Introduction

Shape Memory Alloys are a broad class of functional materials that can recover large strains under stress or thermal cycles1,2,3. Due to this exceptional property, SMAs are used in various applications, such as actuators4,5, aircraft morphing6, and devices for minimally invasive surgery7. Transformation twinning is the key microstructural process in SMAs that leads to the shape-memory effect and controls its reversibility and functional fatigue1,2. In particular, it is the reversible, self-accommodation mechanism of athermal nucleation and migration (or detwinning) of these twin interfaces that influences the thermo-mechanical performance of SMAs2,8.

Among existing SMAs, NiTi has attracted most of the attention due to its superior damping capacity, biocompatibility, as well as corrosion, abrasion and fatigue resistance9. Well-established theories, such as the phenomenological theory of martensite crystallography10 and the energy minimisation theory (EMT)11, have been successfully used to predict twinning systems in NiTi, including the magnitude of the transformation shear9,12,13. However, crystallographic theories are elusive regarding the origin of the twin formation hierarchy in NiTi. For example, [011] Type II twin forms much more frequently than its conjugate (011) Type I twin, even if both require the same amount of shear12. Grain boundary energy is typically regarded as the primary factor that drives microstructure formation. As a result, minimum energy interfaces are considered to be the most likely to form14,15. However, previous studies suggested that factors beyond interfacial energy may also play a role in twinned microstructure formation, such as Eshelby’s driving force16,17 and the propagation mechanisms for twin interface motion18,19. Recently, Shilo et al.19 proposed a general model for twin interface migration, that connects the mobility to interface disconnections, identified by the topological model (TM)20. Distinct propagation mechanisms are envisioned for Compound, Type I and Type II twins, yet it is unclear how atomistic mechanisms control disconnection-mediated interface motion, and how interface kinetics influences the twin formation hierarchy and detwinning. Molecular Dynamics (MD) simulations can enable the atomic-scale analysis of these mechanisms, yet past work21,22,23,24,25,26,27 lacks a holistic integration of atomistic simulations with established crystallographic theory, TM, and crucial validation with experimental evidence of the atomistic interface structure.

Here, we use such an integrated approach (see Fig. 1) to determine the hierarchy of twin formation in NiTi. We perform MD simulations of Compound, Type I and Type II twins and obtain atomistic interfaces that are fully consistent with available high-resolution transmission-electron-microscopy (HRTEM) evidence. The atomistic twin interface structure, energy, driving force for twin boundary motion and propagation mechanism are revealed. The twin interface motion is carried by interface disconnections and the driving force scales with the magnitude of the Burgers vector. A series of Type II twin stepped/terraced interfaces is obtained, whereby the interface propagation occurs by athermal gliding of disconnections on the \((11\overline{1})\) terrace plane, while all the other twinning interfaces require thermal activation. This finding settles an open question regarding the Type II interface propagation mechanism. Consistent with experiments, our analysis reveals that the twin formation frequency correlates with the driving force rather than the twin boundary energy, highlighting the importance of kinetics in martensite formation and twin selection.

Fig. 1: Integrated approach to twin interface modelling in SMAs.
figure 1

a Crystallographic theory (i.e. EMT11) is used to identify the characteristics of the twinning systems in NiTi (twinning shear magnitude s, twinning direction η1 and twinning plane normal K1). b Guided by EMT, fully 3D MD configurations are relaxed to obtain the atomistic interface structure, and compared with HRTEM. In the Figure, the example of a Compound twin is provided. c The interface motion is analyzed by MD, and the correlation between driving force g0 and twin formation frequency is discovered. d Based on MD input, TM is used to identify the interface disconnections, and the consistency between TM and EMT is assessed. HRTEM figure is reproduced after permission of Elsevier33.

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Results

Atomistic structure of NiTi twin interfaces: MD simulations guided by EMT and validated with HRTEM

In NiTi, self-accommodated twin structures arise due to martensitic transformation from the cubic B2 austenite to the twinned monoclinic B19’ martensite1. The reported experimental observations of twins in near-equiatomic NiTi are \((11\overline{1})\) and (011) Type I twins; (001) and (100) Compound twins; and [011] Type II twins12,28,29,30,31,32,33. These observations are HRTEM micrographs that provide information regarding the key crystallographic features of the twin interfaces: the alignment of the crystallographic planes and of the atomic columns, as well as the existence of interface steps/disconnections. However, these experiments are inconclusive regarding the possible presence of terraces e.g. in Type II twins. Moreover, they cannot provide complete knowledge of the 3D interface structure and its mechanism of motion. Here, we use EMT to determine the initial atomistic twinned configuration, that is subsequently relaxed using MD (see Supplementary Information 1, “Construction of the atomistic models for twin interfaces”). Figures 2 and 3 show 2D slices of the relaxed atomistic twin interfaces compared to the available HRTEM micrographs (see “Methods” section)30,31,32,33. For all twin interfaces, consistent agreement between HRTEM and atomistic simulations is demonstrated, with minor angular discrepancies between crystallographic and twin planes arising from MD lattice parameters departing slightly from experiments34. In particular, in all cases, the crystallographic planes of MD simulations align with the HRTEM planes, and correspondence with the atomic columns in simulations and experiments is also found (see Supplementary Information 1, “Comparison of atomistic simulations with HRTEM”).

Fig. 2: Comparison between HRTEM30,33 and atomistic models of Compound and Type I twin interfaces.
figure 2

a (100) Compound twin, visualized along the [010] direction. Both atomistic simulations and HRTEM show that the (100) twin plane is at the intersection of the (001)A and (001)B planes, where A and B indicate the two martensite variants. This result is qualitatively consistent between HRTEM and atomistic simulations, with minor angular discrepancies between (001) planes and the twin plane arising from MD lattice parameters departing slightly from experiments34. b (001) Compound twin, visualized along the [010] direction. The (001) twinning plane results from the intersection of the \({(10\overline{1})}_{A}\) and \({(10\overline{1})}_{B}\) planes. This result conforms nicely with previous atomistic work67. c (011) twin plane, viewed along the [100] direction. Note the usage of a different convention with respect to HRTEM when labeling planes. d \((11\overline{1})\) twin plane, viewed along the [011] direction. The \((11\overline{1})\) twinning plane occurs at the intersection of the \({(\overline{1}1\overline{1})}_{A}\) and \({(\overline{1}1\overline{1})}_{B}\) crystallographic planes. HRTEM micrographs show the correspondence of crystallographic planes and atomistic positions, and are reproduced after permission of Elsevier.

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Unlike other twinning systems, Type II twins (Fig. 3) show a range of interface orientations30,31, which crystallographic theory predicts to be irrational. Here, three different orientations are relaxed to encompass the experimental observations, by approximating the irrational orientations with rational ones (see Table S2 and Supplementary Information 1). The twin variants share common crystal planes, including the \((11\overline{1}),(01\overline{1})\), and (100) planes. Remarkably, this crystallography is intrinsic to the twin interface and independent of the twin interface orientation or approximation. Nevertheless, the twin interface is undistinguishable along the [011] rational shear direction (Fig. 3a). However, more recent work showed that martensite variants are distinguishable along the [101] crystal direction31 (Fig. 3b). It is then possible to observe that the twin interface irrationality can be accommodated by a stepped/terraced geometry, irrespective of the twin interface orientation or approximation. After relaxation, every approximated Type II twin reported in Table S3 (Supplementary Information 2) results in the same semi-coherent configuration as the one shown in Fig. 3, differing only by the terrace length. Thus, the atomistic Type II twin can achieve multiple interface orientations. The interface structure agrees with a model with terraces along the \((11\overline{1})\) plane and steps parallel to the \((01\overline{1})\) plane28.

Fig. 3: Comparison between HRTEM30,31 and atomistic models of Type II interfaces.
figure 3

Type II twin exhibits a crystallographic shear direction, enabling a direct comparison between atomistic models and experiments along the [011] direction. a Comparison between HRTEM micrograph30 and atomistic model of \((34\overline{4})\) Type II twin interface, viewed along [011] direction. In both simulations and HRTEM, the twin variants share common crystal planes, including the \((11\overline{1}),(01\overline{1})\), and (100) planes30. b Comparison between HRTEM micrograph31 and atomistic model of \((9\,10\,\overline{10})\) Type II twin interface, viewed along [101] direction. HRTEM micrographs reproduced after permission of Elsevier30,31.

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The simulated Compound, Type I and Type II twin interfaces show quantitative agreement with all the available HRTEM micrographs reported in the literature for NiTi SMAs: this is the first main result of this paper.

Identification of the interface defect structure by TM, and consistency with EMT

We now turn to determine the defect structure of the twin interfaces by using the topological model (TM)35. According to TM, twin interfaces result from the generation and motion of twinning disconnections along rational planes. These disconnections have both a dislocation nature (Burgers vector b) and a step character (height h). The Burgers vector is the difference between the translation vectors of the twin variants, b = vB − vA (A and B indicate the two variants). The step height is a multiple of the atomistic interplanar spacing. Disconnection glide yields a shear strain γ = ∣b∣/h. Consistency of the atomistic simulations with the crystallographic theory (EMT) is ensured by verifying that γ computed by applying TM to atomistic interfaces equals the transformation shear s obtained by EMT

$$\gamma =\frac{| {{{\bf{b}}}}| }{m\,{h}_{{{{{\bf{K}}}}}_{1}}^{MS}}\approx s,$$
(1)

where \({h}_{K1}^{MS}\) is the (elementary) interplanar spacing between two consecutive \({K}_{1}^{MS}\) planes measured from the relaxed atomistic models, and m is the multiplicity of \({h}_{{{{{\bf{K}}}}}_{1}}^{MS}\) so that the step height is \(h=m\,{h}_{{{{{\bf{K}}}}}_{1}}^{MS}\). Table 1 lists the TM parameters computed based on the simulated twin structures, and the comparison between the TM shear and EMT predictions (Supplementary Information 2). For all the simulated interfaces, a set of disconnections is found such that the TM shear γ is consistent with the EMT shear s. Notably, Type II twin prediction gets closer to EMT as the model interface is closer to the irrational interface orientation. TM recovers exactly EMT predictions when the interplanar spacing is measured in the bulk, far from the atomistic interface.

Table 1 Identification of twinning disconnections by TM, and validation with EMT predictions
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The TM results show some surprising features. First, all twin interfaces emerge from a single array of disconnections with non-zero Burgers vector and step height. This outcome is not obvious, since interface steps are apparent only in the case of Type II interface (Fig. 3). Second, there is a striking difference between Burgers vectors belonging to conjugate twinning systems (e.g. A1-A2 and B1-B2), exhibiting the same twinning shear. For example, (001) Compound twin has a larger Burgers vector than its conjugate (100). Similarly, (011) Type I twin shows a larger Burgers vector than [011] Type II twin. Overall, Compound and Type I twins require much larger Burgers vectors than Type II twins. These significant differences in the Burgers vector may affect both the driving force and the propagation mechanism of the twin interfaces19, as it will be shown in the next Section. Third, for TM and EMT shears to match, \((11\overline{1})\) Type I step height equals twice the elementary interplanar spacing \({h}_{{{{{\bf{K}}}}}_{1}}^{MS}\). This seemingly ad-hoc TM assumption will be verified in the next section by direct atomistic simulation of the interface mobility.

The identification of the interface disconnections by TM and the demonstrated consistency with EMT is the second main result of this paper.

MD predictions of interface motion mechanisms and driving force

The surprising differences among disconnection characteristics of the five twin interfaces suggest implications for their mobility that can be studied by applying a shear stress to the validated atomistic configurations (see Fig. 4 and Table S2 in the Supplementary Information 1). The athermal (T = 0 K) stress vs strain curves (Fig. 4a) for an applied shear along the twinning direction display an initial elastic behavior up to a threshold stress value τ0. Beyond this stress limit, a jagged response occurs up to large deformations, that are accommodated by detwinning mediated by interface motion. Simulations reveal no significant τ0 variations among different Type II twin approximations. Remarkably, the stress τ0 to move Compound and Type I twin interfaces is much larger than the one of Type II twin interface, which moves at  ~ 1 MPa applied stress. Moreover, the driving force for twin boundary motion g0 (see Supplementary Information 3) of conjugate twinning systems differs significantly and correlates with the Burgers vector magnitude (see Fig. 4b). For example, within the A1-A2 pair, (100) Compound twin has the smallest Burgers vector and requires a driving force g0 that is  ~ 10 times smaller than its (001) conjugate. Even more strikingly, in the B1-B2 pair, the Type I twin has a larger Burgers and three orders of magnitude larger g0 than its conjugate, the Type II twin. This considerable difference in driving forces (and therefore in stresses) is consistent with experimental measurements36.

Fig. 4: Activation stress and driving force for twin interface motion.
figure 4

Strain-controlled simulations are performed at T = 0 K and T = 300 K (see “Methods” section). a Shear stress τ vs shear strain plot, for τ applied along the shear direction η1. When the activation stress τ0 is reached, the twinning system can accommodate large deformations. The simulations show no dislocation activity during this process. Rather, the large deformations are accommodated by detwinning due to the twin interface motion. b T = 0 K and T = 300 K driving force (note the logarithmic scale) compared with the interface disconnection Burgers vector magnitude ∣b∣ for the twin interfaces considered in this work. c Polar plot of the stress at which twin interface starts to propagate vs the angle ω indicating the applied shear direction. The simulation results (markers) match the predictions based on Eq. (2) (lines). d Lattice energy barriers for twin boundaries propagation. Note that the energy barrier for the irrational interface is negligible compared to that for rational interfaces.

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To verify that the EMT/TM shear direction η1 is also the minimum stress direction, shear is applied along multiple in-plane directions (see Fig. 4c). The measured direction-dependent stress τ is compared with the Schmid law

$$\tau =\frac{{\tau }_{0}}{\cos (\omega -\alpha )},$$
(2)

where, for each twinning system, the direction-dependent stress τ is predicted as a function of the stress τ0 along η1. The angle ω, which takes values in the [0, π] interval, is the angle between the direction of the applied external stress τ and the simulation cell vector a1 (see Table S2, Supplementary Information 1), while α is the angle between the shear direction η1 and a1. Therefore, the difference ω − α represents the angle between the direction of the applied external stress τ and the shear direction η1 (see Extended Data Figure 6). Simulations show that the interface moves according to Eq. (2). Moreover, the simulated stress τ approaches the asymptotic behavior (no twin boundary propagation) when ω − α = π/2, i.e. shear applied along a direction orthogonal to η1, for which detwinning cannot be accommodated at any applied finite stress. Simulations capture this behavior, since detwinning is not observed for shear orthogonal to the twinning direction. Thus, the activation stress for detwinning (Eq. (2)) is the critical stress τ0, and the twinning shear direction η1 is indeed the minimum stress direction.

The role of temperature on interface motion is further investigated (see Methods and Supplementary Information 3). Overall, Type I and Compound twins show decreased activation stresses as temperature increases, consistent with experiments37, while Type II twinning activation stress is unchanged. Notably, at T = 300 K only (100) Compound twin shows a significant drop of τ0 (one order of magnitude), while the stress reduction is milder for the other interfaces (Fig. 4b). As shown in Supplementary Video 2, at both 0 K and 300 K, (001) Compound detwinning is achieved by nucleating new twin-related variants of nanometric thickness until the complete transformation into a single variant is achieved. Interestingly, (001) Compound has been frequently observed in NiTi nanocrystals32,33, hence the simulated nanotwin formation is consistent with experiments. Instead, (100) Compound and Type I twin interfaces move by coordinated local atomic shuffling and flat interface motion (see Supplementary Videos 1, 3, and 4). The atomistic simulation of \((11\overline{1})\) type I twin interface motion also confirms our TM model assumption of a double-interplanar spacing step propagation (see Supplementary Video 4).

Gliding of [011] Type II twin interface shows completely different features. First, because of the irrational nature of the interface, twinning disconnections are arranged in the form of a “stepped” interface. These twinning disconnections are spread on the terrace plane and carry a displacement jump and a long-range displacement field (see Supplementary Information 3, “Gliding of irrational twin interfaces”): as such, Type II twinning disconnections show similar features as interface dislocations (see Supplementary Videos 5 and 6). Moreover, the Type II interface moves at negligible stress τ0 at 0 K, thanks to the glide of these disconnections along the \((11\overline{1})\) terrace planes (see Supplementary Video 7). Hence, thermal activation is not required to overcome the energy barrier for twin boundary motion, and the process can be regarded as athermal for Type II in NiTi. Twinning disconnections are, therefore, the carriers of the transformation of one variant into another because their propagation along the terrace shuffles the atoms into the motif of either the twin or matrix variant, depending on the direction of the twinning shear (see Supplementary Information 3). The observed athermal nature of the interface motion can then be understood in terms of the classical Peierls-Nabarro model19,38, according to which a small Burgers vector and the significant spreading of the dislocation core result in a small barrier to dislocation glide. As such, Type II twin interface disconnections have a dislocation character consistent with grain boundary kinetic theory15,19.

Simulations at 300 K do not show with clarity the occurrence of a thermally activated mechanism causing the glide of rational interfaces, such as the nucleation of disconnections and/or kink-pairs as envisioned in previous works19,39, possibly because of thermal noise adding to local atomic shuffles. To further investigate the interface propagation mechanisms, we conduct simulations using climbing-image Nudge Elastic Band (NEB) method and determine the energy barrier and the minimum-energy path associated with the motion of each twinning interface (see Fig. 4d)40. These simulations reveal that rational twin interfaces need to overcome a finite energy barrier to move. Temperature can facilitate this movement by thermal activation. In contrast, the energy barrier for Type II propagation is negligible, confirming that Type II twin motion is athermal, unlike the propagation of rational interfaces that requires thermal activation.

Additionally, the energy barrier for Type I twins corresponds to the nucleation of disconnections, which glide subsequently as straight lines on the twin boundary plane (see Supplementary Videos 10–12). No secondary barrier is observed along the minimum energy path, indicating that the propagation of disconnections is athermal. Therefore, the shear γ predicted by TM is accommodated by interface motion, carried by glide of disconnections (see Supplementary Information 3, “Gliding of rational twin interfaces”). The observed mechanism of nucleation and propagation of steps with disconnection/dislocation character is consistent with the picture underlying unified approaches of grain boundary motion, that envision disconnections as defects with dislocation character (see Supplementary Information 3, “Process of twin interface motion, according to Shilo et al.”)15,19. Interestingly, Compound rational interfaces behave differently from Type I twins. In the case of the (100) Compound twin, the NEB minimum energy transition path confirms that the interface glides as a whole by coordinated local atomic shuffling (see Supplementary Videos 8), while in the case of (001) Compound twin, simulations confirm that the interface is immobile and that detwinning is achieved by nucleating new twin-related variants. After nucleating, the disconnections glide on the twin plane as straight lines and give rise to new twin interfaces (see Supplementary Videos 9), increasing the system energy. Finally, for Type II twins, NEB simulations confirm propagation of twinning disconnections as straight lines, without kink-pair mechanism: this confirms the athermal nature of Type II twin interface propagation (see Supplementary Videos 13).

The quantification of the stress τ0, along with the driving force g0 and energy barriers for twin boundary motion, and clarification of the atomistic interface glide mechanisms across temperatures for all NiTi twinning interfaces is the third main result of this paper.

Discussion

Several HRTEM experiments28,30,31 report evidence of the [011] Type II twin interface in NiTi, yet two contrasting opinions regarding its structure exist due to lack of consensus in interpreting the available HRTEM micrographs. One hypothesis, which is confirmed by our atomistic work, suggests that the irrational twin interface consists of ledges on the \((11\overline{1})\) plane and steps parallel to the \((01\overline{1})\) plane28,31. In contrast, another hypothesis envisions the existence of an atomistically irrational twin plane, that is “randomly curved”, and that gives rise to a local elastic strain that is relaxed by gradual atomistic displacements (not estimated nor measured30 by the proponents of this hypothesis)29,30. This second hypothesis is not confirmed by our atomistic simulations.

Moreover, seemingly contradicting measurements of the Type II interface orientations have been reported. Recent experimental work31 provides clear micrographs of stepped/terraced geometry and interface orientation close to \((89\overline{9})\)41. This orientation is however at odds with several previous experiments12,28,30 and with previous crystallographic theory predictions12,13, which conclude that Type II twin plane orientation is close to \((34\overline{4})\). Our work (see Fig. 3 and Figure S2 in Supplementary Information 1) shows that the irrational Type II atomistic interface can be approximated by locally rational orientations encompassing those observed in the literature and that the intrinsic interface stepped/terraced defect structure is independent of the imposed approximated orientation of the Type II twin plane. As such, our work is consistent with the recently proposed Evolving Interface theory, which states that the orientation of the twin plane might be affected by the microstructural strain and the local twin volume fraction42. This theory predicts the well-known \((34\overline{4})\) orientation at 0% strain, and the \((89\overline{9})\) orientation at −5% strain. Thus, the Type II twin interface may have locally varying orientations and can accommodate its irrationality by stepped/terraced structures with different terrace lengths, as proposed by the Evolving Interface theory42. Over length-scales larger than atomistic simulations/HRTEM micrographs, the average interface orientation predicted by the crystallographic theories could then be recovered10,11. Finally, TDs gliding on the \((11\overline{1})\) terraces are consistent with the TM developed by Mohammed et al.43 Therefore our simulations do not confirm the TM developed by Pond et al.44, which envisioned TDs gliding on the conjugate plane (011) (see Supplementary Information 3, “Gliding of irrational twin interfaces”).

Numerous experimental evidence shows that [011] Type II twin is the most frequently observed transformation twin in NiTi28,29,30,31, and that there is a hierarchy in the formation of twinning modes in the martensite microstructure. Grain boundary energy is usually understood to be the main driver of microstructure formation and minimum energy interfaces the most likely to form14,15. However, here (Fig. 5a) we find no correlation between twin boundary energy ETB, computed with MD, and relative observation frequency f of the experimentally observed twinning modes29. Interestingly, the [011] Type II twin shows the largest twin boundary energy compared to the other twinning systems, yet it glides athermally, at negligible stress τ0. On the contrary, rational interfaces show low interfacial energy but much larger stress τ0. Instead, the twin observation frequency correlates with the driving force for twin boundary motion (Fig. 5b). Hence, the driving force for twin boundary motion rather than the twin boundary energy appears to drive the formation of twins in the martensite microstructure, where systems with low g0 or τ0 are the most likely to form. This finding explains why the frequency of observation of Type II twin (B2) is much larger than the conjugate Type I twin (B1), despite their transformation shear being the same. The consistent correlation between twinning shear stress and twinning hierarchy is a key implication of this paper, that can be incorporated in mesoscale models to predict microstructure formation45,46.

Fig. 5: Correlation between observation frequency f and driving force for interface glide.
figure 5

(1 − f) measures how infrequent a twinning system is, according to experiments29. a Comparison with the twin boundary energy ETB. b Comparison with the driving force g0 at T = 300 K (solid contour) or T = 0 K (dashed contour). The arrow indicates the drop in driving force with temperature.

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As a final remark, it is now well understood that the possible twinning systems for the B19’ martensite lattice are (011) and \((11\overline{1})\) Type I, [011] Type II, (001) and (100) Compound twins47. Depending on the transformation details (direct or two-step, see Supplementary Information 1, “NiTi interatomic potential and twinning systems”), all these twinning modes can be possible solutions (i.e., lattice invariant shears) of EMT for the B19’ lattice47. Therefore, available HRTEM micrographs for other SMAs, where the B19’ martensite is a relevant phase (such as NiTiCu48, NiTiHf49, NiTiFe47, and CuZr50), may be used for qualitative comparison with our simulations. As a case example, the (011) and \((11\overline{1})\) Type I and the [011] Type II have been repeatedly reported in the ternary NiTiHf systems49. Since the lattice parameters, which determine the quantitative crystallography of the interface, depend on the chemistry, any such comparison can only be qualitative. For example, in NiTiHf (011) Type I twin interface49, we find a good alignement of planes, since the twin interface lies at the intersection of the \({(11\bar{1})}_{A}\) and \({(11\bar{1})}_{B}\) planes, where A and B represent the two martensite variants. Moreover, consistently with the HRTEM49, the twin boundary is a flat interface without the occurrence of any step and terrace. Nonetheless, the correspondence of the atomic columns does not match, but this is related to the fact that, despite having the same unit cell (i.e. B19’), NiTi and NiTiHf have a different atomic motif. Quantitative comparison could be performed by using interatomic potentials (IAPs) for these systems. Currently, IAPs are available for NiTiCu51, NiTiHf52, and CuZr53. Nevertheless, the suitability of these IAPs to simulate martensite twinned microstructures should be rigorously addressed, as in the case of the NiTi system34.

The proposed integrated approach holds promise also for simulating systems beyond the SMAs B19’ martensite, such as deformation twinning in Zn54, and Mg55. It is important to note that when considering systems other than B19’, the atomistic features of the twin boundaries are likely to change, which is a crucial factor to consider when analyzing these systems and drawing comparisons. Interestingly, NiMnGa has a monoclinic structure similar to B19’ (i.e. 10M martensite)56,57,58, which exhibits Compound, Type I and Type II twins56. Analogous to the NiTi results shown here, extensive experimental work has established that Type II TBs in NiMnGa ferromagnetic SMA have exceptionally high mobility and low twinning stress57, and a stepped/terraced structure with terraces parallel to the (011) planes59.

To conclude, using an integrated array of methods, including crystallographic theory, topological model, and MD simulations validated on HRTEM, the present work demonstrates that twin interface mobility can control the twin formation hierarchy in SMAs. With the recently introduced combinatorial approach to materials design based e.g. on High Entropy Alloys60,61, the mechanistic understanding of twin interface motion achieved in this work can be used in combination with recent solute-strengthening theories62 and established crystallographic theory63 to search for alloys with improved functional fatigue, expanding the design and application space of SMAs.

Methods

Crystallographic theories of twinning and interface structure

In this work, we use EMT and notation consistent with the classical theory of twinning64 to compute the twinning systems in NiTi. Twinning systems are defined by the twinning shear s, the shear direction η1 and the twin plane normal K1. The triplet (s, η1, K1) uniquely defines the twinning mode and the associated twin interface orientation. As a convention, if both η1 and K1 coincide with crystallographic/rational directions, the twin is called “Compound”. In Type I twin, only the twin plane normal is rational. In contrast, only the twinning shear is rational in Type II twins. Calculations are performed by solving the Twinning Equation, which reads (with respect to the martensite reference system)

$${{{\bf{F}}}}-{{{\bf{I}}}}=s\,{{{{\boldsymbol{\eta }}}}}_{1}\otimes {{{{\bf{K}}}}}_{1},$$
(3)

where F is the relative shear deformation between the two martensite variants, I is the identity tensor, and (s, η1, K1) are the twinning elements. In Equation (3), ⊗ is the dyadic product between two vectors. The formal proof of the general solutions of Equation (3) has been shown in several works11, and we refer to them for an exhaustive treatment of the topic. In this work, solutions to Equation (3) are found by using Mallard’s law8, see for additional details Section “EMT: Predicting Twinning Systems by the Twinning Equation” in Supplementary Information 1.

TM provides an essential bridge between the twin interface structure and its mobility. In TM, interfaces are described as an array of twinning disconnections35. On top of the step height h, TDs are characterized by the Burgers vector b like classical Volterra dislocations. Therefore, nucleation and gliding of TDs have been treated according to classical dislocation theory19. In this work, we use the TM for twin interfaces by retrieving the defect content − to use as input in Equation (1) − directly from the relaxed twin interfaces shown in the Result section. Additional information on how to calculate the topological input parameter can be found in Supplementary Information 2.

Atomistic simulations

All atomistic simulations (MD/MS) were performed using the LAMMPS package65,66. Here, we use the 2NN-MEAM interatomic potential67, which outperforms FS-EAM interatomic potentials for the prediction of both the crystallography (lattice parameters and monoclinic angle) and the martensite microstructure34. Details regarding the interatomic potential and benchmarking details are provided in Supplementary Information 1, “NiTi Interatomic Potential and Twinning Systems”. Throughout this work, crystallographic visualizations use OVITO68, and express all the crystal directions and planes in the monoclinic martensite reference system. We use Equation (3) to construct the initial simulation configurations to model twinning systems in NiTi69. Here, we compute the twinning systems based on the 2NN-MEAM interatomic potential lattice parameters67, which are: a = 2.878 Å, b = 4.129 Å, c = 4.659 Å and β = 99. 4∘. The results are shown in Table S1 (Supplementary Information 1). Creating the atomistic models for twin interfaces also integrates with how LAMMPS defines simulation boxes. The procedure details are in Section “Construction of the atomistic models for twinned interfaces” in Supplementary Information 1. Table S2 in Supplementary Information 1 summarizes all the modelling parameters used in this work.

Once the LAMMPS configurations are constructed using EMT, they are relaxed according to the conjugate gradient (CG) and FIRE algorithms. At the first stage, we use the FIRE algorithm70, with fixed box dimensions, to ensure that the atoms reach a local minimum configuration. We relax the box thereafter using the CG algorithm, allowing box dimensions and tilts to change, and we alternate CG box relaxation with the fixed-box FIRE algorithm until convergence of CG and FIRE is reached. The procedure is repeated by assigning different initial configurations over a 20 × 20 grid in the twinning plane and picking the twin boundary with the minimum energy, relaxed configuration. The twin boundary energy is determined from the relaxed structures by calculating the energy difference between the supercell containing a twin interface, and a supercell without the interface71, as reported in Section “Construction of the atomistic models for twinned interfaces” in Supplementary Information 1. The force tolerance set for the FIRE algorithm is 10−4 eV Å−1, with maximum 2 × 105 iteration/evaluations. For the CG algorithm, the energy tolerance is 10−10 eV, and the force tolerance is 10−10 eV Å−1, with a maximum number of evaluations 106. This minimization scheme allows all the twin structures constructed to reach the minimum potential energy. The equilibration of the twin interfaces at finite temperatures is conducted as follows. The heating process is performed under isothermal-isobaric (NPT) conditions. The cell size for each twinning system is increased along the in-plane coordinates (see Supplementary Information 1) to avoid size effects that might affect the nucleation of twin disconnections. A random velocity distribution is assigned to the atoms to perform a first equilibration at T = 10 K for 1 ns. After this preliminary equilibration, the temperature is increased to T = 300 K with 0.5 Kps−1 thermal rate. The system is then relaxed in isothermal conditions at 300 K for 1 ns. Throughout the heating process, we keep the target stress components of the system fixed to 0 MPa.

Strain-controlled simulations at T = 0 K are performed as follows. An external shear deformation is applied by tilting the box out of the XY plane. This deformation is enforced by incrementally changing the xz and yz tilt factors, and by imposing an external shear strain of 0.01% at each increment. Configurations of constant applied strain are computed by relaxing atomic positions while holding the applied box deformation fixed. Simulations are performed up to 20% applied shear strain. To assess the influence of temperature on the gliding mechanism of rational interfaces, we also perform strain-controlled simulation at T = 300 K. The simulations are conducted by incrementally increasing the simulation box tilts xz and yz in 0.01% increments, and by equilibrating the system for 0.02 ns at each iteration under isothermal-isochoric (NVT) conditions at constant temperature T = 300 K. Simulations are performed up to 3% total strain.

To conduct climbing-image nudged elastic band (NEB) calculations40, we compute the relaxed initial state as described earlier, however by keeping the Z direction non-periodic. The final state has the same structure as the initial state, but the twin interfaces are displaced along the Z direction by a distance equal to the step height predicted by the TM (refer to Table 1). We create an initial path of intermediate configurations (replicas) by linearly interpolating the atomic positions between the relaxed initial and final configurations. NEB computations are then carried out under the constraint of stress-free boundaries. The force tolerance is set at 5 × 10−3 eV Å−1, and the NEB inter-replica spring constant is set to 10−2 eV Å−2. This choice does not impact results but helps optimize the convergence of the calculations. The total number of replicas is increased until convergence is achieved.