Introduction

There is an ongoing demand for high-temperature shape memory alloys (SMAs) as novel applications continue to be developed in the areas of designing mechanical and aerospace SMA-actuated structures for better system performance. TiPd and TiPt intermetallic systems are among the high-temperature alloys of high interest. To adjust the shape memory and mechanical properties of these intermetallics, alloying elements are added to these systems that alter their transformation temperatures. Phase Transformation temperature of an SMA is an important characteristic, since it is the temperature at which the SMA material recovers from apparent permanent deformation through a reversible martensitic transformation. This shape recovery behavior is often referred to as the shape memory effect and has been adapted in various aerospace, structural, and biomedical applications [1]. Elastic properties of the matrix crystal before martensitic transformation and microstructural features such as precipitates were shown to be important factors affecting the transformation temperatures [2]. To understand the factors that control the transformation temperatures in the SMAs system, Zarinejad and Liu suggested the need to acquire in-depth knowledge of how the atomic bonding strength of the matrix crystal can be influenced by the chemical composition of the alloys. Dependence of transformation temperatures on electron variables in SMAs was introduced [3,4,5]. It was shown that variation of elastic properties of the parent crystal (austenite) because of composition change influence the resistance against shear, responsible for martensitic phase transformation. This understanding was achieved based on the knowledge that prior to martensitic transformation a softening of elastic moduli occurs with the lowering temperature. The experimental results, mainly in Cu-based and NiTi-based SMAs [6,7,8,9], suggest that there is a critical value of elastic constant at which transformation takes place, which is not sensitive to alloy compositions [10], and is only slightly dependant on temperature [2]. Based on this, the elastic moduli of the alloy, which are dependent on both composition and temperature are the key parameters influencing the transformation temperatures, [2, 11]. Since in metallic materials the delocalized valence electrons dominate the strengths of bonds and elastic properties [12,13,14], it is of particular importance to study the electronic parameters of alloys.

In the present work, the investigation is extended to the vast TiPd-based and TiPt-based high-temperature SMA family of alloys in detail to reveal in-depth understanding of the dependence of transformation temperatures (Ms, As) on the electronic parameters of the alloy. The electronic parameters of nearly all the TiPd-based and TiPt-based alloys made or studied to date are examined. Three parameters are paid attention to in the current study: (1) the average atomic number of the alloys; (2) the number of valence electrons, and; (3) Valence Electron Ratio (VER) of the alloys.

We reveal and explain the dependence of the transformation temperatures of important TiPd and TiPt family of alloys on VER. The correlations found in the current work enable an important understanding towards effective alloy design.

Materials and Methods

Different Ti/Pd or Ti/Pt atomic ratios and alloying elements such as Ni, Fe, Co, Cr, Mn, Cu, V, Hf, W, Ta, Ir, etc., have been utilized by researchers to alter the transformation temperatures and characteristics of TiPd and TiPt SMAs. A survey of the literature reveals many TiPd- and TiPt-based shape memory alloys was performed to extract the phase transformation temperatures and temperature hysteresis. The data [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64] were extracted from the relevant literature as listed in tables or presented in graphs. Almost all the data correspond to solution-treated alloys to minimize the effects of precipitates and mechanical work. These possible effects, if existing, were not considered in the present work. The average atomic numbers of the alloys were calculated based on the atomic fractions and atomic numbers of the elements comprising the alloys as follows:

$$Z = f_{{{\text{Ti}}}} Z_{{{\text{Ti}}}} + f_{{\text{Pd(Pt)}}} Z_{{\text{Pd(Pt)}}} + f_{{\text{T}}} Z_{{\text{T}}} + f_{{\text{Q}}} Z_{{\text{Q}}}$$
(1)

where Z is the fraction-averaged atomic number of the alloy. ZTi, ZPd(Pt), ZT, and ZQ, etc. are the atomic numbers Ti, Pd (or Pt), the ternary, quaternary, quinary elements and so on, respectively. The fx (x = Ti, Pd or Pt, the ternary, quaternary, quinary elements, or even quinary element, etc.) is the corresponding atomic fraction of each element in the alloy. To study the dependence of the transformation temperatures on the chemistry of alloys the basic electron configurations of the alloys were analyzed in the following section.

The number of valence electrons is usually considered as the number of d and s electrons for an atom of transition metals. The valence electrons per atom of TiPd- and TiPt-based alloys can be calculated based on the atomic fractions of the elements in the alloy by the following equation:

$$\frac{{e_{{\text{v}}} }}{a} = f_{{{\text{Ti}}}} e_{{\text{v}}}^{{{\text{Ti}}}} + f_{{{\text{Pd}}\left( {{\text{Pt}}} \right)}} e_{{\text{v}}}^{{{\text{Pd}}\left( {{\text{Pt}}} \right)}} + f_{{\text{T}}} e_{{\text{v}}}^{{\text{T}}} + f_{{\text{Q}}} e_{{\text{v}}}^{{\text{Q}}} + \cdots$$
(2)

where similarly fTi, fPd(Pt), fT and fQ represent the atomic fractions of Ti, Pd (or Pt), the ternary quaternary elements, etc., in the alloy and, \(e_{{\text{v}}}^{{{\text{Ti}}}}\),\(e_{{\text{v}}}^{{\text{Pd(Pt)}}}\),\(e_{{\text{v}}}^{{\text{T}}}\) and \(e_{{\text{v}}}^{{\text{Q}}}\) are the corresponding numbers of valence electrons of elements Ti, Pd (Pt), ternary and quaternary elements, respectively. The valence electron ratios (VER) of the alloys were calculated. VER is defined as the ratio of the number of valence electrons to the total number of electrons of the alloy, VER = (ev/et), which can be simply calculated as follows:

$${\text{VER}} = \frac{{e_{{\text{v}}} }}{{e_{{\text{t}}} }} = \frac{{f_{{{\text{Ti}}}} e_{{\text{v}}}^{{{\text{Ti}}}} + f_{{{\text{Pd}}}} e_{{\text{v}}}^{{{\text{Pd}}\left( {{\text{Pt}}} \right)}} + f_{{\text{T}}} e_{{\text{v}}}^{{\text{T}}} + f_{{\text{Q}}} e_{{\text{v}}}^{{\text{Q}}} }}{{f_{{{\text{Ti}}}} Z_{{{\text{Ti}}}} + f_{{{\text{Pd}}\left( {{\text{Pt}}} \right)}} Z_{{{\text{Pd}}\left( {{\text{Pt}}} \right)}} + f_{{\text{T}}} Z_{{\text{T}}} + f_{{\text{Q}}} Z_{{\text{Q}}} }}$$
(3)

where ZTi, ZPd,(Pt) ZT, ZQ etc., represent the atomic numbers of Ti, Pd (Pt), the ternary and quaternary elements, etc., respectively. Similarly fTi, fPd(Pt), fT, fQ represent their corresponding atomic fractions in the alloy.

Theory (Elastic Properties of Crystal and Phase Transformation)

In the martensitic transformation in TiPd- and TiPt-based alloys shear causes shape changes to the austenite parent crystals. The elastic response coefficients are the most fundamental of all the properties of solid crystals and the most important subset of them are the shear and bulk moduli. Bulk modulus is a measure of the resistance of a solid to volume change and shear modulus is a measure of resistance to shape change. Shear moduli have the highest influence on the mechanical properties of the crystals [1, 2, 9, 65]. Bonding type (metallic, covalent, ionic, and molecular), and solidity index of a crystal (which is simply the ratio of the shear modulus to the bulk modulus, multiplied by a coefficient of the order of unity), have been used to classify crystals [12, 65]. Therefore, shear and bulk moduli and their ratio in the parent phase (austenite) are of paramount importance for this athermal martensitic phase transformation [2, 65]. The chemical factors that affect these parameters in the TiAu- and TiIr-based alloys under study in this work are discussed in the following.

Results and Discussion (Chemical Factors Influencing Bulk and Shear Moduli of Alloy Crystals)

Average Atomic Number

Table 1 presents more than 120 TiPd and TiPt alloy compositions [2, 14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64] that include nearly all alloy compositions studied so far, together with their forward and reverse transformation temperatures (Ms and As) in ascending order, and the thermal hysteresis where available. The maximum martensite start temperature (Ms = 713 °C) for TiPd-based system corresponds to a Pt and Zr alloyed composition, i.e., Ti45Zr5Pd5Pt45, whereas in the TiPt system, Ir-alloyed compositions exhibit the highest Ms temperatures, with Ti50Pt12.5Ir37.5 having a Ms of 1184 °C. The lowest transformation temperatures correspond to alloys with minimum Pd or Pt quantity in their compositions.

Table 1 Martensite start temperature (Ms), Austenite start temperature (As), temperature hysteresis, number of valence electrons per atom (ev/a), valence electron, ratio (VER) and average atomic number (Z) of TiPd- and TiPt-based shape memory alloys
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The average atomic numbers of the alloys under study are tabulated in Table 1. The transformation temperatures of these alloys clearly increase with Z of the alloys, as plotted in Fig. 1. The transformation temperature in the alloy systems studied shows increasing trends with the increasing average atomic number of the alloys (Fig. 1). In transition metals and intermetallics it is known that when the average atomic number is halfway through in the transition metals rows of the periodic table of elements, i.e., either Z = 25–26, or Z = 43–44, and so on., the martensitic transformation temperature can be reduced to some extent. This is because the elastic properties of the parent crystal hit maxima at or near these atomic numbers [65], and martensitic transformation should start at a lower temperature. This effect (if present) is masked by data scatter in the current work (Fig. 1). As shown, the existing TiPt alloys can have higher transformation temperatures than TiPd alloys, though the trend of change of Ms and As temperatures with atomic number is similar. These temperatures increase with the increasing atomic number of the alloys. In other words, the transformation temperatures of the alloys increase with increasing the total number of electrons (average atomic number) of the alloy. Non-valence electrons do not contribute to bonding and together with the protons comprise the ion kernel in the metallic bonding of the transition metal alloys. Increasing the non-valence electrons or the size of the kernels at constant (or nearly constant) valence electron numbers leads to weaker elastic bonding [65, 66]. This, in turn, means the resistance to shape and volume change of the crystal will be lower (at least in some crystallographic directions), and therefore, the phase change from austenite to martensite may occur at higher temperatures. The presence of Pt (Z = 78) and Pd (Z = 46) in the alloys causes the total number of non-valence electrons to extensively increase, with the influence of the former more than the latter due to higher Z (Table 1).

Fig. 1
figure 1

Transformation temperatures vs. atomic number: variations of Martensite start (Ms) and Austenite start (As) temperatures with average atomic number of TiPd- and TiPt-based alloys (Z)

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Number of Valence Electrons

Alloys with Ms (As) temperatures from as high as 1184 (1175) °C down to 20 °C (32 °C) were included in this investigation. In the categorization of the shape memory alloys (SMAs) with respect to their number of valence electrons per atom, a wide range was previously observed (4 ≤ ev/a ≤ 12) by Zarinejad and Liu [5]. The alloys were divided into low (e/a < 5), medium (5 ≤ e/a ≤ 7.50) and high (e/a > 7.50) valence electron groups. On this basis, TiPd-, and TiPt-based alloys all belong to the medium valence electron group with their e/a (6.65–7.30) in a range equal to or near 7 (Table 1). A considerable number of non-magnetic SMAs happen to belong to the medium range [5]. Majority of the highest Ms in shape memory alloy systems are usually observed in the medium electron group (5 ≤ ev/a ≤ 7.50) [5], especially when ev/a is 7.00 or near 7.00 (6.6 ≤ ev/a ≤ 7.3). A wide scatter of Ms temperatures of the alloys in this narrow range of valence electron per atom is observed (Fig. 2a). Similarly, a scatter of austenite start temperatures, As, with a relatively narrow number of valence electrons per atom variation is observed for (Fig. 2b). A wide range of transformation temperatures is observed, even though in TiPd- and TiPt-based alloy systems, the numbers of valence electrons per atom that contributes to the bonding strengths are close to each other. This indicates that although the number of ev/a is a factor in bonding, its effect must be studied along with other influencing factors such as the average atomic number (Z). With the increasing number of non-valence electrons because of increasing the average atomic number of the alloys at an almost constant bonding power (valence electrons), the interatomic bonding that keeps the ion kernels in the metallic bonds together is affected by the size and density of the kernels, [4, 65]. Higher numbers of non-valence electrons and protons in this condition reduce the bonding effects of the valence electrons and, therefore, reduce the elastic moduli at least in some shear directions.

Fig. 2
figure 2

Transformation temperatures vs. number of valence electrons: variations of Ms and As with the number of valence electrons per atom (ev/a) of TiPd- and TiPt-based alloys: a Martensite start (Ms); and b Austenite start (As) temperatures

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Valence Electron Ratio (VER) and Transformation Temperatures

Depending on the alloying elements, atomic fractions of elements comprising an alloy, different VER values have resulted. These values for the alloys examined are tabulated in Table 1. The variations of Ms and As temperatures vs. VER are plotted in Fig. 3. In both cases, a main trend is observed with increasing the electron ratio. Ms and As both, decrease consistently from temperatures as high as 1184 (1179) °C with VER values as low as 0.134 down to temperatures as low as 20 (32) °C with VER values of as high as 0.270. The general trends of variation of Ms and As with VER are similar (Fig. 3). Furthermore, Fig. 4 illustrates the combined influence of VER and ev on the transformation temperatures. It is evident that the variation of these temperatures with VER is more pronounced and follows a descending trend with increasing VER. The influence of ev is masked by the dominance of the effect of VER (Fig. 4). The relationship between VER, elastic properties, and transformation temperatures of the alloys are discussed. In metallic bonding, the valence electrons act like ‘glue’, bonding non-valence electrons and nuclei units together [4, 5, 66]; whereas non-valence electrons contribute to the total atomic volume of the alloy. Increasing VER could mean thickening of the ‘glue’ bonding the ionic kernels together. An established empirical relationship between the valence electron density and bulk modulus of metallic materials and intermetallic compounds is known. Higher VER usually results in higher bulk and, therefore, higher shear moduli [65, 66]. Increasing VER of the TiPd-based and TiPt-based alloys, therefore, is expected to result in higher elastic and shear moduli. The change of VER by replacing alloying elements for Ti, and Pd (Pt) is accompanied by a change in the elastic properties as the interatomic bonding is affected. A key factor controlling the bonding is the VER of the alloy. Before the martensitic transformation, the elastic moduli decrease during cooling and reach a critical value in a phenomenon known as the pre-curser effect [2]. If the elastic moduli of B2 austenite crystal of these alloys are enhanced because of higher VER, the cooling should continue to lower temperatures before a critical elastic constant is reached because of pre-martensitic softening of bulk and/or shear moduli, hence Ms is decreased. This is the case in nearly all alloying elements added in the present alloys that result in the transformation temperatures to reduce from that of the highest Ms alloy to the lowest (Table 1). On the contrary, if the alloy has lower elastic bulk and shear moduli in the austenite phase, the critical elastic constant can be reached at higher temperatures and Ms is increased. Any alloying elements that increase the VER of the alloy, decrease the transformation temperatures. Ni, Cu, Sc, etc. are examples of alloying elements that increase VER and lower the transformation temperatures. In case the addition of the alloying elements results in a reduction of VER, the transformation temperatures are elevated. For instance, the addition of Ir, Pt, Hf, Pd, Hf, etc. reduces the VER of the alloy and elevates the transformation temperatures. Ir and Pt are more effective in the increase of Ms and As due to higher Z (Table 1, Figs. 3 and 4). For each shape memory alloy, Ms is determined based on the elastic moduli of austenite and As follows Ms due to reversibility of the thermoelastic shape memory alloys in this study. Ms is controlled by VER, and the reverse transformation starting at (As) which is necessitated by reduction of the energy of the system due to increased vibrations of electrons and atoms in the crystal (by the increase of the temperature), is influenced by the factors that determine the thermal hysteresis, i.e., relaxation of stress arising from the forward transformation, defect structure changes inherited from the austenite to martensite transformation, or formed during reverse transformation (the level of bonds breakage and fluctuation of VER at dislocations). These factors determine the degree of reversibility of the transformation. Hence As (influenced by reversibility factors) follows Ms as the empirical data trend shows (Fig. 3).

Fig. 3
figure 3

Transformation temperatures vs. valence electron ratio: dependence of Ms and As on the valence electron ratio (VER) of the TiPd- and TiPt-based alloys: a Martensite start (Ms); and b Austenite start (As) temperatures

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Fig. 4
figure 4

Transformation temperatures vs. ratio and number of valence electrons: combined influence of valence electron ratio (VER) and number of valence electrons (ev/a) on Ms and As Temperatures of the TiPd- and TiPt-based. The dominance of the influence of VER over ev can be observed

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These discussions explain why some of the TiPd- and TiPt-based SMA compositions have high, and some others have low transformation temperatures. Moreover, in austenite to martensite transformation in these alloys, where shear is responsible for phase change from cubic to orthorhombic, the change of VER in austenite may affect the shear moduli of the crystals in different directions (C44 and C* as two main shear constants of the austenite B2 cubic crystal) differently, and therefore, the Zener’s anisotropy factor A = C44/C* will be affected by VER [65]. The anisotropy factor in turn facilitates shear and the martensite start temperature to be changed. The shown trends are general and indicative of the major underlying influence of VER on the transformation temperatures.

A comparison of the effect of ev/a, and VER on transformation temperatures in magnetic and non-magnetic shape memory alloys can be useful. The ev/a parameter is of paramount importance in all systems of shape memory alloys. However, the level of their influence on the transformation temperature is different and is dependent on: (a) what range of ev/ and VER we are dealing with; and (b) presence or lack of presence of magnetization and anti-ferromagnetism in the alloys. It was shown that when 5 ≤ ev/a ≤ 7.5, the effect of ev/a is less pronounced, and VER is dominant [5]. The alloys studied in the present work belong to this category and their transformation temperatures are dominated by VER. In other words, it is the change of VER that controls the transformation temperature and ev/a is less influential, as shown in Fig. 4. On the other hand, in magnetic shape memory alloys such as Heusler alloys and half Heusler alloys (containing Mn, Cr, Co, etc.), the transformation temperature characteristics are more complicated. Most of the alloys belong to the high valence electron category (ev/a more than 7.5), although some of them may have an ev/a parameter of 7–7.5. More Importantly, their VER value is high (VER > 0.27) [5]. In high VER alloys, the number of electrons per atom for bonding seems to be too high and, further increase of it has a negative effect on bonding and in turn on elastic properties of the austenite crystal. As a result, higher ev/a in these alloys results in increased transformation temperatures. Hybridization of d orbitals with s and p orbitals of common elements in magnetic shape memory alloys such as Ga, In, Al, Sn, etc., also influence the stability of austenite vs. martensite. Moreover, when VER of the alloy is high (VER > 0.27) by increasing the concentrations of elements such as Mn or Cr and possibly some other elements beyond a certain limit the elastic properties of austenite are negatively affected due to the antiferromagnetic behavior that these elements can exhibit and Ms and As temperatures both increases. Magnetization makes austenite more stable and decreases the transformation temperature and anti-ferromagnetism can reduce the strength of the bonds (at least in some shear directions) and make austenite less stable, hence increases the Ms and As temperatures.

Therefore, in magnetic shape memory alloys, by increasing ev/a, because of the excessive amount of valence electrons, the transformation temperatures increase, and its effect is more pronounced and could be a very influential parameter. This is not seen in non-magnetic alloys, where VER plays a dominant role.

The general trend presented in this work provides an overall criterion for the design of the alloys with the desired service temperature. However, comparisons at small differences in temperature or compositions concerning the overall trend is not recommended owing to the scatter of data (Figs. 3 and 4). Details of the microstructural features and deformation regime of the alloy need to be known for comparisons in small differences between the alloys. However, VER seems to be the most influential factor dominating the transformation temperatures.

Temperature Hysteresis

The available data for transformation hysteresis in the alloys under study show a relatively wide range of transformation hysteresis (7–114 °C) (Table 1). It is shown that by increasing the valence electron ratio (VER) of alloys the transformation temperature hysteresis is generally decreased (Fig. 5a). In addition, temperature hysteresis increases with the average atomic number of the alloys, (Z) (Fig. 5b). These are indicative of the influence of the strengths of the bonds in the thermoelastic transformation. However, how the strengths of the bonds affect it, is complex, since hysteresis is dependent on several factors at microstructure, crystal structure, and (electronic) bonding levels. In principle, in alloys and intermetallic compounds, the dislocation mobility is determined by the pattern and intensities of the electron density fluctuations. VER is, to a major extent, in line with the valence electron density of the material as shown by Zarinejad et al. [4, 5]. It can simply be argued that in general, weaker atomic bonds can be plastically strained more extensively, following thermoelastic martensitic transformation. The relaxation of the energy by plastic deformation mechanisms results in wider hysteresis. The strengths of the bonds are controlled by valence electrons among other factors. Based on this argument, the alloys with higher VER and stronger austenite bonds exhibit narrower temperature hysteresis as is seen in this study (Table 1, Fig. 5a). However, the complexity of hysteresis is further realized as it is controlled by other factors at microstructure and atomic scales. In crystals with non-local bonding electrons, where dislocation lines move concertedly (such as in transition metal alloys), the motion of the dislocation lines is not activated. It is resisted by the viscous drag created by somewhat free electrons (from the sea of electrons in metallic bonding), and by phonons. However, it is also known that shear stress from transformation drives dislocation motion. When the stress is sufficiently high, at localized barriers, it can cause surmounting a barrier or reducing its height by affecting the local bonding electrons, [65]. Hence, the bonding which is dominated by VER is a key factor in hysteresis and is affected by the chemistry of alloys through changes in VER. Separation of the effect of bonding, crystallographic, and microstructural factors on temperature hysteresis was found to be complex and not possible in current work. Temperature hysteresis requires deeper study at these scales.

Fig. 5
figure 5

Transformation temperature hysteresis vs. valence electron ratio and atomic number: variations of transformation hysteresis of TiPd- and TiPt-based alloys with a valence electron ratio (VER), and b average atomic number (Z). The dotted lines in the figure indicates the overall trends

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Conclusions

The main correlations shown in this work reveal clear dependence of the transformation temperatures of TiPd- and TiPt-based alloys on the most dominant factor, i.e., valence electron ratio (VER), of these transition metal shape memory alloys. Alloying elements and in general, alloy chemistry influence the number and ratio of valence electrons of TiPd- and TiPt-based shape memory alloys and, changes the transformation temperatures through their effects on elastic properties of the crystals. Ms and As both significantly decrease with increasing VER from 0.134 to 0.270. This gives rise to transformation temperatures from 1184 °C (1179 °C) to 20 °C (32 °C) in these alloy systems. VER is the dominant intrinsic parameter that can be utilized to take concrete steps toward successful alloy design with the desired service temperatures. Increasing VER, and average atomic number of the alloy contribute to the widening the temperature hysteresis of the alloys. TiPd- and TiPt-based alloys with higher transformation temperatures generally show wider temperature hysteresis.