Mapping deformation mechanisms in lamellar titanium aluminide

Accepted Manuscript Mapping deformation mechanisms in lamellar titanium aluminide Zong-Wei Ji, Song Lu, Qing-miao Hu, Dongyoo Kim, Rui Yang, Levente Vitos PII:

S1359-6454(17)30971-0

DOI:

10.1016/j.actamat.2017.11.028

Reference:

AM 14199

To appear in:

Acta Materialia

Received Date: 4 October 2017 Revised Date:

6 November 2017

Accepted Date: 14 November 2017

Please cite this article as: Z.-W. Ji, S. Lu, Q.-m. Hu, D. Kim, R. Yang, L. Vitos, Mapping deformation mechanisms in lamellar titanium aluminide, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.11.028. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Mapping deformation mechanisms in lamellar titanium aluminide Zong-Wei Jia,b , Song Lu∗,b , Qing-miao Hu∗,a , Dongyoo Kimb , Rui Yanga , Levente Vitosb,c,d a Shenyang

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National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China b Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE-100 44, Sweden c Department of Physics and Astronomy, Division of Materials Theory, Uppsala University, Box 516, SE-75121, Uppsala, Sweden d Research Institute for Solid State Physics and Optics, Wigner Research Center for Physics, Budapest H-1525, P.O. Box 49, Hungary

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Abstract

Breakdown of Schmid’s law is a long-standing problem for exploring the orientation-dependent deformation mechanism in intermetallics. The lack of atomic-level understanding of the selection rules for the plastic deformation modes

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has seriously limited designing strong and ductile intermetallics for high-temperature applications. Here we put forward a transparent model solely based on first principles simulations for mapping the deformation modes in γ-TiAl polysynthetic twinned alloys. The model bridges intrinsic energy barriers and different deformation mechanisms and beautifully resolves the complexity of the observed orientation-dependent deformation mechanisms. Using the model, one can elegantly reveal the atomic-level mechanisms behind the unique channeled flow phenomenon in lamellar TiAl alloys.

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Key words: stacking fault; twinning; slip; first principles

1. Introduction

As an innovative class of advanced engineering materials, TiAl-based alloys have been extensively studied because

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of the low density, decent corrosion and creep resistance, and a balanced combination of strength and ductility at elevated temperatures, showing great potential in automobile and aerospace industries. [1–4] The polycrystalline TiAl

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alloys have been successfully applied in gas turbine to replace Ni-based superalloys for a higher specific strength. [5– 9] However, the major concern about polycrsyalline and single-crystalline γ-TiAl alloys is their poor ductility and low fracture toughness at ambient and low temperatures, which significantly limit their applications. [10–12] The so-called polysynthetic twinned (PST) single crystals with well-aligned layered structure composed of α2 -Ti3 Al and γ-TiAl variants have greatly improved the mechanical properties, e.g., ductility at room temperature. [13, 14] Recent progress shows that at ambient temperature the tensile ductility and yield strength of the PST TiAl alloys loaded along the direction parallel to the lamellar interface reach as high as 7.6% and 735 MPa, [15] respectively, while in polycrystalline TiAl alloys the ductility is usually less than 2-3%. The pronounced activity of twinning in the PST ∗ The

author to whom the correspondence should be addressed to; Email: [email protected], [email protected]

Preprint submitted to Acta Materialia

November 6, 2017

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TiAl alloys is acknowledged for producing simultaneously high strength and high ductility. However, the detailed mechanisms for deformation twinning and the selection of deformation modes with respect to loading orientation in the PST TiAl alloys have remained unclear. [4, 16–18] There are three major competing deformation modes in γ-TiAl with L10 structure: 1/2h110] ordinary dislocation 1

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(OD), h101] superlattice dislocation (SD) and {111}h11-2] true twin which preserves the chemical order of γ-phase. Compared to the twelve twinning systems in the fcc structure, there are only four available twinning systems in

L10 . [13] The operation of deformation modes in γ phase in PST crystals depends strongly on orientation and the sense of loading (compression or tension) due to the unidirectional nature of twinning [19, 20] and polarization of SD [21], and as a result the mechanical properties are strongly anisotropic. [13, 22] The measured yield stress is the

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highest when loading axis is perpendicular to the lamellar interfaces (N orientation), however, the tensile ductility is almost zero. When the lamellar interface is at an intermediate angle to the loading axis (B orientation), the strength

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is low and ductility is high. The best combination of strength and ductility is obtained for samples with the lamellar interface orientated parallel to the loading axis (A orientation), which is due to a peculiar macroscopic deformation phenomenon - the so-called channeled flow. [13, 15, 20, 23] That is to say no macroscopic shape change is observed in the direction perpendicular to the lamellar boundaries while the material flows parallel to the lamellar boundary. This phenomenon is intrinsically connected to the operating deformation modes. It was observed that some γ lamellar deform by SD or OD slip with Burger vectors parallel to the lamellar interfaces while the others deform by the co-plane activity of OD slip and twinning with shear vectors inclined to the lamellar interfaces. Either way the deformation

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occurs, no macroscopic strain is produced in the direction perpendicular to the lamellar interfaces. [13, 20, 24–26] According to the general wisdom, dislocation slip and twinning are usually exclusive to each other on the same slip plane under a given mechanical load. [27] In fact, the channeled flow challenges the Schmid’s law by showing macroscopic plastic flow on the plane with zero Schmid factor. Furthermore, the activation of deformation modes in

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the four {111} slip planes were also shown often violating the Schmid’s law. [13, 20, 22, 24] Usually, the discrepancy between the observed deformation modes and the ones predicted by Schmid’s law was simply ascribed to local stress concentration. [28] Naturally, the ease and propensity of twinning are expected to

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be connected to the intrinsic properties of materials, i.e., generalized stacking fault energies (GSFEs). GSFEs have been proved significant for understanding the mechanical behavior of materials, in particular dislocation core [29, 30] and deformation modes in fcc metals. [31–33] However, a systematic picture describing the relationship between the selection of deformation modes, the sense of loading (compression or tension), crystal orientation, and GSFEs in intermetallics has not yet been established. In the present work, we present the missing piece of puzzle by rationalizing the plastic deformation modes in γ-TiAl using the orientation dependent effective energies barriers. The rest part of present paper is organized as follows. The computational details and the development of plasticity 1 The

existence of 1/2 h112] superlattice dislocations was observed due to the decomposition of h101] SDs. [13] They are not considered in

the present work. As it is commonly utilized in the literature, the cubic notation with the mixed parentheses {hkl) and huvw] (the third index corresponds to the tetragonal axis) will be employed also in this paper.

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model in γ-TiAl are presented in Section 2. Results for the theoretical deformation map is given in Section 3, followed by a detailed comparsion with experimental observations under various loading conditions. A summary is drawn in the last section.

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2. Methodology and computation details 2.1. Total energy method

The total energy calculations presented in this work are carried out using the projector augmented waves (PAW) method [34, 35], as employed in the Vienna ab intito simulation package (VASP) [36–38] based on density functional

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theory (DFT) [39]. For Ti, we adopted the 3p semicore electrons as valence electrons. The exchange-correction energy functional is treated within the generalized gradient approximation (GGA) as parameterized by Perdew and Wang [40]. We set the energy cut off to 400 eV, which guarantees good convergence, and use the k-points spacing smaller than

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2π*0.03 Å−1 in the reciprocal space according to the Monkhorst-Pack [41] sampling scheme. The conjugate gradient method is used in the structural optimization. In the self-consistent loop, the total energy is converged to within 10−6 eV, and the force on the atoms is less than 0.01 eVÅ−1 .

A supercell composed of six-layer of {111} planes is adopted for the calculation of GSFE surface. The three lattice vectors of the supercell for the fcc lattice, a, b, c, are 1/2[1-10], 1/2[11-2] and 2[111]. The GSFE surface is therefore calculated by monitoring the energy change with the shifting vector v = {x, y, 0} which is added to lattice vector c,

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c = c + v. [42] All the atoms are relaxed along the direction perpendicular to the fault plane. Twinning energies and barriers are calculated in a similar way by slipping on the adjacent {111} plane of one existing stacking fault. Experimentally, it has been established that the easy slip planes are the {111} planes. [43] The present results for GSFEs on the 111 planes are shown in Fig.1(b) and (c). Here γsisf , γusisf , γcsf , γucsf , γtw , γutw , γapb , and γuapb stand for the stable and unstable formation energies of the superlattice intrinsic stacking fault (SISF), complex stacking

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fault (CSF), twin (TW), and antiphase boundary (APB), respectively. The corresponding numeric values are listed in Table 1. All calculations are carried out for the theoretical equilibrium volume at 0 K. Quantitatively, the ab initio

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results are in nice agreement with each other. Here we emphansize that γ-TiAl is crystallized into L10 structure with c/a ≈ 1.02. In the present work, however, all the calculations are performed assuming c/a = 1, which causes negligible effect on the stacking fault energies. Due to the symmetry of L10 structure, APB, CSF and SISF are not necessarily stable or placed themselves at the ideal positions as envisioned geometrically. [44] From the calculated γ-surface, SISF is found at the ideal position, while CSF moves towards the APB by 0.17× 61 [-1-12] relative to its ideal position. γcsf is correspondingly relaxed by ∼16%, from 423 J m−2 at the ideal CSF position to 358 J m−2 at the minimum. APB is found to be unstable. Our results agree well with previous theoretical studies. [45] We notice that, when the GSFE surface is calculated at the equilibiurm c/a ≈ 1.02, the corresponding positions for SISF and CSF moves along [11-2] direcion. CSF is, therefore, closer to the geometrical postion, while SISF moves away.

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Figure 1: (Color online) (a) Schematic of the slip/twinning systems on the (111) plane in γ-TiAl with L10 structure. The Burgers vectors for the [10-1] SD, [0-11] SD, 1/2[11-2] SD, 1/2[1-10] OD, and 1/6[11-2] twin are shown, together with the full-dissociation paths. The different sizes of circles for atoms (empty for Ti and solid for Al) correspond to three consecutive (111) layers. τ is the projection of the external stress on the (111) plane. (b) and (c) are the calculated generalized stacking fault energies of γ-TiAl for the SISF leading and the CSF leading paths, respectively. The

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γ-curves for pseudotwins are plotted using dashed lines. See text for notations.

However, this small changes do not affect the deformation modes and thus the present results and conclusions remain unchanged.

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2.2. Development of plasticity model

In the following, we discuss the competition among different deformation modes as formulated on the basis of activation of leading and twinning/trailing partials (Fig.1 (a)). From the energetic point of view, the activation of

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partials (1/6h112i) is always easier than OD or SD. We emphasize that the activation of deformation modes depends significantly on the source/nucleation of dislocations. For example, mechanical twinning was observed as an intensive deformation mode in polycrystalline Ti46 Al54 alloys at various temperatures, [47] while in single crystals it was only observed at high temperatures [48]. Both grain and lamellar boundaries were observed as prominent sources of partials. [20, 24, 28, 49] Transmission electron microscopy (TEM) studies have revealed that the growth of twins occurs mainly through the propagation of 1/6h112] Shockley partials on successive {111} planes. [20, 24, 28, 47, 49] In fcc materials, assuming the nucleation of a partial dislocation from a grain boundary, slip deformation means that a leading Shockley 1/6h112i partial dislocation is followed by a trailing partial dislocation on the same slip plane. The leading and trailing partial Burgers vectors add up to make a full 1/2h110i Burgers vector. For twinning, the same leading Shockley partial dislocation is followed by a twinning Shockley partial which slips on an adjacent {111} plane. 4

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Table 1: The calculated stacking fault energies, together with previous ab initio results. γsisf , γusisf , γcsf , γucsf , γtw , γutw , γapb , and γuapb stand for the stable and unstable formation energies for the superlattice intrinsic stacking fault (SISF), complex stacking fault (CSF), twin (TW), and antiphase boundary (APB), respectively. Note that APB in stoichiometric γ-TiAl is not a local stable (meta-stable) structure. [45] γapb is calculated

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for ideal crystallographic APB structure.

γsisf

γusisf

γcsf

γucsf

γtw

γutw

γapb

γuapb

181

316

358,

560,

177

410

657

735

184a

321a

355a

522a

182a

409a

Ref. [45]

a

Ref. [46]

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b

363b

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The competition between deformation twinning and dislocation slip is, thus, the competition between the nucleation of a twinning partial and a trailing partial. Therefore, the energy barriers for twinning and slip are γutw − γisf and γusf − γisf , respectively. [32, 50]

Analogously to the definition of critical resolved shear stress by projecting the external stress to the slip direction, the energy barrier for a deformation mode to the effective energy barrier by considering the angle between the pro-

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jected shear stress τ and the slip direction. [32] Since the angle between the trailing and the twinning partials is 60o , the effective energy barriers for twinning and slip are γtw (θ) =

γutw − γisf γusf − γisf and γsl (θ) = , cos θ cos(60o − θ)

(1)

symmetry of fcc. [32]

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respectively. θ measures from the stacking fault easy direction [11-2] and is in the range of 0o -60o as decided by

In L10 structure, we adopt a similar analysis as above. However, because of its lower symmetry compared to fcc,

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on the (111) plane, the angle between the projected shear stress and the [11-2] (twinning) direction is in the range of 0o -180o and being taken as a reference for the other slip directions (Fig.1(a)). θ is related to the angles λ and φ (Fig.2). φ is the angle between the applied stress and the normal of slip plane, λ is the angle between the applied stress and twinning direction. cos φ cos λ is therefore the Schmid factor for twinning. The three angles are related by cos θ =

cos λ . sin φ

(2)

We first study the competition between leading partial dislocations (p1 and p2 ) with Burgers vectors, 1/6[11-2] and 1/6[1-21], because they lead to the formation of SISF and CSF, respectively. The activation of these two types of leading partials is a critical step for different deformation modes which depend on the following partials. The energy 5

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Figure 2: (Color online) Schematic of the relationship between θ, φ and λ when (-111) is the slip plane.

barriers for them are γusisf and γucsf , respectively, and the corresponding effective energy barriers are γp1 (θ) =

γusisf γucsf and γp2 (θ) = . cos(θ) cos(120o − θ)

(3)

Notice that here we have neglected the effect of the deviation of CSF or SISF from their ideal positions. By comparing

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γp1 and γp2 (Fig.3), we can see that for 0o ≤ θ . 70o , γp1 < γp2 , indicating the activation of 1/6[11-2] partial; while for 70o . θ ≤ 180o , γ p1 > γp2 , meaning that the 1/6[1-21] partial is the leading one. Accordingly, depending on which partial is leading, we may divide the whole orientation into two ranges.

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Range1: 0o ≤ θ . 70o , 1/6[11-2] partial leading

With a 1/6[11-2] leading partial, the two-layer nucleus of true twin can be formed if the following partial with identical Burgers vector slips on an adjacent (111) plane. Nucleus for pseudotwin may also form if the following par-

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tial is 1/6[1-21] on an adjacent (111) plane, which, however, destroys the L10 chemical environment. The pseudotwin nucleus may be viewed as the overlapping of SISF and CSF. The stable and unstable energies are remarkably high (Fig.1 (b), dashed line), and thus this scenario is very unlikely to happen. Indeed, no deformation pseudotwin was observed in experiments. In the picture of Shockley partials and stacking faults, the formation of [10-1] SD (SDI ) needs three consecutive partials with Burgers vectors, 1/6[2-1-1], 1/6[11-2], and 1/6[2-1-1], respectively (Fig.1 (a)), which slip on the same <111> plane as the leading partial. The highest energy barrier for SDI is identified for the first following partial with Burgers vector 1/6h2-1-1] which transforms the stacking fault from SISF to APB. Once this barrier is overcome, the projected stress is readily enough to activate the second and third following partials. Actually, there is no barrier for the third partial (APB → CSF). The energy barriers for true twin and SDI are (γutw − γsisf ) and (γuapb − γsisf ), respectively, as seen from the GSFE surfaces (Fig.1 (b)). Correspondingly, the effective energy barriers 6

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for twinning and SDI are γtw (θ) =

γuapb − γsisf γutw − γsisf and γsdI (θ) = , cos(θ) cos(600 − θ)

(4)

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respectively. Range 2: 70o . θ ≤ 180o , 1/6[1-21] partial leading

In this range of orientation, 1/2[1-10] OD, [0-11] SD (SDII ) and 1/2[-1-12] SD may form and compete with each other, depending on the activation of following partials (Fig.1 (a)). With the existence of CSF, theoretically pseudotwins may also form if the twinning partials are 1/6[11-2] or 1/6[1-21] and slip on an adjacent (111) plane

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to the CSF. The γ-curves for these pseudotwins are shown in Fig.1 (c) by dashed lines. Since the effective energy barriers for them are significantly higher than other deformation modes (not shown), we do not further discuss them here. The energy barrier for 1/2[1-10] OD, (γucsf − γcsf ), is the one for the 1/6[2-1-1] trailing partial which restores the

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lattice from CSF to L10 structure. It deviates by 60o from the [11-2] direction. Notice that the unstable CSF energies for the partials with Burgers 1/6[2-1-1] and 1/6[1-21] are the same. The effective energy barrier for OD is γod (θ) =

γucsf − γcsf . cos(60o − θ)

(5)

The effective energy barrier for SDII is decided by the highest barrier for the following partials. Since the projection angles to the projected stress are different for them, we explicitly calculate the effective energy barriers for the three

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following partials as a function of θ, viz.

γapb − γcsf , cos(180o − θ)

(6)

γapb→sisf (θ) =

γuapb − γapb , cos(120o − θ)

(7)

γsisf→sdII (θ) =

γusisf − γsisf . cos(180o − θ)

(8)

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γcsf→apb (θ) =

It is interesting to note that when θ equals 180o the partial that brings the lattice from APB to SISF can also have the Burgers vector of 1/6[-211] instead of 1/6[1-21] (Fig.1 (a)). The effective energy barriers for these two partials are exactly the same. This path leads to the formation of 1/2[-1-12] SD. Therefore, from this point of view the 1/2[-1-12] and [0-11] SDs are degenerate at θ = 180o . 3. Results and discussion 3.1. Effective energy barrier The competition among the different deformation modes in terms of the effective energy barriers as a function of θ is summarized in Fig.3. In the range of 0o ≤ θ . 65o with 1/6[11-2] partial leading, twinning is the most favorable 7

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Figure 3: (Color online) Competition among different deformation modes in terms of effective energy barriers as function of θ. The deformation mode with smaller effective energy barrier is preferred to be activated. The variation of colors indicates the activation of different deformation modes in the range of orientation.

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deformation mode. SDI occurs only within a very narrow range of angle (65o . θ . 70o ), which is consistent with the very high energy barrier (Fig.1 (b)). We emphasize that the significantly narrowed activation range for SDI

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immediately leads to the breakdown of the Schmid’s law, e.g., at θ around 30o , where the Schmid factor for SDI is larger than that for twinning. For θ in the range of [70o , 180o ] with 1/6[1-21] partial leading, the primary competing modes are 1/2[1-10] OD and [0-11] SDII . From Fig.3 we see that with the existence of CSF, the effective energy barrier for OD is the lowest one for θ . 125o . The effective energy barrier for SDII is dominated by γcsf→apb . In the high angle range from ∼125o to 180o , [0-11] SDII is most favorable. Notice that the above discussion is based strictly on the lowest energy path. Due to the remarkably high formation energy for APB and CSF, the following partial for SDI may likely be 1/6[5-1-4] without splitting into three 1/6h112i partials or with a very compact core structure. [20, 24, 51] The case is similar for SDII , the leading partial could be 1/6[1-5-4] in order to avoid APB and CSF. [51] Considering this situation, the boundaries that separate different deformation modes in Fig.3 are however, only slightly modified. 8

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It is well recognized that twinning in L10 structure is unidirectional. The operation of twinning systems in samples under uniaxial load is not only dependent on the axis orientation, but also on the sense of the load. Specifically, out of the four twinning systems, some twin modes should operate only in compression, whereas others should operate only in tension. Fig.3, however, indicates that the other deformation modes are also strongly polarized in the sense of

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loading direction. Without considering the change of ease of the deformation modes on other slip planes, reversing the loading direction that activates OD may cause the operation of twinning, SDI or OD, depending on θ. Reversing the direction of load that produces SDI or SDII leads to the activation of OD or twinning, respectively. We would also like to emphasize that the asymmetric core structure of h011] SDs with SISF or CSF as leading planar faults may lead to different cross-slip behaviors under compression or tension, and consequently the yielding behaviors under tension

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and compression are asymmetric. [21, 48, 52]

3.2.1. Orientation A2 in compression

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3.2. Comparison with experimental observations

Now we connect the theoretical deformation map to the observed deformation modes in PST TiAl alloys. Even though the above deformation map is obtained with the γ-surface at 0K, the results may still be valid at room temperature since the γ-surface is not expected to change significantly. As an illustration, we focus on the case of compression parallel to lamellar boundaries and along the [11-2] direction (A2 ) which leads to channeled flow in lamellar TiAl alloys. [13, 15, 27] For various orientations, the activated deformation modes in each γ variant have been carefully

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established by Kishida et al. [13] The orientation relationship between the TiAl and Ti3 Al phases in the lamellar structure is (111)γ //(0001)α2 and h1-10iγ //h11-20iα2 . Since the three h11-20i directions on the basal plane in α2 phase with D019 structure are all equivalent while the [1-10] and h01-1] directions on the (111) plane in γ phase with the L10 structure are not equivalent, there are six possible orientations for γ lamellar relative to α2 . [22] In order to make a direct comparison between our predicted deformation modes and the experimental observations, we consider three

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γ variants, IM , IIM and IIIM , following the notation from Ref.[13]. Variants IIM and IIIM are rotated by 120o and 240o around [111] relative to IM , respectively. Therefore, under A2 compression, the loading directions in IIM and IIIM

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are [1-21] and [-211], respectively (Fig.4). The deformation structures in the other three types of variants (IT , IIT and IIIT in Ref.[13]) are the same as in these variants. The Schmid factors for all the potential deformation modes in the three slip planes in the three variants are listed in Table 2. Using the established mapping for the deformation modes and calculated θ (in Fig. 4 and Table 2) together with the Schmid factors, we can now rationalize the activation of deformation modes in each variants. For the present orientation, in the IM variant, the slip systems in the (-111) and (1-11) slip planes are symmetric about the compression axis. The calculated θ values are 41o in the (-111) and (1-11) planes and 180o in the (11-1) plane, respectively. According to Fig.3 we predict the activation of [-11-2] twinning in the (-111), [1-1-2] twinning in the (1-11) and [011] SD in the (11-1) plane, respectively, before considering the competition among deformation modes on different slip planes. Notice that the Schmid factor for twinning (0.314) is much smaller than that for SD 9

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Figure 4: (Color online) The calculated θ in (-111), (1-11) and (11-1) planes for compression along A2 [11-2] direction, shown in the Thompson tetrahedron. The corresponding projected shear stresses and θ values in three γ variants, IM , IIM and IIIM , are colored in red, blue and green, respectively. Aδ, γB, βC, and αD are the allowed twinning shear vectors in (111), (1-11), (-111), and (11-1) planes, respectively.

(0.408) in both (-111) and (1-11) planes. Experimentally, one of the two symmetric twinning systems is observed, which may indicate that the activation of one twinning system is enough to accommodate the deformation, given that the specimens are only deformed by 2-3%. [13] Similar observation was made in the early stages of deformation in L10 CuAu where only one deformation twin system is operative within one lamella. [53] The [011] SD in the (11-1) plane does not operate because of the smaller Schmid factor (0.272). In addition to the [1-1-2] twinning on the (1-11) plane, the co-plane 1/2[110] OD with very small Schmid factor (0.272) was also observed. [13] At first glance, it is 10

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Figure 5: (Color online) Illustration of deformation twin (b=1/6[1-1-2]) hitting a lamellar boundary ((111) plane) in lamellar TiAl alloys loaded under compression in the A2 direction. Schematic of Thompson tetrahedron is plotted for better referring to deformation vectors. Aδ, γB, βC, and

[11-2] and [-1-1-1], respectively.

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αD are the allowed twinning shear vectors in (111), (1-11), (-111), and (11-1) planes, respectively. A2 and N are the uniaxial loading directions,

surprising to see the co-activation of 1/2[110] OD and twinning on the same plane according to Fig.3, given that the leading partials for them are different. We will return to this question below.

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In IIM , θ are 161o , 60o , and 80o in the (-111), (1-11), and (11-1) planes, respectively. The predicted slip system in (-111) is the [0-11] SD, in agreement with the experimental observation. [1-1-2] twinning in the (1-11) plane resulting

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tension violates the macroscopic deformation and is forbidden. The secondary slip system [-101] SDs have zero or small Schmid factors. In the third plane, 1/2[1-10] OD is predicted, which has the same Schmid factor (0.408) as the observed [0-11] SD in the (-111) plane. It seems that one of the most favorable slip systems is picked. In IIIM , we predict the activation of 1/2[-110] OD and [-101] SD in the (11-1) and (1-11) planes, with θ of 80o and 161o , respectively. These two modes have the same large Schmid factor (0.408) and one of them (OD) is observed being chosen to operate. In the (-111) plane, θ is 60o and twinning is not activated because it leads to tension. Other slip modes on this plane have very small Schmid factors compared to the deformation modes in oblique planes. To understand the cooperative deployment of slip and twinning, we recall the fact that twinning leads to rotation of the twin portion relative to the matrix (Fig.5). In the A2 orientated lamellar, the twinning vectors 1/6h1-1-2] (γB and βC in the Thompson tetrahedron) are inclined to the lamellar interface and a step at the interface is expected. Note 11

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that due to the lamellar structure, the twinning shear in one lamella can not be directly translated into the neighboring variants. In other words, the lamellar boundaries resist the transmission of twin. Among the different types lamellar boundaries, the 120o and 240o rotational γ/γ boundaries and γ/α2 phase boundary show much stronger resistance to twin transmission than the 180o γ/γ boundary (twin boundary). [22, 24, 27] Experimentally, the propagation

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of twinning were often observed being stopped at the lamellar interfaces. [22, 54] The resistance against twinning propagation at the lamellar interfaces leads to a reflected concentrated local stress which has a projection perpendicular to the interface. It points to the [111] direction (opposite to N, Fig.5). The calculated θ for this reverse stress at the twin tip is 120o in both (-111) and (1-11) planes (Table 2), and therefore the 1/2[110] OD is predicted to operate according to Fig.3. We emphasize that the effective energy barrier for OD at θ = 120o is about 10% smaller than

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that for twinning at θ = 60o (Fig.3). Therefore, the stress concentration at the position where twin hits the lamellar boundary should be readily enough to activate OD. This is consistent with recent molecular dynamic simulations

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showing that OD and twinning are activated by Peierls stress of similar magnitude. [27] It was further demonstrated that when an ordinary dislocation is introduced within an existing twin it can eliminate the step caused by twin in the lamellar boundary. It was proposed that the combination of one ordinary dislocation and three twinning dislocations whose Burgers vectors are all inclined to the (111) lamellar boundaries leads to a total Burgers vector parallel to the lamellar boundaries. [25] Therefore, the strain component normal to the interfaces is eliminated as observed in the channeled flow. It was suspected that the nucleation of ordinary dislocation into microtwins may be realized by the reaction of twinning partials with interfacial dislocations at the lamellar boundaries. [24, 27] There are also

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experimental evidence showing that these ordinary dislocations are originated from the positions where twins are in contact with lamellar boundaries, giving further support for the above scenario. [22, 54] 3.2.2. Orientation A2 in tension

When the specimens are deformed by tension, the angle θT in one specific plane is simply given by (180o − θC ).

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The Schmid factors for the observed deformation modes in tension are indicated by shades. [13] The θT values in IM are 139o , 139o , and 0o in the (-111), (1-11), and (11-1) planes, respectively. The predicted

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modes in (-111) and (1-11) are [0-11] SD and [-101] SD, respectively. They have the same large Schmid factor. In the third plane, twinning leads to compression and the secondary SD mode has smaller Schmid factor than the SDs in the other two oblique slip planes. Considering the decomposition reactions that produce 1/2[-1-10] OD, 1/2[1-12] SD and 1/2 [-1-1-2] SD by [0 − 11] → 1/2[−1 − 10] + 1/2[1 − 12] and [−101] → 1/2[−1 − 10] + 1/2[−112], the above predicted deformation modes are in prefect agreement with the observations in Ref.[13]. We emphasize that in this case the two symmetric SDs about the tensile direction in the (-111) and (1-11) planes were observed at the same time, which is different from the case of compression, i.e., only one of the two symmetric twinning systems in the (-111) and (1-11) planes was observed. [13] In the IIM variant, the θT values are 19o , 120o , and 100o in the (-111), (1-11), and (11-1) planes, respectively. [-11-2] twinning together with 1/2[110] OD are predicted in the (-111) plane (see the discussion for the activation of 12

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OD in the main text). 1/2[-110] OD in the (11-1) plane with the largest Schmid factor is predicted. In the (1-11) plane, 1/2[110] OD may or may not operate depending on the intensity of deformation. In the third variant, the operating deformation modes are very similar as in the second variant. We mention that it was observed that the operative deformation modes in the A2 [11-2] tension are similar to those

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in the A1 compression ([1-10] direction), since the two directions are normal to each other. [13] Our prediction is also consistent with this observation. 3.2.3. Orientation N in compression

All possible slip and twinning systems in the three variants have the same geometry with respect to this loading

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axis and the operative deformation modes are the same. [13] The θ values are 60o , 60o , and 180o in the (-111), (1-11), and (11-1) planes (Table 2), respectively. The predicted deformation modes are therefore twinning in the (-111) and (1-11) planes together with the 1/2[-1-10] ODs due to the stress reflection (pointing to [111]) caused by intersection

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of twin with (111) lamellar interfaces (see discussion in section 3.2). Note that the 1/2[-1-10] ODs have negative Schmid factors, whose movement leads to compression. There is no way to combine these ODs with the deformation twins to eliminate the strain normal to the lamellar interfaces, which is in accordance with the observed isotropic macrodeformation for this orientation. No deformation structure was reported for tension along N direction because of the almost zero tensile elongation. [13]

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4. Summary

The disclosed atomistic model of plasticity is solely based on intrinsic energy barriers obtained by first principles calculations and has successfully explained the observed deformations modes in the PST TiAl alloys for various loading directions and lamellar variants. Remarkably, when combined with the local stresses developing near the

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lamellar interfaces upon twinning, the microscopic mechanism behind the long-standing mysteries of co-activation of two deformation modes within the same slip plane and the channeled flow in lamllar TiAl alloys is completely revealed. The present work unambiguously demonstrates the decisive role of γ-surface in the operation of plastic

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deformation mechanisms. This development can easily be extended to other intermetallic compounds with L10 , L12 or similar structures. It offers a novel avenue for optimizing and designing intermetallics with enhanced mechanical properties by tailoring the deformation modes using first principles calculations.

Acknowledgements

Discussions with S. K. Kwon are greatly acknowledged. Work is supported by the Swedish Research Council, the Swedish Foundation for Strategic Research, the Carl Tryggers Foundations, the Swedens Innovation Agency (VINNOVA), the Hungarian Scientific Research Fund (OTKA 109570) and the China Scholarship Council. Q.M.H. and R.Y. are also grateful to the National Key Basic Research Program (No. 2014CB644001) and the National Key 13

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Research and Development Program of China (No. 2016YFB0701301) for financial support. We acknowledge the Swedish National Supercomputer Centre in Link¨oping for computer resources.

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[52] M. Zupan, K. Hemker, Yielding behavior of aluminum-rich single crystalline γ-TiAl, Acta Mater. 51 (2003) 6277 – 6290. [53] D. W. Pashley, J. L. Robertson, M. J. Stowell, The deformation of CuAu I, Philo. Mag. 19 (1969) 83–98. [54] J. B. Singh, G. Molnat, M. Sundararaman, S. Banerjee, G. Saada, P. Veyssire, A. Couret, The activation and the spreading of deformation in

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a fully lamellar TiAl alloy, Philo. Mag. 86 (2006) 2429–2450.

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Table 2: Schmid factors for the deformation modes in three differently oriented γ variants in PST crystals with orientations A2 [11-2] and N [-1-11]. T in the table indicates that twinning leads to tension. Variants IIM and IIIM are simply rotated by 120o and 240o around [111] relative to IM . θ under compression in each slip plane relative to the twinning direction is given in parenthesis. The Schmid factors for the observed deformation

A2 [11-2]

Schmid factor slip plane

slip direction

IM

IIM

OD

(-111)

[110]

-0.272

0.136

SDI

[-10-1]

-0.136

0.272

SDII

[0-11]

0.408

TW

[-11-2]

-0.314

[110]

SDI

[0-1-1]

SDII

[-101]

TW

[1-1-2]

OD

(11-1)

[-110]

-0.272

0.272

-0.408

(41o )

-0.272

0.393T

0

(161o )

0.157T

-0.272

0.136

-0.136

0.272

0.272

0.408

0

-0.408

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(1-11)

IIIM

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Deformation mode

OD

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modes in compression and tension are indicated by boxes and shades, respectively. [13]

-0.314

(41o )

0.157T

(60o )

0.393T

0

0.408

-0.272

0.136

-0.272

0.272

-0.136

SDII

[-10-1]

0.272

[112]

-0.314

[110]

0.272

0.272

0.272

[-10-1]

-0.272

-0.272

-0.272

[0-11]

0

0

0

OD SDI SDII

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TW

(-111)

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N [-1-1-1]

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[011]

OD

(1-11)

(180o )

(60o )

-0.079

-0.157

(80o )

(60o )

-0.079

[-11-2]

-0.157

[110]

0.272

0.272

0.272

-0.157

SDI

[0-1-1]

-0.272

-0.272

-0.272

SDII

[-101]

0

0

0

TW

[1-1-2]

OD

(11-1)

-0.157

(60o )

-0.157

(60o )

-0.157

[-110]

0

0

0

SDI

[011]

0.272

0.272

0.272

SDII

[-10-1]

-0.272

-0.272

-0.272

TW

[112]

0.314T

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(180o )

0.314T

(161o )

-0.408

SDI

TW

(60o )

(180o )

0.314T

(80o )

(60o )

(60o )

(180o )