This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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 [

The ferromagnetic and metamagnetic shape memory alloys (FSMAs and MetaMSMAs) form a special class of the multifunctional materials (see [

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 [

The early experiments with the SMAs under hydrostatic pressure _{0} [_{0}/d

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., [_{C}_{MF}_{C}_{MF}

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 [

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

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 _{ik}

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
_{α,β} (α, β = 1,2,…6) of the crystal in the cubic phase, the coefficients of third-order and fourth-order energy terms are the linear combinations of the third-, and fourth-order elastic modules in the cubic phase [

The Gibbs potential of SMA is presented by the equation
_{ik}

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 _{α}_{α}_{α}_{α}_{2}, _{3} ≠ 0, _{4} = _{5} = _{6} = 0 may exist in this case. These states can have the tetragonal symmetry and the values _{2}, _{3} form the two-component order parameter of the cubic-tetragonal MT [_{α}_{4} = _{5} = _{6}. Due to this, the rhombohedral state with _{4} = _{5} = _{6} ≠ 0, _{1 }≠ 0, _{2} = _{3} = 0 may exist [_{2,3} the concomitant nondiagonal strains

Second, the series Equation 4 involves the terms of _{1} is the trace of strain tensor, and therefore, it is related to the volume change of the alloy as
_{MT}

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 _{4} and _{5}). As so, all MTs are the first-order phase transitions (see [

In the case of cubic-tetragonal MT the non-diagonal strain tensor components _{4}, _{5} and _{6} 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

The nondiagonal stress components are not conjugated with the order parameter of cubic-tetragonal MT and are omitted in Equation 9 hence. The coefficient _{2}(_{1} = 0 results in the following relationship:

Using Equation 10, the _{1} value can be excluded from Equation 9 and the Gibbs potential can be expressed through the order parameter components as
_{7}/_{2} are introduced.

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

For the sake of definiteness, let us consider the transformation of cubic phase into _{2} = _{3} = 0). For this variant of martensitic phase _{xx}_{yy}_{2}(

As far as the volume change _{MT}_{0} of the order parameter _{3}, an approximate equality _{xx}_{yy}_{zz}_{0} and _{M}_{[001]} is the specific elongation/contraction of the unit cell of cubic lattice caused by MT; this value is referred to as the martensitic transformation strain.

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 _{2}(_{1}(_{1 }> _{2} is held.

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
_{2},_{2},

In accordance with Equation 12, the

The expression
_{2} and Ω. It may be assumed that all third-order and fourth-order energy coefficients are in roughly the same ratio to each other. This assumption results in the estimation: _{2} can be determined from the approximate relationship
_{11}, _{12}, _{M}_{MT}

The cubic-rhombohedral MT is accompanied by the appearance of non-zero values of shear strains _{4}, _{5}, _{6} while the diagonal strain tensor components _{2} and _{3} are equal to zero in both cubic and rhombohedral phases. The tendency to volume conservation during MT is expressed by inequality

The coefficient _{3}(_{44} 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 _{1} = 0 establishes the following interrelation between the volume change and order parameter components:

The substitution of the _{1} value (Equation 23) into Equation 22 results in the following Landau expansion for the Gibbs potential
_{8}/_{3} is introduced.

The dependencies of the order parameter components on the temperature and pressure values obey the equations _{4–6} = 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 [111], [_{4-6} = 0). For [111]-variant of rhombohedral lattice _{4} = _{5} = _{6} = _{R}

An equilibrium value of the order parameter is related to the specific elongation/contraction of the cubic unit cell in [111] direction as

The high-temperature phase is stable if _{2}(_{1}(_{1} > _{2} holds.

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,

In accordance with Equation 25, the ^{*} value is independent of pressure and hence Equation 32 predicts that the elastic modulus _{44} 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.

The expression
_{3} and Λ.

If all third-order and fourth-order energy coefficients are in roughly the same ratio to each other, the estimation _{3} can be estimated by the order of magnitude from the relationship
_{11}, _{12}, _{M}_{MT}

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 _{2}(_{3}(

This follows from Equations 12 and 21 for _{2} and Equations 25 and 34 for _{3}. 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.

The hydrostatic compression stabilizes the martensitic phase if the volume of alloy decreases during the forward MT (_{MT}

an increase/decrease of the characteristic MT temperatures [

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 _{1} and _{2}. In the simplest approach, the influence of pressure on MT can be characterized by the change of the average temperature _{0} ≡ (_{1} + _{2})/2. This change proves to be strictly proportional to the pressure value [_{0}/_{0}/_{MT}T_{0}/

The MT of cubic-tetragonal or cubic-rhombohedral type is accompanied by the pronounced softening of the shear elastic modulus _{44}, respectively. The pressure influence on the value of shear modulus in the _{MT}

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 [

The Ti–Ni alloys exhibit both cubic-monoclinic (B2→B19) and cubic-rhombohedral (B2→R) transformations [_{[111]}, Equation 26 is modified as

The solution of Equation 39 can be presented as a sum of the spontaneous shear strain _{0} (Equation 27) and elastic strain

It is quite natural that the difference of the elastic modules of two phases tends to zero in the limiting case _{0} → 0, because in this case the phases become physically equivalent. Using Equation 26, one can express the elastic modulus of martensitic phase as

Substituting

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 [

For the sake of definiteness, let us consider the experimentally observed cubic-rhombohedral MT in Ti_{49.5}Ni_{50.5} alloy [_{44} modulus of Ti_{49.5}Ni_{50.5} alloy measured in [

Experimental values of shear elastic modulus of the Ti_{49.5}Ni_{50.5} alloy [

The lability temperatures of martensitic and austenitic phases derived from the experimental values of soft modulus are _{1} = 311 K and _{2} = 300 K, respectively (see arrows in _{44}(_{2}) = 18 GPa. In this case Equation 42 results in the value _{44}(_{1}) = 4.5 GPa. The energy coefficients ^{*} = 2.9 × 10^{7} GPa can be found from Equation 32 using experimental value _{[111]} = 2_{0}(_{2}) ≈ −1% [_{MT}^{−4} resulting from the Clausius-Clapeyron relationship (for more details see [_{3}/_{1} and coefficients _{3}(

Solving of Equation 41 with respect to _{0} value enables the quantitative description of the pressure influence on the MT strain _{[111]} if the parameter Λ, energy coefficients ^{*}, and function _{44} (_{49.5}Ni_{50.5} alloy. The _{44} (_{2} < _{1}.

_{44} (

(

The shifts of MT temperatures caused by a hydrostatic pressure are measured in experiments (see, e.g., [_{0}(_{1}(_{2}(_{1}(_{2}(_{1}(_{2}(

(_{0} on the hydrostatic pressure. (

It should be remembered that the conclusions derived from _{[111]}. If the MT strain is positive, the sign of parameter Λ should be changed to keep in force these conclusions.

A measurement of the shear elastic modulus _{44} 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 [

(

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 [

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 [

a considerable difference in the stress–strain dependencies measured before and after martensite aging [

a gradual change of the lattice parameters during martensite aging [

a martensite stabilization [

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 [

Stress tensor _{ik}^{(i)} is interpreted as the "internal" pressure, which is responsible for the volume change accompanying the evolution of defect subsystem during the martensite aging. An axial part of the internal stress is described by the tensor

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 [

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

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 [^{(i)}(

The isotropic mechanism of a martensite stabilization can be illustrated by the consideration of martensitic transformation of Au-Cd alloy.

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
^{(i)}(^{(i)} value is constant in the equilibrium state, which corresponds to the minimal energy of crystal with defects; the rate of change of ^{(i)} is roughly proportional to the deviation from the equilibrium value ^{(i)}(^{(i)}(

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 _{2} ≈ 300 K [_{44}(_{1}) ≈ 40 GPa [_{M}_{R}_{MT}^{−3} [_{3}/_{1}, _{5}(0) = 1.6 × 10^{5} GPa and ^{*}(0) = 4 × 10^{7} GPa from Equations 31–34. The Clausius–Clapeyron relationship with the values ^{−1} and ^{−3} [_{1} = 306.8 K.

The values presented above prescribe a time evolution of the

The dependence of austenite start and finish temperatures on aging time ^{(i)}(_{AF}

Time evolution of the austenite start (dotted line), austenite finish (solid line), and two-phase temperature range (dashed-dotted line). The experimental values (circles) for the austenite finish temperature are taken from [

It was shown in the

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

The values presented above result in the following estimation

Let us sum up the new aspects of a martensite stabilization effect considered in [

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 [

The quantitative theoretical analysis of the experimental results in Ref. [

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 Ni_{57.5}Mn_{22.5}Ga_{20.0} alloy has been studied [_{0} = 0.587 nm to a non-modulated tetragonal phase with lattice parameters

The Sample 1 with dimensions of 0.19 × 0.5 × 9.1 mm^{3} and Sample 2 with dimensions of 0.27 × 0.28 × 8.5 mm^{3} were cut from the same Ni_{57.5}Mn_{22.5}Ga_{20.0} single crystal. Both samples were oriented in [100] crystallographic directions. The details of specimen preparation are reported in [

Experimental temperature dependence of the elastic modulus of Sample 1 is shown in _{MF}_{AS}_{MS}_{AF}

Elastic modulus of Sample 1 of Ni_{57.5}Mn_{22.5}Ga_{20.0} single crystal.

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 _{n}_{AF}_{MS}

(

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

(

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

The influence of mechanical cycling on the superelastic behavior of Sample 2 of Ni_{57.5}Mn_{22.5}Ga_{20.0} single crystal.

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 _{R}_{AS}_{AF}_{MS}

The influence of thermal cycling on the forward (bottom curve) and reverse (upper curve) MT temperatures of Sample 2.

The experiments demonstrate a noticeable softening of the elastic modulus of martensite after thermomechanical cycling (

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,

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 [

Physical considerations, which are essentially similar to those presented in _{0} is the characteristic integer prescribed by a rate of the defect subsystem reconfiguration, ^{(i)}(_{MT}

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 ^{(i)}(

The further analysis of problem is similar to that presented in the _{1}(_{2}(

The value of this modulus in the martensitic phase ^{(M)} can be expressed through the equilibrium value of the order parameter _{0} as explained in the

Now, let us use the experimental temperature dependence of the elastic modulus, measured before cycling of the Ni–Mn–Ga alloy (^{(A,M)}(^{(A,M)}(

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 _{0}(_{MS}_{0}(_{AF}_{0}(_{MF}_{0}(_{AS}^{(A)}(_{MS}^{(A)}(_{AF}^{(M)}(_{MF}^{(M)}(_{AS}

The coefficients _{2}(_{4} and _{1,2}) ≈ _{MS,MF}_{MT}_{0}(_{MF}_{4} = −10.1 GPa, _{2}/_{1} = 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 _{1} = _{MS}_{2} = _{MF}_{1} = _{AF}_{2} = _{AS}

Using values obtained above, the functions _{0} = 4 was used for computations. Characteristic values of the internal pressure ^{(i)}(^{(i)}(

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 (_{R}_{AS}_{AF}_{0} (_{1} (_{2} (_{0} 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 _{R}_{1}(_{MS}_{MS}

It can be summarized now that the destabilization of the Ni_{57.5}Mn_{22.5}Ga_{20.0} 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 (_{MT}_{57.5}Mn_{22.5}Ga_{20.0} 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 (

a decrease of the MT temperature (

Experiment shows that the cycling increases noticeable the stress value, which in turn triggers the stress–induced MT. As it was shown in [

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 [

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 [

The Landau expansion for the free energy, Equation 4, involves the terms _{5}_{4}_{5}_{6}, 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 _{4} or _{5} 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 [

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

Let some internal process or factor induce an internal pressure ^{(i)}. The positive/negative internal pressure decreases/increases the volume of an alloy. The negative/positive value _{MT}^{(i)} and _{MT}^{(i)}_{MT}^{(i)}_{MT}

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 _{4} and Ω or _{5} and Λmust be of the opposite signs. In the same time, the signs of parameters

It should be expected that _{MT}

The energy coefficient _{0} ≈ −0.12) in [_{0}^{−1} and _{MT}_{MT}^{−2}, while the values estimated from thermodynamic measurements are smaller by almost one order of magnitude (see [

The influence of internal pressure on the coefficient of the third-order energy term, which predetermines the first-order character of martensitic transformation in the Ni-Mn-Ga alloy considered in [

In the case of intermediate and large volume changes, the internal pressure can reduce the ^{(i)}(

The influence of internal pressure on the transformational and elastic properties of the rhombohedral Au-Cd martensite and tetragonal Ni-Mn-Ga martensite with

Influence of internal pressure on the coefficient of the third-order energy term, which predetermines the first-order character of martensitic transformation in the rhombohedral Au-Cd martensite and tetragonal Ni-Mn-Ga martensite, considered in [

The stabilizing pressure ^{(i) }= 3.8 GPawas estimated for the aged Au-Cd alloy in

The destabilizing pressure of about –6 GPa was induced by thermomechanical cycling of Ni-Mn-Ga martensite with _{M}^{4} for the Ni-Mn-Ga alloys with

It should be stressed that in view of an ambiguity in Ω, Λ and ^{(i)} values, the data presented in

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 Δ_{M}_{C}_{M}_{C}_{M}

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 _{el}_{2}, _{3} of diagonal strain tensor components by Equation 11 with _{2} = _{3} = 0:

The magnetic part of free energy _{k}_{C}_{C}_{C}

In accordance with the general principles formulated in _{ex}

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 [

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 [_{A}_{M}

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 _{1} 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.

Subtracting _{A}

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 _{2} = _{2} and elastic modulus of martensitic phase is more complicated (see Equation 52). However, the _{2}(_{2}(_{MF}_{MS}

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 Δ_{mag}

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 ^{(me)}(_{1} = 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 ^{−1} value and so, can be verified easily in the experiment.

Another way of the determination of magnetic entropy change is proposed in [_{MS}_{C}_{0} = _{C}_{1} variable is equal to zero before the forward MT and to the constant value _{MT}

The temperatures _{C}_{C}_{C}

The parameter

The experiments with different FSMAs demonstrate a stepwise change of magnetization value in the narrow temperature interval below _{MS}_{C}_{MS}_{MF}_{B}T_{C}

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

The ordinary magnetostriction of FSMA is substantially smaller than the MT strain _{MT}, and the axial magnetostriction is much smaller than the spontaneous volume magnetostriction ^{(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 _{MT}^{(me)}(_{MF}

Not only the magnetic entropy change during MT but also the magnetization jump is caused by the magnetoelastic coupling and interrelated with the product _{MT}^{(me)}(_{MF}

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 _{52.6}Mn_{23.5}Ga_{23.9} alloy (Alloy 1) that undergoes MT to the 5-layered tetragonal phase with _{MF}_{C}_{0} = 715 G [_{mag}^{−1}K^{−1}. This entropy change corresponds to the heat exchange _{mag}_{0}Δ_{mag}^{−1}. It is astonishing that the theoretical estimation of the “magnetic” heat exchange is very close to the experimental value of ^{−1} reported for Alloy 1 [^{−1}K^{−1}.

The solid line in _{mag}_{MF}_{C}_{MF}_{mag}^{−1} K^{−1} for the alloys with _{C}_{MF}_{C}_{MF}

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

Theoretical values of magnetic entropy change (line) and experimental values of total entropy change (circles) at MTs in the Ni-Mn-Ga alloys.

The temperature dependence of saturation magnetization of Ni_{53.5}Fe_{19.5}Ga_{27} alloy (Alloy 2) with _{0} ≈ 480 G, _{C}_{MF}_{MF}_{C}_{0} = 480 G and _{C}_{MF}_{M}_{S}

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., [_{mag}

The experimental values of lattice parameters reported for Alloy 2 [_{MT }^{−2}, which corresponds to the value _{1} ≈ −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 _{0} ≡ _{ex}^{2}(_{0}) is approximately equal to _{0} = −0.4 GPa [

Theoretical values of magnetic entropy change computed using the experimental values of saturation magnetization (solid line) and the solution of Equation 73.

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 TA_{2} [110] 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 Δ_{el}

The elastic part of the entropy change is proportional to the squared tetragonal distortion of the unit cell, (1−^{2}), which is about of 4 × 10^{−2} for the Ni-Mn-Ga alloys with _{MS}_{C}^{−3} for the alloys with_{MS}_{C}_{MS}_{C}_{MS}_{C}

In the case of FSMAs with _{MS}_{C}_{C}_{MF}

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

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 inverse proportionality of (_{MS}_{MF}

The experimental verification of the inverse proportionality of two-phase temperature interval to entropy change [

The following important remark about the value of volume change during MT is appropriate. The agreement between the theoretical and experimental values, illustrated by

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 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 |Δ_{1}|~|_{MT}^{−3} ≤ _{MT}^{−2} are true in most cases. The spontaneous shear of crystal lattice during MT results in the energy density change |Δ_{1}|~|_{M}^{−2} ≤ _{M}^{−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 Δ_{1} and Δ_{2} are in general of the same order of magnitude, and therefore, the volume effect of MT cannot be neglected

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.

_{2.18}Mn

_{0.82}Ga Heusler alloys with a coupled magnetostructural transition

_{2}′-martensite of AuCd alloys

_{51}Ni

_{47}Si

_{2}shape memory alloy