Anisotropic Size-Dependent Plasticity in Face-CenteredCubic Micropillars Under Torsion

ILL RYU,1,4 WEI CAI,2 WILLIAM D. NIX,3 and HUAJIAN GAO1

1.—Present address: School of Engineering, Brown University, 182 Hope Street Barus & HolleyBldg., Rm. 737, Providence, RI 02912, USA. 2.—Department of Mechanical Engineering, StanfordUniversity, Stanford, CA 94305-4040, USA. 3.—Department of Materials Science and Engineer-ing, Stanford University, Stanford, CA 94305-4040, USA. 4.—e-mail: ill_ryu@brown.edu

Three-dimensional dislocation dynamics (DD) simulations are performed toinvestigate the size-dependent plasticity in submicron face-centered cubic(FCC) micropillars under torsion. By using a previously implemented surfacenucleation algorithm within DD, we show that the plastic behavior of FCCmicropillars under torsion is strongly affected by the crystallographic orien-tation: In h110i oriented submicron pillars, coaxial dislocations nucleate andpile up near the axis, leading to homogeneous deformation along the pillars. Incontrast, in h100i and h111i oriented pillars, heterogeneous plasticity has beenobserved due to the formation of localized dislocation arrays. As a result of theexistence of a coaxial slip plane in h110i oriented pillars, stronger size-de-pendent plasticity is observed in this case compared with those in other ori-entations.

INTRODUCTION

The size-dependent mechanical properties ofmetals at small scales have been of great interest asthe feature dimensions of modern devices are con-tinuously getting smaller. Compared with theirbulk counterparts, micron- and submicron-scalemetallic samples have shown elevated flow stress invarious loading conditions, including micropillarcompression/tension,1–10 wire torsion,11–14 foilbending,15–17 and micro- and nano-indentation.18–20

However, conventional plasticity theories areinherently size independent and unable to capturethe observed size effect. In addition, it can be diffi-cult to set up experiments to measure small-scalemechanical responses of materials and to investi-gate the detailed microstructure during deforma-tion. Recently developed experimental techniquesbased on in situ transmission electron microscope(TEM)21–23 and Laue micro diffraction24,25 couldprovide a large amount of unprecedented informa-tion, but it remains challenging to employ thesetechniques to track evolving dislocation structuresin microscale samples.

When materials deform under inhomogeneousloading, the observed size effects are generallyattributed to plastic strain gradients associated

with geometrically necessary dislocations(GNDs).26,27 According to strain gradient plasticity(SGP) theories, GNDs are accumulated to accom-modate plastic strain gradients, and this leads toadditional hardening through GND interaction withstatistically stored dislocations (SSDs). However,the SGP theories are usually formulated in a phe-nomenological way without accounting for thedetailed dislocation microstructure.

In this article, we perform dislocation dynamics(DD) simulations to investigate the plastic responseof single crystalline metallic submicron pillarsunder torsion by taking into account the collectivebehavior of dislocations.28–37 Our study builds onexisting research on the mechanical response ofsingle crystalline metallic micropillars under uni-axial loading. Due to the high surface-to-volumeratio at a small scale, it is necessary to account fordislocation sources not only from dislocation inter-actions in the interior of the samples but also fromnucleation at free surfaces. Most existing DD mod-els consider only the internal sources while ignoringthose associated with dislocation nucleation at thesurface. To avoid this deficiency, we have previouslydeveloped an algorithm within the DD framework toaccount for dislocation nucleation at free surfacesbased on both atomistic modeling and reaction rate

JOM

DOI: 10.1007/s11837-015-1692-1! 2015 The Minerals, Metals & Materials Society

theory.38 From our DD simulation results, it hasbeen observed that the plastic response ofmicropillars under torsion is strongly influenced bytheir crystallographic orientation: In h110i orientedsubmicron pillars, Eshelby-like coaxial dislocationsnucleate at the surface and pile up near the pillaraxis, whereas twist boundaries are formed by arrayof dislocations in h100i and h111i oriented pillars. Asa result, homogeneous plastic deformation isobserved along the h110i pillars, while localizedplasticity is seen in other orientations. In addition, astronger size dependence is observed for plasticityin h110i oriented pillars compared with those inother orientations. The dislocation microstructuresfrom our DD models show qualitative agreementwith previous molecular dynamics (MD) simulationof nanowires under torsion.39,40

SIMULATION METHODS

A three-dimensional DD model is used to studythe size effects of plasticity in FCC submicron pil-lars. To explore the orientation dependence, weconsider micropillars with axes in three high sym-metry directions, h100i, h110i, h111i, and diametersranging from 150 nm to 1000 nm. Isotropic elasticproperties are assumed for all orientations for sim-plicity. The initial dislocation density was chosen tobe about 1013 m!2. In our DD models, dislocationsare represented as a collection of straight segmentsconnected by nodes. The Peach-Koehler force oneach dislocation segment is used to update itsposition through a linear mobility law (for disloca-tion motion in the viscous drag regime) subject tothe glide constraint. Finally, the evolution of dislo-cation microstructures is obtained by consideringtopological changes and remesh requirements. Theinitial dislocation structures consist of randomlydistributed dislocation glide loops with junctionsnaturally formed through a relaxation processdescribed by Motz et al.33 The height (h)-to-diame-ter (D) aspect ratio of the pillar is fixed at 5, and thematerial properties and controlling parameters aresummarized in Table I. Periodic boundary condi-tions are imposed along the axial direction of thepillar. The torque-controlled torsion is applied along

the cylinder axis, and detailed torsion loadingmechanism will be described in a later section. Formore detailed explanation, the reader is referred tosome of the previous publications.28,38 To obtainreasonable statistics, five simulations are performedfor each sample.

Dislocation Nucleation at the Free Surface

At the submicron scales of interest here, plasticityis generally controlled by surface nucleation of dis-locations, single-arm operation, and pinning pointsby voids or impurities. Since mechanical propertiesat a small scale can be highly sensitive to individualdislocations, it is necessary to take all possiblesource mechanisms into account in the model. Givenample internal sources from dislocation interactionsinside the sample, little attention has been paid, sofar, to dislocation sources at the surface. To thisend, we have recently developed a simple surfacenucleation algorithm within the DD framework.38

To accommodate surface nucleation, the dislocationnucleation rate (m) is adopted from reaction ratetheories based on atomistic models for Cu nanor-ods,41,42 as follows:

m ¼ m0 exp !Q r;Tð ÞkBT

! "ð1Þ

where m0 is the attempt frequency, kBT the thermalenergy, and Q the activation free energy dependingon both stress and temperature. Note that r is theexternally applied stress and the internal stressfields of the existing dislocations are neglected inthis simple nucleation model. In this study, we focuson plastic flow at room temperature. At a givennucleation rate, the nucleation site is selected fol-lowing the kinetic Monte Carlo algorithm amongrandomly distributed N possible nucleation sites onthe surface with different stress concentration fac-tors, mimicking surface roughness. Nucleation isallowed to occur by creating a dislocation half loopwhose slip system has the maximum resolved shear

stress among all 1=2 110h i= 111f g type slip systems.As mentioned,38 although this simple surfacenucleation scheme could provide valuable insightsinto the governing mechanisms for plastic defor-mation under uniaxial loading, its direct applicationto torsion is questionable due to the highly localizedcharacter of surface nucleation (because the nucle-ation model does not account for the internal stressof existing dislocations); dislocation nucleationunder uniaxial loading occurs at only a few prefer-ential sites with highest stress concentration fac-tors. In uniaxial loading, nucleated dislocationseasily move out of the opposite side of sampleswithout interaction with others, and the plastic flowbehavior is not significantly influenced by thelocalization of nucleation. However, since disloca-tion pile-ups under torsion and bending have beenobserved in other DD work and experiments,17,43,44

Table I. Material properties (Cu) and operatingparameters

Material andparameters Dimension Value

Shear modulus [GPa] 48Poisson ratio + 0.34Edge mobility [Pa!1 s!1] 105

Screw mobility [Pa!1 s!1] 105

Burgers vector length (b) [m] 2.556 9 10!10

Temperature [K] 300Attempt frequency (m0) [sec!1] 1 9 1013

Coefficient (c) + 0.2

Ryu, Cai, Nix, and Gao

the highly localized nucleation could give rise tooverestimation of strain hardening from excessivedislocation pile-up.

To overcome this deficiency, we modified thenucleation algorithm to delocalize nucleation sites.Following the mechanistic model to explain thelength-induced ductile–brittle transition by Wuet al.,45 we adjust the stress concentration factorunder torsion with respect to a nucleation event asfollows:

Knew

Kold¼ 1! c

DhNucP

hE

!pR2

pR2 ! dS

! "ð2Þ

where Kold,Knew are the stress concentration factorsfor the selected nucleation site before and after thenucleation; DhNuc

P and hE are the incremental plastictwist angle per length from the nucleation andelastic twist angle per length, respectively; R is theradius of the micropillar and dS is the projectedarea change due to the surface step from nucleation;and c is a constant that describes how nucleationeffectively decreases the total torque. The firstparenthesis results from a decrease in torsionalstress induced by the dislocation nucleation, and thesecond one describes the stress increase from thecross-sectional area change. In this work, we focuson torsional loading only, whereas delocalizednucleation under tension could be treated in asimilar way, as follows:

Knew

Kold¼ 1! c

EDeNucP

rapplied

! "pR2

pR2 ! dS

! "ð3Þ

where DeNucP is the incremental plastic strain from

the nucleation, rapplied is the externally appliedstress, and E is Young’s modulus. We choose c = 0.2,which is found to induce delocalized nucleationunder torsion and, at the same time, and not tointroduce an artificial hardening under uniaxialloading by lowering the stress concentration factor.

Loading Mechanism for Twist

To apply torsional loading, a constant torqueincrement is applied until the rate of surface plasticstrain overcomes the given tolerance of 5 9 104 s!1,which is chosen to distinguish real strain burst fromnoise.38,46 The incremental torque is set to raise thefollowing normalized torque ( !T) by an amount equalto 0.05 MPa for all sized samples:

!T ¼ T

D3¼ lIPhE

8R3¼ l

p16

R

LuE ¼ l

p16

AuE ð4Þ

where uE is the elastic end-to-end twist angle and IPis the polar moment of inertia of the cylindricalpillar; D and L are the diameter and length of thecylinder, respectively;l is the shear modulus; andA % R=L is the aspect ratio. The incremental torqueis thus applied by imposing the same increment intwist angle for all sized samples. The length of the

cylinder is fixed by imposing an elastic strain thatcancels the plastic strain induced by the disloca-tions. Under these load conditions, the obtainedstress–strain responses are reasonably insensitiveto the loading rate under torsion.

To obtain the normalized torque-surface straincurve during torsion, it is necessary to compute theplastic twist due to dislocation motion at each timestep. Following Eshelby’s analysis of a screw dislo-cation in a cylinder,47 we can derive the followingexpression for the incremental plastic twist per unitlength (DaP) by dislocation motion in a cylinder:

DaP ¼ 1

L & Ip

X

k

rk bkznkh þ bkhn

kz

# $DAk

swept ð5Þ

where bk, nk, and DAkswept are a Burgers vector, slip

plane, and swept area of the kth dislocation seg-ment, respectively; and rk is the distance from thepillar axis to the kth dislocation segment. Thedetailed derivation will be published elsewhere.48

SIMULATION RESULTS

The normalized torques for 110h i orientedmicropillars are plotted against the surface totalstrains and surface plastic strains in Fig. 1a and bwith the results clearly showing strong size-depen-dent plastic flow. As the sample diameter decreasesfrom 1 lm to 150 nm, the normalized torque (whichis proportional to the shear stress at the surface) isroughly 50 MPa to 200 MPa at the surface plasticstrain of 0.2%. In smaller micropillars, plasticdeformation starts at higher stress and exhibitsmore pronounced strain burst or serrated plasticflow than that in larger pillars, whereas in largersamples, gradual strain hardening takes placeunder increasing applied torqued, as is typicallyobserved in micro-mechanical measurements.Figure 1c shows that the dislocation densitiesincrease approximately linearly with respect to thesurface plastic stain in all sized samples. In h110iorientated micropillars, a slip plane parallel to thecylinder axis exists that has the maximum resolvedshear stress under torsion. According to our nucle-ation scheme in which the slip plane of a nucleateddislocation is chosen to have the maximum resolvedshear stress, screw dislocations parallel to the pillaraxis could nucleate at the free surface and glidetoward and pile up at the center of the pillar wherethe resolved stress is almost zero. Due to the mod-ified nucleation algorithm described in the previoussection, the nucleation sites change on the freesurface every few operations. Figure 1d shows theevolution of dislocation microstructure of a 1-lmpillar as it reaches the plateaued state in the nor-malized torque-surface plastic strain curve. In allfigures showing dislocation microstructures, thecolors of the segments indicate the Burgers vectorsof the various dislocations, among which the redsegments indicate dislocation junctions.

Anisotropic Size-Dependent Plasticity in Face-Centered Cubic Micropillars Under Torsion

For h111i oriented pillars, the normalized torquesare plotted against the surface total strains andsurface plastic strains in Fig. 2a and b. Comparedwith h110i oriented pillars, the size dependence ofplastic flow in h111i pillars is smaller: The flowstress at 0.2% plastic strain increased from about150 MPa to 200 MPa as the sample size is reducedfrom 1 lm to 150 nm. Interestingly, the linearrelation between dislocation density and plasticsurface strain holds, as shown in Fig. 2c. Althoughthe overall trend of all plots is generally similar tothat of h110i oriented samples, a very differentdislocation microstructure has been observed in thisorientation. During the initial stage of torsion,preexisting dislocations are reoriented withoutcausing noticeable changes in the normalized tor-que-surface strain curves. Subsequently, surfacenucleation occurs on 111f g slip planes with themaximum resolved shear stress. Planar dislocationarrays form on the plane perpendicular to the pillaraxis, and dislocations are accumulated by the stress

gradient associated with torsion, which is in con-trast to the Eshelby-like coaxial dislocation networkshown in h110i oriented micropillars. Figure 2dshows the evolution of a dislocation microstructureof a 1-lm pillar in which, in the deformed configu-ration, the angle between dislocation arrays in dif-ferent slip planes are around 60", which is similar tothe formation of hexagonal twist boundaries asshown in MD simulation results in this orienta-tion.39,40 The fact that hexagonal dislocation arraysdo not form on the same slip plane in the DD sim-ulations is caused by the neglect of the internalstress of existing dislocations in the calculation ofdislocation nucleation rates.

The corresponding results of h100i oriented pillarsare shown in Fig. 3a and b. In this orientation, thesize dependence of plastic flow is smaller than thatin h110i pillars, but it is comparable with that inh111i pillars: Flow stress is at 0.2% plastic strain,which increases from about 100 MPa to 200 MPa asthe sample size is reduced from 1 lm to 150 nm.

Fig. 1. (a) Normalized torque versus total surface strain curves, (b) normalized torque versus plastic surface strain curves, and (c) dislocationdensity evolution for the 150-nm, 300-nm, 600-nm, and 1000-nm pillars oriented in a h110i direction under torsion. (d) Dislocation microstructureof a h110i oriented 1-lm pillar before/after the deformation when viewed along the pillar axis and from the side of the pillar. Microstructure of thedeformed pillar is obtained at a plateaued state in the torque-surface plastic strain curve.

Ryu, Cai, Nix, and Gao

The dislocation density is again linearly dependenton the plastic surface strain, as shown in Fig. 3c.The plastic flow behaviors and dislocation densityevolution in h100i pillars looks qualitatively similarto those in h111i pillars. During the initial stage oftorsion, the preexisting dislocations rearrange andpile up near the pillar axis, leading to homogeneousplasticity. After surface nucleation occurs at severalsites over the free surface, some rectangular (whenviewed along the pillar axis) screw dislocation net-works start to form, as shown in Fig. 3d.

To see the orientation-dependent size effect ofmicropillars under torsion more clearly, the nor-malized torques at 0.3% surface plastic strain aredetermined and plotted against the correspondingpillar diameters in Fig. 4. The log–log plot shows astrong size dependence in h110i oriented pillarswith an exponent of !0.75, whereas h100i, h111iorientated samples show a smaller size effect withexponents of !0.22 and !0.17, respectively.

DISCUSSION

For FCC micropillars under uniaxial loading, thesize-dependent plasticity is generally believed to becontrolled by dislocation sources, depending on thesample size and dislocation density. In micropillarswith diameters less than 150 nm, plastic deforma-tion is mainly governed by dislocation nucleation atthe surface, whereas for large pillars with diametersclose to 1 lm, the operation of truncated sourcesformed through dislocation interactions is believedto be the governing mechanism. In between thesetwo regimes, both surface nucleation and truncatedsource operation take place simultaneously.21,38

However, in torsion, the truncated source operationis effectively hindered by the back stress from dis-location pile-up near the pillar axis and cannot playa major role in plastic deformation. Instead, theGNDs from the strain gradient could facilitate dis-location interaction through the pile-up, which in

Fig. 2. (a) Normalized torque versus total surface strain curves, (b) normalized torque versus plastic surface strain curves, and (c) dislocationdensity evolution for the 150-nm, 300-nm, 600-nm, and 1000-nm pillars oriented in a h111i direction under torsion. (d) Dislocation microstructureof a h111i oriented 1-lm pillar before/after the deformation.

Anisotropic Size-Dependent Plasticity in Face-Centered Cubic Micropillars Under Torsion

turn contributes to an additional size effect. The sizeeffect from the GNDs could be represented by thesize exponent in h100i and h111i pillars withoutcoaxial dislocations, as shown in Fig. 4. Surpris-ingly, our DD results show that the size exponentunder torsion is actually smaller than that undertension at the same size scale, ranging from 0.5 to1.0.21,49,50 This might be understood by the fact thatplastic deformation in FCC micropillars is governedby different strengthening mechanisms under uni-axial loading and torsion, namely, single arm oper-ation in tension and dislocation pile-up in torsion,whereas surface nucleation contributes to sizeeffects under both loading conditions.

Our DD simulation results with the modifiedsurface nucleation algorithm show orientation-de-pendent microstructures during torsion. In h110ioriented pillars, a coaxial dislocation network isformed after surface nucleation, whereas planar

Fig. 3. (a) Normalized torque versus total surface strain curves, (b) normalized torque versus plastic surface strain curves, and (c) dislocationdensity evolution for the 150-nm, 300-nm, 600-nm, and 1000-nm pillars oriented in a h100i direction under torsion. (d) Dislocation microstructureof a h100i oriented 1-lm pillars before/after the deformation.

Fig. 4. Size dependence of the normalized torque in h110i, h111i,and h100i oriented micropillars under torsion. Flow stresses from DDsimulations are collected at a surface plastic strain of 0.3%.

Ryu, Cai, Nix, and Gao

dislocation arrays are formed in h111i, h100i pillars.In the latter cases, the angle between dislocationarrays is around 60" for h110i pillars and 90" forh100i pillars. These dislocation microstructures arebroadly consistent with MD results by Weinbergerand Cai.39,40 However, surface nucleation in our DDmodels generally occurs at multiple locations andthe nucleated dislocations are in parallel with eachother, whereas in MD results, dislocation of differ-ent Burgers vectors are nucleated on the sameplane, leading to the formation of twist boundaries.This discrepancy results from the fact that disloca-tion nucleation in this simple DD model is notinfluenced by the stress field of existing dislocations,whereas in MD, that effect is naturally taken intoaccount. However, since the micropillars of interestin this study are significantly larger than thenanowires considered in MD simulations, micropil-lars may have many nucleation sites on the freesurfaces. In addition, the strain rates used in MDmodels ((108 s!1) are much higher than those inexperiments and in DD. Admittedly, our presentnucleation scheme in DD will likely require furtherimprovements to give accurate predictions of thenucleation sites. To understand the nucleationprocess better, it would also be useful to performexperiments on single crystalline micropillars within situ electron microscopy.

We could observe stronger size dependence inh110i orientated micropillars compared with thosein other orientations, and the orientation depen-dence of flow stress is more pronounced in largerpillars, as shown in Fig. 4. It can be attributed tothe existence of a slip plane parallel to the pillaraxis, on which the pile-ups of coaxial or Eshelbydislocations give rise to a larger amount of plasticdeformation due to the larger swept area in thisorientation. Especially for the larger pillars, whereTaylor hardening can contribute more, the ‘‘easyglide’’ contribution for the h110i pillars leads to alower yield stress compared with the h100i andh111i pillars.

CONCLUSION

Three-dimensional DD simulations have beenperformed to investigate the governing mechanismsof size-dependent plasticity in FCC micropillarsunder torsion. By using a modified surface nucle-ation scheme, our DD simulations reveal an orien-tation-dependent size effect in plasticity andprovide a detailed dislocation microstructure undertorsion. In micropillars oriented along the h110idirection, plasticity occurs though nucleation ofcoaxial dislocations, leading to dislocation pile-upsnear the pillar axis, while only localized dislocationpile-ups are observed in the inclined slip planes inh100i, h111i oriented micropillars. Due to the exis-tence of a slip plane for easy glide of coaxial disloca-tions, a strong size effect is observed in h110i orientedmicropillars. The dislocation microstructure from

ourDD simulations are in broad agreementwith thatfrom MD simulations of much smaller nanowiresunder torsion.39,40

ACKNOWLEDGEMENTS

The authors gratefully acknowledge funding fromthe U.S. Department of Energy through the DOEEPSCoR Implementation Grant No. DE-SC0007074. W.C. and W.D.N. gratefully acknowl-edge support from the Office of Science, Office ofBasic Energy Sciences, of the U.S. Department ofEnergy under contracts No. DE-SC0010412 and No.DE-FG02-04ER46163, respectively.

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Ryu, Cai, Nix, and Gao