Analysis of double cross-slip of pyramidal I 〈c+a〉 screw dislocations and implications for ductility in Mg alloys

Analysis of double cross-slip of pyramidal I hc + ai screw dislocations and implications for ductility in Mg alloys

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Analysis of double cross-slip of pyramidal I hc + ai screw dislocations and implications for ductility in Mg alloys Rasool Ahmad, Zhaoxuan Wu, W.A. Curtin PII: DOI: Reference:

S1359-6454(19)30725-6 https://doi.org/10.1016/j.actamat.2019.10.053 AM 15623

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Acta Materialia

Received date: Revised date: Accepted date:

1 August 2019 8 October 2019 27 October 2019

Please cite this article as: Rasool Ahmad, Zhaoxuan Wu, W.A. Curtin, Analysis of double cross-slip of pyramidal I hc + ai screw dislocations and implications for ductility in Mg alloys, Acta Materialia (2019), doi: https://doi.org/10.1016/j.actamat.2019.10.053

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Graphical Abstract

a

b

Pyr I Pyr II

Pyr I

c

d

1

Analysis of double cross-slip of pyramidal I hc + ai screw dislocations and implications for ductility in Mg alloys Rasool Ahmada,∗, Zhaoxuan Wub , W. A. Curtina a

Laboratory for Multiscale Mechanics Modeling, Institute of Mechanical Engineering, EPFL, 1015 Lausanne, Switzerland b Department of Materials Science and Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China

Abstract Solute accelerated cross-slip of pyramidal hc + ai screw dislocations has recently been recognized as a crucial mechanism in enhancing the ductility of solid-solution Mg alloys. In pure Mg, crossslip is ineffective owing to the energy difference between the high energy pyramidal I and low energy pyramidal II hc + ai screw dislocations. A small addition of solutes, especially rare earth (RE) elements, can reduce this energy difference and accelerate cross-slip, thus enabling enhanced ductility. With increasing solute concentrations, the pyramidal I dislocation can become energetically favorable, which switches the primary hc + ai slip plane and alters the cross-slip process. Here, the transition path and energetics for double cross-slip of pyramidal I hc + ai dislocations are analysed in the regime where the pyramidal I dislocation is energetically more favorable than the pyramidal II. This is achieved using nudged elastic band simulations on a proxy MEAM potential for Mg designed to favor the pyramidal I over pyramidal II. The minimum energy transition path for pyramidal I double cross-slip is found to initiate with cross-slip onto a pyramidal II plane followed by cross-slip onto a pyramidal I plane parallel to the original pyramidal I plane. A previous mechanistic model for ductility is then extended to higher solute concentrations where pyramidal I is favorable. The model predicts an upper limit of solute concentrations beyond which ductility again becomes poor in Mg alloys. The model predictions are consistent with limited experiments of Mg-RE alloys at high concentrations and motivate further experimental studies in the high concentration regime. Keywords: Cross-slip, pyramidal dislocation, Mg-RE alloy, NEB method, molecular dynamics

1. Introduction High ductility and high fracture toughness are key attributes for metal forming, drawing, extrusion and forging. Both high ductility and high fracture toughness require the material to yield and strain harden at appropriate levels so as to resist strain localization (necking and shearing) and ∗

corresponding author Email address: [email protected] ( Rasool Ahmad )

Preprint submitted to Acta Materialia

November 6, 2019

fracture. For crystalline metals, yielding and strain hardening are usually dictated by the underlying dislocation slip and twinning systems. A necessary condition for generalized plastic flow is the von Mises criterion for strain compatibility, which requires five independent slip systems to operate for continuous plastic flow. The von Mises criterion is easily satisfied in high symmetry cubic metals, but less so in other structures. In the family of HCP metals, plastic slip in the hci axis direction is essential for strain compatibility but generally much harder relative to slip in the hai direction. Plastic strain in the hci axis is primarily provided by dislocations of hc + ai Burgers vector on pyramidal planes, since slip via hci dislocations is even harder (if not essentially impossible [1]). hc + ai dislocations on the pyramidal planes are thus the only active slip systems able to satisfy the von Mises criterion. However, hc + ai slip is generally difficult, requires excessively high stresses to operate at size scales relevant to engineering applications [2, 3, 4, 5], and is the ductility-limiting factor across the entire HCP family. Understanding and controlling the hc + ai slip behaviors are thus particularly important for developing wrought products of HCP metals including Ti, Zr and Mg. For HCP Mg in particular, poor ductility and low fracture toughness at room temperature limit its current light weight structural applications in the automobile and aerospace industries [6, 7]. The low ductility has recently been attributed to a thermally-activated transition of the glissile hc + ai dislocations on either pyramidal I or II planes into a sessile basal-dissociated structure [2, 3, 4, 5, 8, 9]. This leaves pure Mg effectively devoid of any glissile dislocation systems to accommodate hci axis plastic strain. Achieving good room temperature ductility in Mg thus requires some mechanism(s) to mitigate the detrimental hc + ai pyramidal-to-basal (PB) transition. Mg alloys with dilute rare earth (RE) solutes show enhanced room temperature ductility and a concomitant high activity of hc + ai slip [10, 11, 12, 13, 14, 15]. Several possible hypotheses have been put forth to explain these observations. For example, Sandlöbes et al. [16] and Agnew et al. [17] proposed that basal I1 stacking faults can act as heterogeneous sources of hc + ai dislocations, and that there exists a correlation among solute, basal I1 stacking fault energy, hc + ai slip activity and final ductility. However, this mechanism requires a prior high density of I1 faults, very high stresses to operate, and is not consistent with the trend of solute effects as revealed by quantitative and comparative DFT calculations [18]. Ahmad et al. [19] analysed, through molecular dynamics (MD) simulations, the possibility that Y solutes stabilize the hc + ai dislocation on the pyramidal glide plane against the PB transition but found essentially no effect on the detrimental transition rate. These mechanisms are thus unlikely the primary mechanism responsible for the increased ductility in Mg-RE alloys. Recently, a mechanism and theory, with accompanying experimental evidence, was proposed wherein certain solutes can accelerate the hc + ai dislocation cross-slip and double cross-slip on the pyramidal II planes so as to enable the generation of new hc + ai dislocation loops at rates fast enough to defeat the immobilization of hc + ai dislocation due to the PB transition [15, 20]. Cross-slip on the pyramidal II planes of the HCP lattice is subtle, since no two pyramidal II planes share a common hc + ai Burgers vector. Cross-slip thus requires that the hc + ai screw dislocation on a pyramidal II plane must first cross-slip onto an intersecting pyramidal I plane with a different core structure and near core energy. The energy barrier for cross-slip is then very sensitive to the energy difference per unit length between the screw hc + ai dislocations on the pyramidal II and I planes [21]. In pure Mg, the cross-slip energy barrier is high and the cross-slip rate is low 3

(relative to those of the PB transition). In Mg alloys, however, even dilute solute additions can shift the pyramidal II-I energy difference sufficiently to change the cross-slip barrier significantly. Based on this mechanism, a theory combining solute/stacking-fault interactions computed using first-principles DFT and statistical considerations for random solid solutions was developed and successfully applied to rationalize the ductility and observed texture trends in dilute solid solution Mg alloys across a wide range of solute types and concentrations. This ductility-enhancing mechanism and theory provide a computational path forward to design of new ductile Mg alloys [20]. However, the proposed mechanism operates at low solute concentrations where the hc + ai screw dislocation is energetically more favorable on the pyramidal II plane than on the pyramidal I plane. For certain solutes, the theory predicts that, above some solute concentrations, the hc + ai dislocation will become more stable on the pyramidal I plane than on the pyramidal II plane. This is especially relevant for the RE solutes, which have both high solubility in Mg and the strongest effects on reducing the near-core energy of the dislocation on the pyramidal I plane relative to that on the pyramidal II plane. For example, experiments [14] and theory [15, 20] indicate that at around 0.6 at.%Y, the hc + ai screw dislocation becomes energetically equally favorable on the pyramidal I and II planes. At higher Y concentrations, the pyramidal I plane becomes increasingly more favorable and the dominant hc + ai slip can switch to the pyramidal I planes. The overall ductility of the alloy is then controlled by the competition between hc + ai cross-slip and the PB transition on the pyramidal I planes. Since the energy barrier for the pyramidal I PB transition is even lower than that of pyramidal II [8], ductility could become poor again in Mg alloyed with high concentrations of seemingly ductilizing solutes. This has practical implications for the design of new ductile Mg alloys. Increasing the solute concentration increases the alloy strength, and so would seem to be an avenue for creating ductile and strong alloys. If increasing solute concentration in fact leads to lower ductility, such a feature is essential to be identified quantitatively and qualitatively. Recognizing that there are two regimes for hc + ai slip, pyramidal I favorable and pyramidal II favorable, the cross-slip transition paths and solute effects can be substantially different. When pyramidal I slip is favorable, hc + ai cross-slip may not necessarily require cross-slip onto the higher-energy pyramidal II planes because different pyramidal I planes do share a common hc + ai Burgers vector. In the pyramidal I plane favorable regime, there are thus four possible hc + ai double cross-slip paths, as shown in Fig. 1. The first possible path (Fig. 1(a)) requires cross-slip onto a higher-energy pyramidal II plane, glide by some distance on the pyramidal II plane, and then cross-slip back to a pyramidal I plane parallel to the original plane. A second plausible cross-slip path (Fig. 1(b)) is similar to the first except that the dislocation cross-slips onto the other accessible pyramidal I plane after some glide on the pyramidal II plane. The third possibility is that the dislocation gliding on a pyramidal I plane cross-slips directly onto the next adjacent parallel pyramidal I plane shifted by only the pyramidal I interplanar distance, as shown in Fig. 1(c). In this cross-slip mechanism, the dislocation thus has to form a pair of atomic-scale jogs on the pyramidal II plane connecting the two pyramidal I planes. Unlike the case in Fig 1(a), the dislocation does not fully dissociate onto the pyramidal II plane during the cross-slip process. Therefore, the energy barriers for the cases in Fig. 1(a) and 1(c) are different. The fourth possibility (Fig. 1 (d)) involves hc + ai cross-slipping directly from the original pyramidal I plane to the other accessible pyramidal I plane by forming a jog/kink pair. Since the cross-slip paths are different, solute 4

effects could change dramatically along the different paths and in the two regimes. Quantitative prediction thus first requires detailed calculations of these possible transition paths and associated energy barriers, followed by analysis of the rates of the cross-slip and PB transition in the regime where pyramidal I slip is dominant (i.e., in Mg alloys with some sufficiently high concentrations of ‘favorable’ solutes). In this work, we examine these possible cross-slip paths using nudge elastic band (NEB) calculations with a new MEAM potential for pure Mg adjusted so as to make the pyramidal I hc + ai screw dislocation more favorable (lower energy) than the pyramidal II hc + ai . We show that the first path (Fig. 1(a)) has the lowest cross-slip barrier, so that the cross-slip process is simply the reverse of the process in the pyramidal II favorable regime. We then extend the previous mechanistic theory for ductility to encompass this new regime where pyramidal I is favorable. The model predicts that at some sufficiently high RE solute concentrations, the pyramidal I hc + ai becomes too energetically favorable, relative to pyramidal II, such that pyramidal I cross-slip becomes too slow to circumvent the associated pyramidal I PB transition, leading to a return to low ductility for such alloys. We predict the upper limit to RE concentrations for achieving good ductility to be roughly 1.5 at.% across the range of RE solutes. The limited available experiments in this higher range of RE concentrations are evaluated in light of our predictions.

a

b

Pyr I Pyr II

c

d

Pyr I

Figure 1: Schematic illustration of possible cross-slip or double cross-slip paths for hc + ai screw dislocation between two pyramidal I planes. The crystallographic relation among the different pyramidal I and pyramidal II planes accessible through cross-slip process, i.e. containing a common hc + ai direction, is also shown. The color code is: pyramidal I (red), pyramidal II (blue). Dashed lines in (a), (b), (c) and (d) represent the configuration of hc + ai dislocation lying on the primary pyramidal I plane just before cross-slip, and solid lines show the dislocation lines during cross-slip process. (a) and (b), respectively, show the possibility of hc + ai dislocation cross-slipping from one pyramidal I plane to the parallel and other accessible pyramidal I planes after gliding on high energy pyramidal II plane. (c) and (d) depict the accomplishment of the cross-slip from one pyramidal I plane to the parallel and other accessible pyramidal I planes, respectively, without going through the pyramidal II plane.

5

The remainder of the paper is arranged as follows. Section 2 presents the details of the simulation along with the new MEAM potential for pure Mg which favors pyramidal I hc + ai screw dislocation over that on the pyramidal II plane. Energetics of different cross-slip processes are described and analysed in Section 3, and the implications are presented in Section 4. Section 5.1 briefly outlines the mechanistic theory for ductility prediction. The results for Mg-RE alloys over the entire range of compositions are then presented in Section 5.2, followed by discussions on relevant experimental results and recapitulation of the findings of the present work in Section 6. 2. Pyramidal I-favorable interatomic potential for Mg and simulation details The lack of sufficiently accurate interatomic potentials for Mg alloys of interest renders it impossible to directly simulate the cross-slip process of hc + ai dislocation in such alloys at any composition [22, 23, 24]. For instance, the MEAM potential for Mg-Y developed by Kim et al. [23] predicts a solute misfit volume much smaller than DFT values, and also yields very small solute interaction energy with basal and pyramidal I stacking faults [22]. Moreover, relevant to the present study, the Kim potential predicts, quite contrary to the DFT result, that Y solute atoms have smaller effects on pyramidal I stacking fault energy than on pyramidal II stacking fault energy. The EAM interatomic potential for Mg-Y developed by Pei et al. [24] predicts, according to our calculations, the FCC phase of Mg to be more stable than the HCP phase (EAM potentials also tend to show unstable hc + ai core structures on the pyramidal II plane contrary to DFT results [25]). The MEAM interatomic potential for Mg-Y developed in Ahmad et al. [22] improves the prediction of misfit volume and interaction energies with various stacking faults and pyramidal II hc + ai edge dislocation. However, it fails to capture the relative strength of solute interactions with pyramidal II and I stacking faults with the needed quantitative accuracy. Since this is the crucial feature for pyramidal hc + ai dislocation cross-slip, this potential is also unsuitable for the present study. Therefore, we resort to re-parameterization of the existing MEAM potential for Mg [26] that describes plasticity related properties in good agreement with DFT/experiments. By tweaking the parameters of the original MEAM potential of Wu et al. [26], we can obtain better agreement with DFT in stable stacking fault energies of the pyramidal planes while keeping every other property essentially unchanged. In this way, we have generated two new nearly identical potentials (see Table 1) that differ only in their predictions of the relative energy/stability of the pyramidal I and pyramidal II hc + ai screw dislocations. This energy difference is subtly coupled to the pyramidal I and II stable stacking fault energies. For the mechanism of interest in this work, we are interested in the potential for which the pyramidal I screw is favorable relative to pyramidal II screw. In a real alloy, this condition is created by the average effects of the solutes on the two pyramidal stacking fault energies [18]. Thus the second MEAM potential in Table 1 is the most suitable one for this I−II work. This new MEAM potential yields an energy difference ∆EMg ≈ −28 meV/nm, as compared I−II to the value ∆EMg ≈ +27 meV/nm given by the original MEAM potential. The parameters for this potential are presented in Table 2; the role of each parameter is discussed in Refs. [27, 28]. The potential files in LAMMPS format are provided as supplementary data. The core structures of the pyramidal I and II hc + ai screw dislocations as predicted by the new MEAM potential used here are shown in Fig. 2, and are essentially the same as those obtained using the original MEAM 6

potential as well as the other new potential (presented in Appendix A) introduced here, both of which have the pyramidal II screw more stable than the pyramidal I screw. Table 1: Properties of three MEAM interatomic potentials for Mg including lattice constants (a and c/a), elastic constants (C11 , C12 , C13 , C33 , C44 ), various stable stacking fault energies(SFE) and dislocation energy difference between I−II pyramidal I and pyramidal II dissociated hc + ai screw dislocations (∆EMg ). The first potential is from Ref. [26]; second and third MEAM potentials are developed by adjusting the original MEAM potential parameters to obtain better agreement with DFT stacking fault energies of pyramidal planes. In this work, we use the second potential for which the pyramidal I screw dislocation has the lowest energy. Properties

MEAM (Ref. [26])

MEAM (this work)

MEAM (pyramidal II stable)

DFT[1]

a (Å) c/a Ec (eV/atom) C11 (GPa) C12 (GPa) C13 (GPa) C33 (GPa) C44 (GPa) SFE (basal I2 ) (mJ m−2 ) SFE (pyramidal I) (mJ m−2 ) SFE (pyramidal II)(mJ m−2 ) I−II ∆EMg (meV/nm)

3.187 1.623 -1.51 64.3 25.5 20.3 70.9 18.0 23 169 200 27

3.187 1.622 -1.51 65.4 24.8 21.9 69.4 19.5 21 161 188 -28

3.185 1.622 -1.51 64.2 26.7 22.0 70.1 17.8 20 154 169 54

3.189 1.627 -1.51 61 26 20 63 18 34 161 165 —

Table 2: The parameters of MEAM potential for pure Mg used in this study. Ec (eV)

re (Å )

α

A

β(0)

β(1)

β(2)

β(3)

t(1)

t(2)

t(3)

Cmin

Cmax

rc (Å)

∆r (Å)

1.52

3.18

5.61

0.79

2.35

1.46

2.30

1.15

6.82

1.50

-7.09

0.36

2.38

7.89

6.60

Introducing solutes into pure Mg has two effects on the cross-slip barrier. The first effect, menI−II tioned above, is a change in the average dislocation energy difference ∆EMg which scales linearly with solute concentrations [18]. The interatomic potential used here is thus equivalent to an alloy I−II with solutes at concentrations that would give an average effect of ∆EMg ≈ −28 meV/nm. One such alloy would be, according to the DFT predictions by Wu et al. [15], Mg-1.3at%Y. The second effect of solutes is associated with fluctuations in the local solute concentration that lead to an additional change in the local cross-slip barrier. This is not captured by the present interatomic potential, but in fact need not be. The path and the intrinsic barrier of the underlying cross-slip process can be identified only by the average solute effects. The effects of fluctuations in the random alloy are added subsequently as discussed in Section 5.1. We also note that the applicability of an effective single atom potential to represent a random alloy has already been demonstrated and applied to study dislocation cross-slip in random FCC solid solution alloys [31]. Typical NEB calculations to obtain the minimum energy path (MEP) and associated energy barrier require the configurations of the initial and final states. Here, the initial state is a fullyrelaxed hc + ai screw dislocation dissociated on the initial pyramidal I plane. The final state 7

a

c

b

I pyr

pyr II

pyr

I

Figure 2: Core structures of hc + ai screw dislocations which are accessible through cross-slip, i.e. having common Burgers vector, as determined from the MEAM potential presented in Table 2. Dislocation cores shown in (a) and (c) are of equal energy and dissociated on the two distinct intersecting pyramidal I planes; and dislocation core in (b) is dissociated on the only intersecting pyramidal II plane. Blue atoms denote atoms with HCP lattice environment and yellow atoms represent non-HCP environment as identified from common-neighbor analysis [29, 30]. The magenta line in all three figures indicates the pyramidal I or pyramidal II slip plane on which dislocation is dissociated. The employed interatomic potential is designed to yield pyramidal I hc + ai screw dislocation more stable than pyramidal I−II II with energy difference ∆EMg = −28 meV/nm.

is a fully relaxed hc + ai screw dislocation dissociated on the final pyramidal I plane. For the processes shown in Fig. 1(a) and (b), the final plane could be at some arbitrary glide distance away from the initial plane. NEB calculations are performed at temperature T = 0 K, and hence the MEP and barriers will also involve the (high) Peierls barrier for the glide of the pyramidal II screw dislocation at T = 0 K. Such barriers are not small, but are not relevant for the real problem of interest. Under realistic conditions, glide of the pyramidal II dislocation occurs over a Peierls barrier that is reduced by whatever applied resolved shear stress needed to enable plastic flow at experimental strain rates and temperatures. The reduced glide barriers are much smaller than the cross-slip barriers. Therefore, in our NEB computations at T = 0 K, the final pyramidal I plane is chosen such that the glide distance on the pyramidal II plane is minimal. In NEB calculations for the case of Fig. 1(a), the glide of pyramidal II hc + ai dislocation is restricted to the adjacent pyramidal I planes. Furthermore, the glide of the pyramidal II dislocation in Fig. 1(b) is eliminated entirely, and cross-slip of the hc + ai dislocation from the initial pyramidal I to pyramidal II, and then to the final pyramidal I plane, is accomplished at the same position. Finally, the cross-slip path shown in Fig. 1(d) does not involve glide on the pyramidal II plane. Since the cross-slip paths in Fig. 1(b), after elimination of pyramidal II glide, and Fig. 1(d) do not entail dislocation glide and occur at the same position, we study these two cross-slip cases in a simulation supercell with periodic boundary conditions in the dislocation line direction and fixed boundary conditions in the other two directions. We first create the pyramidal II hc + ai screw dislocation in an effectively infinite elastic medium by applying the anisotropic displacement field of the corresponding Volterra dislocation, followed by minimizing the system energy using the conjugate gradient algorithm while keeping the boundary atoms (within twice the potential cutoff radius from the outer boundary) fixed. We then perform MD simulations at 100 K and the hc + ai screw dislocation cross-slips to the lower energy pyramidal I plane as given by the reparameterized MEAM potential. The hc + ai dislocation on the other accessible pyramidal I plane is obtained by running MD simulations at 100 K and with a non-glide shear stress of ≈200 MPa 8

applied in the direction that promotes dissociation of the dislocation on the other pyramidal I plane orientation [21]. The pyramidal I hc + ai dislocation cores thus obtained are further relaxed using the same initial boundary conditions as that of the pyramidal II dislocation. Thus we obtain three hc + ai screw dislocations which are shown in Fig. 2: one on the high-energy pyramidal II plane and two of equal energy on the low-energy pyramidal I planes. The three screw dislocations have the same Burgers vector, share the same boundary conditions and are thus accessible to each other by cross-slip processes. The minimum energy paths and associated energy barriers of the cross-slip processes are next probed using the NEB method as implemented in LAMMPS [32, 33, 34, 35, 36]. The size of the simulation box for these cases is l x × ly × lz ≈ 40 × 40 × 21 nm3 , where the x axis is along the ¯ direction, z axis along the dislocation line direction [1123], ¯ and y axis along the normal of [1010] the pyramidal II plane. For the case in Fig. 1(b), with pyramidal II glide eliminated, we use 128 replicas in which two pyramidal I hc + ai screw dislocations are at the initial and final replicas with the intermediate replicas initialized by linearly interpolating the atomic positions of the two end replicas. In the case of Fig. 1(d), we use 64 replicas that are created by splicing the initial and final configurations of different lengths such that the replicas gradually transform from the initial configuration to the final configurations. We run the NEB minimization in the two cases until the maximum force on any atom across all replicas falls below 2 meV/Å. On the other hand, the cross-slip process shown in Fig. 1(a) requires gliding of the dislocation along the pyramidal II plane. To eliminate the image forces due to the fixed boundary conditions, we investigate this cross-slip path in a simulation box periodic in both the dislocation line and glide directions, i.e. along the x − z pyramidal II glide plane, and with traction free boundary conditions in the direction (y) perpendicular to the pyramidal II plane. The coordinate system used in this case is the same as that described above, and the size of the simulation box is l x ×ly ×lz ≈ 50×32×21 nm3 . We first create two configurations each containing a hc + ai dislocation on the pyramidal II plane √ shifted by the minimum glide distance of 3a, a being the lattice constant, along the x direction. The initial and final configurations, each with a hc + ai screw dislocation on the pyramidal I plane, are obtained by performing MD simulations of the two pyramidal II configurations at 100 K. The two pyramidal I hc + ai screw dislocations obtained in this way are dissociated on the same type √ of parallel adjacent pyramidal I planes, and are connected by glide of 3a on the pyramidal II plane. For the NEB calculations here, we use a total of 64 replicas with intermediate replicas initialized by linearly interpolating the atomic positions of the initial and final replicas. This setup of the NEB calculation also enables us to gauge the feasibility of the cross-slip path shown in the Fig. 1(c), since the minimum energy path will select whether to first cross-slip onto the pyramidal II plane or directly cross-slip to the adjacent parallel pyramidal I plane. All atomic structures are visualized and analyzed using the Open Visualization Tool (OVITO) [37]. 3. Cross-slip transition paths and barriers The results of the NEB calculations are presented as the total excess system energy versus the reaction coordinate (RC) along the MEP path. The reaction coordinate is a measure of the cumulative distance of the replicas along the MEP starting from 0 at replica k=1. For replica 9

annihilation

spreading

pyrII - pyrI cross-slip nucleation

annihilation

spreading

nucleation

annihilation

nucleation

spreading

0.8

annihilation

spreading

annihilation

1.0

nucleation

pyrI-pyrII cross-slip transfer of pyrII dislocation to the upper adjacent pyrII plane spreading

1.2

nucleation

Energy difference ∆E (eV)

P k > 1, RC(k) = kj=2 |r j − r j−1 | where r j is the 3N dimensional vector of the atomic positions in replica j and N is the total number of atoms in each replica.

0.6 0.4

RC: 44

RC: 86

RC: 146

RC: 187 RC: 231

RC: 0

0.2 0.0 0

20

40

60

80 100 120 140 160 Reaction coordinate (˚ A)

180

200

220

240

Figure 3: Energy difference versus reaction coordinate, as determined in NEB calculation of the minimum energy path of the cross-slip process between two pyramidal I planes accomplished through the route where the pyramidal I dislocation cross-slips to other accessible pyramidal I plane via a high energy pyramidal II plane. The cross-slip process is accomplished in three stages as mentioned at the top of the figure. Three steps involved in each transformation stages are shown: nucleation is the formation of the incipient jog/kink pairs; spreading is the lateral motion of the kink/jog pair along dislocation length and energy change in this step is dislocation length dependent; annihilation is the mutual cancellation of the kink/jog pair forced by the periodic boundary along dislocation length. Atomistic pictures of intermediate metastable dislocation core structures encountered during the cross-slip process −with their reaction coordinates − are shown by drawing only non-hcp near core atoms identified using common neighbor analysis. The brown cross-hairs plotted on the dislocation cores are for the purpose of indicating the motion/glide/transformation of dislocation cores against a static background. A schematic diagram of the cross-slip process, i.e. Fig. 1(b), is depicted in the upper right corner.

Fig. 3 shows the MEP and corresponding energies of the intermediate dislocation structures for the case shown in Fig. 1(b) with glide on the pyramidal II plane eliminated. In this case the hc + ai screw dislocation cross-slips from one pyramidal I plane to the other accessible pyramidal I plane via the pyramidal II plane. The cross-slip occurs in many stages and involves several intermediate metastable dislocation structures. Fig. 3 also identifies the nucleation, spreading along dislocation line length and annihilation of kink/jog pair which accompany every stage. The intrinsic barrier is involved in these nucleation and annihilation steps. Energy changes in the nucleation and annihilation steps are dislocation length independent, while that in spreading step may depend on the dislocation length (controlled by the energy difference of the two connected metastable core structures). Furthermore, the enthalpy change in spreading part can be reduced by applying a resolved shear stress which does work in the bowing out of the dislocation. In the first stage, the pyramidal I dislocation transforms onto the pyramidal II plane (at RC ≈ 44). This 10

Energy difference ∆E (eV)

process is essentially the same as that of the cross-slip of the low energy pyramidal II hc + ai dislocation to the high energy pyramidal I hc + ai dislocation in pure Mg [21]. The energy barrier in this stage depends on the dislocation length and is ≈ 0.9 eV for the dislocation length used here (21 nm). This dislocation core obtained at RC 44 could glide on pyramidal II plane under resolved shear stresses which is not considered in the present calculations. The next stage involves the shuffling of atoms in order to migrate the pyramidal II hc + ai dislocation from one pyramidal II plane to the adjacent pyramidal II plane just above ( RC 44 to 187). The energy barrier (≈ 0.55 eV) for this stage depends weakly on the dislocation length as it goes through slightly higherenergy metastable dislocation structures (at RC ≈ 86, 146). In the final stage, the pyramidal II hc + ai dislocation cross-slips back to the pyramidal I plane with an associated length-independent, intrinsic energy barrier of ≈ 0.3 eV required to form the cross-slip jog. For cross-slip of the current dislocation of length 21 nm, the total energy barrier is ≈ 1.15 eV.

1.2

RC: 32

RC: 55

RC: 84

1.0 nucleation

0.8

spreading

annihilation RC: 96

0.6 RC: 13

0.4

RC: 104 RC: 0

0.2 0.0 0

10

20

30

40 50 60 70 Reaction coordinate (˚ A)

80

90

100

110

Figure 4: Energy difference versus reaction coordinate, as determined in NEB calculation of the minimum energy path of the cross-slip process between two pyramidal I planes accomplished through the route where the pyramidal I dislocation cross-slips directly to other accessible pyramidal I plane by nucleating a pair of jogs and without ever residing on the high energy pyramidal II plane. Three steps of the cross-slip process are shown: nucleation is the formation of the incipient jog/kink pairs; spreading is the lateral motion of the kink/jog pair along dislocation length and energy change in this step is dislocation length dependent; annihilation is the mutual cancellation of the kink/jog pair forced by the periodic boundary along dislocation length. Atomistic pictures of intermediate metastable dislocation core structures encountered during the cross-slip process −with their reaction coordinates − are shown by drawing only non-hcp near core atoms identified using common neighbor analysis. A schematic diagram of the cross-slip process, i.e. Fig. 1(d), is depicted in the upper right corner.

Fig. 4 shows the second possible MEP and associated energetics for cross-slip between two different pyramidal I planes as shown Fig. 1(d). In this MEP, the first step (RC 1-20) is the nucle11

right partial - cross-slip left partial - cross-slip + glide

right partial - glide

right partial cross-slip+glide

right partial - glide left partial-cross-slip+glide

0.25

RC: 0

RC: 107

annihilation

RC: 25

spreading

0.5

RC: 130 nucleation

0.75

spreading

RC: 66

annihilation

1.0

spreading

1.25

annihilation nucleation

1.5

nucleation

Energy difference ∆E (eV)

ation of a pair of dislocation jogs which transforms a small dislocation segment from the primary pyramidal I plane to the cross-slip pyramidal I plane. During the second step (RC 20-84), the crossslipped jogs expand laterally along the dislocation line and gradually transforming the dislocation in the process. In the third and last step (RC 84-105), due to the periodic boundary conditions, the two jogs come very close and finally annihilate each other, transforming the whole dislocation to the cross-slip pyramidal I plane. The energy barrier of this MEP is the intrinsic energy of formation of the initial cross-slip jogs, ≈ 1.0 eV, and hence is independent of the dislocation length.

RC: 172

0.0 0

25

50

75 100 125 Reaction coordinate (˚ A)

150

175

Figure 5: Energy difference versus reaction coordinate, as determined in NEB calculation of the minimum energy path of the cross-slip process between two pyramidal I planes accomplished through the route where the pyramidal I dislocation first cross-slips to a high energy pyramidal II plane, and goes back to a parallel pyramidal I plane after gliding some distance on a pyramidal II plane. The cross-slip process is accomplished in four stages as mentioned at the top of the figure. The first stage (cross-slip of right partial, and cross-slip and glide of left partial) is completed by multiple nucleation events which are not well separated from each other. Three steps involved in other three transformation stages are shown: nucleation is the formation of the incipient jog/kink pairs; spreading is the lateral motion of the kink/jog pair along dislocation length and energy change in this step is dislocation length dependent; annihilation is the mutual cancellation of the kink/jog pair forced by the periodic boundary along dislocation length. Atomistic pictures of intermediate metastable dislocation core structures encountered during the cross-slip process −with their reaction coordinates − are shown by drawing only non-hcp near core atoms identified using common neighbor analysis. The brown cross-hairs plotted on the dislocation cores are for the purpose of indicating the motion/glide/transformation of dislocation cores against a static background. A schematic diagram of the cross-slip process, i.e. Fig. 1(a), is depicted in the upper right corner.

Fig. 5 shows the variation of the energy and the transformation of the dislocation structures for the cross-slip path in Fig. 1(a), where the initial pyramidal I dislocation cross-slips onto, and glides on, a pyramidal II plane and then cross-slips back onto a pyramidal I plane parallel to the 12

original pyramidal I plane. The whole process is accomplished in multiple stages as indicated in Fig. 5. We observe that during the first stage (RC from 0 to 66), the right partial of the hc + ai screw dislocation cross-slips to the pyramidal II plane, and the left partial dislocation cross-slips to and glides on pyramidal II plane. This stage involves multiple nucleation events which are not sharply demarcated from each other, thus we cannot determine the intrinsic barrier for this forward process. This nucleation process is length-dependent with an intrinsic barrier of at least ≈ 0.23 eV which is the energy change during the RC 63-66. In the second stage of the process (RC from 66 to ∼ 107), the right partial glides on the pyramidal II plane. Because the dislocation goes from a low-energy configuration to high-energy configurations, the energy barrier in this stage is dislocation-length dependent with an intrinsic barrier of ≈ 0.23 eV. The third stage (RC from 107 to 130) involves the transfer and a little glide of right partial on the final pyramidal I plane. The energy barrier in this stage is length dependent with a small intrinsic barrier of ≈ 0.06 eV. In the last stage (RC from 130 to 172), the right partial further glides on the pyramidal I plane, and the left partial glides on the pyramidal II plane followed by transfer onto the pyramidal I plane. The intrinsic barrier of this last stage is ≈ 0.06 eV. The total energy barrier for the overall process is ≈1.49 eV. 4. Implications for double pyramidal I cross-slip As discussed in the Introduction, the ductility-enhancement mechanism proposed by Wu et al. [15] envisions that high ductility can be achieved by sustained plastic strain in the hci direction through the generation of hc + ai dislocation loops at a rate fast enough to counter the immobilization of hc + ai dislocations due to the PB transition. The generation of hc + ai dislocation loops is accomplished by the double cross-slip of the screw segments of each dislocation loop. Here, we analyse the cross-slip barriers when the solute concentration is high enough such that for the hc + ai screw dislocation, the pyramidal I plane is energetically more favorable than the pyramidal II plane. In this case, cross-slip occurs by one of the processes examined using NEB in the previous section. The cross-slip process shown in Fig. 3 involves cross-slip onto the higher-energy pyramidal II plane followed by cross-slip back onto another available pyramidal I plane. The barrier for this process includes an intrinsic barrier of at least ≈ 0.45 eV, associated with the initial cross-slip to the pyramidal II plane and the migration of the pyramidal II dislocation one layer above the original plane, plus an additional barrier arising from the pyramidal II/I energy difference per unit dislocation length. The latter length-dependent part of the barrier can be reduced by the application of a resolved shear stress on the glide plane but the intrinsic barrier remains. Therefore, this double cross-slip path is not effective in alleviating the deleterious effects of the PB transition that has a barrier of only 0.3 eV [8]. Two distinct pyramidal I planes intersect each pyramidal II plane along a common hc + ai direction, as shown in Fig. 1. Therefore, starting from a pyramidal II hc + ai dislocation, there are two different pyramidal I planes available for the cross-slip process. It can be inferred from Fig. 3 that the two pyramidal I planes associated with a pyramidal II plane are not equivalent with respect to the cross-slip of the hc + ai screw dislocation. For instance, beginning from the pyramidal II hc + ai dislocation at RC 44, the energy barrier to the pyramidal I dislocation at RC 0 (≈ 0.3 eV) 13

is significantly lower than that on the other pyramidal I plane at 230 (≈ 0.5 eV). In the same way, for the pyramidal II dislocation at RC 187, the energy barrier to the pyramidal I dislocation at RC 230 is smaller (≈ 0.3 eV) than that on the pyramidal I plane at RC 0 (≈ 0.5 eV). This should be a general conclusion regardless of the specific interatomic potential because of the fact that pyramidal II lacks mirror symmetry across the slip plane. The direct cross-slip between two pyramidal I planes without ever residing on a higher-energy pyramidal II plane (Fig. 4) has an energy barrier of ≈ 1 eV, which is more than three times that of the corresponding PB transition barrier on pyramidal I planes. Therefore, this path is not effective in circumventing the detrimental PB transition either. The cross-slip process shown in Fig. 5 involves cross-slip onto a pyramidal II plane followed by cross-slip back onto a parallel pyramidal I plane. The processes of cross-slip and glide on pyramidal II are accomplished simultaneously for each partial successively. This particular energy path includes the the rather high Peierls barrier of the screw hc + ai dislocation on the pyramidal II plane at 0 K temperature. A separate NEB calculation (not shown here) computes a Peierls barrier of ≈ 1 eV for the screw hc + ai dislocation of ≈ 21 nm gliding on the pyramidal II plane. At nonzero temperatures under applied glide shear stresses, the glide of the screw dislocation is expected to have much lower energy barriers. The independent cross-slip of the individual partials is also caused by the nearly identical Burgers vectors and cores of the partials dissociated on the pyramidal I and pyramidal II planes [38]. Thus, the barrier for cross-slip onto the pyramidal II plane is the controlling barrier for double cross-slip with an intrinsic barrier of ≈ 0.3 eV. This process is basically the same as that occurring when the pyramidal II plane is energetically favorable, and so is suitable for mitigating the detrimental effects of the PB transition. To be effective, however, this process must be assisted by a net resolved shear stress acting on the pyramidal II plane to remove the length-dependent portion of the energy barrier. Since the initial pyramidal I dislocation does not go directly to the adjacent parallel pyramidal I plane, this computation further indicates the implausibility of the cross-slip path shown schematically in Fig. 1(c). On the basis of the above discussion we conclude that, in the regime where solutes make the pyramidal I hc + ai dislocations energetically more favorable, cross-slip to generate new hc + ai dislocation loops occurs by the process shown in Fig. 6. Specifically, the pyramidal I dislocation first cross-slips onto a pyramidal II plane following the path of RC from 0 to 44 (pyramidal I to pyramidal II hc + ai cross-slip) of Fig. 3. Subsequently, after an arbitrary amount of thermally activated glide along the pyramidal II plane, hc + ai screw dislocation cross-slips back onto the similarly orientated pyramidal I plane, which is essentially the reverse of the forward cross-slip process from pyramidal I to pyramidal II plane.

14

thermally activated glide on pyr II plane

RC: 44

RC: 0

10

20

annihilation

nucleation RC: 44

0

spreading

cross-slip from pyr II to pyr I

annihilation

spreading

nucleation

Energy difference ∆E (eV)

cross-slip from pyr I to pyr II

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

RC: 0

30

43.96 43.96 Reaction coordinate (˚ A)

30

20

10

0

Figure 6: The energetics and minimum energy path of the most favorable cross-slip path between pyramidal I planes, as inferred from the NEB results shown in Figs. 3, 4 and 5. The pyramidal I hc + ai dislocation cross-slips first to the pyramidal II plane and after gliding some distance on a high energy pyramidal II plane, again cross-slips back to a parallel pyramidal I plane.Three steps involved in the two cross-slip stages are shown: nucleation is the formation of the incipient jog/kink pairs; spreading is the lateral motion of the kink/jog pair along dislocation length and energy change in this step is dislocation length dependent; annihilation is the mutual cancellation of the kink/jog pair forced by the periodic boundary along dislocation length. Atomistic pictures of intermediate metastable dislocation core structures encountered during the cross-slip process −with their reaction coordinates − are shown by drawing only non-hcp near core atoms identified using common neighbor analysis. A schematic diagram of the cross-slip process, i.e. Fig. 1(a), is depicted in the upper right corner.

5. Prediction of ductility via enhanced cross-slip for Mg-RE binary alloys Having established the double cross-slip path, we can now apply the mechanistic model for enhanced ductility to predict the range of RE solute concentrations over which Mg alloys are ductile, including both the regimes where pyramidal II is favorable relative to pyramidal I [15] and pyramidal I is favorable relative to pyramidal II. 5.1. Mechanistic model The mechanistic model follows the works of Wu et al. [15] and Ahmad et al. [20], which associates a “ductility index” χ with alloy compositions. Consider a hc + ai dislocation loop of length L × L residing on a pyramidal II or I plane, as shown in Fig. 7 for the pyramidal II case. Enhanced ductility requires that the total cross-slip rate RXS occurring over the screw part of length L must be greater than the total rate of PB transition RPB happening over edge or mixed part of length L. Using the Arrhenius law for thermally-activated processes, this condition at temperature

15

T can be written as L lXS

! ! ∆GXS L ∆EPB exp −  exp − , kT lPB kT

(1)

where lXS and lPB are the critical lengths of the cross-slip and the PB transition processes, respectively; ∆GXS and ∆EPB are the corresponding energy barriers of the cross-slip and PB transition; and k is Boltzmann’s constant. We introduce the ductility index χ such that the cross-slip rate is 10χ times the PB transition rate, i.e. RXS = 10χ RPB . Using Eq. 1, χ is obtained as " ! # lPB ∆EPB − ∆GXS 1 χ= ln + . (2) ln 10 lXS kT The value of χ reflects the potency of cross-slip in circumventing the PB transition. We consider χ > 1 sufficient to achieve high ductility, and χ < 0 to be indicative of poor ductility. For 1 < χ < 0, predictions of the model are less definitive, and ductility can be particularly influenced by other factors, such as the precise thermo-mechanical processing, texture and grain sizes [39, 15]. In Eq. 2, χ can be increased either by increasing the energy barrier of the PB transition ∆EPB or by decreasing the cross-slip barrier ∆GXS . The PB transition for the pyramidal I mixed dislocation is driven by an energy difference of ∼ 0.3 eV/Å between the initial and final dislocation structures [5, 8] and is so energetically favorable that solute elements are not likely to impede the PB transition [19]. The cross-slip process is controlled by the energy difference between pyramidal I and pyramidal II hc + ai screw dislocations [21, 40] and so can be altered by the addition of solute atoms. In solid solution alloys, the random spatial distribution of solutes gives rise to an average barrier change h∆GXS i plus local fluctuations in the barrier characterized by a standard deviation σ [lXS ] evaluated at the critical cross-slip nucleation length lXS . Including the average and fluctuation effects of solute atoms on the cross-slip barrier, χ becomes " ! # lPB ∆EPB − (h∆GXS i − σ [lXS ]) 1 ln + . (3) χ= ln 10 lXS kT Below, we briefly outline the model of Wu et al. [15] that is used to determine the quantities above; a more detailed description is provided in Refs. [15, 20]. When the solute concentration is such that the pyramidal II plane is more favorable, the rate of cross-slip from the pyramidal II plane to the pyramidal I plane determines the ductility index χ since the subsequent cross-slip from pyramidal I to pyramidal II is faster. Similarly, if the solute concentration is sufficient to make the pyramidal I more favorable, the barrier of cross-slip from pyramidal I to pyramidal II plane is the controlling factor. In other words, the cross-slip process from the lower energy plane to the higher energy plane is the critical process for achieving high ductility. The cross-slip from the lower energy plane to the higher energy plane is accomplished by a net resolved shear stress ∆τ driving the expansion of the dislocation on the higher energy plane, and the net cross-slip barrier is given by [21, 40, 15] ∆GXS = ∆GXS,i + ∆ElXS + Γ∆s − ∆τbA, 16

(4)

Figure 7: Schematic representation of the mechanistic model for the solute concentration regime favoring pyramidal II hc + ai dislocation. Edge part of the dislocation loop undergoes deleterious pyramidal to basal (PB) transition arresting the growth of dislocation, while cross-slip and double cross-slip of screw part of the dislocation loop tend to negate the effect of PB transition by multiplying the hc + ai dislocation loop. Ductility is then determined by the relative rates of the two thermally activated processes. lPB and lXS are the critical lengths associated with the PB transition and the cross-slip process. Atomistic pictures shown are obtained from the MD simulations of the processes. Atoms are drawn based on their crystal structures as determined by the common neighbor analysis: fcc:green, nonhcp/others:yellow. Reprinted with permission from AAAS.

where lXS = lCXS + lnuc is the critical cross-slip length at maximum barrier, with lCXS being the bowed out length of the cross-slipped part and lnuc the nucleation/jog length which does not bow out under the influence of ∆τ; b is the Burgers vector of the dislocation; ∆GXS,i is the intrinsic energy associated with nucleation of the cross-slip jogs; ∆E is the energy difference per unit length between the higher and lower energy screw dislocations and is a function of solute concentration; −1 Γ = 0.25µb2 is the dislocation line tension p with µ the shear modulus; ∆s = 2r sin (lCXS /2r) − lCXS and A = r2 sin−1 (lCXS /2r)−(rlCXS /2) 1 − (lCXS /2r)2 are the change in dislocation length and area swept due to circular bow out with radius r = Γ/∆τb of the cross-slipped dislocation, respectively. For binary Mg alloys with solute concentration c, the average dislocation energy difference h∆Ei is calculated using anisotropic elasticity [41] and considering only the solute-stacking fault 17

interaction energy, and is derived to be     I I II II     K12 k K12 k   h∆Ei = abs ∆EMg +  I − II  c , γMg γMg

(5)

where ∆EMg is the energy difference between the higher energy pyramidal I and lower energy I I pyramidal II hc + ai screw dislocations; K12 , γMg and kI are the energy prefactor for pyramidal I screw dislocation partials calculated from anisotoropic elasticity [41], pure Mg pyramidal I stacking fault energy, and the average effect of solutes on the pyramidal I stacking fault, respecII II tively [18, 22]; K12 , γMg and kII are the same quantities corresponding to the pyramidal II plane. The cross-slip length lCXS is calculated at maximum ∆GXS in the average material, leading to √ lCXS = 2 h∆Ei(2Γ − h∆Ei)/b∆τ. The fluctuation in energy difference, i.e. the standard deviation over the cross-slip length lXS , due to solute concentration fluctuations, is expressed as s clXS X  I−II 2 ∆Ui , (6) σ [lXS ] = b i where ∆UiI−II is the difference of interaction energy of a solute at site i with the pyramidal I and pyramidal II dislocations, and the summation is performed over unique atomic sites within a line length of one b around the dislocation cores (for details, see Refs. [15, 20]). Putting together the average and fluctuation effects of solutes, the activation barrier for crossslip is finally obtained as s clXS X  I−II 2 ∆Ui . (7) ∆GXS,i = h∆GXS i − σ [lXS ] = ∆GXS,i + h∆EilXS + Γ∆s − ∆τbA − b i This result applies to both (i) low solute concentrations where pyramidal II has a lower energy than pyramidal I and (ii) higher concentrations where the average effect of solutes has reversed the order of absolute stability. The absolute value of the average energy h∆Ei in Eq. 5 handles both cases automatically. The sign of fluctuation term is always negative because cross-slip will always nucleate at favorable sites with the lowest barriers, whether going from II to I or I to II. Eq. 7 is then used in Eq. 3 to compute the ductility index χ for given alloy compositions. The above results are valid at solute concentrations for which there are more than one solute present in the cross-slip nucleation length; lower concentrations, the “very dilute” limits, are dealt with in Refs. [15, 20] and are not relevant for the main focus here at high concentrations where pyramidal I dislocations become favorable due to solutes. 5.2. Predictions We now apply the model of the preceding Section 5.1 to predict the range of concentrations of RE solutes (Er, Y, Gd, Nd, Ce) for which Mg alloys are ductile. Parameters of the model used in this work are presented in Table 3, and are calculated by density functional theory (DFT), molecular dynamics/statics (MD/MS) simulations, or inferred from experimental observations [20]. The 18

analysis for the net resolved shear stress ∆τ acting on the cross-slip pyramidal I hc + ai screw dislocation, in the case of pyramidal II stable regime, is estimated to be in the range of 16-27 MPa for strong basal texture at applied uniaxial stresses of 100-200 MPa along the rolling direction in an earlier work [20]. A similar analysis of the net resolved shear stress ∆τ acting on the cross-slip pyramidal II hc + ai screw dislocation, corresponding to the pyramidal I stable regime, is presented in Appendix B, and provides an estimate in the range of 7-19 MPa. The magnitude of the driving stresses for both cases are fairly close, and so we use the same value of ∆τ = 20 MPa to analyse the cross-slip processes in both regimes. We further remark that this value of driving I−II stress, in conjunction with the energy difference ∆EMg , was chosen to better describe the wide range of experimental observations [20]. Table 3: Properties of Mg relevant to pyramidal I and pyramidal II stacking faults. b, γMg , and A0 are calculated using DFT [1]. ∆GXS,i and lnuc are determined from atomistic simulation using MEAM potential [26]. ∆EMg = 25 meV/nm is inferred from comparing the model prediction with the broad range of experiments, as discussed in Ahmad et al. [20]. K12 is determined in the framework of anisotropic elasticity utilizing DFT data, and ∆EPB and lPB are computed employing MD simulations using MEAM potential. Properties

b (Å)

µ (GPa)

∆EMg (meV/nm)

∆GXS,i (eV)

∆τ (MPa)

γMg (mJ/m2 )

K12 (eV/Å)

A0 (Å2 )

∆EPB (eV)

lPB (Å)

lnuc (Å)

NT

Pyramidal I Pyramidal II

6.09 6.09

18 18

25 25

0.23 0.23

20 20

163.648 167.572

0.159 0.180

18.759 16.833

0.3 0.5

20 20

25 25

45 45

Fig. 8(a) shows the variation of average energy difference between pyramidal I and pyramidal II dislocations with RE solute concentrations as computed using Eq. 5 (without taking the absolute value) and using the DFT-computed solute/stacking-fault interaction energies [15, 20]. In Fig. 8, c¯ is the concentration at which the energies of both dislocations become equal in the binary alloys, and in Fig. 8(b), the concentration is normalized with respect to c¯ for each solute. As we gradually increase the solute concentration up to c¯ (Er : 0.62 at.%, Y : 0.61 at.%, Gd : 0.56 at.%, Nd : 0.53 at.%, Ce : 0.53 at.%), the dislocation energy difference decreases, which in turn increases the rate of cross-slip process. Fig. 8(b) shows the predicted cross-slip energy barrier versus solute concentrations. As shown in earlier works [15, 20], the RE elements lead to a rapid decrease in the pyramidal hc + ai crossslip barrier at low solute concentrations. At concentrations for which χ > 1, the generation of new pyramidal II dislocation loops is deemed sufficiently fast to enhance ductility in spite of the on-going detrimental PB transitions. These predictions were shown to be consistent with a host of experimental studies performed on Mg-RE alloys [10, 11, 12, 13, 14, 15, 42, 43]. At concentrations c > c¯ , further addition of RE solutes makes pyramidal I more stable and increases the average pyramidal I-II energy difference. The average solute effect will thus drive the cross-slip barrier higher. However, the effect of solute fluctuations is increasing and continues to reduce the overall cross-slip barrier. Ultimately, however, with increasing solute concentrations, the increasing average effect of the solutes counteracts the fluctuation effect and the cross-slip barrier begins to increase again. Since the PB transition of the mixed pyramidal I dislocation has a much lower energy barrier than that of the pyramidal II edge dislocation, the increasing cross-slip barrier reenters the less-ductile (χ < 1) domain at concentrations ≈ 3−3.5 times the concentrations at which 19

a

30 Er: c¯ = 0.62 at.% Y: c¯ = 0.61 at.% Gd: c¯ = 0.56 at.% Nd: c¯ = 0.53 at.% Ce: c¯ = 0.53 at.%

h∆E I-II (c)i (meV/nm)

20 10 0

-10 -20 -30 0.0

b

0.7

0.2

0.4

0.6

II-I

0.5

1.6

1.8

Mg-RE

Pure Mg

0.6 h∆GXS i − σ[lXS ] (eV)

0.8 1.0 1.2 1.4 Solute concentration c (at.%)

2.0

χ I-II

0.4 -2 -1 0 1 2

0.3 0.2 0.1 0.0

-0.1 -0.2 0.0

0.5

1.0

1.5 2.0 2.5 Relative solute concentration c/¯ c

3.0

3.5

4.0

Figure 8: Prediction of ductility in Mg-RE binary alloys using cross-slip based model with ∆EMg = 25 meV/nm, ∆τ = 20 MPa. (a) Variation of the average energy difference between the pyramidal I and pyramidal II screw hc + ai dislocations as a function of solutes concentrations. (b) Cross-slip barrier, including fluctuation, and ductility index χ as a function of concentrations normalized by c¯ at which the energies of the pyramidal I and pyramidal II dislocations become equal. Value of c¯ (at.%)- Er : 0.62, Y : 0.61, Gd : 0.56, Nd : 0.53, Ce : 0.53. Mg alloys with χ > 1 (in the green region) is predicted to be ductile, and not ductile with χ < 0 (in the red region). At concentrations lower than c¯ , pyramidal II is more stable and cross-slip from the pyramidal II to pyramidal I planes competes with the PB transition of the pyramidal II hc + ai edge dislocation (with barrier of ∆EPB = 0.5 eV). Beyond c¯ , pyramidal I becomes more stable and now cross-slip from the pyramidal I to pyramidal II planes competes with even faster PB transition of the pyramidal I mixed hc + ai dislocation (with barrier of ∆EPB = 0.3 eV).

20

pyramidal I and pyramidal II are energetically equal. Ductility is predicted to be rapidly reduced with further addition of solutes, as χ decreases toward 0 and further to χ = −1. Therefore, we predict an upper limit to the ductilizing effects of the RE solutes, in the vicinity of 1.5 at% for the RE elements, beyond which the alloys are predicted to have low ductility. This is the main result of the current work, which has practical implications in future design of new ductile Mg alloys. 6. Discussion and conclusion The detailed predictions of the model depend on some parameters that can not be easily and firmly established, mainly the dislocation energy difference ∆EMg in pure Mg and the net resolved shear stress ∆τ on the cross-slip plane. However, even treating these as slightly adjustable parameters (e.g. as in Ahmad et al. [20]) to match key observed features in the ductility trends versus solute concentration, the overall predicted trends of the model in the low concentration regime are consistent with broad experiments. The new finding here that ductility can be lost with increasing RE solute concentrations is thus also expected to be robust, even if the precise concentration at which ductility begins to decrease significantly depends on these parameters. We now compare our prediction with some recent experiments on high concentration Mg-RE alloys. Rikihisa et al. [14] investigated the dominant slip systems in designed single crystal MgY alloys at 0.6 at.%, 0.9 at.%, 1.1 at.%, and 1.3 at.% Y concentrations. Their findings suggest that pure Mg yields by pyramidal II slip while all these Mg-Y alloys yield by pyramidal I slip except the Mg-1.3 at.% Y alloy where yield is evidently dominated by prismatic slip. This is in agreement with the result shown in Fig. 8(a), where pyramidal I dislocation becomes more favorable at around 0.6 at.% Y. Furthermore, deformation studies of polycrystal Mg-0.9at.%Y alloy indicate higher activity of pyramidal I hc + ai dislocations, with concomitant good ductility, in accordance with the results shown in Fig. 8(b). Gao et al. [44] studied deformation of solution treated polycrystal Mg-Gd cast alloys and found decreasing ductility with increasing Gd contents. In their experiments, as the Gd concentration is increased from 0.5 at.% to 3.63 at.%, the alloy ductility is reduced with increase in yield strength. This loss of ductility may, however, be due to normal ductility reduction with increasing strength, according to the Considère criterion. However, the experiments are not inconsistent with our findings. Data from the same work shows low ductility in ternary Mg-1.82Gd-1.02Y(at.%), Mg1.48Gd-1.13Y(at.%), and Mg-1.07Gd-1.77Y(at.%). Since Gd and Y have similar effects [15], our binary predictions are approximately appropriate for these tenaries at the total Gd+Y content. All of these alloys have a Gd+Y concentration around 1.6-1.8 at.%, which is above the level where we predict a transition back to low ductility. Nonetheless, caution is needed in interpreting the observation because ductility can also be influenced by other factors such as texture, grain boundary segregation and precipitation, and grain sizes. For example, in these experiments, cast alloys were studied and these microstructures presumably would not have strong texture and presumably basal hai dislocation could accommodate some plastic strain and thus lead to some ductility. Finally, we discuss the experiment performed by Wang et al. [45] on rolled and extruded Mg1.8Gd-0.9Y-0.1Zr(at.%) alloys. Again, interpretation of ductility measurements in experiments is not straightforward, since ductility is not an intrinsic material property and depends on many factors. Nevertheless, the experiment shows that the as-rolled and as-extruded alloys have low 21

ductility. This is consistent our theory since the amount of RE elements is 2.7 at.%, well beyond the ductility limit according to Fig. 8(b). This low ductility may also arise due to hard precipitate formation, since excessive formation of hard precipitates would increase yield/flow strength and decrease ductility. However, the alloy ductility increased after ageing, which might be due to the depletion of RE solutes in the matrix when forming the precipitates, and restoring the matrix concentration of RE elements back into the ductile range. Thus, the experimental observations here are broadly consistent with the present theory. Some further implications of our findings are as follows. Increasing the alloy strength by increasing the RE content would be counterproductive in the pursuit of high strength and high ductility alloys. Emerging alloying elements that can enable ductility, such as Zr, Ca, and Mn [15, 20], are not soluble at high concentrations and so would not be affected by the present findings. Other high-solubility non-RE elements such as Li and Sn do not cause a transition to pyramidal I dominance at any moderate concentrations (10% or less). These solutes could be particularly useful additions for providing good ductility along with some solute strengthening at moderate concentrations. The use of such solutes at moderate concentrations plus other dilute solutes that enable precipitate formation might enable strengthening due to precipitation while maintaining ductility due to high remaining Li and Sn concentrations in the Mg lattice. Our analysis is based on the solid solution model and does not consider the solubility of solute elements in Mg. For the cross-slip mechanism to be effective, the solutes at concentrations indicated in Fig. 8 must be in solid solution state. Whether this solid solution concentration range is attainable is determined by alloy thermodynamics. Thermodynamics calculation can be performed to estimate the solute solubility limits in Mg alloys at various temperatures and processing stages, such as before and after precipitation. In summary, we have extended a previous ductility model [15, 20] to include the higher solute concentration range where the pyramidal I hc + ai screw dislocation becomes energetically more favorable than the pyramidal II hc + ai dislocation. In this regime, we have studied various transition paths for the double cross-slip process of the pyramidal I screw dislocation. The lowest-barrier process is that in which the pyramidal I dislocation first cross-slips onto a pyramidal II plane and then cross-slips back to a parallel pyramidal I plane, after some arbitrary amount of glide on the pyramidal II plane. The ductility-enhancing model has been extended to the high solute concentration regime where pyramidal I hc + ai is favorable. We find that increasing the RE solute concentration, which increases the stability of the pyramidal I hc + ai relative to the pyramidal II hc + ai , ultimately increases the barrier for cross-slip to levels that are too high for double cross-slip to overcome the detrimental effects of the pyramidal I mixed dislocation transitions to immobile basal-dissociated structures. Thus, there is an upper limit of approximately 1.5 at.% concentration of RE solutes beyond which high ductility in Mg-RE alloys is lost. This is perhaps a surprising result, but not inconsistent with limited experimental data. Our findings motivate further experiments on the well-studied Mg-RE alloys at higher concentrations and in the solutionized state that have not been probed thoroughly, and where some implications of our findings for the design of strong and ductile Mg alloys have been identified.

22

Acknowledgements RA and WAC acknowledge financial support of this work through a grant from the Swiss National Science Foundation entitled “Control of Atomistic Mechanisms of Flow in Magnesium Alloys to Achieve High Ductility” (project #162350). The authors also acknowledge support from EPFL to the Laboratory for Multiscale Mechanics Modeling that enabled the required highperformance computing provided by Scientific IT and Application Support (SCITAS) at EPFL. Appendix A. Pyramidal II favorable MEAM potential The parameters of the MEAM interatomic potential, which favors pyramidal II dissociated hc + ai screw dislocation over pyramidal I dislocation, are presented in Table A.1. Furthermore, the core structures of the two pyramidal hc + ai screw dislocations are shown in Fig. A.1. Table A.1: The parameters of MEAM potential for pure Mg which predicts the pyramidal II hc + ai screw dislocation to be relatively more stable than the pyramidal I. Ec (eV)

re (Å )

α

A

β(0)

β(1)

β(2)

β(3)

t(1)

t(2)

t(3)

Cmin

Cmax

rc (Å)

∆r (Å)

1.52

3.18

5.61

0.69

2.45

1.51

2.30

1.15

6.52

-0.10

-8.09

0.36

2.36

8.29

7.60

a

b

pyr II

I r y p

Figure A.1: Core structures of (a) pyramidal I and (b) pyramidal II dissociated hc + ai screw dislocations, as determined from the MEAM potential presented in Table A.1. Blue atoms denote atoms with HCP lattice environment and yellow atoms represent atoms in non-HCP lattice environment as identified from common-neighbor analysis. The magenta line in all three figures indicates the pyramidal I or pyramidal II slip plane on which dislocation is dissociated. This potential predicts the pyramidal II hc + ai screw dislocation to be more stable than the pyramidal I with energy I−II difference ∆EMg = 54 meV/nm.

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x3

a

b

x3

+ hc

φ

ai

x2 x1

σ x2

φ0 ψ

x1

Figure B.2: (a) Schematics of the pyramidal I and pyramidal II planes accessible to the hc + ai dislocation cross-slip. (b) Coordinate system and direction of applied stress σ used in the calculation of the net driving stress. The coordinate system is same as that used in (a).

Appendix B. Net driving stress ∆τ for pyramidal I to pyramidal II cross-slip We follow the same analysis as that for cross-slip from pyramidal II to pyramidal I plane [20]. In particular, we assume the pyramidal I glide to happen in equilibrium, i.e., at zero net stress. At equilibrium, the total shear stress on a pyramidal I dislocation segment due to applied stresses and backstresses τb is equal to the Peierls stress, τI = τapp − τb = τI,P

(B.1)

As shown in Fig. B.2(a), we consider the triplet of one pyramidal II and two cross-slip pyramidal ¯ I planes described in a coordinate system where x1 , x2 and x3 are oriented along a([21¯ 10]), c× ¯ a([0110]) and c([0001]) directions, respectively. Fig. B.2(b) shows a general applied uniaxial load σ − source of the resolved shear stresses τapp and τb − which is at an angle φ from x3 (hci ) axis and whose projection on basal plane (x1 − x2 ) is at angle ψ from the x1 (hai ) axis. The resolved shear stress on the shown pyramidal II plane is   (B.2) τII = σ 0.446 sin2 φ cos2 ψ − 0.452 sin φ cos φ cos ψ − 0.446 cos2 φ ,

and the resolved shear stresses acting on the cross-slip pyramidal I planes accessible to this pyramidal II segment are  τI =σ 0.4 sin2 φ cos2 ψ − 0.405 sin φ cos φ cos ψ − 0.4 cos2 φ (B.3)  ±0.231 sin2 φ sin ψ cos ψ ∓ 0.376 sin φ cos φ sin ψ , (B.4)

Substituting Eq. B.4 and τI = τI,P into the ∆˜τ = τII − τII,P , and denoting the pyramidal II Peierls stress as τII,P , lead to the net driving stress acting on the pyramidal II plane, for the shown triplets, 24

as a function of orientation of load axis (φ, ψ) ∆˜τ(φ, ψ) = 1.125τI,P − τII,P + 1.125σ 0.231 sin2 φ sin ψ cos ψ − 0.376 sin φ cos φ sin ψ

(B.5)

where we consider the pyramidal I plane with the larger of the two possible stresses, leading to the absolute value term. Furthermore, we note that there are other five similar triplets of pyramidal I/pyramidal II planes, situated π/3 degrees apart about x3 axis. Thus for a given load orientation (φ, ψ), the net resolved shear stress is the maximum of all six triplets, which is ∆τ(φ, ψ) = max{∆˜τ(φ, ψ + kπ/3) : k = 0..5}.

(B.6)

Next, we estimate the net driving stress in the presence of different textures, generally observed in Mg alloys. We consider a polycrystals with a strong basal texture, typical of rolled Mg [46, 43], where hci axes of grains is predominantly aligned along the normal direction of the sheet. The load σ is applied along the rolling direction which lies along a random direction contained in the basal plane. For this case, φ = π/2 and ψ ∈ (−π, π] is randomly chosen for each grain. The net driving stress is then estimated by averaging over the angle ψ, which due to symmetry of six triplets of pyramidal II/I planes reduces to Z 3 π/6 ∆τ = ∆τ(φ = π/2, ψ)dψ π −π/6 (B.7) = 1.125τI,P − τII,P + 0.124σ. For σ = 100 MPa, which is comparable to experimental yield stresses of Mg alloys, the net resolved shear stress on pyramidal II plane is ∆τ = 1.125τI,P − τII,P + 12.4. The MEAM Peierls stresses for the screw dislocation at T =300 K are τII,P ≈ 45 MPa and τI,P ≈ 35 MPa [20], leading to ∆τ ≈ 7 MPa. At higher stress level of σ = 200 MPa, the net resolved shear stress becomes ∆τ ≈ 19 MPa. Furthermore, solute strengthening on dislocations in Mg-Y alloy would modify the net resolved shear stress. Since the interaction of RE solutes with pyramidal I stacking fault is stronger than that with pyramidal II stacking fault, we expect the net resolved shear stress to increase with solute concentrations. The average net resolved shear stress ∆τ = 20 MPa used in the calculations is reasonable. We next consider the polycrystal with basal texture split into two lobes and tilted towards the rolling direction, which is observed commonly in Mg alloys [47, 48, 49, 11, 43]. For the basal pole tilted ±15◦ towards the rolling direction, φ = 75◦ . Again averaging over ψ ∈ (−π/6, π/6] yields ∆τ = 1.125τI,P − τII,P + 0.19σ

(B.8)

For the tilted basal pole using τII,P ≈ 45 MPa, τI,P ≈ 35 MPa and σ = 100 MPa, we obtain ∆τ = 13 MPa. Thus the presence of this tilted basal pole texture increases the driving force for the crossslip process and would lead to the better ductility even for some unfavorable solutes, which is in accordance with the experimental observations. 25

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