Study of ageing and size effects in Nickel–Titanium shape memory alloy using molecular dynamics simulations

ABSTRACT

The effect of ageing process at 1073.15 K on the athermal phase transformation in different nanograins (4.5, 6 and 9 nm) of Nickel–Titanium shape memory alloy is studied using molecular dynamics simulations. The martensitic transformation start temperature and transformed martensite volume decrease with decreasing grain size. The two-stage phase transformation with intermediate phase formation is observed only in the smallest grain (4.5 nm) where a single martensite variant is formed. On the contrary, larger grains show a self-accommodated twinned martensite variants with one-stage transformation from austenite to martensite. The ageing process hardly influences the morphology of the martensite phase, however, it rises the martensite start temperature and reduces the austenite start temperature. The nano grain size models demonstrate the constrained transformation due to the transformation barrier and the accumulation of internal stress, while the ageing process shows the tendency for stress relaxation.

KEYWORDS:

• Shape memory alloy
• molecular dynamics
• size effect
• ageing

1. Introduction

The Nickel–Titanium (Ni–Ti) alloy is widely acknowledged as a shape memory alloy (SMA) which has the ability to recover its shape after deformation. The shape memory effect (SME) and super elasticity properties in this alloy are extensively applied in medical and mechanical engineering applications [1]. In several engineering applications, e.g. actuator, extractor, structure, the Ni–Ti alloy can deform and restore structures as required by controlling the local temperature. When Ni–Ti alloy undergoes the ageing process, the austenite (B2) and martensite (B${19}^{\mathrm{\prime }}$) phase transformation temperatures and path are affected [2] and it becomes difficult to predict the microstructure behaviour in Ni–Ti SMA.

The ageing process had proved to contribute to the SME and the mechanical properties of Ni–Ti alloys in the early experiments [3,4]. They addressed that the influence of the ageing treatment is caused by the existence of precipitates which also lead to the dislocation of phase structure and initiate the formation of the B${19}^{\mathrm{\prime }}$ martensite phase [5]. The precipitates (Ni${}_4$Ti${}_3$ and Ni${}_3$Ti) formed in the material can lead to changes in the microstructure, such as high dislocation density caused plastic deformation in B2 phase and formation of grain boundaries, which can affect the material properties of SMA and derives multistage transformation in ageing processing [6,7]. In addition, the multiple stages of phase transformation behaviour between austenite and martensite after the ageing process were observed in the experiments [8]. Liu et al. [4] performed low temperature ageing experiments between 590 K and 650 K, where a two-stage transformation was found in heating processes from martensite to austenite, and a three-stage transformation with an interim phase was found in the cooling process for the near-equiatomic Ni–Ti alloy. Liu et al. [9] found multiple transformations after the ageing process around 473 K to 748 K, whereas a specimen with a higher ageing temperature and longer ageing time has more potential to undergo direct phase transformation. Similarly, in the research by Sinha et al. [10], the homogenised specimen is more likely to undergo single-stage transformation in both heating and cooling processes after ageing, which is in agreement with Liu et al. ageing experiment at 1073 K [9].

Grain size has a significant effect on the mechanical properties in metallic alloys [11,12] and is also believed to have large impact on the martensite transformation [13,14]. Fine austenite grain was found to make it difficult for the transformation of martensite in steel caused by the strengthening of austenite [15]. Similarly, the model proposed by Ni et al. [16] showed that the martensite transformation temperature decreases with finer grain size due to the higher energy density in the phase interface. In the experiment by Zhang et al. [17], it was found that the transformation temperature is controlled by the internal stress caused by the refined grain, phase composition and internal defects in the Ni–Ti alloy.

Molecular dynamics (MD) simulations are prevalent in the study of phase transformation. It has been proved that the MD approaches can be used to study various phenomena in Ni–Ti alloy, especially the SME, athermal and stress-induced phase transformations [18,19]. However, the SME has not been studied with the combined effect of size and ageing effects in MD. Due to the limited time scale, simulating the ageing process just for a minute can also require tremendous computational resources. There are several theoretical methods for modelling of ageing in alloys, such as kinetic Monte Carlo (KMC) [20], phase field [21], phenomenological and thermokinetic models using DICTRA [22]. The study of microstructure behaviour at the atomic level has the advantage in using the KMC and MD methods [23]. However, because of the unpredictable microstructure behaviour it may have in quenching (e.g. the precipitate growth, atomic dislocation caused pile-up, pinning and grain boundary sliding), the KMC method may lack well-defined mechanisms and pathways to demonstrate these possibilities. Although the MD has a very limited length scale, it can provide details about the possible phase transformation pathways that can be used in other modelling approaches to continue in larger time-scale simulation models [24]. Nevertheless, a relatively small ageing model may still have the potential to provide some insights into the transformation behaviour. In addition, an optimum size of the model is expected to balance the computational resource consumption and demonstration quality in the ageing simulation. According to Streiz et al. [25], the optimized simulation box can prove its independence in microstructure demonstrating that the size can be relatively small to conserve simulation periods. Different sizes of nanocubes in the simulation domain can have different morphologies [26], however, the expected microstructure phenomenon is able to be observed in the simulation with a model size above a certain level. Considering the grain size effect, different simulation domain sizes are used to study the optimum size for ageing simulation of Ni–Ti alloy as well as to study the size effects on the phase transformation. In this work, the combined effect of grain size and ageing on Ni–Ti alloy during cooling and heating is investigated at an ageing temperature of 1073.15 K using MD simulations.

2. Methodology

In this study, the Ni–Ti shape memory alloy ageing simulations are performed in three different simulation domains with the number of lattices 15, 15, 3; 20, 20, 2; 30, 30, 3, in the X, Z and Y directions, as shown in Figures 1(a–c), respectively. The periodic boundary conditions are applied in the X, Y and Z directions (Figure 1). The initial phase models are single crystal austenite with different grain sizes without grain boundary. The size of the simulation domain represents the grain size of each single crystal. The cubic austenite lattice has the same orientation as the initial model (Figure 1). The model is able to capture four martensite variants which have the $\left(100\right)$ surface orientation towards the $\left[010\right]$ and $\left[010\right]$ directions of austenite lattice [27]. The simulations are initiated with B2 austenite phase that is held at 1073.15 K for 5 $\mu \mathrm{s}$, 10 $\mu \mathrm{s}$ and 2 $\mu \mathrm{s}$, with a time step of 1 fs, 2 fs and 1 fs, respectively. The simulation is performed using the LAMMPS software [28,29] with pressure and temperature controlled by Nose–Hoover thermostat and barostat modules. The second nearest-neighbour modified embedded-atom method (2NN-MEAM) potential is used as the interatomic potential in the Ni–Ti alloy simulations in LAMMPS. This potential has been widely used in the research for studying phase transformation in Ni–Ti alloys [30–33]. The energy equation for the modified embedded-atom method potential is given by Ko et al. [34] as (1) $E=\sum _i[F_i\left({\overline{\rho }}_i\right)+\frac{1}{2}\sum _{j\ne i}{\varphi }_{ij}\left(R_{ij}\right)]$(1) where ${\overline{\rho }}_i$ is the background electron density, $F_i$ is the embedded energy and ${\varphi }_{ij}\left(R_{ij}\right)$ is the pair interaction between atoms i and j with a range $R_{ij}$ which is a value computed from the total energy and the embedding function. The potential parameters of the interaction potential of Ni–Ti originally published by Ko et al. [34] are archived in OpenKIM [35], which are summarized in Tables 1 and 2, which are the sublimation energy $E_c$ (eV/atom), the equilibrium nearest-neighbour distance $r_e$ (Å), the bulk modulus B (${10}^{12}dyne/c{m}^2$), the scaling factor for the embedding energy A, the exponential decay factors for the atomic densities ${\text{β}}^{\left(i\right)}$, the weighting factors for the atomic densities $t^{\left(i\right)}$ and the smooth cut-off function $C_{min}$ and $C_{max}$. The simulation results are visualized using the software OVITO [36].

Figure 1. The orientation of cubic austenite lattice is shown in the initial Ni–Ti ageing models with distinct sizes containing (a) $4.5\times 4.5\times 0.9{\mathrm{n}\mathrm{m}}^3$ with 1350 atoms, (b) $6\times 6\times 0.6{\mathrm{n}\mathrm{m}}^3$ with 1600 atoms and (c) $9\times 9\times 0.9{\mathrm{n}\mathrm{m}}^3$ with 5400 atoms.

Three initial state of Ni–Ti MD models with a 3D parallel view.

Table 1. The MEAM parameters for pure Ni and Ti alloys.

Atom	$E_c$	$r_e$	B	A	${\text{β}}^{\left(0\right)}$	${\text{β}}^{\left(1\right)}$	${\text{β}}^{\left(2\right)}$	${\text{β}}^{\left(3\right)}$	$t^{\left(1\right)}$	$t^{\left(2\right)}$	$t^{\left(3\right)}$	$C_{min}$	$C_{max}$
Ni	4.45	2.490	1.8586	0.79	2.48	1.94	3.46	2.56	2.84	$-1.20$	2.49	0.95	1.75
Ti	4.75	2.850	1.0735	0.24	2.20	3.00	4.00	3.00	$-18.0$	$-32.0$	$-44.0$	0.25	1.58

Table 2. The MEAM parameters for Ni–Ti alloy; 1 and 2 represent Ni and Ti.

$E_c$	$r_e$	B	A	$C_{min}^{1,1,2}$	$C_{min}^{1,1,2}$	$C_{min}^{1,1,2}$	$C_{min}^{1,1,2}$	$C_{min}^{1,1,2}$	$C_{min}^{1,1,2}$	$C_{min}^{1,1,2}$	$C_{min}^{1,1,2}$
4.96	2.612	1.2818	0.025	0.25	1.70	0.09	1.70	0.49	1.40	1.60	1.70

3. Results and discussion

The ageing simulation results show that a higher ageing temperature leads to an unstable phase compared to a lower temperature, as shown by the presence of white atoms in the austenite phase at 1073 K in Figure 2. The unstable atoms could be caused by the abnormal free energy variation from a higher local internal stress due to the constrained grain model. The 4.5 nm grain model shows a two-stage transformation in cooling and single stage transformation in heating, while the 6 nm and 9 nm grain models show single-stage transformation in heating and cooling. With respect to different proportions of unstable atoms in 4.5 nm, 6 nm and 9 nm sized grains are 2.2 at.%, 0.9 at.% and 1.4 at.%, respectively, the model with an interim phase (4.5 nm grain) has about 100 % more unstable atoms than the models with single-stage transformation. Therefore, it can be construed that the unstable phase in ageing has the potential to increase the internal stress and the B${19}^{\mathrm{\prime }}$ transformation energy barrier which yields an interim phase.

Figure 2. Phase transformations during quenching after ageing for 1 µs. Plots of local volume and temperature for different simulation system sizes containing (a) 4.5 nm with 1350 atoms, (b) 6 nm with 1600 atoms and (c) 9 nm with 5400 atoms. The green and grey atoms denote Nickel and Titanium atoms, respectively. The blue and red regions denote the austenite (B2) and martensite (B${19}^{\mathrm{\prime }}$) phase structures, respectively. The blue region in the martensite phase (red) indicates the retained austenite at the interfaces of martensitic twins. The red lines indicate the direction of martensite variants.

The phase transformation behaviours in three different size Ni–Ti alloy models with temperature variation in the quenching process.

As shown in Figure 2, the sudden change of volume at lower temperatures indicates the start point of phase transformation. In the calculation by Waitz et al. [37], the energy barrier grows exponentially with the decreasing grain size below 50 nm in Ni–Ti SMA. Consequently, the phase transformation from austenite to martensite is constrained due to the high energy barrier. Therefore, the interim phase is more likely to emerge before the martensite due to the much lower corresponding energy barrier for the transformation of austenite to interim than that for austenite to martensite. Correspondingly, the martensite transformation is suppressed by the smaller grain size and the martensite start temperature ($M_{\mathrm{s}}$) tends to decrease with decreasing grain size as shown in Figure 2.

The twinned martensite structure in the model sizes 6 nm (Figure 2b) and 9 nm (Figure 2c) is in agreement with the experimental results [38] demonstrating the herringbone twinned martensite due to its accommodation of the high transformation strain. A relatively small-sized model of 4.5 nm demonstrates a nanograin formed within the simulation box with retained austenite near martensite interface (Figure 3a,b), which is comparable to the structure observed in a near-equiatomic Ni–Ti alloy using transmission electron microscopy (Figure 3c) by Waitz et al. [37]. The nanograin and the grain boundary form to self-accommodate the transformation strains attribute to the local high energy barrier generated by the limited strain and interfacial energy in the region. Due to the interim phase formation, which reduces the internal strain energy and increases the retained austenite volume, the martensite will no longer be twinned to accommodate the stress. On the other hand, without interim phase formation in larger grains (Figure 2b,c), the martensite phase has the trend to minimize the internal strain energy through twin formation.

Figure 3. Martensite formation in nanograins of Ni–Ti SMA. (a) Growth of a single variant of martensite inside the nanograins of the 4.5 nm sized system. Nickel (green) and Titanium (grey) atoms where martensite variants and the single grain boundary are indicated by red lines and black lines, respectively. (b) Phase distribution map of the nanostructure shown in Figure 3(a), with martensite (red) and austenite (blue) phases and black line is the grain boundary. (c) Transmission electron microscopy (TEM) image of the single martensite variant in Ni–Ti SMA by Waitz et al. [37].

A single variant martensite nanograin forms in the 4.5 nm sized model which has a similar pattern with the experimental figure shown here.

In the quenching simulation of 4.5 nm sized model shown in Figure 4, the interim phase emerges inside the B2 phase and is fully replaced by the B${19}^{\mathrm{\prime }}$ martensite phase after cooling down below the martensite finish temperature ($M_{\mathrm{f}}$ ). The phase transformation in this model shows a two-stage transformation with the interim phase B19. As recognized in Figure 4, the yellow frames indicate the structure of the monoclinic product phase and its parent phase (tetragonal structure) as well as the interim phase (orthorhombic structure) corresponding to the same group of Ni–Ti atoms. According to Zheng et al. [39], the interim phases can grow in the low-temperature ageing model with precipitates developing along the interface or grain boundary, and it will also dissolve in the domain with a higher ageing temperature above 833 K. In this study, the ageing process is performed by holding the system at 1073.15 K, where the formation of interim phase is not expected [39]. However, in the 4.5 nm sized model, the limited grain size suppresses the growth of martensite initially and stimulates the intermediate B19 phase formation (Figure 4b) and thereafter martensite formation is observed with retained austenite in the interface region between martensite grains (Figure 4 c). According to Chen et al. [40], martensite phase transformation can be completely suppressed in small-sized grains. In this study, complete suppression of martensite formation is not observed in small grains. However, the fraction of retained austenite (RA) at 100 K is higher in smaller grains, where RA is 17.7%, 3.6% and 2.1% in 4.5 nm, 6 nm and 9 nm grains, respectively. These results show that smaller grains can strengthen austenite and lead to incomplete martensitic transformation.

Figure 4. Martensite and interim phase formation in the 4.5 nm sized system with 1350 atoms after ageing for 1 µs. Through the quenching process from (a) Austenite (B2) phase (300 K) to (b) Interim (B19) phase (200 K), (c) Martensite (B${19}^{\mathrm{\prime }}$) phase (100 K). The blue and red regions are defined as austenite and martensite phases, respectively. The yellow blocks indicate a local lattice transformation path from B2 to B19 to B${19}^{\mathrm{\prime }}$.

Two stages of martensite formation are demonstrated in the quenching process, the lattice structure frames indicate different phases structures before and after the phase transformation.

The decreased model size has the potential to achieve higher surface strain energy as this small model has a large region of retained austenite at the interface between martensite twins (Figure 4) with the density value above 6.482 g cm⁻³ compared to all the other models at about 6.477 g cm⁻³. Similarly, from the calculation by Waitz et al. [37], the low boundary energy can promote the high density in the phase to overcome the energy barrier for transformation, which leads to a special transformation path for the single variant martensite grain similar to the model shown in Figure 4. Compared to the experimental results, the Ni–Ti alloy model is more homogenous with perfect B2 structure in the domain without any external strain from the existing grain interface and precipitates. Consequently, the critical size of the model which is showing a suppressed phase transformation is smaller than the observed grain size in experiments.

The nanograins of size 4.5 nm, 6 nm and 9 nm with 1350, 1600 and 5400 atoms, respectively were aged at 1073.15 K for different durations, with an interval of 100 ns, and quenched to predict the $M_{\mathrm{s}}$ and $A_{\mathrm{s}}$ temperatures. Due to the extended relaxation in the ageing models, the results show an increasing trend of the $M_{\mathrm{s}}$ and a decreasing trend of the austenite start temperature ($A_{\mathrm{s}}$) for individual cases based on the logarithmic prediction (Figure 5). By following the prediction to 1 h, the $M_{\mathrm{s}}$ for the medium (6 nm) and large models (9 nm) are predicted up to 300 K and 380 K respectively, which are close to the phase transformation temperature of 330 K to 350 K for near-equiatomic Ni–Ti alloy after ageing for 1 h, measured by Prokoshkin et al. [41]. Constrained by the grain size, the transformation of austenite to martensite is suppressed up to considerably low temperature in the small sized model (4.5 nm), where the $M_{\mathrm{s}}$ is predicted as 215 K after ageing for 1 h. Moreover, the high temperature ageing process contributes to austenite stabilization, in agreement with experiments on SMA [42], where the $M_s$ trends to decrease. The low temperature ageing promotes martensite stabilization in the SMAs and will lead to the growth of A${}_{\mathrm{s}}$ [43]. However, in Ni–Ti alloy, the precipitates formation and crystal size may have additional influence on the phase transformation temperature [37].

Figure 5. Variation of martensite and austenite start temperatures of simulated samples aged for different durations, with 100 ns interval, in different system sizes of (a) 4.5 nm with 1350 atoms, (b) 6 nm with 1600 atoms and (c) 9 nm with 5400 atoms, respectively. The dashed lines are the trend lines based on a logarithmic prediction.

The scatter plots of martensite and austenite start temperature for the three different size models demonstrate the variation trend along with the simulation ageing time.

The transformation barrier is one of the key aspects that controls the formation of phases, as discussed by Fischer and Reisner [44]. The total free energy difference per unit volume between austenite and martensite is the driving force and triggers the phase transformation. As calculated from the simulation, the total free energy difference per lattice for the 4.5 nm, 6 nm and 9 nm sized models are 0.07852, 0.04375 and 0.0337 eV/lattice, respectively. These values show that a large driving force is required to overcome the large transformation barrier that exists in the 4.5 nm sized model where the interim phase was observed. The driving force required for the transformation in the small system (4.5 nm) is double than that in the large size model (9 nm), which is also reflected in the predicted significant decrease in the $M_{\mathrm{s}}$ temperature for smaller grains. Accordingly, the interim phase is more likely to form before the final martensite starts to form in the small size model, similar to the R-phase [37]. The R-phase was also found as an interim phase and has considerably lower transformation strains as well as substantially smaller energy barrier compared to martensite [37]. Moreover, Sewak and Dey [45] had recognized the intermediate phase as the B19 phase, according to the crystal structure parameter and the perturbed angular correlation (PAC). Therefore, the interim phase of both R and B19 phases is possible to be observed in the phase transformation. Similarly, the interim phase exists after quenching in samples aged for different durations. Although the ageing simulation timescale is negligible compared to the duration of the experimental ageing process, however, with an optimum model, the phase transformation temperature variation through ageing shows a tendency that can be predicted in the case of nanograins in equiatomic Ni–Ti alloy.

4. Conclusion

MD simulations of three Ni–Ti alloy grains of different sizes (4.5 nm, 6 nm and 9 nm) are performed by subjecting the grains to ageing and quenching process. The results show that in a smaller grain, the martensite transformation is constrained and leads to lower temperatures of the transformation and reverse transformation. As the ageing time increases, $M_{\mathrm{s}}$ tends to grow and the $A_{\mathrm{s}}$ shows the opposite tendency. The 4.5 nm sized model demonstrates a single variant of twinned martensite with retained austenite formed at the grain boundaries, while larger grains exhibit fine twinned martensite resembling the herringbone structure. Due to the formation of retained austenite and interim phase in the limited grain size, the local energy is minimized in the martensite region for the B${19}^{\mathrm{\prime }}$ phase transformation which only forms one martensite variant. On the contrary, the large size models demonstrate the martensite self-accommodation to the twinned structure for releasing the internal strain and local energy. In the model size of 4.5 nm, a two-stage transformation of B2–B19–B${19}^{\mathrm{\prime }}$ is observed and is concluded that the smaller nanograins are more likely to form an interim phase during quenching after ageing at 100 K. The study of complete phase transformation phenomena with different ageing temperatures and adequate ageing periods, which require significant computational resources, are yet to be pursued.

Acknowledgments

The authors wish to acknowledge CSC–IT Centre for Science, Finland, for computational resources.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix. Total free energy data

Table A1. Total free energy value in three size models.

Size	$4.5\times 4.5\times 0.9{\mathrm{n}\mathrm{m}}^3$ with 1350 atoms	$6\times 6\times 0.6{\mathrm{n}\mathrm{m}}^3$ with 1600 atoms	$9\times 9\times 0.9{\mathrm{n}\mathrm{m}}^3$ with 5400 atoms
Total free energy (eV) / Energy per lattice at 1073.15 K (eV)	−6450 / −9.55556	−7642 / −9.55250	−25815 / −9.56111
Total free energy (eV) / Energy per lattice at $M_{\mathrm{s}}$ (B19${}_{\mathrm{s}}$) (eV)	−6725 / −9.96296	−7978 / −9.97250	−26881 / −9.95593
Total free energy (eV) / Energy per lattice at $M_{\mathrm{f}}$ (B${19}_{\mathrm{f}}^{\mathrm{\prime }}$) (eV)	−6778 / −10.04148	−8013 / −10.01625	−26972 / −9.98963
Total Energy difference per lattice between $M_{\mathrm{s}}$ and $M_{\mathrm{f}}$ (eV)	0.07852	0.04375	0.03370
Density at 100 K / ${\mathrm{g}\mathrm{c}\mathrm{m}}^{-3}$	6.482	6.478	6.477