Twinning in metastable high-entropy alloys

Twinning is a fundamental mechanism behind the simultaneous increase of strength and ductility in medium- and high-entropy alloys, but its operation is not yet well understood, which limits their exploitation. Since many high-entropy alloys showing outstanding mechanical properties are actually thermodynamically unstable at ambient and cryogenic conditions, the observed twinning challenges the existing phenomenological and theoretical plasticity models. Here, we adopt a transparent approach based on effective energy barriers in combination with first-principle calculations to shed light on the origin of twinning in high-entropy alloys. We demonstrate that twinning can be the primary deformation mode in metastable face-centered cubic alloys with a fraction that surpasses the previously established upper limit. The present advance in plasticity of metals opens opportunities for tailoring the mechanical response in engineering materials by optimizing metastable twinning in high-entropy alloys.

D esigning strong, and at the same time, ductile metals has been among the most ambitious goals for metallurgists. The apparently conflicting mechanisms often resulted in grades compromising between strength and ductility. In spite of the early ideas revolving around Fe-Mn alloys 1 , simultaneous improvement of the two properties has been very challenging. The discovery of the advanced high-strength steels near the turn of the century was a pioneering step towards reaching such goals 2 . Recently, the strength-ductility tradeoff was also overcome by designing dual-phase high-entropy alloys (HEAs) 3 , which turned attention towards non-conventional alloys containing multi-principal elements in equal or nearly equal concentrations 4,5 . Successful single-phase solutions also exist such as the fracture-resistant HEAs 6 and the exceptional damagetolerance medium-entropy alloys (MEAs) 7 . Further, HEAs with surprising and unanticipated properties are being continuously put forward [8][9][10][11][12][13][14][15][16][17][18] . Parallel with the experimental efforts, computational modeling was also advanced to optimize compositions and reveal the atomic-level mechanisms 19 behind the strength and ductility improvements.
The common denominator of the most promising routes over the strength-ductility dilemma is the tailored plastic deformation mechanism. Face-centered cubic (fcc) metals normally deform by dislocation glide resulting in good ductility and reduced strength. But in the aforementioned special alloys, the leading deformation modes are twinning and transformation induced plasticity. In particular, twinning was found to play an important role for the enhanced ductility and high strain hardening rate 2,3,6,7,20,21 . There are several phenomenological plasticity models 22,23 developed to understand the activation of twinning deformation in fcc metals. These models are based on the stacking fault energy (SFE) defined as the excess energy due to the stacking fault connecting the Shockley partial dislocations. For instance, austenitic steels with SFE higher than 10-16 mJ m −2 are predicted to show twinning [24][25][26] , whereas alloys with SFE below this critical value should exhibit deformation-induced phase transformation. Theoretical works also addressed the question of twinnability [27][28][29] , and they indicate twinning for fcc stable metals with twinning fraction below 50% 27 .
Within the simplest model, the SFE corresponds to the energy difference between the hexagonal close-packed (hcp) and fcc structures 30 . Thus, the sizable positive SFE required for twinning by the phenomenological models 22,23 implies a thermodynamically stable fcc phase. Traditional alloys like austenitic steels 31 belong to this group. Since the discovery of the first fcc HEAs based on 3d transition metals 4,5 , it was expected that these multi-component systems also follow a similar plastic deformation scenario. However, recent experimental works reported a pressure-induced fcc-hcp transition in CrMnFeCoNi, and the hcp phase was retained after decompression to ambient pressure 32,33 . The observed equilibrium phase boundary between the fcc and hcp phases was placed near 600 K 33 . For the multicomponent alloys in question 3,6,7,20,[34][35][36][37] , including CrMnFe-CoNi, ab initio theory fully confirms the experimental findings by predicting a thermodynamically unstable fcc lattice at low temperature. Hence, both experiment and theory suggest that the metastable fcc CrMnFeCoNi alloy should possess very small positive or negative SFEs at ambient and cryogenic conditions. But such SFE values in connection with the phenomenological models 22,23 clearly fail to explain the observed twinning 6 . Thus, the available experimental and theoretical evidences severely challenge the validity of the widely-used SFE-based plasticity models in the case of metastable HEAs 22,27 . Although this fundamental problem was recognized by several groups 19,34-38 , so far no unified solution has been put forward.
Here, we employ first-principle quantum mechanical tools to resolve the twinning mechanism in metastable multi-component alloys. We select the representative Fe 80−x Mn x Co 10 Cr 10 (30 ≤ x ≤ 45) and CrMnFeCoNi HEAs 3,6 , and CrCoNi MEA 7 , for which reliable experimental data for the deformation mechanisms exist. Based on the intrinsic energy barriers, we show that it is the particular chemistry that is primarily responsible for the emergence of twinning in metastable alloys, and decreasing temperature efficiently drives the system across various deformation modes. The inherent affine shear strain prior to slip, which is associated to the relatively large critical twinning stress characteristic to the present systems, further promotes twinning. Our results are expected to help optimizing the properties of engineering materials involving HEAs towards excellent strength-ductility combinations.

Results
Deformation modes. The leading deformation modes in fcc metals are stacking fault (SF), twinning (TW), and fullslip (SL). The active mode is decided by the competition between the corresponding effective energy barriers (EEBs): γ SF θ ð Þ ¼ γ usf = cos θ, γ TW θ ð Þ ¼ ðγ utf À γ isf Þ= cos θ, and γ SL θ ð Þ ¼ ðγ usf À γ isf Þ= cosð60 À θÞ for SF, TW, and SL, respectively 27 . In these expressions, θ varies between 0°and 60°and reflects the effect of grain orientation relative to the resolved shear direction, γ isf is the intrinsic SFE, γ usf the unstable SFE, and γ utf the unstable twin fault energy. These intrinsic material parameters (IMPs) describe the generalized SFE (γ-surface) 39 and are determined from a series of ab initio calculations. According to the above EEBs, twinning is possible for γ TW θ ð Þ γ SF θ ð Þ, otherwise martensitic transformation occurs. Furthermore, due to the shear direction dependence, SF and SL or TW and SL can coexist in a homogenous material, but SF and TW are exclusive to each other independently of θ (they possess similar Schmid factors). We mention that SF and TW can coexist in special alloys (e.g., interstitial HEAs [16][17][18] due to the broadening of their boundary. For the sake of simplicity, in the following we do not always mention SL when discussing the SF and TW mechanisms. Using the calculated EEBs, first we explore the deformation modes of the Fe 80−x Mn x Co 10 Cr 10 alloys as a function of temperature and composition. These alloys are reported to exhibit twinning or martensitic transformation depending on Mn content 3 . The room-temperature γ SF À γ TW À Á γ SF 0 ð Þ À γ TW 0 ð Þ values are plotted in Fig. 1 as a function of Mn content. We find that γ SF À γ TW À Á changes sign with increasing Mn concentration. Namely, for x ≥ 33, TW is favored against SF, whereas for x < 33 SF is activated. In addition, SL is present in both regimes with θ increasing towards 60°(not shown). These theoretical predictions are in perfect agreement with observations 3 and confirm that the present theory correctly captures the observed trends of the deformation modes as a function of composition. Figure 1 also shows the free energy difference ΔF ðF hcp À F fcc Þ for Fe 80−x Mn x Co 10 Cr 10 by varying x. Negative ΔF indicates a thermodynamically unstable fcc lattice, which turns out to be the case for all compositions considered here. By combining the behavior of ΔF with that of γ SF À γ TW À Á in terms of Mn content, we arrive to a very interesting and unprecedented phenomenon. Namely, in Fe 80−x Mn x Co 10 Cr 10 HEAs, the room-temperature twinning is associated with a metastable fcc phase for 33 ≤ x ≤ 50 (upper limit is estimated by extrapolating the slope of ΔF towards high Mn content). We emphasize that since these alloys possess negative or very small positive γ isf (as shown below), the empirical plasticity models based on SFE 22,23 would dictate SF rather than TW mechanism, opposed to the present theoretical and former experimental findings 3 . Actually, in these special alloys, γ TW is lower than γ SF , which induces twins instead of stacking faults toward martensitic transformation, despite of the negative ΔF. Below, we refer to this phenomenon as metastable twinning (MTW).
Next, we consider the effect of temperature on the deformation modes. We recall that tuning the deformation mechanisms by temperature is a well-known routine to alter the plasticity of materials 6,7 . Figure 2a (solid symbols) displays γ SF À γ TW À Á and ΔF as a function of temperature for the Fe 40 Mn 40 Co 10 Cr 10 HEA. Alloys with different Mn-levels follow qualitatively the same behavior and thus they are not included in the figure. At high temperature, the system is in the positive ΔF and TW + SL regime. Similar normal behavior is observed at very low temperature, where negative ΔF is associated with SF + SL. However, there exists a temperature interval where the Fe 80−x Mn x Co 10 Cr 10 HEAs behave unexpectedly: TW is activated in spite of the negative ΔF. For x = 35, 40, and 45, the abnormal interval includes room-temperature, whereas for x = 30 it is shifted above room-temperature. Similar calculated γ SF À γ TW À Á and ΔF versus temperature data for CrMnFeCoNi and CrCoNi alloys are shown in Fig. 2b,c (solid symbols), respectively. According to the free energy differences, for these two alloys, the hcp structure is more stable than the fcc one at low temperatures. Our findings regarding the structural stability are fully consistent with the previous theoretical works [34][35][36]40,41 (see also Supplementary Note 1 and Supplementary Figs. 1 and 2). When comparing the trends for γ SF À γ TW À Á and ΔF, it turns out that the peculiar MTW appears in the CrMnFeCoNi alloy between~300 and~460 K, and the metastable CrCoNi alloy shows no twinning in the temperature range considered here. Before contrasting these theoretical predictions with the experimental observations, we investigate the calculated generalized SFEs.
Generalized stacking fault energy. The room-temperature γsurfaces of Fe 40 Mn 40 Co 10 Cr 10 , CrMnFeCoNi, and CrCoNi alloys are shown in Fig. 3a. Remarkably, all three alloys possess γ usf % γ utf with small (or negative) γ isf . Similar IMPs were reported by several former theoretical woks 34,36 (see also Supplementary  Fig. 1). This particular shape of the γ-surface differs from that of conventional fcc alloys 27,31,42 and has important implication on the deformation mechanism as explained in the following paragraphs. The γ-surfaces in Fig. 3a were obtained from alias shear (inset in Fig. 3a), i.e., by shifting the upper part of the fcc 111 f g layers along the h11 2i direction. In practice, this process requires a critical resolved shear stress, which raises a small but finite affine shear: all atomic planes are actually elastically strained along the shear direction (inset in Fig. 3b) 43 . The affine shear strain-dependence of γ SF À γ TW À Á is presented in Fig. 3b. Interestingly, increasing affine shear strain increases γ SF À γ TW À Á and thus favors twin nucleation for all alloys considered here. In particular, MTW is predicted in CrCoNi alloys as well with increasing affine shear strain. Therefore, in addition to the intrinsic materials properties, the strain conditions also sensitively affect the twinnability of these alloys. The sensitivity of the deformation mode to the shearing conditions should be attributed to the special shape of the γ-surfaces in Fig. 3a. We note that the beneficial effect of strain on the twinnability is mentioned in connection with the empirical deformation maps as well 22 . For CrMnFeCoNi alloy, the measured critical twinning stress is around 235 ± 10 MPa 44 . Combining this value with the resolved shear modulus (48 GPa), for the affine shear strain we obtaiñ 0.8% as the characteristic value 45 . The Fe 40 Mn 40 Co 10 Cr 10 and CrCoNi alloys are expected to possess slightly smaller and larger strains, respectively, as suggested by their EEBs (Fig. 3a). The impact of the strain on the temperature-dependent γ SF À γ TW À Á for the considered alloys is illustrated in Fig. 2 7 . In the sheared CrMnFeCoNi (strains ≳0.7%), MTW occurs for all temperatures below~460 K, which explains the reported stress-strain behavior at cryogenic condition 6 . On the other hand, for CrMnFeCoNi, the calculation without shear strain predicts a martensitic transformation below~300 K, which is consistent with the pressure-induced fcc-hcp phase transition 32,33 .
Mechanism of metastable twinning. So far we defined MTW as twinning deformation when ΔF ≤ 0. Below, we reformulate the MTW condition using the SFE and related IMPs. An intrinsic stacking fault in an fcc lattice can be treated as a two-layer embryo with the hcp structure embedded in the fcc matrix. Accordingly, the SFE is often expressed as 46 where A is the area of the stacking fault and σ * is the pseudointerfacial energy between the hcp embryo and the fcc matrix. Hence, γ isf 2σ Ã can be taken as the condition for the thermodynamic stability of the hcp lattice. Using our calculated γ isf and ΔF values, for the present alloys we obtain σ * ≈ 4-9 mJ m −2 , which compares well with 8-9 mJ m −2 reported for Fe-Cr-Ni alloys encompassing about 14-20% Cr and 16-20% Ni 47 . In applications, σ * is often considered to be independent of temperature 46,48 .
In terms of IMPs, twinning can occur if γ isf > γ utf À γ usf . Elemental fcc metals and austenitic steels 31,39,42,49 have γ utf > γ usf , and thus a positive SFE is required to activate TW. The opposite case γ utf < γ usf is characteristic to hcp metals 27,50 , for which however the large negative γ isf forbids TW. These two cases provide the physical background for the phenomenological correlations between SFE and the deformation modes 22 . However, twinning can take place also when γ utf ≲ γ usf if γ isf remains larger than a small (negative) critical value. Numerically, for twinning one should have γ isf > 2δ where δ γ utf À γ usf À γ isf =2 expresses the deviation relative to the so-called universal scaling law 51 . Theoretical calculations show that for fcc metals and alloys, δ can be both positive and negative 42 , and its absolute value is usually around a few mJ m −2 . Combining this constraint for twinning with the one for the lattice instability, for the condition of MTW, we obtain δ < γ isf =2 < σ Ã . Hence, MTW can only be realized in systems where δ is below the pseudointerfacial energy.
The above condition excludes from the MTW family, all metals and alloys, which possess γ isf > 2σ Ã at all relevant temperatures. But for the present multi-component alloys 38 , as well as for the paramagnetic Fe 52 or the low-Ni austenitic stainless steels 31 , γ isf is negative at static conditions and increases with increasing temperature. In such systems, γ isf =2 usually scans the interval between δ and σ * when changing the temperature, and thus the MTW mechanism might become effective within a specific temperature range. The sign of ðδ À σ Ã Þ in turn is decided by the chemistry and its magnitude is sensitive to the affine shear strain preceding the deformation. Since σ * is usually small (typically below 10 mJ m −2 ), mostly alloys with very small or negative δ are expected to enter the MTW regime upon changing the temperature. Alloys with positive δ (but still below σ * ), such as Fe-Cr-Ni 31 , are predicted to have a very narrow temperature interval for MTW. Finally, alloys with δ > σ Ã transform directly from the normal TW region of the stable austenitic phase to the SF region with lowering the temperature.
The two critical parameters for MTW, ðγ isf =2 À σ Ã Þ and ðδ À σ Ã Þ, are shown in Fig. 4 for Fe 40 Mn 40 Co 10 Cr 10 , CrMnFe-CoNi, and CrCoNi as a function of affine shear strain and temperature. It is remarkable that all three alloys satisfy the ðδ À σ Ã Þ < 0 condition at all temperatures and strains from the figure and therefore they are prone to MTW. At the same time, the calculated relative SFEs ðγ isf =2 À σ Ã Þ increase with temperature but remain negative below 410 K for Fe 40 Mn 40 Co 10 Cr 10 , 460  Fig. 4 Twinning parameters as a function of the affine shear strain. a-c show the strain-dependence of (γ isf =2 À σ Ã ) and (δ À σ Ã ) for three different temperatures for the Fe 40 Mn 40 Co 10 Cr 10 , CrMnFeCoNi, and CrCoNi alloys, respectively. Metastable twinning corresponds to ðγ isf =2 À σ Ã Þ>ðδ À σ Ã Þ and ðγ isf =2 À σ Ã Þ < 0 (green area), martensitic transformation to ðγ isf =2 À σ Ã Þ < ðδ À σ Ã Þ (yellow area), and normal twinning to ðγ isf =2 À σ Ã Þ > 0 (red area). The latter occurs in Fe 40 Mn 40 Co 10 Cr 10 and CrMnFeCoNi alloys at high temperature but not in CrCoNi. Metastable twinning is present in all three alloys within certain strain-temperature windows K for CrMnFeCoNi, and at all temperatures considered here in the case of CrCoNi. Hence, MTW should appear within certain temperature and strain intervals in all three multi-component alloys. These findings are in perfect line with the present theoretical predictions in Fig. 2 as well as with the available observations 3,6,7,20 .
Stacking fault energy and metastable twinning. In order to express the deformation modes in terms of the experimentally accessible SFE, in Fig. 5a we plot the relationship between γ SF À γ TW À Á and γ isf for Fe 40 Mn 40 Co 10 Cr 10 for temperatures between 100 and 500 K and three affine strain levels. In the stressfree case, MTW is present for γ isf between −8 and 10 mJ m −2 , i.e., both negative and small positive SFE values are associated with TW. Increasing the affine shear strain, the maximum allowed γ isf for MTW is shifted slightly towards positive values due to the weak stress-induced increase of σ * . Furthermore, the lowest SFE for which twinning is observed decreases strongly with strain and thus broadening the range for MTW. For 0.8% strain, which is close to the estimated value for CrMnFeCoNi, the lowest SFE where TW is the active deformation mode (in addition to SL) is −16.5 mJ m −2 . The room-temperature SFE values for strains up to 0.8% vary between −1.3 and 2.3 mJ m −2 , and lie well inside the MTW region. These SFEs are far below the empirical limit for twinning (10-16 mJ m −2 ) and confirm that classical SFE-based deformation models 22,23 fail to account the observed twinning in this HEA 3 . In Fig. 5b, we propose a schematic deformation mode diagram based on the theoretical data obtained for Fe 40 Mn 40 Co 10 Cr 10 . We mention that the deformation maps for CrMnFeCoNi and CrCoNi (not shown) are qualitatively the same as that in Fig. 5b, but the boundaries between SF and MTW are slightly shifted towards larger SFE values. Like in the phenomenological diagram 22 , in Fig. 5b, the SF mode occurs for negative and the TW mode for positive SFE. Interestingly, the present theoretical SFE values for normal TW are above 7-14 mJ m −2 (depending on the affine shear level), which agree well with 6-12 mJ m −2 obtained by correcting the empirical critical values of twinning [24][25][26] by the estimated strain contribution near the partial dislocations 53 . However, in the deformation mode map for MEAs and HEAs, there is a broad MTW region, which broadens towards both positive and negative SFE values with increasing affine strain. In contrast, in the case of traditional alloys like austenitic stainless steels, the MTW region is very narrow or completely missing.
The present theoretical exploration opens opportunities to fine tune the plastic flow mechanisms in HEAs. One important example is the enhanced twinning probability (f TW ), which means that an increased number of randomly oriented grains exhibit twinning. In ideal metals (δ = 0), we obtain f TW = 50% for γ isf = 0 mJ m −2 , meanwhile in metals with δ > 0, the twinning fraction turns below 50%. For instance, for Fe 68 Cr 20 Ni 12 alloy 31 , δ = 7 mJ m −2 and γ usf = 276 mJ m −2 give f TW = 45%. However, in alloys exhibiting MTW, f TW can reach values substantially higher than the previously established upper limit. For instance, the present results for Fe 40 Mn 40 Co 10 Cr 10 yield δ = −4 mJ m −2 (neglecting the affine strain) and thus f TW = 53%. Hence the TW fraction for the present HEA is~18% larger than the value obtained for steels.
We have investigated the plastic deformation in multicomponent alloys and presented a transparent atomic-level theory for the MTW mechanism. The theory is based on the EEBs derived from first-principle calculations. We focus on the high strength and high ductility alloys, Fe 80−x Mn x Co 10 Cr 10 , CrMnFeCoNi, and CrCoNi, in which twinning was reported to be a fundamental deformation mechanism. Both theoretical and experimental results suggest that the studied alloys possess a thermodynamically unstable fcc lattice at ambient and cryogenic conditions, which questions the phenomenological plasticity models. In contrast to those models, the present theory demonstrates that twinning can be the primary deformation mode for metastable fcc alloys with a twinning fraction that surpasses the previously established upper limit. It is found that small affine shear strain can further amplify the MTW phenomenon. The disclosed behavior of the present MEAs and HEAs calls for a critical revision of the previously adopted design criteria regarding twinning-induced plasticity. Our quantum theory aided understanding of the plasticity shows that a precise control of the deformation mechanisms is feasible in these new families of multi-component alloys, which can ultimately facilitate the optimal harvesting of their properties.

Methods
Theoretical methodology. The present calculations were based on densityfunctional theory formulated within the generalized gradient approximation 54 and the exact muffin-tin orbitals formalism 55 . The Kohn-Sham equations were solved within the scalar-relativistic and soft-core schemes. Since the Curie temperatures estimated via the mean field approximation 56 are below 77 K, all alloys were described in the paramagnetic state modeled by the disordered local magnetic moment approximation 57 . The chemical and magnetic disorders were taken into account using the coherent potential approximation 55 . The free energies included the lattice expansion and magnetic entropy terms. For a randomly oriented polycrystalline alloy, the fraction of grains showing TW was estimated from the IMPs via 27