Influence of Crystallographic Orientation on Twin Nucleation in Single CrystalMagnesium

C.D. Barrett1, M.A. Tschopp1, H. El Kadiri1, B. Li1

1Center for Advanced Vehicular SystemsMississippi State University, Starkville, MS USA

Keywords: Magnesium, Twin Nucleation, Molecular Dynamics, Single crystal

Abstract

Experimental plasticity on single crystals has found sub-stantial non-Schmid effects in both twinning and non-basal slip in pure magnesium. The deviation fromSchmid’s law has been attributed to the strong sensitivityof both twinning and slip to small lattice heterogeneities[1] and the effect of pre-slip and non-planar dislocationdissociation [2]. However, most molecular dynamics sim-ulations use heterogeneities so the effect of slip on twinnucleation and vice-versa has been shrouded. This hasmotivated us to investigate the influence of crystal load-ing orientation on homogeneous slip and twin nucleationusing molecular dynamics. These simulations allowedus to appreciate the propensity and nature of twin nu-cleation when pre-existing defects are absent. Analysesof deformation mechanisms and stress-strain responsesshows that homogeneous dislocation nucleation on thebasal slip system is correlated with the highest Schmidresolved shear stress, while homogeneous nucleation oftensile twins did not always correlate with the highestSchmid resolved shear stress.

Introduction

Twinning readily occurs in double lattice structures suchas hexagonal-close-packed (HCP) structures primarily be-cause the close-packed directions contained within thebasal planes cannot accommodate c-axis deformation.Other slip modes having a Burgers’ vector with a non-zero component along the c-axis, such as pyramidal 〈c+a〉slip are at least five times harder than basal slip in poly-crystals. Although twinning is believed to be athermal,the critical stress to activate prismatic and pyramidal slipdecreases with increasing temperature, thereby minimiz-ing the contribution of twinning at higher temperaturesunless the strain rate is increased. Twinning sharply re-orients the parent crystal, transmutes parent dislocations[3], and introduces substantial interfaces, thereby causingstrong crystallographic anisotropy and a rapidly increas-ing hardening rate behavior, particularly when twinning

is able to consume the entire parent grain in sharply tex-tured materials. As high-strength wrought magnesiumalloys are systemically textured, twinning is highly soughtto be prohibited or mitigated so to promote forming atcost-effective rates without risking distortion of compo-nents. In part, these have motivated renewed efforts to-wards a fundamental understanding of the mechanismsunderlying twin nucleation in HCP lattices.

While some insight into twin nucleation behavior maybe able to be obtained via in situ high resolution trans-mission electron microscopy (TEM) experiments, theseexperiments are often very difficult to perform and arenot without complications. The use of molecular dynam-ics (MD) simulations to probe plasticity in HCP struc-tures has received much interest in the last decade. Theprimary objective of these simulations has been to eluci-date the mechanisms of twin nucleation as well as someinterest dedicated to non-basal slip activities. In thesesimulations, the twin nucleation problem tends to be ahighly-debated subject. The influence of the interatomicpotentials for HCP metals on twinning is important in theresults of MD simulations. Also, the use of constrainedfree surfaces introduces a planar defect into the systemthat influences twinning in many MD simulations. Planardefects, such as grain boundaries, and line defects, suchas dislocations, play an important role in twin nucleation.Twins systemically nucleate at low angle grain boundariesin polycrystals [4]. Furthermore, recent experiments haveshown that even under profuse twinning conditions, twinsnucleate only after 1.3% plastic strain. These studiesshow that prior slip plays a key role in twin nucleationand have actually substantiated the theories developedearlier by Mendelson [2] on the role of prior slip in twinnucleation. This slip-twin interaction may have also animportant influence on the non-Schmid behavior observedwhen single crystals have been experimentally tested. Forinstance, Kelley and Hosford [1] showed a surprising be-havior of non-basal slip and twins nucleating in samplesdesigned to favor only one single deformation mode basedon assumptions from the Schmid law classically used insingle lattice structures such as FCC and BCC. This had

led to considerably conflicting values of threshold stressesof non-basal slip and twinning systems.

So, how can the influence of resolved stress compo-nents and heterogeneities be addressed using moleculardynamics simulations? Molecular dynamics simulationsthat use pre-existing defects in a crystal lattice makeit difficult to distinguish what is the effect of resolvedstress components and what is the effect of the intro-duced heterogeneities. However, previous simulations onhomogeneous and heterogeneous dislocation nucleationin FCC crystals may provide the framework for answer-ing this question. Tschopp, Spearot, and McDowell [5]approached this problem in FCC Cu by: 1) analyzingthe influence of resolved stresses on homogeneous dislo-cation nucleation in single crystal Cu, 2) formulating aphenomenological nucleation that accounted for resolvedstress components, 3) analyzing the influence of addeddefects on heterogeneous dislocation nucleation, and 4)inserting a damage-like parameter to capture the result-ing influence due to lattice hetereogeneities. While twinand dislocation nucleation in HCP metals is considerablydifferent from dislocation nucleation in FCC crystals, themethodology can be applied to HCP metals. The key firststep in this methodology is understanding how resolvedstresses impact homogeneous nucleation in single crystalsand developing a fundamental phenomenological modelthat can capture this. In this paper, we have used MDsimulations to explore homogeneous dislocation and twinnucleation in single crystal Mg under uniaxial loadingconditions as a function of crystallographic orientation.The crystallographic orientation of loading will impactthe resolved shear stresses on various dislocation and twinsystems within Mg. This will enable us to explore the re-lationships between the actual mechanisms, multiple po-tential slip/twin systems, and the corresponding Schmidand non-Schmid resolved shear stresses to try to developbetter phenomenological models for twin nucleation as afunction of crystallographic orientation.

Simulation Methodology

A parallel molecular dynamics code (LAMMPS, [6]) thatincorporates domain decomposition is used to deform thesingle crystal atomistic models. A similar methodologyto that previously used to examine homogeneous disloca-tion nucleation in single crystal FCC crystals is used here[7; 8; 9]. First, the configuration is equilibrated using MDin the isobaric-isothermal (NPT) ensemble at a pressureof 0 bar and a temperature of 100 K for 10 ps. Next, theconfiguration is deformed in uniaxial tension at a con-stant strain rate of 109 s−1 with a stress-free condition

Figure 1: An example stress-strain curve showing twinnucleation at the maximum tensile stress.

for the other two boundaries. For mechanical properties,the system stress is calculated using the virial definition.The stress required for twin or dislocation nucleation isdefined as the maximum uniaxial tensile stress. Figure 1shows a stress-strain curve for the 〈2116〉 loading orien-tation at a strain rate of 109 s−1 and a temperature of100 K. As shown in Fig. 1, visualization of selected ten-sile axis orientations along the appropriate planes showedthat twins/dislocation are nucleated at a displacementvery close to the maximum tensile stress for all singlecrystal models. In most cases, twins appeared to nucle-ate slightly before the maximum tensile stress is reached(<1.0% below the maximum tensile stress). However, inlight of the difficulty of visually ascertaining exactly whenthe twin nucleates (i.e., how many spatially clustered,distorted atoms on the twinning plane constitute a singletwin embryo), the maximum tensile stress provides an ac-curate indication of the stress required to homogeneouslynucleate a twin in each single crystal model.

The Sun et al. [10] embedded-atom method potentialfor Mg is employed in this study. The Sun et al. potentialwas fit to crystal, liquid, and melting properties of Mgand dislocation core structures compare well comparedto ab initio and experimental values [11]. Furthermore,Yasi and colleagues [11] found that the Sun et al. potentialbetter captured the splitting distance of dissociated edgeand screw partial dislocations on the basal plane thanthe Liu et al. [12] potential when compared to ab initio

results. Additionally, the Peierls stresses for basal (0.3and 3.6 MPa for edge and screw) and prismatic slip (13and 44 MPa for edge and screw) are in agreement with ex-periments. For all these reasons, the aforementioned po-tential is deemed sufficient to characterize homogeneoustwin nucleation in single crystal Mg.

A 3D periodic computational cell is used to investigatehomogeneous twin nucleation in single crystal Mg underuniaxial tension at a constant strain rate. Cell dimensionsare chosen to properly enforce the 3D periodic boundaryconditions for each orientation, with a minimum length of20 nm in all directions. This length is chosen to minimizeboth the effect of periodic boundaries on twin nucleationand the number of atoms in the system for computationalefficiency. The non-integer nature of the c/a ratio forthe interatomic potential complicates the generation of a3D periodic cell in HCP metals. Although not explicitlyshown in this work, simulations investigating the effectof the periodic length validate that the stress requiredfor twin nucleation is essentially unaffected by further in-creases in the size of the simulation cell. This minimumperiodic length results in cells with sizes on the order of105-106 atoms. Therefore, the minimum length of 20 nmis deemed sufficient to avoid significant effects of periodicboundaries on the 3D dislocation nucleation dynamics.

Figure 2 shows the tensile axis for the 13 crystallo-graphic orientations examined in this work, within the ba-sic stereographic triangle with 〈0001〉, 〈101̄0〉, and 〈12̄10〉vertices. Each single crystal model is deformed underuniaxial tension at a temperature of 100 K. The expectedtwinning system is the {101̄2} 〈101̄1̄〉 twin system for alltensile axis orientations in the 〈0001〉-〈101̄0〉-〈12̄10〉 tri-angle.

The computational methodology used to investigatetwin nucleation in this work differs from MD simulationsfor HCP metals described in previous literature. First,we have chosen to use a larger number of orientations toexplore the anisotropy in twin nucleation throughout thestereographic triangle. Second, the simulation boundaryconditions are different than previous simulations. Manyprevious simulations have used a fixed boundary to applytension or simple shear, thereby admitting heterogeneoustwin nucleation at the free surface and fixed regions. Thepresent simulations incorporate a 3D periodic computa-tional cell with no free surfaces or fixed regions and twinnucleation occurs homogeneously as a function of the lo-cal stress state. The current uniaxial deformation simu-lations of twin nucleation take into account the resolvedshear stress, the resolved normal stress, and natural cou-plings that exist between resolved stresses on the twinningand slip planes.

Figure 2: Uniaxial loading orientations and conditions forsingle crystal deformation simulations.

Simulation Results

The stress-strain curves for the 13 crystallographic load-ing orientations are shown in Figure 3. As in Fig. 2,the single crystals are loaded in tension and the stressincreases until a peak stress, at which either a dislo-cation, twin, or void/crack nucleates within the singlecrystal. For each loading orientation, the deformationmechanism was characterized by examining the relevantslip/twin planes and directions to ascertain what defectcorresponded to the peak nucleation stress. The stress-strain curves are colored according to deformation type.In many loading orientations, the stress required to nucle-ate dislocations tended to be lower than that for twins. Inthe curve with the highest nucleation stress, neither dis-locations or twins were observed to nucleate. It should benoted that these stresses are the uniaxial tensile stressesrequired for the nucleation events; the resolved stressesare lower.

Of the 13 crystallographic orientations studied, fiveorientations nucleated twins and seven orientations de-formed via slip. The final orientation was loaded in thebasal 〈0001〉 direction and showed no evidence of twinningor slip. Interestingly, this orientation has the highest like-lihood of twinning under tensile stress, yet no twinningis observed. Again, recall that the single crystal modelsused here are absent of point defects, pre-existing disloca-tions, or free surfaces. Perhaps the lack of twin nucleation

Figure 3: Stress-strain curves for all single crystal models.

suggests that other point, line, or planar defects need tobe present in the lattice in order to spur the twin nu-cleation process. Additionally, analysis of resolved stresscomponents may shed light on this phenomenon.

The crystallographic loading orientations which nucle-ated dislocations by slip were all characterized as basalslip, as would be expected from the Schmid resolved shearstress. The twins were easily identified, but further clas-sification of twinning plane and directions is still ongoingwork. It appears that a number of twin modes were ob-served in these simulations.

The influence of temperature (100 K, 200 K, 300 K,400 K) and strain rate (108-1010 s−1) was examined fora few loading orientations. In the loading orientationsexamined, the temperature range examined affected thenucleation stress and strain, but the mechanism tendedto be unaffected. Higher temperature simulations at 400K showed a lower elastic modulus, nucleation stress, andstrain at nucleation. Interestingly, the rate of propagationof the twinned structure appeared more rapid in simula-tions run at higher temperatures. Simulations at strainrates from 108 s−1 to 1010 s−1 showed very little vari-ation in results. These conditions nucleated the samedeformation mechanism, at near the same stress, withthe nucleating stress decreasing slightly as strain rate de-creases. The choice of a strain rate of 109 s−1 appearsvalid. Interestingly, in many of the loading orientationswhich nucleated twins, one variant was observed to nucle-ate with multiple twin variants observed thereafter as thestrain was increased. In only one loading orientation, the〈101̄0〉 orientation, were multiple twin variants nucleated

at the peak stress.

The resolved stress components may shed light on whyeach loading orientation nucleated either twins or dislo-cations. Figure 4 shows the resolved shear stresses cal-culated from the nucleation stress and the Schmid fac-tor for various twinning/slip systems. Here, we calcu-lated the resolved shear stresses for {101̄2} 〈101̄1〉 twins,{0001} 〈112̄0〉 basal dislocations, {101̄0} 〈112̄0〉 prismaticdislocations, {101̄1} 〈112̄0〉 pyramidal dislocations, and{112̄2} 〈112̄3〉 pyramidal 〈c+a〉 dislocations. The resolvedstress for all systems are plotted along with the observedtwin/dislocation system (circled in black). The observedresolved shear stress varies with loading orientation, butin nearly all the loading orientations the observed mech-anism had the highest Schmid factor and, therefore, thehighest resolved shear stress. All of the loading orienta-tions characterized as nucleating dislocations on the basalor prismatic slip systems predicted that that slip systemhad the highest resolved shear stress. The 〈101̄0〉 loadingorientation nucleated twins at a very low resolved shearstress, but the ‘2’ indicated on the plot refers to the simul-taneous nucleation of two twin variants. In this case, thelow Schmid factor may be accommodated by the ability tosimultaneously nucleate on two twin systems. Addition-ally, while the observed twinning system did not alwayshave the highest resolved shear stress, the prismatic andpyramidal slip systems are known to have higher critical

Figure 4: Resolved shear stresses on the twinning andslip systems for different loading orientations and the ob-served twin/dislocation system (circled in black).

resolved shear stresses, possibly explaining the nucleationof twins for these orientations. The 〈101̄2〉 loading orien-tation appeared to nucleate twins even though the basalslip mechanism has a higher resolved shear stress. It isnot known why this occurs; future work will investigatethe potential influence of non-Schmid resolved stresses onnucleation as well as the role of the interatomic potential.It is clear that only calculating resolved stresses via theSchmid factor may not be able to completely capture thecomplicated slip-twin nucleation event. Furthermore, ex-amining in greater detail the atomic positions and crys-tallography corresponding to the twin nucleation eventitself may shed light on whether dislocation nucleationplays a key initial role in the nucleation of twins on thesesystems.

Summary

The objective of this research is to elucidate the mech-anisms of dislocation/twin nucleation in single crystalsand grain boundaries for Mg. Here, we concentrate onsingle crystal calculations, where we used molecular dy-namics simulations to study the influence of crystallo-graphic orientation on dislocation/twin nucleation underuniaxial loading. Multiple crystallographic loading ori-entations from around the stereographic triangle weredeformed under uniaxial tension to examine the role ofresolved stresses on twin/dislocation nucleation. The re-sults of this work detailed the influence of crystallographicloading orientation in single crystals on dislocation andtwin nucleation mechanisms in magnesium. Here, wehave shown that the Schmid resolved shear stress ade-quately predicts basal or prismatic slip in all orientationswhich were observed to nucleate dislocations on the basaland prismatic slip planes. However, in orientations wheretwins nucleated, the Schmid resolved stresses may notalone be able to completely capture the complicated slip-twin nucleation event.

Acknowledgments

This work was performed at the Center for Advanced Ve-hicular Systems (CAVS) at Mississippi State University.The authors would like to acknowledge funding from theDepartment of Energy.

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