Mechanism and energetics of ⟨c + a⟩ dislocationcross-slip in hcp metalsZhaoxuan Wua,b and W. A. Curtina,1

aInstitute of Mechanical Engineering, École Polytechnique Fédérale de Lausanne, Lausanne CH-1015, Switzerland; and bInstitute of High PerformanceComputing, Singapore 138632

Edited by L. B. Freund, University of Illinois at Urbana–Champaign, Urbana, IL, and approved August 12, 2016 (received for review March 10, 2016)

Hexagonal close-packed (hcp) metals such as Mg, Ti, and Zr arelightweight and/or durable metals with critical structural applicationsin the automotive (Mg), aerospace (Ti), and nuclear (Zr) industries.The hcp structure, however, brings significant complications in themechanisms of plastic deformation, strengthening, and ductility, andthese complications pose significant challenges in advancing thescience and engineering of these metals. In hcp metals, generalizedplasticity requires the activation of slip on pyramidal planes, but thestructure, motion, and cross-slip of the associated ⟨c+ a⟩ dislocationsare not well established even though they determine ductility andinfluence strengthening. Here, atomistic simulations in Mg reveal theunusual mechanism of ⟨c+ a⟩ dislocation cross-slip between pyrami-dal I and II planes, which occurs by cross-slip of the individual partialdislocations. The energy barrier is controlled by a fundamental step/jog energy and the near-core energy difference between pyramidal⟨c+a⟩ dislocations. The near-core energy difference can be changedby nonglide stresses, leading to tension–compression asymmetry andeven a switch in absolute stability from one glide plane to the other,both features observed experimentally in Mg, Ti, and their alloys.The unique cross-slip mechanism is governed by common featuresof the generalized stacking fault energy surfaces of hcp pyramidalplanes and is thus expected to be generic to all hcp metals. An ana-lytical model is developed to predict the cross-slip barrier as a func-tion of the near-core energy difference and applied stresses andquantifies the controlling features of cross-slip and pyramidal I/IIstability across the family of hcp metals.

metallurgy | hcp | dislocation | cross-slip | atomistic simulation

Plastic deformation of crystalline materials occurs mainly by slipalong low-index atomic planes. Slip regions are bounded by line

defects called dislocations (1), which are characterized by thecrystallographic slip increment known as the Burgers vector b andthe local line direction ξ. The transition from slipped to unslippedregions is resolved at the atomic scale by local atomic rearrange-ments creating a dislocation “core” structure. Slip occurs by motionof this core, and the surrounding elastic fields, driven by an appliedstress. Plasticity is thus controlled by the details of the core region,which differ significantly among face-centered cubic (fcc), body-centered cubic (bcc), and hexagonal close-packed (hcp) metals (2).The motion of the dislocation core is usually confined to glidewithin the slip plane defined by the normal vector n= b× ξ. Screwdislocations have b× ξ= 0, however, and so do not have a uniqueslip plane and can therefore “cross-slip” among glide planes thatshare a common Burgers vector. Cross-slip is a crucial process inplasticity, acting to spread slip spatially on multiple nonparallelplanes and to enable dislocation multiplication and annihilationprocesses; all of these processes affect ductility and ultimate strengthof the material. Whereas the atomistic mechanisms, energy barriers,and rates of cross-slip are well established in cubic metals (3), themechanisms in hcp metals are not well known and are under activestudy (4, 5), especially for the technologically important metals Mg,Ti, and Zr.In hcp metals, only hc+ ai dislocations, existing on the pyra-

midal (Pyr.) I and II planes (Fig. 1A), enable plastic slip in the hcidirection (1, 6). The hc+ ai dislocations are thus essential to

achieving generalized plastic flow and, consequently, high ductility.Moreover, experiments demonstrate that the activation of hc+ aicross-slip has strong effects on plastic flow evolution and strength-ening of hcp metals (7–10). Frequent cross-slip of hc+ ai dislocationscan occur in early stage, room temperature deformation (10–15).However, the propensity and nature of hc+ ai cross-slip differ amongdifferent materials and can also depend on the applied stress andtemperature in the samematerial (12, 16–22) (Fig. 2 and refs. 23–44).The range of observed behavior arises in part because, unlike fcc andbcc metals where cross-slip is possible among equivalent slip systems,cross-slip of hc+ ai dislocations can occur only from the structurallydifferent Pyr. I and Pyr. II planes or vice versa. Although hc+ aidislocation core structures are emerging (45, 46) and elastic analysescan help identify which slip system is energetically favorable (47, 48),there are no studies, criteria, or mechanisms to rationalize, much lesspredict, the material, stress, and/or temperature dependence of cross-slip behavior for the critical hc+ ai slip in hcp metals (2, 49).Here, we study the mechanism, energy barrier, and stress de-

pendence of the hc+ ai screw dislocation cross-slip between Pyr. Iand II planes in Mg, using the nudged elastic band (50, 51) (NEB)method with a density functional theory (DFT) validated empiri-cal potential (6). We identify a mechanism of cross-slip, show thatthe key features enabling this mechanism are common across hcpmetals, and develop an analytical model to predict hc+ ai dislo-cation cross-slip behavior. The insights and model rationalize theexperimentally observed behavior noted above (Fig. 2) and pro-vide a framework for design of improved hcp alloys.

ResultsIn Mg, the hc+ ai screw dislocations dissociate into a pair ofpartial dislocations of primarily pure screw character separatedby a stacking fault, for both Pyr. I and II planes (6, 45, 46). These

Significance

For all hexagonal close-packed (hcp) metals, the ability to plasti-cally deform in the crystallographic c axis is crucial for achievinghigh ductility and high fracture toughness. ⟨c+a⟩ dislocation slipis the only sustainable mechanism to accommodate c-axis strain,but is difficult across the family of hcp metals. We reveal themechanism, energy barrier, and unusual stress dependence of⟨c+a⟩ dislocation cross-slip in Mg, using atomistic simulations.Our results provide mechanistic insights into ⟨c+ a⟩ cross-slip be-havior, rationalize observed changes in pyramidal I/II slip stability,enable predictions of slip trends across the family of hcp metals,and suggest that applied stresses and/or precise solid solutionalloying can optimize cross-slip and enhance c-axis strain capacity,which can ultimately guide design of improved hcp alloys.

Author contributions: Z.W. and W.A.C. designed research; Z.W. performed research; Z.W.and W.A.C. analyzed data; and Z.W. and W.A.C. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: william.curtin@epfl.ch.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1603966113/-/DCSupplemental.

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dissociations are consistent with the positions of the metastablestacking faults on the generalized stacking fault energy (γ) sur-faces (52) of the two pyramidal planes as calculated by both DFT(45, 48) and modified embedded-atom method (MEAM) (6)(Fig. 1 B and C and Pyramidal γ Surface of hcp Metals). Thedissociated structure leads to a unique cross-slip mechanism,which is revealed by computing the transition pathway for ascrew dislocation to move back and forth between Pyr. I and Pyr.II planes (Figs. S1 and S2). Fig. 3A shows the atomic configu-rations (53) along the minimum energy path (MEP) for cross-slipfrom Pyr. II onto Pyr. I under stress-free conditions. The cross-slip process has three distinct stages: nucleation, propagation,and annihilation. In the first stage, cross-slip is initiated byforming a pair of jogs/steps on one of the partial dislocationcores/stacking fault planes. Because the Burgers vectors of thepartials are not identical on Pyr. I and II planes, part of theBurgers vector of the migrating partial is also transferred tothe other partial through local atom rearrangements. The neteffect of this process is the transformation of one segment ofscrew dislocation on the Pyr. II plane into a segment on the Pyr. Iplane. In the second stage, cross-slip proceeds simply by lateralmotion of the jogs/steps along the dislocation line, thus shiftingmore segments onto the Pyr. I plane. In the third stage, the leftand right jogs/steps become sufficiently close that they begin toannihilate. For cross-slip from Pyr. I onto Pyr. II, the path isreversed so that the third stage shown in Fig. 3A is the nucleationstage. The unique constriction-free hc+ ai dislocation cross-slipmechanism shown in Mg is the first main result of this paper.Unlike dislocations in fcc and bcc metals, the hc+ ai disloca-

tions on the initial and final slip planes have different corestructures, stacking faults, and relative partial orientation and,therefore, different near-core energies per unit length Estruc. Thefar-field elastic energies are the same (1, 48, 54). Cross-slip be-tween Pyr. I and II planes thus leads to a net change in totaldislocation energy, which influences the cross-slip energy barrier.For Mg, our simulations show that the hc+ ai screw dislocation

has a lower energy on the Pyr. II plane compared with the Pyr. Iplane. The energy difference is small, ∼30 meV/nm, but has acrucial effect on cross-slip: The energy barrier for cross-slip of adislocation segment depends on the segment length. To illustratethis feature, we perform NEB calculations for three differentdislocation lengths (l = 146 Å, 219 Å, 292 Å) under stress-freeconditions. Fig. 3B shows the atomic configurations along theMEP for a longer length, demonstrating that the underlyingthree-stage process is length independent. However, the ener-gies along the MEP are length dependent, as shown in Fig. 3 Cand D. In Fig. 3C, the energy difference is shown relative to thelow-energy Pyr. II screw, whereas Fig. 3D shows the energy dif-ference relative to the higher-energy Pyr. I screw. The reactioncoordinate of replica j is the accumulated distance (in angstroms)along the path in terms of the replica coordinate x; i.e.,dj =

Pji=1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi − xi−1Þ · ðxi − xi−1Þp

. In stage 1 (nucleation, 0< d< 17),there is a sharp rise in energy that is similar for all three lengths(Fig. 3C). During stage 2 (propagation, 17 < d <dmax − 12),the energy difference increases linearly with increasing d, duepurely to the difference in Pyr. I and Pyr. II dislocation near-coreenergies. The extent of the linear region, and the associatedincrease in energy barrier, scales directly with the dislocationlength l. In stage 3 (annihilation, dmax − 12 < d <dmax but alsothe nucleation stage starting from the Pyr. I screw), the energydecreases to the total energy of the straight Pyr. I dislocationand is the same for all three lengths (Fig. 3D). The cross-slipenergy barrier ΔGII−I for slip from Pyr. II to Pyr. I is thus de-pendent on the total length of the cross-slipping segment,whereas the barrier for the reverse process ΔGI−II, from Pyr. Ito Pyr. II, is independent of length and much smaller.Therefore, under stress-free conditions, a typical long Pyr. II screw

dislocation (l > 20 nm) is highly unlikely to cross-slip entirely onto thePyr. I plane because of the very high barrier (ΔGII−I > 0.8 eV),whereas a Pyr. I screw dislocation of any length will cross-slip rapidly(∼1 ns at room temperature) over the low barrier (ΔGI−II ∼ 0.24 eV)onto the Pyr. II plane. Note that room-temperature molecular

A B C

Fig. 1. (A) Schematic of the Pyr. I and II planes in the hcp unit cell and the hc+ ai Burgers vector. (B and C) Generalized stacking fault energy surfaces of Pyr.I and II planes, respectively, indicating the Burgers vector, minimum energy path for slip, and energies and positions (‘‘×’’ symbols) of the metastable stackingfaults under full atomic relaxation. The energies are calculated using the MEAM potential for Mg (6).

Fig. 2. Experimentally observed hc+ ai dislocation slip planes (Pyr. I and II) for various hcp metals and alloys under a/c-axis compression and tension. In eachpanel, hexagon and circle symbols represent single- and polycrystal tests, respectively, whereas colors indicate temperature (green, below room temperature;cyan, at room temperature; and red, above room temperature).

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dynamics (MD) simulations using short dislocation lengths (l < 5 nm)show frequent cross-slip that is not pertinent to real behavior. Weconclude that at low stresses and low temperatures, hc+ ai slip onPyr. II planes is expected to be dominant in Mg, consistent with datain Fig. 2 and with past (11, 16) and very recent (14, 25) transmissionelectron microscopy studies reporting (sessile) dislocations primarilyaligned with the ½�1010� direction that can come only from Pyr. IIhc+ ai slip. The general control and asymmetry of Pyr. I/II cross-slipenergy barriers caused by the near-core energy difference, and thespecific preference for Pyr. II slip in Mg, constitute the second mainresult of this paper.The near-core energy difference per unit length ðEI −EIIÞ

between Pyr. II and I dislocations can, however, be changed by

applied nonglide stresses through interactions with the disloca-tion cores and stacking faults, with a possible Escaig-stress effect(55) on the small edge components of the Pyr. I partials. Fig. 4Ashows the energy difference EI −EII as a function of appliedstress σyy normal to the Pyr. II plane. The stress σyy is related tothe standard single crystal a/c-axis loading directions via a geo-metrical factor as indicated in Fig. 4A. Compressive stresses in-crease the energy difference, making Pyr. II even more favorable,whereas tensile stresses on the Pyr. II plane decrease the energydifference, making Pyr. I less unfavorable. In fact, at high tensilestresses (>660 MPa), absolute stability of the hc+ ai dislocationchanges: The hc+ ai screw dislocation has a lower energy on thePyr. I plane. Related effects are seen for other stress states

A

B

C D

Fig. 3. Atomic configurations and energy profiles along the minimum energy path for hc+ ai screw dislocation cross-slip from Pyr. II to I planes (left to right), ascalculated by NEB under stress-free conditions. (A) dislocation length l = 146 Å; (B) dislocation length l = 292 Å. On top of each image, the number indicates thecorresponding reaction coordinate d shown in C and D. (C) Energy vs. path coordinate for hc+ ai screw dislocations at three different lengths of cross-slip, withreference to the lower-core-energy Pyr. II dislocation. (D) The same as in C but with reference to the higher-core-energy Pyr. I dislocation. In C and D, the energybarrier for cross-slip from the Pyr. II to the Pyr. I plane is denoted as ΔGII−I and depends on dislocation length, whereas that for cross-slip from Pyr. I to Pyr. II isdenotedΔGI−II and is length independent. Atoms are colored on the basis of common neighbor analysis (53): purple, bcc; yellow, others. Dislocation cores are shownwith only non-hcp atoms.

A B C

Fig. 4. (A) Dislocation energy differences EI − EII as a function of applied normal stress σyy . (B and C) Energy profiles on the minimum energy path for hc+aiscrew dislocation cross-slip from Pyr. II to I planes calculated by NEB under applied normal stress σyy .

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(Supporting Information and Fig. S3). The cross-slip energybarrier then also depends on these applied stresses. Fig. 4 B andC shows the energy along the MEP at one length (l= 292 Å) forhc+ ai screw dislocation cross-slip under applied stresses σyy.Consistent with the absolute changes in relative near-core energy,compressive stresses increase ΔGII−I whereas tensile stresses de-crease it. The change in ΔGII−I is mainly manifested in changesin the slope in stage 2. The applied stress gives a similar trendfor the energy barrier in stage 1 (nucleation process). Thus, tensilestresses on the Pyr. II plane also lower the energy barrier forforming a segment of dislocation on the Pyr. I plane. In contrast,σyy has little effect on ΔGI−II. Therefore, tensile σyy increases thenet rate of cross-slip onto Pyr. I while not affecting the rate of thereverse process. This normal-stress dependence of the near-coreenergy difference and its influence on cross-slip are the third mainresult of this paper.Our atomistic analyses were executed on Mg, due to the ex-

istence of a well-validated interatomic potential. But the ob-served phenomena and mechanisms are expected to be commonto hcp metals. First, for hcp Mg, Ti, Zr, Co, Cd, and Zn, both thePyr. I and Pyr. II γ surfaces show the existence of metastablefault vectors generally aligned with the hc+ ai Burgers vector(Fig. S2 and Pyramidal γ Surface of hcp Metals). These faultvectors are largely dictated by crystal symmetries. Pyr. I and Pyr.II near-core energies are also generally different, so that theasymmetry in cross-slip is also expected. Finally, experimentalobservations indicated that relative stability between Pyr. I and IIcan change with loading (Fig. 2), consistent with a dependence ofnear-core energies on applied stresses.Despite the difficulty of cross-slip under zero glide stress,

cross-slip can be facilitated by resolved shear glide stresses on thecross-slip plane. So, when a dislocation on the primary slip planeis pinned by other metallurgical features (precipitates, solutes,forest dislocations) such that local screw segments on the pri-mary plane experience resolved shear stress below their Peierlsstresses, resolved shear stresses above the Peierls stress can stillexist on the cross-slip plane. This shear stress provides a drivingforce that reduces the energy barrier for cross-slip. Incorporatingall mechanistic factors, we can develop a general analyticalmodel for pyramidal dislocation cross-slip in hcp materials asfollows. Because the Pyr. I and II dislocations glide readily at lowapplied stresses at room temperature, we use a line-tension frame-work and study a configuration with a cross-slipped segment bowingout on the cross-slip plane, as shown in Fig. 5A (Semicircular Dis-location Bow Out). For cross-slip between two structurally differentplanes, the cross-slip energy barrier has a number of contri-butions (3),

ΔG=Es +ΔEðσÞl+TΔsðT, b, τ, lÞ− τbAðT, b, τ, lÞ, [1]

where Es is the intrinsic energy associated with the jogs/stepsconnecting the cross-slipped and uncross-slipped dislocation

segments (corresponding to stage 1 or stage 3), ΔEðσÞ is thestress-dependent near-core energy difference between the twoglide planes, l is the chord length of the putative cross-slippeddislocation, T is the line tension, Δs is the additional dislocationline length due to bow out, τ is the resolved shear stress on thecross-slip plane, and A is the area swept by the dislocation duringthe bow out. Crucially, the work done by the applied stress, thelast term in Eq. 1, can overcome the length-dependent near-coreenergy difference that prohibits cross-slip at zero stress.At stress τ, the cross-slipped segment can expand unbounded

if it can reach a critical length lc determined by ∂ΔG=∂l= 0.Approximating the bow-out geometry as a circular arc with arclength s and area A (Semicircular Dislocation Bow Out) gives lc as

lc = 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔEðσÞð2T −ΔEðσÞÞ

p �ðbτÞ. [2]

Inserting lc into Eq. 1 yields the activation barrier for nucleatingonto the cross-slip plane. The result is not analytic and so weshow a numerical example below. In general, however, increasingthe shear stress on the cross-slip plane makes both the criticallength lc and the activation energy barrier smaller, increasing therate of thermally activated cross-slip.As a concrete example relevant to Mg, we use atomistically de-

rived values Es = 0.23 eV and ΔEðσÞ (Fig. 4A) and approximateT = 0.5μb2, where μ is the shear modulus. The cross-slip energybarrierΔG as a function of l vs. applied shear stress τ is shown in Fig.5B; the critical length lc and energy barrier correspond to the max-imum value of ΔG at each stress value. The barrier cannot decreasebelow Es, but even at moderate stresses (∼80 MPa), the barrier isnot significantly higher (∼0.35 eV) and at an accessible nanoscalelength (lc ≈ 7 nm). Thus, cross-slip from Pyr. II to Pyr. I can becomequite frequent under conditions typical of Mg processing and/orapplications. Easy cross-slip thus also makes it difficult to experi-mentally determine the primary/dominant slip plane (15).Moreover, Fig. 5C shows the activation energy vs. resolved

shear stress τ for a range of applied nonglide normal stresses σyy,using Mg values. For all τ, tensile σyy reduces the energy barrier forcross-slip whereas compressive σyy increases the energy barrier.This is consistent with experimental observations where Pyr. II slipis commonly seen in c-axis compressive loading (11, 14, 16) butwhere Pyr. I slip becomes more important/frequent at high tem-peratures or in tensile loading (16, 17) (Fig. 2). At τ= 50 MPa andσyy within ±100 MPa [typical values at the onset of yield at roomtemperature in single-crystal Mg under a/c-axis deformation (8)],we find ΔG is ∼0.42 ± 0.04 eV. At laboratory strain rates, cross-slip is thus predicted to be particularly active at room temperaturebut unlikely at 77 K. However, the Pyr. I screw hc+ ai can glideonly for a short distance before cross-slipping back to the Pyr. IIplanes. Thus, frequent double cross-slip is expected, consistentwith various experiments (8, 14, 23, 24) with Pyr. II hc+ ai slipdominant. At the same time, under c-axis compression loading at

A B C

Fig. 5. (A) Schematic of dislocation cross-slip configuration. (B) Cross-slip energy barrier ΔG as a function of cross-slip length l under resolved shear stress τ oncross-slip plane. (C) Cross-slip energy barrier ΔG as a function of resolved shear stress τ and normal stress σyy .

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room temperature, single-crystal Mg strain hardens rapidly (11,14, 15, 56) with the flow stress reaching ∼300 MPa within a fewpercent of strain. The increase in flow stress increases τ and σyyproportionally (Fig. 4), so that the net effect, per results in Fig. 5C,is to decrease the cross-slip energy barrier and increase the dis-location cross-slip rate. This enables more Pyr. I slip at later stagesof deformation. At even higher stresses, such as those during coldrolling of polycrystals (57), hc+ ai screw dislocations in somegrains may have similar energies on Pyr. I and Pyr. II so thatΔE= 0 (Supporting Information). Then, the barriers for cross-slipfrom Pyr. I to II planes and vice versa may become comparable,making hc+ ai slip equally favorable on Pyr. I and II planes. Theabove conclusions are all in general agreement with experimentson Mg where deformation mechanisms are most clearly discern-able and rationalize the experimental ambiguities (15, 57–59) inthe competition between Pyr. II and Pyr. I slips in Mg.

Discussion and ConclusionLooking more generally across the family of hcp metals shown inFig. 2, differences among them arise due to differences in Pyr. Iand Pyr. II stacking fault, core, and jog/step energies and theirdependencies on applied stresses. For Ti, hc+ ai slip on Pyr. I ismore common, indicating that the Pyr. I screw has lower energythan the Pyr. II screw; this is consistent with our DFT calcula-tions showing the metastable stacking fault energy on Pyr. I to beless than half of that on Pyr. II planes. However, hc+ ai cross-slipis frequently observed in Ti and Ti alloys at high temperatures orunder a/c-axis compressive loadings (Fig. 2); this indicates thatboth the energy difference EI −EII and the cross-slip activationenergy barrier ΔGI−II are relatively low in Ti and can be modifiedby stresses [glide/nonglide (12)] in the same way for Mg. Fur-thermore, the high flow stresses in Ti alloys may also help reducethe cross-slip activation energy barrier for cross-slip onto Pyr. IIplanes. In fact, new experiments, using microcantilever bending(10) to induce both tension and compression in the same sampleunder the same conditions, clearly show Pyr. I slip in tension butwavy Pyr. I + II slip in compression. In contrast to Mg and Ti,hc+ ai cross-slip is not common in hcp metals like Zn, Cd, andZr, nor is there evidence of any switch in dominant slip systembetween Pyr. I and Pyr. II (Fig. 2). This result indicates that thedifferences in Pyr. I and II dislocation near-core energies and/orthe step/jog energy are relatively larger in Zn, Cd, and Zr andthus with larger energy barriers, but the mechanisms should re-main the same because their γ-surface profiles are similar tothose of Mg and Ti (Supporting Information).Our analysis shows that the cross-slip frequency and relative

stability depend on the near-core energy difference between Pyr.I and II screw dislocations. This finding suggests that alloyingwith solutes, which interact with the dislocation cores andstacking faults, can shift the near-core energy difference and,therefore, shift the relative stability of Pyr. I and II and changethe frequency of cross-slip. Indeed, experiments show that the

dominant slip system (and thus the cross-slip) is influenced byalloying in Mg and Ti (Fig. 2). This further suggests that precisesolid solution alloying could maximize/optimize cross-slip in Mgand Ti to achieve a desired evolution of plasticity. Of course,such alloying will have many other effects such as strengtheningof all of the various hcp slip systems, so that optimization ismultidimensional. For Mg, with other strategies aimed at en-hancing c-axis strain capacity under active research (57) andsome apparently exhausted (54, 60), optimizing cross-slip mayprovide fresh opportunities. In contrast, because Zr, Zn, and Cdlikely have larger near-core energy differences between Pyr. Iand II screw cores, leading to high dominance of one of the twopyramidal slip systems, solid solution alloying would be less likelyto alter the dominant slip system. But alloying could still signif-icantly influence the rate of double cross-slip. There are alsorelatively fewer direct observations in Zn (37, 38) and Cd (8, 36).In summary, the crystallographic features of hcp pyramidal

planes drive hc+ ai screw dislocation dissociation into near-screwpartials, which allows for a unique mechanism of cross-slip. How-ever, the screw dislocations on Pyr. I and II have different ener-gies, greatly affecting the cross-slip barriers and rates. This energydifference also depends on nonglide stresses, leading to tension/compression asymmetry in cross-slip and slip plane dominance.These insights rationalize a wide body of experimental literature inhcp metals and provide a mechanistic basis for modifying hc+ aislip via targeted alloying, for instance. The unusual features of hcppyramidal cross-slip must also be incorporated into higher-scalemodeling such as dislocation dynamics and crystal plasticity models,if such models are to accurately reflect real plasticity mechanisms inhcp metals. The work here provides quantitative results for Mg,which can be made even more quantitative with additional first-principles studies, and provides a firm basis for future quantitativework on Ti, Zr, and other hcp metals. Combined with other recentworks (54, 61, 62), computational metallurgy is now revealing therich, distinct, and complex behavior in a class of materials, hcpmetals, that have high technological value.

Materials and MethodsAtomistic simulations and NEB calculations are performed using large-scaleatomic/molecular massively parallel simulator (LAMMPS) (63) with Mg atominteractions described by a modified embedded-atom method (64) potentialparametrized by Wu et al. (6). In the current NEB calculations, 64, 96, and 128replicas are used for dislocation of lengths 146 Å, 219 Å, and 292 Å. Unlessotherwise stated, convergence is assumed when the maximum force on atomsis below 10−4 eVÅ−1. More details on the simulation cell and boundary con-ditions are described in Supporting Information.

ACKNOWLEDGMENTS. Z.W. acknowledges financial support from theAgency for Science, Technology and Research (A*STAR), Singapore. W.A.C.acknowledges support of this work through a European Research Council(ERC) Advanced Grant, “Predictive Computational Metallurgy,” ERC Grant339081-PreCoMet. Part of the computer time was supported by a grant fromthe Swiss National Supercomputing Centre (CSCS) under project ID s631.

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