Introduction

Applications of high-temperature shape memory alloys (HTSMAs) may encounter serious challenges such as unstable cyclic behavior and large plastic deformations at high stress and temperature [1, 2]. There are a few approaches such as thermomechanical processing, solid solution hardening, and precipitation hardening that can be employed to overcome these challenges and improve the shape memory and mechanical properties of HTSMAs [3,4,5,6]. However, since the HTSMAs are often ordered intermetallics with limited ductility at low and intermediate temperatures, applying thermomechanical processing can be difficult and expensive [7]. Therefore, precipitation hardening is considered the most practical and cost-effective method among other strengthening approaches. Introducing nanoscale precipitates into the material can increase the critical shear stress required for slip in the matrix, resulting in improved shape memory properties [8]. In addition, the inclusion of these nanoparticles can act as barrier against dislocation motion leading to enhanced fatigue life and cyclic stability [8,9,10].

Among the HTSMAs, NiTi-based alloys have been studied the most since they exhibit the highest transformation temperatures (TTs). There are a limited set of elements, including Pt, Pd, Au and Hf, which can be employed for alloying with NiTi to create ternary and quaternary HTSMAs [11,12,13]. It has been reported that the addition of a similar content of Hf has a more significant effect on the TTs in comparison with Pd and Au [1, 4]. Furthermore, unlike NiTiAu, NiTiPd and NiTiPt which are limited to high priority applications due to being expensive, NiTiHf alloys can be employed in a broad range of high-temperature applications [14,15,16]. Additionally, NiTiHf exhibits superior transformation strains compared to NiTi in both tension and compression. Adding even minor quantities of Hf to NiTi improves the transformation characteristics. Under compression, NiTiHf alloys can achieve transformation strains of up to 7%, and under tension, these strains can reach up to 15%, while those values for binary NiTi alloys are 5.5% in compression and 10% in tension [17].

Based on previous studies, it has been found that NiTiHf alloys can contain different types of precipitates, which are influenced by several factors such as alloy composition—specifically whether it is Ti-rich or Ni-rich—and heat-treatment schedule, among others. For instance, König et al. [18] investigated a wide range of Ti-rich compositions by fabricating thin films of NiTiHf and observed four distinct types of precipitate, i.e., HfNi(Ti), Ti2Ni(Hf), Hf2Ni(Ti), and the Laves phase. Their findings demonstrated that with an increase in the Ti content within the NiTiHf alloy, there was a concurrent rise in the occurrence of the secondary Laves phase. The Ti2Ni(Hf) precipitates have also been observed in Ni-lean NiTiHf alloys [19, 20]. It has been reported that increasing the Ni content decreases the volume fraction of Ti2Ni(Hf) precipitates, however, they can still be found in slightly Ni-rich compositions [19, 21]. Fine Ti2Ni(Hf) precipitates act as vital agents in strengthening and improving the shape memory and superelasticity properties of NiTiHf alloys. It should be noted that the size of the Ti2Ni(Hf) precipitates significantly influences and controls the patterns of martensite.

The characterization of precipitate phase in Ni-rich NiTiHf was performed by Han et al. [22] where they conducted an aging heat treatment at 600 °C for 150 h on Ni48.5Ti36.5Hf15 alloy. By performing comprehensive Transmission Electron Microscopy (TEM) analyses, they identified the presence of a specific type of precipitate in the alloy with an approximate composition of Ti0.6Hf0.4Ni. This precipitate exhibited a face-centered orthorhombic lattice structure and demonstrated a close association with the B2-type matrix. Furthermore, the precipitate possessed a distinct oblate spindle-like shape, which was then called the “H-phase” precipitate. To identify the exact composition of the precipitate, Yang et al. [23] utilized advanced techniques such as high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM), atom probe tomography (APT), and first-principles calculations. They subjected an alloy with a composition of Ni50.3Ti29.7Hf20 to an aging process at 600 °C for 815 h. By extending the aging time, the H-phase precipitates coarsened, which facilitated the precise characterization of them. Upon examination, it was revealed that the formed H-phase precipitate possessed F 2/d 2/d 2/d space group with a composition of Ni3TiHf2. Nonetheless, it is essential to emphasize that both prior research [24,25,26] and the observations made by Yang et al. [23] have demonstrated that the composition of H-phase precipitates can differ depending on the initial alloy’s composition, even when subjected to similar heat treatment processes. This is in contrast to Ni4Ti3 precipitates, which form with a fixed composition regardless of the initial composition of NiTi.

It is important to highlight that the effectiveness of the strengthening achieved by precipitates depends on various factors, including their size, volume fraction, interparticle spacing, and coherency with the matrix [27]. Additionally, factors such as aging time, temperature, and lattice mismatch between the matrix and precipitates introduce local stress fields that can modify the strengthening behavior, martensite nucleation, and shape memory properties. In spite of many variables involved in the alloy design by precipitation, limited research has been conducted to investigate the influence of precipitates on the behavior of NiTiHf alloys. To gain further understanding of the properties of alloys, atomistic simulations are considered a viable and cost-effective method to explore the microstructural features of the precipitates in NiTiHf [28]. In this study, molecular dynamics (MD) simulations were employed to investigate the effect of precipitates on the temperature- and stress-induced phase transformation behaviors of Ni-rich NiTiHf Alloy. Additionally, direction-dependence of superelastic and martensitic response of NiTiHf models was studied by applying compressive stress in various crystallographic directions of the models containing one or more precipitates.

Computational Method

In order to simulate the effect of H-phase precipitate on the temperature- and stress-induced phase transformation of NiTiHf, MD simulations were employed. To describe the interaction of atoms in the ternary alloy, a newly developed 2NN MEAM potential for NiTiHf was used [29]. It has been shown that the computed lattice parameters of NiTiHf with varying compositions for both cubic B2 (austenite) and monoclinic B19′ (martensite) structures closely agree with experimental values when employing the specified interatomic potential. Additionally, this potential has effectively replicated the high-temperature superelastic and low-temperature martensitic behavior of Ni-rich NiTiHf in prior simulations [29]. LAMMPS [30] and Ovito [31] packages were utilized for MD simulations and visualization of results, respectively. For the atomic arrangement, models of B2 crystal structure of a widely used Ni-rich composition, i.e., Ni50.3Ti29.7Hf20, were generated using Atomsk [32]. To generate NiTiHf models with the desired composition, an equiatomic NiTi system was created. Subsequently, Hf atoms were introduced into the system and replaced by Ti atoms in quantities that matched the desired final composition. This introduction of Hf atoms was done randomly. Finally, the distribution of atoms was visually examined to verify that there were no clusters or blocks of Hf atoms, ensuring a uniform dispersion of Hf throughout the model. According to the crystal structure reported by Yang et al. [23] for Ni-rich NiTiHf, spindle-shaped H-phase precipitates were modeled of Ni3TiHf2 composition and an orthorhombic structure. It is known that shorter aging time at a lower temperature leads to formation of fine and coherent H-phase precipitates. In contrast, at elevated aging temperatures and longer time, precipitates grow, resulting in increased interparticle distances. To investigate the influence of each of these cases, two scenarios were examined for models containing precipitates. In the first scenario, the model featured a single, large precipitate positioned at the center of the matrix, corresponding to an alloy subjected to longer aging. Conversely, in the second scenario, a model of the same size consisted of eight smaller precipitates evenly dispersed within each of the octants of the box. The percentage of precipitation was similar in both scenarios.

In order to model NiTiHf matrix with one precipitate (referred to as single-precipitate model), the oblate spindle-like precipitate was embedded into the center of a NiTiHf cube with 250 × 250 × 250 Å dimensions. The lattice correspondence of the precipitate (subscript H) to the matrix (subscript B2) was: \({\left[100\right]}_{H} \to {\left[001\right]}_{B2}\); \({\left[010\right]}_{H} \to {\left[110\right]}_{B2}\); \({\left[001\right]}_{H} \to {\left[\overline{1 }10\right]}_{B2}\). Moreover, the minor axis of the precipitate was along [001] orientation of the matrix. For the other scenario with multiple precipitates (referred to as multi-precipitate model), first a similar procedure was employed on a smaller model with 125 × 125 × 125 Å dimensions, and a proportionally smaller precipitate was embedded into the center of the model. Then, this model was replicated consecutively in all the three dimensions, to generate the final model with eight precipitates. In both scenarios, the models contained almost one million atoms and the fraction of the precipitates was 2.6%. The aspect ratios (the ratio of major to minor axis length) of precipitates were based on the experiments of Evirgen et al. [33].The details of the dimensions and the crystal structure of the modeled precipitates are presented in Table 1. In addition, Fig. 1a shows the single-precipitate model (cut in the middle for better visualization) and Fig. 1b presents the distribution of the fine precipitates in the multi-precipitate model (matrix atoms are not shown). It should be noted that for comparison purposes, in this study the precipitate arrangement was considered to be the same and parallel in both single-precipitate and multi-precipitate models. More cases will be considered in the future for the cases of intersecting precipitates as reported by [34]. During the simulation, periodic boundary conditions were applied to the models in all the three dimensions and the timestep was taken as 2 fs. To control the temperature and pressure, Nose–Hoover thermostating and barostating were applied on all atoms.

Table 1 Details of the modeled orthorhombic precipitates
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Fig. 1
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a Single-precipitate model (cut in the middle for better visualization), b distribution of fine precipitates in multi-precipitate model (matrix atoms are not shown)

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Temperature-Induced Phase Transformation

To capture the effect of H-phase precipitates on the temperature-induced phase transformation of NiTiHf, a cycle of cooling followed by heating was applied to models with and without precipitates. Before applying the thermal cycle, the models were equilibrated at 800 °C to obtain similar initial condition of austenite. Following equilibration, the models were cooled down from 800 to 0 °C and then heated back to 800 °C at a constant rate of 2.75 °C/ps. The rate of martensitic transformation (computed through measuring the martensite fraction in the model in each step) during thermal cycle was extracted for all models, where its variation has been presented in Fig. 2a with respect to the temperature change. In these graphs, the sudden jump in the rate of martensitic transformation during cooling indicates the start of transformation from austenite to martensite (thus, martensite start temperature, Ms) and the end of subsequent drop in the rate of transformation indicates the completion of martensitic transformation (thus martensite finish temperature, Mf). Similarly, the sudden changes in the rate of transformation during heating are the indication of martensite to austenite transformation, As, austenite start and Af, austenite finish temperatures. It can be seen that the As and Af increased in the models that contained precipitate compared to the precipitate-free model. While the extent of this change is significant for the single-precipitate model, the multi-precipitate model showed a lower increase. Karaca et al. [35] also reported the same trend of change in the TTs, by conducting heat treatment at different temperatures on Ni50.3Ti29.7Hf20 alloy. Based on the Differential Scanning Calorimetry (DSC) curves of their experiments (Fig. 2b [35]), aged samples showed higher TTs as opposed to as-extruded sample that exhibited the lowest TTs. Among heat-treated samples, the ones that aged at higher temperature formed larger precipitates leading to yet higher TTs.

Fig. 2
figure 2

a Variation of the rate of martensitic transformation with temperature change, indicating the transformation temperatures of the material, b DSC curves of Ni50.3Ti29.7Hf20 obtained from experiments for different aging conditions [36]

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It should be noted that the composition difference between the matrix and the precipitate can change the overall composition of the model compared to the initially stated composition. This could be a potential reason for the difference of TTs among various models. Table 2 shows the overall compositions of the final state of different models. As can be seen in this table, the overall composition across different models will remain close to the intended composition. On the other hand, the small difference between the TTs of multi-precipitate and precipitate-free models suggests that the composition change cannot be considered the main reason for the increase in the TTs of the single-precipitate model. To investigate the difference between TTs of precipitated and precipitate-free models, a cross-section of each model showing the present austenite and martensite phases at the end of cooling are presented in Fig. 3. Color coding based on Polyhedral template matching (PTM) algorithm [37] was employed to identify different phases in each of the models. In Fig. 3, the martensite phase (B19′ structure) is shown as red, and the austenite phase (B2 structure) is presented as blue. As can be seen in this figure, in all the models most of the initially austenite material has undergone phase transformation into martensite. However, the blue bands of atoms, which signify the twinning planes, distributed densely in the precipitate-free model, whereas in the single-precipitate model these bands are lumped into a large area. It should be noted that during the reverse phase transformation from martensite to austenite, these twinning planes act as nucleation zones for austenite. Therefore, the reverse transformation from martensite to austenite during heating occurs sooner (i.e., at a lower temperature) in the precipitate-free model, leading to lower As and Af. Conversely, in the single-precipitate model, which only exhibits one rather thick twinning plane, due to the existence of smaller potential nucleation zones, the reverse phase transformation happens at a higher temperature and, thus, higher As and Af will be observed for this case. In the multi-precipitate model, the number of twinning planes is larger than the single-precipitate model, yet less than the precipitate-free model, which accordingly results in significantly lower TTs compared to the single-precipitate model, and slightly higher TTs than the precipitate-free model.

Table 2 Final composition of the models after adding the precipitate(s)
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Fig. 3
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Present austenite and martensite phases at the end of cooling in different models (red and blue regions denote martensite and austenite phases, respectively) (Color figure online)

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Stress-Induced Phase Transformation

In order to analyze how the presence of precipitates affects the superelastic and martensitic responses of NiTiHf alloy, a compressive stress was applied to the aforementioned precipitated and precipitate-free models at two temperatures: one below Mf and one above Af. The temperature above Af for superelastic behavior was chosen in a way that the resulting stress–strain responses could be compared. Since according to the Clausius–Clapeyron equation there is a linear relationship between transformation stress and temperature, stress-induced phase transformation was studied at Af + 50 °C for each model. Hence, using an isobaric-isothermal NPT ensemble, the models were equilibrated and loaded at 0 °C as a martensitic temperature. Using the same procedure, the precipitate-free, single-precipitate, and multi-precipitate models were subjected to equilibration and loading at 675, 780, and 685 °C (as austenitic temperatures), respectively. These temperatures were obtained according to the TTs in Fig. 2. A uniaxial compressive load was gradually applied along the [100] direction from 0 to 2 GPa, followed by unloading of the model back to 0 GPa. Before studying the NiTiHf models, an equivalent model composed solely of the precipitate (a box with the crystal structure of the precipitate, Ni3TiHf2) was analyzed. This preliminary investigation aimed to gain insight into the stress–strain behavior of the precipitate. Figure 4 presents the stress–strain response of the precipitate at 0 °C and 675 °C. As can be seen in Fig. 4a at 0 °C, the model reaches more than 0.03 strain with a residual strain of 0.01 at the end of unloading. Furthermore, an elastic response was observed at 675 °C, reaching a maximum strain of 0.024 (Fig. 4b).

Fig. 4
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Stress–strain response of solely precipitate model subjected to compressive load: a 0 °C, and b 675 °C

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The compressive stress–strain response of the NiTiHf models with and without precipitate along the [100] crystallographic direction and at martensitic and superelastic temperatures are presented in Fig. 5a and b, respectively. A martensitic behavior with residual strain at the end of unloading, can be observed for all the models at 0 °C in Fig. 5a. The single-precipitate and multi-precipitate models exhibit higher stress plateaus and maximum strains (under 2 GPa of stress) compared to the precipitate-free model. Although, previous research suggest that the H-phase precipitates are fully coherent with the austenitic matrix [26], after cooling the model to a martensitic temperature (i.e., 0 °C), this coherency between the H-phase precipitates and the matrix becomes disrupted due to the transformation strains. Because of the presence of H-phase precipitates, the martensitic matrix undergoes adjustments to accommodate the non-transforming particles, and this accommodation results in higher levels of strain. In Fig. 6a, the loss of coherency can be observed, where the models are characterized by Ovito’s defect mesh analysis. This algorithm identifies those parts of the model that the atomic arrangement does not resemble a perfect crystal. Accordingly, insets (I) and (III) show the defect mesh (precipitate surface) of the single-precipitate and multi-precipitate models before equilibration, while insets (II) and (IV) represent the defect mesh after equilibration at martensitic temperature. It can be seen how the precipitates’ surface became uneven and rough, which is the indication of coherency loss between precipitate and the matrix.

Fig. 5
figure 5

Stress–strain response of the models subjected to compressive load along: a [100] direction at 0 °C, b [100] direction at Af + 50 °C, c [110] direction at 0 °C and d [110] direction at Af + 50 °C

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Fig. 6
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a Coherency of precipitates before and after equilibration at 0 °C, b formed martensite variants in single- and multi-precipitate models at Af + 50 °C

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Figure 5b presents the superelastic stress–strain curves of the models loaded along [100] direction at the temperature of Af + 50 °C. During the loading phase, the austenitic alloys initially experienced elastic deformation. The stress-induced phase transformation was triggered around 0.025 strain in all the models. As the loading continued, the models underwent some transformation strains, characterized by a plateau stage observed in the stress–strain curves. The superelastic stress–strain response observed in the multi-precipitate model, exhibited a reduction in the energy dissipation capacity (i.e., reduction in the size of hysteresis loop). Such a reduction in the size of the hysteresis loop can be favorable for higher fatigue resistance [38, 39]. Additionally, the transformation strain and maximum strain in the multi-precipitate model were lower compared to the other models since it is more challenging to select and grow martensite variants at the presence of many precipitates. Visualization of different martensite variants formed in the precipitate-free, single- and multi-precipitate models are presented in Fig. 6b. The annotation of variants was based on the distortion of each area compared to the original structure. As compared in the insets (VI) and (VII), the single-precipitate model exhibits more types of variants during stress-induced phase transformation.

Previous studies have highlighted the substantial influence of crystallographic orientation on the superelastic and martensitic behavior of Ni-rich NiTi alloys [28, 40,41,42]. In order to investigate the direction-dependence of these responses in NiTiHf and the impact of H-phase precipitates on them, a compressive load was applied along [110] direction (as opposed to previous case which was loaded along [100] direction) on the precipitated and precipitate-free models. Similar to [100] direction, the models were subjected to load at 0 °C and Af + 50 °C. Figure 5c shows the stress–strain response of all the models at 0 °C. Both precipitated models and the precipitate-free model exhibited elastic behavior along [110] direction without any phase transformation, unlike the loading along [100] direction. It has been reported that in NiTi, grains orientated in [110] direction inhibit the stress-induced phase transformation and detwinning [43, 44]. The elastic behavior of all models in Fig. 5c can be attributed to the same reasoning since the models are at martensitic temperature, and under the applied load, they cannot transform from twinned to detwinned martensite. As can be seen in Fig. 5d, the stress–strain response of models at Af + 50 °C was different from [100] direction. In models loaded along [110] direction, the maximum strain achieved was slightly over 0.03, whereas in the [100] direction, the maximum strain was twice as large. Additionally, the energy dissipation in [110] direction was considerably lower compared to [100] direction. Again, the martensitic transformation was hindered due to the crystallographic orientation leading to smaller transformation strains compared to [100] case. Furthermore, the single-precipitate and multi-precipitate models showed higher stress plateaus compared to precipitate-free model, similar to [100] direction.

Conclusions

In this study, MD simulations were employed on Ni-rich NiTiHf to investigate the effect of H-phase precipitates on the temperature- and stress-induced phase transformations. Three scenarios were considered for the simulations of the alloy: containing a single large precipitate, multiple fine precipitates and no precipitate. Calculation of the TTs of the models indicated higher austenite start and finish temperatures for the models with precipitate(s), which was attributed to the smaller number of twinning planes in these models, that can act as nucleation sites during the reverse phase transformation leading to higher As and Af. By applying a compressive load, the precipitated models exhibited larger transformation strains at martensitic temperature, as a result of disruption of coherency between the precipitate and the matrix at low temperatures, which causes the matrix to undergo adjustments to accommodate the non-transforming precipitate leading to larger transformation strains. Finally, by subjecting the models to the same compressive load along [110] crystallographic orientation, it was found that the maximum strain and energy dissipation are significantly smaller than the models that were subjected to the load along [100] direction.