the sign of volume change during MT the hydrostatic pressure increases or decreases MT temperatures. The positive sign of dT0/dP is observed, e.g., for the Au-Cd, Ti-Ni and Ni-Mn-Ga alloys [24,25,28], while Fe-Ni-Co-Ti SMAs show a strong negative slope of MT temperature as a function of hydrostatic pressure [26,27].
Generally, the volume changes caused by MT, hydrostatic pressure or external magnetic field are considered mainly as a subject of academic interest. Nevertheless, these issues applied to SMAs have also practical importance as already mentioned above. No wonder, the relevant studies are still continuing (e.g., ). In particular, the intriguing results were obtained recently for the MTs in FSMAs and MetaMSMAs. First, the decisive role of volume magnetostriction in the kinetics of a coupled magnetostructural phase transformation of Ni-Mn-Ga alloy was demonstrated . Second, the large contribution of the magnetic subsystem to the entropy change ΔS accompanying the MTs of the SMAs was discovered: it has been found that ΔS strongly decreases with the distance between the MT temperature and the Curie point TC [31,32,33]. As far as this behavior of the entropy change appeared to be inherent to the magnetic SMAs belonging to the different alloy systems, the spontaneous volume magnetostriction arising at the Curie point was assumed to be the universal factor that stipulates the magnetic contribution to the entropy change . This assumption was confirmed by the quantitative theoretical analysis of the experimental data available for the Ni-Mn-Ga FSMAs. Third, the drastic influence of strong magnetic field (1T < H < 20T) on the MTs of Ni-Mn-In and Ni-Mn-Sn MetaMSMAs was discovered (see [32,34] and references therein). In particular, the so-called “magnetic arrest” of forward MT and magnetically induced reverse MT were observed. Inasmuch as the driving force (the difference of the Gibbs potential of two phases) for the thermally induced MT is given by the product ΔSΔT, where ΔT is the supercooling below the phase equilibrium temperature, the arrest of forward MT is promoted by the combination of two physical effects, which are the decrease of the forward MT temperature TMF measured for the Ni-Mn-X (X = In, Sn, Sb) alloys under the elevated magnetic field and the decrease of ΔS value caused by the increase of the difference TC − TMF . A substantial contribution of spontaneous volume magnetostriction to the entropy change  suggests an idea about the role of forced volume magnetostriction in the arrest of forward MT and magnetic assistance to reverse MT.
Geometrically, the shear deformation of crystal lattice can be performed without the volume change. However, the rigorous theoretical analysis shows, that the shear deformation of crystal lattice and, in particular, the martensitic transformations are always accompanied by the volume change although the tendency to volume conservation exists in the literature and retards the measurement of volume change (see  and references therein). Furthermore, a consistent theoretical consideration shows that the crystal defects formation/redistribution during/after MT is also accompanied by the volume change. This volume change is thermodynamically conjugated to the “internal pressure” , which can stabilize  or destabilize  martensitic phase, depending on the type of MT and conditions of experiment.
Both stabilization and destabilization of martensitic state changes the functional properties of SMA, and therefore, the consideration of stabilizing and destabilizing pressures induced by the by external forces or internal factors is not only of academic interest but of practical importance as well.
In the present review article, a systematic analysis of physical effects related to the volume change during MT is presented. The analysis is based on the formal resemblance between the hydrostatic pressure produced by the hydrostatic mechanical load and internal pressure created by the crystal defects or thermodynamically conjugated to volume magnetostriction. This resemblance is substantiated in Section 2 using the symmetry conforming theory of cubic-tetragonal and cubic-rhombohedral MTs. The effect of hydrostatic pressure on the MT temperature, MT strain and soft elastic modulus of SMA is considered in Section 3. The martensite stabilization and destabilization caused by the martensite aging and cyclic MTs is considered in Section 4, Section 5, respectively. The role of the internal pressure in the appearance quasi-second-order MTs is discussed in Section 6. The experimental data that are relevant to the cubic-rhombohedral and cubic tetragonal MTs in the Ti-Ni, Au-Cd and Ni-Mn-Ga are used in the Section 3, Section 4, Section 5 for the quantitative estimations. In Section 7, the magnetic contribution to the entropy change during the MT is related to the volume magnetostriction and evaluated for the Ni-Mn-Ga and Ni-Fe-Ga alloys. A reasonable agreement between the theoretical results and experimental data is stated. As a whole, this review uncovers the unprecedented role of volume variation due to MT, defects reconfiguration or magnetostriction in the phenomena exhibited by practically important martensitic multifunctional materials.
2. Quantitative Theory of Thermoelastic Behavior of SMAs
2.1. Symmetry Conforming Approach to the Problem
The SMAs undergo phase transformations from high-symmetry (austenitic) to low-symmetry (martensitic) phase. The tendency to the minimum of elastic energy is commonly considered as the basic principle of transformational behavior of SMAs. To use this principle for the description of functional properties of SMAs, the elastic energy is expanded as a series in the strain tensor components εik. The mathematical structure of series is prescribed by the symmetry of crystal lattice: the series contains all combinations of strain tensor components, which are invariant with respect to the symmetry group of the crystal lattice in the high-symmetry (cubic) phase. The invariant series constructed for the different symmetry groups are presented in .
The elastic part of the Helmholtz free energy of cubic crystal can be expressed in terms of the linear combinations of strain tensor components, which are the basic functions of the one-dimensional, two-dimensional and three-dimensional irreducible representations of cubic group. The basic function of one-dimensional representation
The Gibbs potential of SMA is presented by the equation
To clarify the use of the Gibbs potential for the description of MTs the following remarks about the free energy, Equation 4, are pertinent.
First, the equilibrium strain values satisfy the extremum conditions ∂G/∂uα = 0, which are equivalent to the conditions ∂F/∂uα = 0 if σα = 0. The subscripts α = 1,2,3 and α = 4,5,6 correspond to the diagonal and non-diagonal stress/strain tensor components, respectively. It is of importance that the terms linear in uα values with α = 4,5,6 are absent in the Equation 4 for the free energy. It is easy to see that the equilibrium states with u2, u3 ≠ 0, u4 = u5 = u6 = 0 may exist in this case. These states can have the tetragonal symmetry and the values u2, u3 form the two-component order parameter of the cubic-tetragonal MT [39,40]. The terms linear in uα with α = 2,3 are present in Equation 4, but they vanish if u4 = u5 = u6. Due to this, the rhombohedral state with u4 = u5 = u6 ≠ 0, u1 ≠ 0, u2 = u3 = 0 may exist . In the presence of the terms of type (n = 1,2,3) the cubic-tetragonal MT is not possible, because the extremum conditions for the Gibbs potential show that in the course of phase transformation characterized by the two-component order parameter u2,3 the concomitant nondiagonal strains must appear. The appearance of these strains leads to the appropriate lowering of tetragonal symmetry of the product phase. In presence of the energy terms of type the phase transformation characterized by the three-component order parameter is accompanied by the appearance of the concomitant strains and the appropriate lowering of rhombohedral symmetry of the product phase.
Second, the series Equation 4 involves the terms of type. As so, the cubic-tetragonal phase transition is characterized by the appearance of the concomitant component , while the concomitant component arises in the course of the cubic-rhombohedral phase transition. The role of concomitant components is especially important for the purview of this article, because the value of 3u1 is the trace of strain tensor, and therefore, it is related to the volume change of the alloy as
Finally, it is of importance that the Equation 4 involves the third-order terms in the order parameters components (the coefficients of these terms are a4 and a5). As so, all MTs are the first-order phase transitions (see  and references therein). The detailed consideration of the role of third-order energy terms in the transformational behavior of alloys will be presented in Section 6.2.
2.2. Cubic-Tetragonal MT
In the case of cubic-tetragonal MT the non-diagonal strain tensor components u4, u5 and u6 are equal to zero in both parent and product phases. The tendency to volume conservation during MT is observed for all phase transformations of martensitic type. This tendency is expressed by strong inequality and therefore, the energy terms , and can be considered small as compared with and . In that case the equation for the Gibbs potential takes the following form
The nondiagonal stress components are not conjugated with the order parameter of cubic-tetragonal MT and are omitted in Equation 9 hence. The coefficient c2(T) changes sign when the cubic-tetragonal MT occurs. Due to this, the cubic-tetragonal MT is accompanied by a pronounced softening of the elastic modulus C′, see Equation 5. The volume change that accompanies cubic-tetragonal MT can be interrelated with the order parameter components. The condition ∂G/∂u1 = 0 results in the following relationship:
Using Equation 10, the u1 value can be excluded from Equation 9 and the Gibbs potential can be expressed through the order parameter components as
The expansion of the Gibbs potential in terms of the order parameter components (Equation 11) enables the description of the cubic-tetragonal MT in the framework of Landau theory of phase transitions. The conditions result in the equations describing the dependencies of the order parameter components on the temperature and pressure. A zero solution of these equations corresponds to the high-temperature (cubic) phase. The nonzero solutions correspond to three equivalent variants of tetragonal lattice, which are usually referred to as the x- y- and z-variants of martensitic phase. The principal axis (c-axis) of x-variant is parallel to the crystallographic direction  of cubic phase, the principal axes of y- and z-variants are aligned with  and  directions, respectively (for more details see [39,40]).
For the sake of definiteness, let us consider the transformation of cubic phase into z-variant of tetragonal phase occurring in the absence of anisotropic stressing (i.e., in the case of σ2 = σ3 = 0). For this variant of martensitic phase εxx = εyy and therefore, u2(T,P) = 0 in both austenitic and martensitic phases. The solution of the equation
As far as the volume change νMT is substantially smaller than the equilibrium value u0 of the order parameter u3, an approximate equality εxx + εyy ≈ −2εzz is held. Due to this, the equilibrium value of the order parameter can be evaluated from the experimental values of lattice parameters of cubic and tetragonal phases (a0 and a, c, respectively) using the relationships
In accordance with the general principles of Landau theory, the high-temperature (austenitic) phase is stable if the coefficient of the second-order energy term is positive, that is if . The solution describing the low-temperature (martensitic) phase is stable if . If the coefficients, Equation 12, are known, the equations
It should be emphasized that Equations 15–17 are suitable for a description of the transformational and deformational properties of the imperfect samples of paramagnetic and magnetically ordered shape memory alloys. Both defect formation/reconfiguration and magnetic ordering of the alloy results in the volume change, and therefore, induces an “internal” pressure. The pressure caused by the defects formation/reconfiguration is proportional to their concentration; the pressure that is induced due to the magnetic ordering is strictly related to the volume magnetostriction of an alloy.
For the quantitative description of the cubic-tetragonal MT, the coefficients of the Landau expansion for Gibbs potential (Equation 11) must be expressed through the basic measurable characteristics of SMA. For this purpose the relationships
In accordance with Equation 12, the value is independent of pressure. As so, the Equation 19 predicts that the shear modulus of cubic phase measured just before the start of MT depends on the pressure as the second power of MT strain measured just after the finish of MT. This prediction can be verified experimentally.
2.3. Cubic-Rhombohedral MT
The cubic-rhombohedral MT is accompanied by the appearance of non-zero values of shear strains u4, u5, u6 while the diagonal strain tensor components u2 and u3 are equal to zero in both cubic and rhombohedral phases. The tendency to volume conservation during MT is expressed by inequality . Due to this, the energy terms , and can be considered negligibly small as compared with and , respectively. In this case the expression for the Gibbs potential, Equation 4, is reduced to the form
The coefficient c3(T) changes sign when the cubic-rhombohedral MT occurs. This coefficient is proportional to the shear modulus C44 in the cubic phase (Equation 5), and therefore, the cubic-rhombohedral MT is accompanied by the pronounced softening of this modulus. The extremum condition for the Gibbs potential ∂G/∂u1 = 0 establishes the following interrelation between the volume change and order parameter components:
The substitution of the u1 value (Equation 23) into Equation 22 results in the following Landau expansion for the Gibbs potential
The dependencies of the order parameter components on the temperature and pressure values obey the equations ∂G/∂u4–6 = 0. A zero solution of these equations corresponds to the high-temperature phase. The nonzero solutions correspond to the four equivalent variants of rhombohedral lattice with the principal crystallographic axes parallel to , ,  and  crystallographic directions (for more details see, e.g., ). For the sake of definiteness, let us consider the transformation of cubic phase into the -variant of rhombohedral lattice occurring in the absence of anisotropic stressing (i.e., at σ4-6 = 0). For -variant of rhombohedral lattice u4 = u5 = u6 = uR. The solution of the equation
An equilibrium value of the order parameter is related to the specific elongation/contraction of the cubic unit cell in  direction as
The high-temperature phase is stable if . The solution describing the low-temperature phase is stable if . In the case when the Landau expansion coefficients are determined from the equations Equation 25, the conditions
The Equations 27–30 are suitable for description of the transformational and deformational properties of the imperfect paramagnetic and magnetically ordered shape memory alloys. The coefficients of Equation 25 in the Landau expansion for the Gibbs potential depend on the effective pressure that may be induced by the hydrostatic load, defects formation/reconfiguration, magnetic ordering of SMA, etc. For the quantitative description of the cubic-rhombohedral MT, these coefficients must be expressed through the basic measurable characteristics of SMA. The relationships
In accordance with Equation 25, the b* value is independent of pressure and hence Equation 32 predicts that the elastic modulus C44 of cubic phase measured just before the start of MT depends on the pressure as the second power of MT strain measured just after the finish of MT. The interdependence between the soft elastic modulus and MT strain can be verified in the experiments with the different alloys belonging to the same alloy family.
If all third-order and fourth-order energy coefficients are in roughly the same ratio to each other, the estimation is valid and the energy coefficient a3 can be estimated by the order of magnitude from the relationship
2.4. Summary of Basic Points and Preview of Applications of the Theory
Summarizing the main points of the symmetry conforming theory of MTs in the SMAs, we can conclude that this theory starts from the invariant (with respect to the cubic symmetry group) expression for the Gibbs potential. This expression can be represented as the Landau expansion in the order parameter components (see Equations 11 and 24). The expansion coefficients depend on the pressure produced by the hydrostatic mechanical load or on the “internal” pressure caused by crystallographic defects formation, chemical or magnetic ordering of the alloy, and any other physical factor, which changes the volume of SMA.
Formally, the pressure results in the renormalization of the Landau expansion coefficients. It is especially important for the purview of this article that the renormalization of the coefficients c2(T) and c3(T) of the second-order energy terms is proportional to (i) pressure value and (ii) the ratio of volume change during MT to the squared MT strain, that is
This follows from Equations 12 and 21 for c2 and Equations 25 and 34 for c3. In this line of reasoning, all physical effects, which can be described in terms of the internal pressure, must be closely related to the volume changes that accompany MTs.
The drastic influence of the volume change on the elastic, thermodynamic and magnetic properties of SMAs will be demonstrated in the next sections of this overview. Special attention will be paid to the effects caused by hydrostatic loading, martensite aging and cyclic across forward/reverse MTs. The physical consequences of spontaneous volume magnetostriction of ferromagnetic SMAs will be also described. The Landau expansion coefficients will be estimated for the representative Ti-Ni, Au-Cd and Ni-Mn-Ga alloys. The interdependence between the volume changes caused by the MTs in these alloys, on the one hand, and MT temperatures, MT strains and elastic modules, on the other hand, will be studied. The contribution of volume magnetostriction to the entropy change during MT of ferromagnetic shape memory alloy will be disclosed by consideration of the representative Ni-Mn-Ga and Ni-Fe-Ga alloys.
It is commonly known that the cubic-orthorhombic or cubic-monoclinic MTs are observed in many alloys. In some cases the product phases arising in the course of these MTs are characterized by the slightly distorted tetragonal or rhombohedral crystallographic cells. In these cases Equation 35 approximately relates the volume change to the main component of MT strain. It may be expected therefore that the conclusions derived from this equation are valid for the majority of thermoelastic alloys.
3. Martensite Stabilization/Destabilization under Hydrostatic Pressure
3.1. Interrelation between the Volume Change during MT and Pressure Effects
The hydrostatic compression stabilizes the martensitic phase if the volume of alloy decreases during the forward MT (i.e., if νMT < 0) and destabilizes it otherwise. The stabilization/destabilization of martensitic phase under pressure manifests itself basically as
an increase/decrease of the soft elastic modulus of martensite.
If the transformation goes in the quasiequilibrium way, the shifts of martensite start and martensite finish temperatures under pressure are approximately equal to the shifts of the lability temperatures of the martensitic and austenitic phases T1 and T2. In the simplest approach, the influence of pressure on MT can be characterized by the change of the average temperature T0 ≡ (T1 + T2)/2. This change proves to be strictly proportional to the pressure value [25,26] and the coefficient of proportionality dT0/dP can be estimated from the Clausius–Clapeyron relationship dT0/dP = vMTT0/qm, where m is mass density, q is the latent heat of MT normalized to the unit of mass. Hence, the shift of MT temperature under hydrostatic pressure is expressed as
The MT of cubic-tetragonal or cubic-rhombohedral type is accompanied by the pronounced softening of the shear elastic modulus C' or C44, respectively. The pressure influence on the value of shear modulus in the austenitic phase of alloy can be estimated using the relationship
Every physical factor, which causes the martensite stabilization, noticeably changes the functional properties of SMAs. In the particular case of martensite stabilization by alloy aging, this statement is confirmed by the numerous experiments (see [44,45,46,47] and references therein). To extend this statement to include the other factors of martensite stabilization, the influence of hydrostatic pressure on the MT strain and shear elastic modulus of SMA will be considered below. The Ti-Ni alloy system will be chosen as the illustrative example. The properties of Ti-Ni alloys deserve a consideration because of the broad use of these alloys in the medicine  and engineering [4,5,7].
3.2. Landau Expansion Coefficients Estimated from the Shear Modulus of Ti-Ni Alloy
The Ti–Ni alloys exhibit both cubic-monoclinic (B2→B19) and cubic-rhombohedral (B2→R) transformations . A consistent theory of the cubic-monoclinic phase transformation is not developed yet, while the cubic-rhombohedral transformation can be studied using Equations 26–30. The coefficients of Equation 26 can be estimated from the shear modulus , which describes the elastic response of alloy in the austenitic state to the mechanical loading in  crystallographic direction. In the presence of stress σ = σ, Equation 26 is modified as
The solution of Equation 39 can be presented as a sum of the spontaneous shear strain u0 (Equation 27) and elastic strain . The substitution of the values and into Equation 39 leads to the formula that interrelates the MT strain and the difference of shear modules characterizing the martensitic and austenitic states of an alloy:
It is quite natural that the difference of the elastic modules of two phases tends to zero in the limiting case u0 → 0, because in this case the phases become physically equivalent. Using Equation 26, one can express the elastic modulus of martensitic phase as
Substituting (see Equation 31) into Equation 41, one can derive the relationships
The Equations 28,39,42,43 enable the solution of two following problems:
a theoretical evaluation of the shear modulus from the experimental temperature dependencies of lattice parameters in the martensitic phase;
a restoring of the temperature dependence of MT strain from the temperature dependence of the shear modulus.
An example of the solution of the first problem, the cubic-tetragonal MT was described in . The solution of the second problem is presented below.
For the sake of definiteness, let us consider the experimentally observed cubic-rhombohedral MT in Ti49.5Ni50.5 alloy . The temperature dependence of C44 modulus of Ti49.5Ni50.5 alloy measured in  is shown in Figure 1 by circles.
The lability temperatures of martensitic and austenitic phases derived from the experimental values of soft modulus are T1 = 311 K and T2 = 300 K, respectively (see arrows in Figure 1). Moreover, the experiment shows that C44(T2) = 18 GPa. In this case Equation 42 results in the value C44(T1) = 4.5 GPa. The energy coefficients and b* = 2.9 × 107 GPa can be found from Equation 32 using experimental value ε = 2u0(T2) ≈ −1%  and estimation νMT ≈ −5 × 10−4 resulting from the Clausius-Clapeyron relationship (for more details see ). To describe an influence of the hydrostatic pressure, a dimensionless parameter Λ that is involved in the renormalized energy coefficients, Equation 25, should be estimated to be (see the last paragraph of Section 2.3). The estimated values enable calculation of the ratio a3/c1 and coefficients c3(T,P), from Equations 33 and 25. Due to this, a quantitative description of the transformational behavior of the single-crystalline sample under the action of uniaxial mechanical loading becomes possible (the appropriate theoretical procedure is described in ).
3.3. The Influence of Pressure on the MT Strain and MT Temperature
Solving of Equation 41 with respect to u0 value enables the quantitative description of the pressure influence on the MT strain ε if the parameter Λ, energy coefficients , b*, and function C44 (T,0) are known. The energy coefficients were determined above for the Ti49.5Ni50.5 alloy. The C44 (T,0) function must coincide with experimental temperature dependence obtained for the austenitic and martensitic phases of this alloy and satisfy Equation 42. This function is shown in Figure 1 by the solid line. The figure demonstrates that experimental temperature dependence of shear modulus does not satisfy the condition of Equation 42. A disagreement between theory and experiment is obviously caused by the appearance of mixed two-phase state in the temperature range of T2 < T < T1.
Figure 2(a) shows the solutions of Equation 41 computed for different Λ values using the C44 (T,0) function shown in Figure 1 by the solid line. It is seen that the pressure of 1 GPa noticeably changes the temperature dependencies of the MT strain if Λ = −200 or 400 but leaves it almost unchanged if Λ = 200. It means that the computations show the non-monotone dependence of MT strain on Λ value. If Λ = −200, the theory predicts a monotone dependence of MT strain on pressure; the pressure of 2 GPa increases the absolute value of MT strain by 25% (Figure 2(b)).
The shifts of MT temperatures caused by a hydrostatic pressure are measured in experiments (see, e.g., [25,26,28]). It is observed that the shifts of the forward and reverse MT temperatures are close in value. To describe the influence of pressure on the MT temperatures, an average value T0(P) = [T1(P) + T2(P)]/2 of lability temperatures T1(P) and T2(P) can be computed. This value is shown in Figure 3(a). It is seen that the variation of MT temperature is roughly equal to 10 K/GPa and 2 K/GPa for the values Λ = 200 and Λ = −200, respectively. The difference ΔT(P) = T1(P)−T2(P) (that is the width of two-phase range) also depends on pressure (see Figure 3(b)). The pressure application diminishes the width of two-phase temperature range. This diminution is especially pronounced if the parameter Λ is positive: in this case the width of two-phase temperature range is less than 2 K for the wide interval of pressures.
It should be remembered that the conclusions derived from Figure 2, Figure 3 are valid in the case of negative value of the MT strain ε. If the MT strain is positive, the sign of parameter Λ should be changed to keep in force these conclusions.
3.4. The Influence of Pressure on the Shear Elastic Modulus
A measurement of the shear elastic modulus C44 of SMA in the martensitic phase is a difficult task even in the absence of pressure. In the same time, the value of this modulus strongly affects the temperature dependence of heat capacity  and other thermodynamic properties of SMAs. It stipulates an importance of the calculation of the pressure and temperature dependencies of shear modulus. The results of computations are presented in Figure 4.
The computations show that the pressure of about 1 GPa drastically increases the shear modulus in the martensitic phase and decreases it in the austenitic phase. This result explicitly demonstrates the stabilization of martensitic phase and the appropriate destabilization of the austenitic one. If the parameter Λ is negative, the pressure influence on the shear elastic modulus of austenite is less pronounced than if Λ > 0 but still observable, nevertheless.
It should be stressed, that in view of the ambiguity in Λ values, an uncertainty of the theoretical results arises. However, these results are in a qualitative agreement with experimental data reported in . Therefore, the strong influence of the hydrostatic pressure on the MT strains and soft elastic modules of SMAs should be expected. This influence can manifest itself, in particular, as the change of heat capacity of the alloy in the wide temperature range (see ).
4. Stabilization of Martensitic Phase by Its Aging
4.1. Isotropic Mechanism of Martensite Stabilization
It is commonly known that a spatial distribution of the lattice defects evolves slowly during aging of the shape memory alloys in a martensitic phase [46,53]. This evolution causes the well-pronounced physical effects, which include:
a gradual change of the lattice parameters during martensite aging ;
The works interrelating the effects of aging to the spatial symmetry of defect subsystem can be singled out from the numerous publications related to this subject [44,53,57,58,59]. For the particular case of the point defects, the Symmetry-Conforming Short-Range-Order (SC-SRO) principle has been formulated . According to this principle, the energy of the crystal with defects is minimal when the probabilities of finding point defects in the certain crystallographic positions have the same symmetry as the crystal lattice has. A substantial generalization of SC-SRO principle follows from the special version of a Landau theory of phase transitions, which has been developed very recently for the description of interplay between the deformation of crystal lattice during MT and reconfiguration of defect subsystem [35,36,37]. Firstly, this theory states that the defects of any type induce the “internal” (that is existing even in the absence of external forces) mechanical stress and the energy of every (austenitic or martensitic) phase is minimal when the symmetry of stress tensor corresponds to the symmetry of crystal lattice in this phase. Secondly, the theory takes into account that the tendency of the crystal with defects to the ground state is accompanied by a volume change.
Stress tensor describes an influence of defects on the crystal lattice, and therefore, depends on time t due to a slow reconfiguration of the defect subsystem during martensite aging. This tensor can be subdivided into the isotropic and axial parts as
Time-variation of the axial stress is controlled by the SC-SRO principle and is responsible for the difference in the yield stresses characterizing the stress–strain tests performed before and after the martensite aging. Moreover, the axial internal stress contributes to the martensite stabilization [35,36,60,61].
A volume change caused by the evolution of a defect subsystem after MT does not result in the symmetry change, and therefore, is an extrinsic to the SC-SRO principle. This volume change provides an additional (to symmetry conformation) physical mechanism for the martensite stabilization: if the volume changes arising during and after forward MT are of the same sign, the latter retards the reverse MT and stabilizes martensitic phase therefore. This is the isotropic mechanism of martensite stabilization [35,36,62].
It was shown recently that the martensite stabilization in the representative Au-Cd and Ni-Mn-Ga alloys is caused mainly by the isotropic mechanism [35,36]. Due to this, an evolution of the transformational properties of these alloys is satisfactorily described by the equations presented in the Section 2, Section 3, if in these equations, the hydrostatic pressure P is replaced by the time-dependent internal pressure P(i)(t). In accordance to the arguments presented in Section 2.4, Section 3.1, an influence of the internal pressure on transformational properties of SMA is characterized by a product . It means that the larger is a ratio of the volume change during MT to the MT strain (other parameters being equal), the more pronounced are aging effects.
The isotropic mechanism of a martensite stabilization can be illustrated by the consideration of martensitic transformation of Au-Cd alloy.
4.2. Martensite Stabilization in Au-Cd Alloys
The Au-Cd alloys exhibit MT of the cubic-rhombohedral type. For the theoretical description of martensite stabilization by aging, the Gibbs potential, Equation 24, with time-dependent coefficients
Few thermodynamic characteristics of the Au–Cd alloy are needed for the determination of the time-dependent coefficients of the Gibbs potential, Equation 45. Unfortunately, the complete set of characteristics has never been measured for one certain alloy, and therefore, some representative alloy belonging to the Au–Cd alloy family should be considered. The experimental values of T2 ≈ 300 K , C44(T1) ≈ 40 GPa  and εM = 2uR ≈ −0.8%  are reported for few Au–Cd alloys and the typical value of volume change during the thermoelastic MT, vMT ≈ −3 × 10−3 , can be accepted. A negative value of can be chosen (see below). With known shear modulus, the transformation strain and volume change enable us to evaluate the coefficients a3/c1, a5(0) = 1.6 × 105 GPa and b*(0) = 4 × 107 GPa from Equations 31–34. The Clausius–Clapeyron relationship with the values q ≈ 1 J g−1 and m≈ 14 g cm−3  results in the characteristic temperature value of T1 = 306.8 K.
The values presented above prescribe a time evolution of the coefficient whose temperature dependence leads to the phase transformation. As a consequence, the characteristic MT temperatures determined from the Equation 30 prove to be time-dependent.
The dependence of austenite start and finish temperatures on aging time t can be characterized by the temperature differences and . The value is the temperature range for the two-phase region, that represents a mixture of austenitic-martensitic structures. These temperature differences were computed using the aforementioned set of physical values and Equations 46 and 47. The relaxation time of τ = 50 min and the saturation value of internal pressure P(i)(∞) = 3.8 GPa were determined by fitting ΔAF(t) theoretical curve to the experimental values reported in . The results of computations are shown in Figure 5 together with the austenite finish temperature changes experimentally observed in .
It was shown in the Section 2.3 that the absolute value of the dimensionless parameter Λ can be estimated as . The negative value of Λ was used for obtaining results presented in Figure 5. The negative value was chosen because in this case the theory shows that the martensite aging causes broadening of the two-phase temperature range of the reverse MT (see two-side arrow in Figure 5). This feature of the transformational behavior of aged alloys is indirectly confirmed by the experiments in [67,68,69,70].
The computations performed for the positive Λ values demonstrate a narrowing of the two-phase temperature range by the martensite aging. This type of transformational behavior of an alloy is not typical but, indeed, it was observed for Co-Ni-Ga alloy .
It should be emphasized that the stabilizing pressure renormalizes the parameter , which prescribes that MT must be the first-order phase transition. According to Equation 45 and argumentation presented in Section 2.4, we have:
The values presented above result in the following estimation . This estimation shows that the jumps of physical values observed in the course of MT would be enlarged by factor of 1.4 if the stabilizing pressure of the order of 4 GPa should appear.
The time evolution of the crystal defects subsystem leads to a gradual volume change of the SMA aged in the martensitic state.
According to the general principles of thermodynamics, the time-dependent internal pressure can be defined as the thermodynamic value, which is conjugated to the volume change.
The internal pressure contributes to the martensite stabilization effect but does not change the symmetry of the crystal lattice; therefore, the gradual volume change of the alloy, held in the martensitic phase, provides the isotropic mechanism of martensite stabilization, which is extrinsic to the commonly known SC-SRO principle formulated in [44,53].
The quantitative theoretical analysis of the experimental results in Ref.  shows that the stabilization of rhombohedral martensitic phase of Au-Cd is caused mainly by the isotropic mechanism and the corresponding value of internal pressure is about 3.5 GPa .
5. Destabilization of Martensitic Phase by Thermomechanical Cycling
5.1. Introductory Statement
As we had demonstrated, the reconfiguration of defect system strongly influences the elastic and transformational properties of SMA. In particular, holding an alloy during long time in the martensitic phase leads the crystal lattice with defects to the quasiequilibrium state and causes the martensite stabilization. It suggests an idea that the cyclic thermally- or stress-induced MTs can disturb the crystal lattice with defects and move it out of an equilibrium. As so, the changes in the alloy properties, which are opposite to those described in the previous section, can be observed. This idea was confirmed in the course of cooperative (experimental and theoretical) studies of the cyclic thermally- and stress-induced MTs .
The effect of the thermal, mechanical and combined thermomechanical cycling on the elastic properties and transformation behavior of Ni57.5Mn22.5Ga20.0 alloy has been studied [37,72]. The alloy undergoes MT from cubic phase with lattice parameter a0 = 0.587 nm to a non-modulated tetragonal phase with lattice parameters a = 0.543 nm and c = 0.665 nm (c/a = 1.22).
The Sample 1 with dimensions of 0.19 × 0.5 × 9.1 mm3 and Sample 2 with dimensions of 0.27 × 0.28 × 8.5 mm3 were cut from the same Ni57.5Mn22.5Ga20.0 single crystal. Both samples were oriented in  crystallographic directions. The details of specimen preparation are reported in [37,72]. A TA Instruments Q800 dynamic mechanical analyzer (DMA) was used to measure: (a) the temperature dependence of the low-frequency elastic modulus in the dynamic tensile mode and (b) the tensile stress–strain curves in a quasi-static mode.
Experimental temperature dependence of the elastic modulus of Sample 1 is shown in Figure 6 by the solid line. An abrupt change of elastic modulus in the temperature ranges of the forward (545 K < T < 575 K) and reverse (585 K <T < 625 K) MTs is observed. The horizontal arrows in Figure 6 illustrate that the modulus measured at martensite finish temperature TMF = 545 K or austenite start temperature TAS = 585 K is four times as large as that measured at martensite start temperature TMS = 575 K or austenite finish temperature TAF = 625 K, respectively.
After measurements of the elastic modulus, Sample 1 was subjected to the stress–strain cycles for obtaining a stress–temperature phase diagram. The cycles were performed at different temperatures Tn, where n = 1,2…15 is the cycle number; the sample was heated to 673 K and held at this temperature during 10 min before each cycle. The critical stress values σAF(T) and σMS(T), which correspond to the finish of reverse MT and the start of forward MT, were determined from the stress–strain curves. These values are shown in Figure 7.
The temperature dependence of elastic modulus was measured again after the 15th cycle to study the influence of thermomechanical cycling on the alloy properties. A drastic decrease of elastic modulus after the thermomechanical cycling was observed (see Figure 8).
Sample 2 was used to investigate the influence of the cyclic stress-induced MTs on the superelastic properties. Ten sequential stress–strain cycles have been performed at constant temperature 673 K. The stress–strain loops obtained during the first and tenth cycles are shown in Figure 9.
After mechanical cycling, the unloaded Sample 2 was subjected to 10 heating–cooling runs through MT and a noticeable shift of the reverse MT temperature TR ≡ (TAS + TAF)/2 was observed (see Figure 10). The shift of start of the forward MT temperature TMS proved to be comparable with the scatter of experimental points.
5.3. Qualitative Explanation of Experimental Results
The experiments demonstrate a noticeable softening of the elastic modulus of martensite after thermomechanical cycling (Figure 6). This effect suggests that cycling through the forward and reverse MTs destabilizes the martensitic state of the alloy. The suggestion is confirmed by an elevation of the plateau-like segments of the superelastic stress–strain loops in the course of the cyclic stress-induced MTs of Sample 2 (Figure 9): a destabilization of martensitic phase has to be accompanied by a decrease of MT temperature, and so, the martensite destabilization broadens an interval between the MT temperature and the constant temperature of alloy maintained during the cyclic mechanical loading. The movement of MT temperature away from the alloy temperature retards the stress-induced MT and elevates the critical stress value, which starts this MT [73,74,75,76,77].
The observed destabilization of martensitic phase admits a simple explanation. If the SMA is in the martensitic state during rather long time, the internal stress brings the real crystal lattice, which consists of the atoms situated in the regular crystallographic positions and crystal defects, into its equilibrium state, and therefore, stabilizes the martensitic phase. The cyclic MTs disturb an equilibrium state of the real crystal lattice and destabilize the martensitic phase. So, the martensite destabilization by cyclic MTs and martensite stabilization by aging can be considered as the opposite physical effects.
It is convenient to describe the martensite destabilization in terms of destabilizing internal stress, i.e., to count up the internal stress from its value reached before a thermomechanical cycling. In this case the destabilizing stress caused by cycling and the stabilizing stress that arises during the martensite aging are of the same order of magnitude with the opposite signs. Due to this, the expected absolute value of destabilizing stress can be roughly estimated from the experiments on martensite aging.
The martensite destabilization may be, in principle, caused by both isotropic and axial part of the internal stress, Equation 44. However, the absolute value of axial internal stress (about few tens of megapascals ) observed in the experiments on martensite aging is not sufficiently large to be responsible for the strong influence of cycling on the elastic modulus. In the same time, the elastic modulus can be changed by the internal pressure (see Section 3, Section 4). This assumption is supported by the comparatively large volume changes (about 0.5% ) observed during MTs of Ni-Mn-Ga alloys into a non-modulated tetragonal phase with c/a > 1.
5.4. Quantitative Description of the Martensite Destabilization Effects
Physical considerations, which are essentially similar to those presented in Section 4.2, lead to the following dependence of destabilizing internal pressure on the cycle number n:
The transformational properties of Ni-Mn-Ga alloys are described by minimization of the Gibbs potential, Equation 11. The Gibbs potential coefficients depend on the cycle number as
These dependencies were obtained from Equation 12 by substitution P → P(i)(n).
The further analysis of problem is similar to that presented in the Section 2, Section 3. The austenitic phase is stable if , whereas the martensitic phase is stable for . The equations and prescribe the values of the lability temperatures T1(n) and T2(n). The shear elastic modulus of austenitic phase is related to the coefficient of the second-order energy term as
The value of this modulus in the martensitic phase C(M) can be expressed through the equilibrium value of the order parameter u0 as explained in the Section 3.2, the resultant expression
Now, let us use the experimental temperature dependence of the elastic modulus, measured before cycling of the Ni–Mn–Ga alloy (Figure 6), for a consistent description of the effect of cycling on this modulus (Figure 8) and MT temperatures (Figure 10). To this end, the Gibbs potential coefficients must be evaluated. It can be done assuming that the measured elastic modulus is close in value to the Young's modulus E(T). The latter can be approximately expressed as E(A,M)(T) ≈ 3C(A,M)(T) , because the soft elastic modulus of SMA is substantially smaller than its bulk modulus.
In view that the Landau theory does not describe the hysteresis phenomena properly, Equation 15 are considered to be applicable to both the forward and reverse MT. Therefore, the conditions u0(TMS) = u0(TAF) and u0(TMF) = u0(TAS) are accepted. These conditions together with Equation 52 result in the relationships C(A)(TMS) = C(A)(TAF) and C(M)(TMF) = C(M)(TAS). The experimental temperature dependencies of the elastic modulus validate these theoretical relationships (see Figure 6).
The coefficients c2(T), a4 and correspond to the state of the real crystal before cycling (that is to zero value of destabilizing stress). To evaluate these coefficients, the experimental dependencies of elastic modulus were approximated by the straight dashed lines shown in Figure 6 and Equations 18 and 19 with C' (T1,2) ≈ E(TMS,MF)/3 were used. The approximation was made taking into account Equation 53. In addition, the experimental values of tetragonality ratio c/a = 1.22 and the volume change vMT ≈ −0.5%  were used for computations. These experimental values give rise to the estimations u0(TMF) = 2(c/a-1) = 0.44, a4 = −10.1 GPa, and a2/c1 = 0.017, which are the same for both forward and reverse MTs. Note, that the procedure of obtaining these estimations is similar to that described in Section 3.3, Section 4.2. The difference between the forward and reverse MTs arises when the different experimental temperature dependencies of the elastic modulus (see Figure 6) and correspondent characteristic temperatures (T1 = TMS, T2 = TMF for the forward MT, and T1 = TAF, T2 = TAS for the reverse MT) are substituted into Equations 51 and 52.
Using values obtained above, the functions and were calculated for theoretical description of the influence of thermomechanical cycling on the temperature dependence of elastic modulus of Sample 1. The expression for involves a dimensionless parameter Ω. According to argumentation presented in Section 2.2, the value was accepted. For the sake of definiteness, the value n0 = 4 was used for computations. Characteristic values of the internal pressure P(i)(∞) = −6 GPa (for the forward MT) and P(i)(∞) = −7.5 GPa (for the reverse MT) were adjusted to achieve a reasonable agreement between the theoretical and experimental temperature dependencies of the elastic modulus. Figure 8 shows that an agreement between the experimental and theoretical dependencies takes place in rather wide temperature interval, ~100 K, below the martensite finish (for the forward MT) and austenite start (for the reverse MT) temperatures.
Both experiment and theory shows a significant reduction of the elastic modulus in the martensitic phase and a noticeable decrease of MT temperatures after cycling. These two features reveal a martensite destabilization effect.
Finally, the experimentally observed effect of thermal cycling on the reverse MT temperature (Figure 10) should be described theoretically. For this purpose, the experimentally determined temperature values TR(n) = [TAS(n) + TAF(n)]/2 are compared with the calculated ones T0 (n) = [T1 (n) − T2 (n)]/2 (see Figure 10). The theory demonstrates a noticeable decrease of the T0 temperature in the course of thermal cycling. The agreement between the theoretical and experimental data takes place if the value of internal pressure is put equal to −1.5 GPa. The decrease of TR temperature demonstrates that the thermal cycling destabilizes the martensitic phase and the destabilizing pressure is close in value to the pressure that arises in the course of the mechanical cycling (see ). The theoretical dependence T1(n), which must be compared with the experimental TMS(n) values, is less pronounced. The experiment does not show a noticeable change of TMS temperature in view of the scattering of experimental temperature values, this scattering being comparable in magnitude to the temperature change predicted by theory.
5.5. Summary of Destabilization Effects
It can be summarized now that the destabilization of the Ni57.5Mn22.5Ga20.0 martensite has been discovered in the course of the experimental and theoretical studies of the cyclic MTs. The thermal and/or mechanical cycling of the shape memory alloy disturbs the equilibrium between the crystal lattice and defects. The deviation from the equilibrium is characterized by the internal stress, which destabilizes the martensitic phase. Due to the rather large volume effect of MT (vMT ≈ −0.5%), the majority of experimentally effects of cycling observed in Ni57.5Mn22.5Ga20.0 alloy can be attributed mainly to the influence of negative internal pressure that is the isotropic part of internal stress. Both experiment and theory shows that the cyclic MTs of the alloy result in
a decrease of the elastic modulus of martensitic phase (Figure 8);
a decrease of the MT temperature (Figure 10).
Experiment shows that the cycling increases noticeable the stress value, which in turn triggers the stress–induced MT. As it was shown in , this effect can be attributed to the influence of internal pressure as large as −1.7 GPa but a theoretical description of the stress-strain dependence is obviously impossible as long as the axial stress is disregarded. A consideration of the axial stress is beyond the scope of the present article.
The theoretically estimated negative internal pressure (causing an isotropic dilatation of the crystal lattice) is about –2 GPa, in the case of mechanical or thermal cycling , and –6 GPa, in the case of thermomechanical cycling. A drastic difference between two estimated values of the internal pressure shows that combined thermomechanical cycling destabilizes the martensitic phase stronger than the pure mechanical or thermal cycling procedures performed in the experiments .
6. Quasi-Second-Order Martensitic Transformations
6.1. General Considerations
The numerous experiments show that MTs normally are the first-order phase transitions, but in some cases the quasi-second-order MTs are observed as well . As far as the transformational behavior of SMAs is believed to be controlled by a principle of the minimum elastic energy, an observation of the second-order MT is in contradiction to a Lifshitz criterion (see  and references therein). The latter states that the second-order phase transition is impossible if the symmetry of the crystal allows the presence of cubic term of the order parameter in the Landau expansion for the free energy. It should be remembered that all quadratic, cubic and fourth-order invariant combinations of the order parameter components must be included in the expression for the free energy. This statement cannot be referred to as the “model assumption”; actually it is the basic principle of the theory of various physical phenomena in crystals (optical, ultrasonic, piezoelectric, ferroelectric, magnetic, and so on).
The Landau expansion for the free energy, Equation 4, involves the terms and a5u4u5u6, which are proportional to the third power of the order parameters of the cubic-tetragonal and cubic-rhombohedral MTs. It means that the martensitic transformations must be the first-order phase transition and the jumps of the values of lattice parameters and elastic modules at MT temperature must be predetermined by a4 or a5 values (for the alloys exhibiting cubic-tetragonal or cubic-rhombohedral MT, respectively).
An obvious disagreement between the theory of phase transitions and experimental observations of quasi-second-order MTs can be resolved in two ways.
First, the axial internal stress can be assumed to be present in the alloy specimens. This stress can reduce the cubic symmetry of crystal to tetragonal, rhombohedral, monoclinic or triclinic ones. In this case the MT is not accompanied by the symmetry change and appearance/disappearance of some physical value at MT temperature . As so, the MT can be registered only by the observation of the jumps of the lattice parameters and jump-like changes of relevant physical values. If the axial stress exceeds some critical value, the jumps of lattice parameters vanish and the phase transition disappears; instead, the smooth nonlinear temperature dependence of lattice parameters takes place (for more details see [80,81]).
Second, the internal pressure can be induced by the crystal defects or chemical disorder. These factors are formally described by a renormalization of the energy coefficients and , see Equations 45 and 50, respectively. Using the Equations 20 and 33, the renormalized coefficients can be expressed through the volume effect of MT and MT strain as follows:
Let some internal process or factor induce an internal pressure P(i). The positive/negative internal pressure decreases/increases the volume of an alloy. The negative/positive value vMT corresponds to the spontaneous decrease/increase of the volume during the forward MT. The internal process or factor results in the stabilization of martensitic phase if the spontaneous and forced volume changes are sign-like (see e.g., ), that is if the values P(i) and vMT are of the opposite signs. It means P(i)vMT < 0for the stabilizing pressure and P(i)vMT > 0for the destabilizing one.
The martensite stabilization results in an increase of the absolute value of MT strain, and therefore, denotes an increase of the absolute values of the energy parameters or . If so, the values a4 and Ω or a5 and Λmust be of the opposite signs. In the same time, the signs of parameters and are opposite to the signs of MT strains (see Equations 15 and 28). Therefore, the parameter Ω or Λ must be of the same sign as the strain arising in the course of cubic-tetragonal or cubic-rhombohedral MT. The martensite stabilization accompanies the martensite aging. Therefore, the martensite aging denotes an increase of or magnitude and makes the first-order character of MT more pronounced.
It should be expected that the process or factor that destabilizes martensitic phase (reduces the MT temperature) denotes the decrease of or magnitudes and glosses over the first-order character of MT. The destabilizing internal pressure is of the same sign as the spontaneous volume change vMT. It may be possible to observe a quasi-second order MT in presence of destabilizing internal pressure, which considerably reduces the absolute values of the coefficients and .
6.2. Application to the Ni-Mn-Ga and Au-Cd Alloys
The energy coefficient has been evaluated for Ni-Mn-Ga alloy with a/c < 1 (u0 ≈ −0.12) in . The reported in this work data show that the destabilizing internal pressure of about of −3 GPa reduces the value by one order of magnitude if the values Ω = 2u0−1 and vMT = −0.5% are accepted for estimations. The choice of Ω value is explained in Section 2.3. The values of volume changes during MTs in the Ni-Mn-Ga alloys with c/a < 1 are disputed: the X-ray diffraction data result in the large value vMT ≈ −2·10−2, while the values estimated from thermodynamic measurements are smaller by almost one order of magnitude (see  and references therein). In view of this ambiguity, the ratio is presented in Figure 11 for the low (0.1%), intermediate (0.5%) and high (2%) volume changes.
In the case of intermediate and large volume changes, the internal pressure can reduce the value by one order of magnitude. In this case the MT in the alloy considered in  may bear resemblance to the second-order phase transition. It should be noticed that the theoretical analysis of aging effects in Ni-Mn-Ga martensite with c/a < 1 gave a comparatively low value of stabilizing pressure P(i)(t) < 1 GPa , but larger in the absolute value destabilizing pressure can be induced by the thermomechanical cycling of alloy specimens or some other mode of treatment.
The influence of internal pressure on the transformational and elastic properties of the rhombohedral Au-Cd martensite and tetragonal Ni-Mn-Ga martensite with c/a > 1 was considered in the Section 4, Section 5 and the values of physical parameters inherent to these alloys were listed. It is worth to compute the energy coefficients, Equation 54, for these martensites. The results of computations are shown in Figure 12.
The stabilizing pressure P(i) = 3.8 GPawas estimated for the aged Au-Cd alloy in Section 4.2. The arrows in Figure 12 illustrate that the destabilizing pressure of the same absolute value reduces coefficient by factor of 0.55.
The destabilizing pressure of about –6 GPa was induced by thermomechanical cycling of Ni-Mn-Ga martensite with c/a > 1 (see  and Section 5.3). The arrows in Figure 12 illustrate that this pressure reduces value by factor of 0.93 only. The weak renormalization of the coefficient follows from the large value εM ≈ 15% being inherent to the considered Ni-Mn-Ga martensites: according to Equation 54, the renormalization of is proportional to the ratio of the dimensionless values , which is approximately equal to 300 for the Ni-Mn-Ga alloys with c/a > 1 and is of the order of −104 for the Ni-Mn-Ga alloys with c/a < 1.
It should be stressed that in view of an ambiguity in Ω, Λ and P(i) values, the data presented in Figure 11 and Figure 12 have the illustrative character. Nevertheless, these data and the results reported in [41,81] provide a plausible explanation of the quasi-second-order MTs in some shape memory alloys. Moreover, the consideration presented above shows that the resemblance between MT and the second-order phase transition presumes not only the certain alloy composition but also a special route of preparation of the alloy specimen.
7. Volume Changes Contributing to the Entropy in Magnetic Shape Memory Alloys
7.1. Elastic and Magnetic Components of the Entropy Change during MT
Ferromagnetic and metamagnetic shape memory alloys (FSMAs and MetaSMAs) are attracting considerable interest due to the unique properties they show as a consequence of the coupling between their transformational properties and magnetism. The recent measurements of the entropy change ΔS accompanying the MTs in the different representatives of MetaSMAs, such as Ni-Mn-In and Ni-Mn-In-Co alloy systems, discovered one of the most obvious manifestations of this coupling. It has been found that ΔS strongly decreases with the distance between the MT temperature, TM, and the Curie temperature, TC ≥ TM [31,32]. A similar decrease was observed when the available experimental data about the entropy change and transformation heat were collected and analyzed for the Ni-Mn-Ga FSMAs with TC ≥ TM . It suggests that the magnetic subsystem of the FSMAs and MetaSMAs considerably contributes to the entropy change. Inasmuch as the spontaneous deformation of crystal lattice is considered as the key feature of MT, a coupling between the strain tensor components and magnetic moments of atoms, commonly referred to as magnetoelastic coupling , would be thereafter taken into account to estimate the magnetic contribution to the entropy change that accompanies MT. In the case of ferromagnetic ordering, the magnetoelastic coupling is described by the energy terms that are linear in the strain tensor and quadratic in the magnetization vector components. According to arguments advanced in the Section 2.1, Section 2.4, the magnetoelastic energy can be expressed through the magnetically induced internal stress, which, in its turn, can be subdivided into the isotropic and anisotropic parts. The isotropic part (magnetoelastic pressure) is responsible for volume magnetostriction.
In the absence of external forces, the Gibbs potential of thermodynamic system coincides with the Helmholtz free energy. The expression for the free energy of FSMA that exhibits cubic-tetragonal MT can be presented in the form
Elastic part of the free energy Fel is expressed through the linear combinations u2, u3 of diagonal strain tensor components by Equation 11 with P = σ2 = σ3 = 0:
The magnetic part of free energy is the sum of spin-exchange and magnetic anisotropy energies expressed through the magnetization vector components Mk as
In accordance with the general principles formulated in Section 2.1 the expression for magnetoelastic energy is invariant with respect to the symmetry group of the cubic (paramagnetic austenitic) phase of the alloy. This expression is
The Equation 58 can be presented in the form
It is commonly known that the axial magnetostress must be taken into account when the anisotropic properties of FSMAs or non-scalar physical values (as the axial magnetic-field-induced strain) are considered. However, the volume magnetostriction and magnetoelastic pressure are caused by the spin-exchange interaction, whereas the anisotropic part of magnetoelastic energy and anisotropic part of magnetoelastic stress are mainly related to the spin-orbit interaction. Therefore, the isotropic part of magnetoelastic interaction is much larger than the anisotropic part, and so, the latter can be omitted when the influence of magnetoelastic coupling on the martensitic transformation heat and the relevant entropy change are considered. Noteworthy, in the Ni-Mn-Ga alloys, which are the most studied FSMAs, the magnetoelastic pressure exceeds the anisotropic magnetoelastic stress by two orders of magnitude [41,82,83,84,85], at least, and the disregard of this stress is substantiated by the numerical calculations. The neglect of anisotropic magnetic and magnetoelastic energy terms with respect to isotropic ones results in the expression
In view that we are interested mainly in the magnetic contribution to the entropy change, the FSMAs, exhibiting the martensitic transformations in ferromagnetic state will be considered below.
The procedure of calculation of the entropy change caused by the phase transition is described by L.D. Landau and E.M. Lifshitz . The calculation has to be performed in two steps. At the first step, the partial derivative of free energy has to be calculated considering that the order parameter is temperature-independent. At the second step, the equilibrium values of the order parameter that are inherent to the austenitic and martensitic phases have to be substituted into the obtained expression for the entropy and the difference of the resultant entropy values ΔS ≡ SA − SM has to be calculated.
According to Landau theory, the coefficients of third- and fourth-order terms of elastic energy (Equation 56) are temperature independent. By differentiating Equation 62, one obtains the expression
The free energy, Equation 62, and entropy, Equation 63, are counted of the entropy of paramagnetic cubic phase. In the ferromagnetic austenitic phase the values of the order parameter components and the concomitant u1 value are caused by the ordinary magnetostriction and are much smaller than MT strain and volume change during MT. As so, these values can be put equal to zero. Due to this, the entropy of austenitic phase is approximately equal to
It should be remembered that the “magnetic energy”, Equation 57, is the energy of interaction between the magnetic moments of atoms in the high-symmetry phase, in our case, in the undistorted crystal lattice. By definition, the magnetoelastic energy is the total change of magnetic energy caused by the deformation of cubic lattice. That is why the derivative is substituted in Equation 65 by the value Equation 64, which is inherent to the cubic ferromagnetic phase.
Subtracting SA value from the right side of Equation 65, one can present the entropy change as the sum of elastic and magnetic parts
Equation 67 describes the elastic part of the entropy change through the temperature derivative of the second-order energy coefficient, which is not directly measurable. This coefficient is related to the shear elastic modulus of austenitic phase as c2 = C′/3, but the interrelation between c2 and elastic modulus of martensitic phase is more complicated (see Equation 52). However, the c2(T) is a monotone function and it can be assumed that the slope of c2(T) curve does not vary noticeably in the temperature interval TMF < T < TMS. In this case the elastic part of the entropy change can be estimated from the relationship
A contribution of magnetic subsystem of the alloy, Equation 68, proved to be directly proportional to: (i) the temperature derivative of magnetization value; (ii) the volume change during MT; and (iii) the magnetoelastic constant. These points are physically clear because: (i)' both ΔSmag and ∂M/∂T characterizes the change of degree of magnetic ordering; (ii)' the volume change characterizes the transformation of crystal lattice in the isotropic approximation; and (iii)' the volume magnetostriction constant characterizes the interdependence between the volume of alloy specimen and the intensity of spin-exchange process, which provides the main contribution to the magnetic energy expressed by Equation 57.
The magnetoelastic constant is not directly measurable. Its value can be determined, in principle, from the measurements of a volume magnetostriction. A spontaneous volume magnetostriction v(me)(T) accompanies magnetic ordering. Using condition ∂F/∂u1 = 0 and Equations 4 and 58, it can be expressed as
Therefore, the magnetic entropy change
The evaluation of the magnetic and elastic parts of the entropy change may be hampered by the difficulties in determination of the volume magnetostriction and temperature derivatives from the experimental M(T) and C′(T) values. In this regard, the relationship
Another way of the determination of magnetic entropy change is proposed in  for the case when the experimental temperature dependence of magnetization of FSMA with TMS < TC obeys the standard equations
The temperatures TC and T* satisfy the conditions J(TC) = 0 and J*(T*) = 0, which result in the relationship T* = TC (1 + κ/ζ) involving the dimensionless parameters ζ and
The parameter κ relates the magnetic entropy change to the magnetization of FSMA (see Equation 68).
The experiments with different FSMAs demonstrate a stepwise change of magnetization value in the narrow temperature interval below TMS. The characteristic temperatures TC and T* can be found by fitting the solutions of Equation 73 to the high-temperature (T > TMS) and low-temperature(T > TMF) segments of the experimental M(T) curve. The relationship ζ ≈ nkBTC (n is the amount of magnetic atoms in the unit volume) enables the estimation of the parameter
The findings of general theoretical analysis of the entropy change during MT can be summarized in the following principal points:
The entropy change during MT is inversely proportional to the width of the temperature interval of mixed (austenitic-martensitic) state.
The magnetoelastic energy of a deformable magnetic solid, such as FSMA, is the difference between the magnetic energies of its deformed and undeformed states.
The magnetic contribution to the total entropy change caused by the deformation of magnetic solid is the partial temperature derivative of magnetoelastic energy taken with the opposite sign.
The ordinary magnetostriction of FSMA is substantially smaller than the MT strain vMT, and the axial magnetostriction is much smaller than the spontaneous volume magnetostriction v(me) in the case if the Curie temperature is of the order of room temperature. In this case the magnetic entropy change is directly proportional to the product vMTv(me)(TMF).
Not only the magnetic entropy change during MT but also the magnetization jump is caused by the magnetoelastic coupling and interrelated with the product vMTv(me)(TMF). This follows from the Equations 70,73,75 (for more details see [33,83]).
7.2. Magnetic Entropy Change during MT of Ni-Mn-Ga Alloy
The Ni-Mn-Ga alloys undergoing the cubic-tetragonal MTs in the ferromagnetic state are the most studied to date FSMAs. The magnetization functions of some of these alloys are well described by the Equation 73, and therefore, the virtual Curie temperatures and κ values can be estimated for these alloys by fitting theoretical M(T) curves to the experimental values of magnetization . In particular, the fitting procedure was performed for a Ni52.6Mn23.5Ga23.9 alloy (Alloy 1) that undergoes MT to the 5-layered tetragonal phase with c/a < 1. For this alloy TMF ≈ 275 K, a good fit of the theoretical M(T) curves to experimental values of the magnetization takes place for TC = 375 K, T* = 405 K and M0 = 715 G . For the Ni-Mn-Ga alloy family, the estimation is valid , thus the Equation 76 gives the value κ = 150. In this case the numeric computation of M(T) function and its derivative results in the estimation ΔSmag = 14.3 J kg−1K−1. This entropy change corresponds to the heat exchange qmag = T0ΔSmag = 4.07 J g−1. It is astonishing that the theoretical estimation of the “magnetic” heat exchange is very close to the experimental value of total transformation heat q ≈ 4.2 J g−1 reported for Alloy 1 [78,86]. The experimental value of transformation heat corresponds to the total entropy change of 14.7 J kg−1K−1.
The solid line in Figure 13 shows the magnetic entropy change ΔSmag that was computed from Equation 68 by setting different T = TMF values in the argument of M(T) function keeping fixed the rest of parameters such as those shown above for Alloy 1. The experimental values of the total entropy changes are shown by circles in Figure 13. To present these values, we collected the available experimental data for Ni-Mn-Ga martensites with c/a < 1 [30,87,88,89,90,91]. The arrow points to the experimental value obtained for Alloy 1. The theoretical dependence of the magnetic entropy change on the temperature difference TC −TMF demonstrates that ΔSmag quickly increases when the MT temperature approaches the Curie temperature and is less than 5 J kg−1 K−1 for the alloys with TC − TMF > 200 K. The experimental values obviously correlate with the computed ones in spite of the fact that the different alloys are characterized by different M(T) curves, while the curve used for computations is relevant only to the Alloy 1. The proximity of the experimental points to the theoretical curve suggests that the magnetic part of the total transformation heat exceeds all nonmagnetic parts for the Ni-Mn-Ga alloys with TC − TMF < 150 K, at least. It should be noted, that the same conclusion can be deduced from the experimental data collected for different Ni-Mn-In and Ni-Mn-In-Ga alloys .
The calculated values of the magnetic entropy change are very close or even exceed the values of total entropy changes measured for Ni-Mn-Ga alloys. It suggests that the magnetic entropy change is overestimated. The overestimation can be caused, in particular, by the inaccuracy in the evaluation of parameter ζ, which is absent in the equation for the magnetic entropy change and Equation 75 but is involved in Equation 76. It should be noticed, however, that a clear physical reason for the underestimation of the total entropy change exists. On the one hand, the cubic crystal lattice transforms to the spatially inhomogeneous martensitic state when the latent transformation heat is measured. The martensitic state is not completely self-accommodated and its energy is higher than the energy of thermodynamically equilibrium tetragonal phase considered above. Therefore, the measured transformation heat is smaller than the latent heat, which characterizes the transformation from the cubic to the equilibrium tetragonal phase.
7.3. Magnetic Entropy Change during MT of a Quasi-Stoichiometric Ni-Fe-Ga Alloy
The temperature dependence of saturation magnetization of Ni53.5Fe19.5Ga27 alloy (Alloy 2) with M0 ≈ 480 G, TC ≈ 330 Kand TMF ≈ 250 K is measured in . This dependence is not described by Equation 73 [33,83] and the virtual temperature of ferromagnetic ordering cannot be determined in view of an ambiguity in the extrapolation of experimental M(T) dependence measured in the martensitic phase to the temperature range T > TMF. However, the experimental temperature dependence of magnetization measured in the martensitic phase (that is for TC – T > 80 K) can be used to illustrate that the magnetic entropy change drastically depends on the shape of M(T) curve. For this purpose the computations could be performed in two ways: (i) using the experimental M(T) curve; and (ii) using the solution of Equation 73 with M0 = 480 G and TC = 330 K. The κ and ζ values estimated above for Alloy 1 can be accepted, because in this case Equation 73 gives the magnetization jump M(TMF) − M(TMS) ≈ 15 G which follows also from the experimental M(T) curve presented in [33,83].
It should be emphasized that Equation 73 describes the electron gas in the Weiss molecular field, that corresponds to the widely used Bragg–Williams approximation (see, e.g., ). As it is seen from Figure 14, the Bragg–Williams approximation is not applicable to Alloy 2. Therefore, the magnetic entropy changes, which accompany MTs in Ni-Fe-Ga alloys, cannot be evaluated using this equation. Moreover, the virtual temperature of ferromagnetic ordering cannot be calculated and the parameter κ relating ΔSmag to M(T) dependence cannot be estimated from Equation 76. Therefore, this parameter has to be evaluated from Equation 75. In this case, the magnetoelastic constant must be determined and the volume change must be measured or estimated in some way.
The experimental values of lattice parameters reported for Alloy 2  result in the estimation vMT ≈ −3 × 10−2, which corresponds to the value u1 ≈ −0.01. It is noteworthy that in the case of Alloy 1 this value of the volume change gave rise to the above used value of κ and provided for the excellent match between experimental and theoretical magnetization curves . So, the used above value κ = 150 may be correct for Ni-Fe-Ga alloys if the constant δ0 ≡ δexM2(T0) is approximately equal to δ0 = −0.4 GPa , which was estimated before for Ni-Mn-Ga alloys. To the best of our knowledge, the magnetoelastic constants are still not evaluated for Ni-Fe-Ga alloys.
7.4. Summary on the Entropy Change
The consistent analysis of the entropy change that accompanies MT in FSMA demonstrates the important features of this physical value.
The total entropy change during MT is inversely proportional to the width of the temperature interval of mixed two-phase state.
The elastic part of the entropy change is proportional to the value of temperature derivative of the shear elastic modulus. This feature illustrates that the cubic-tetragonal MT is caused by the instability of crystal lattice with respect to the vibrations corresponding to the TA2  phonon mode and to the softening of this phonon mode which leads to a decrease of C′ value in the vicinity of MT. In practice, the evaluation of ΔSel from the temperature dependencies of shear modules of SMAs is hampered by an uncertainty in the experimental values of these modules.
The elastic part of the entropy change is proportional to the squared tetragonal distortion of the unit cell, (1−c/a)2), which is about of 4 × 10−2 for the Ni-Mn-Ga alloys with c/a > 1, TMS > TC and 3 × 10−3 for the alloys withc/a < 1, TMS < TC. This feature is in line with the fact that the observed in  entropy change is noticeably larger for the Ni-Mn-Ga alloys with TMS > TC than for those with TMS < TC.
In the case of FSMAs with TMS < TC, the Landau theory confirms a crucial role of the interaction between the magnetic and elastic subsystems in the formation of thermodynamic characteristics of the MT. The magnetic part of the entropy change estimated from M(T) curves for Ni-Mn-Ga alloy (Alloy 1) appeared to be close in value to the experimentally observed total entropy change. In this connection, the entropy change demonstrates a sharp decrease when the MT temperature of the alloy moves away from the Curie temperature. The experiments show that this is a common feature of the Ni-Mn-Ga, Ni-Mn-In and Ni-Mn-In-Co alloy systems . Recently, a pronounced dependence of ΔS on TC−TMF has been observed also in Ni-Mn-Ga-Co alloys .
The computations illustrate that the evaluation of the magnetic entropy change from the temperature dependence of magnetization is very sensitive to the character of this dependence (see Figure 14). Moreover, it occurs that the standard equations corresponding to the Bragg–Williams approximation do not describe the magnetization of Ni-Fe-Ga alloys. It can be concluded, therefore, that the careful study of the magnetic structure and magnetization of every alloy system must be carried out for the correct evaluation of magnetic entropy change.
Equation 71 shows that the magnetic entropy change is proportional to the volume magnetostriction. Therefore, a careful theoretical analysis of the effect of spontaneous and forced magnetostriction on the characteristic MT temperatures will be very important.
The following important remark about the value of volume change during MT is appropriate. The agreement between the theoretical and experimental values, illustrated by Figure 13, is achieved for κ = 150. This value corresponds to the magnetoelastic constant [83,84] and volume change of the order of 1% directly given for Ni-Mn-Ga alloys by the X-ray data, while the Clausius–Clapeyron relationship results in a value for the volume change of about 0.1% estimated from the shift of MT temperature by the hydrostatic compression of Ni-Mn-Ga. The reduction of the volume change by one order of magnitude can be compensated only by the proportional increase of the magnetoelastic constant. The value was reported in .
The volume conservation principle is often used for a comprehension of the thermoelastic behavior of shape memory alloys, while a few works include the consideration of equilibrium volumes of different phases. (The excellent first principle calculations of this kind are performed in ). In the same time, the symmetry conforming theory shows that the volume change is an indispensable feature of the cubic-tetragonal and cubic-rhombohedral martensitic transformations. It may be expected that this conclusion is also valid for the MTs to the orthorhombic and monoclinic phases because one of MT strain components often being much greater than others. The theory relates the volume change to MT strain via the certain combinations of the third- and fourth-order elastic modules, but the values of these combinations are not known at present. On the one hand, it shows that the appearance of small MT strain may be accompanied by the comparatively large volume change, and vice versa, the large MT strain may occur with very small volume change. On the other hand, an uncertainty in the used for computations ratios of the third- and fourth order modules (parameters Λ and Ω) retards the quantitative analysis of experimental results.
In spite of difficulties, the combined (experimental and theoretical) study of transformational properties of nonmagnetic and ferromagnetic shape memory alloys indicates the noticeable influence of the volume effect of MT and volume magnetostriction on the elastic and thermodynamic properties of these alloys.
The importance of the volume effect of MT can be explained in a simple manner. The volume change during MT results in the energy density change, which can be estimated by the order of magnitude as |ΔF1|~|vMT| B~0.1−1 GPa, because the bulk elastic modulus B is of the order of 100 GPa and the inequalities 10−3 ≤ vMT ≤ 10−2 are true in most cases. The spontaneous shear of crystal lattice during MT results in the energy density change |ΔF1|~|εM| C′~0.1−1 GPa because the MT strain satisfies the inequalities 10−2 ≤ εM ≤ 10−1 and the shear modulus of shape memory alloy is of the order of 10 GPa in the vicinity of MT temperature. The energy changes ΔF1 and ΔF2 are in general of the same order of magnitude, and therefore, the volume effect of MT cannot be neglected a priory: the disregard of this effect must be substantiated in every case by the numerical evaluations.
We hope that the conclusion about the importance of volume changes will attract the attention of different researches teams and may be useful for planning the further experimental and theoretical studies of shape memory alloys.
V.A.L. is grateful to U. K. Rößler and M. E. Gruner for useful discussions during his stay in IFW, Dresden. A financial support from the Project No. MAT2011-28217-C02-01,02, by the Spanish Ministry of Science and Innovation, Project №0112U001009 by National Academy of Sciences of Ukraine (NASU) and STCU-NASU grant №5715 are greatly acknowledged.
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