MULTISCALE MODEL. SIMUL c - Princeton UniversityMULTISCALE MODEL. SIMUL. c 2005 Society for...

MULTISCALE MODEL. SIMUL. c© 2005 Society for Industrial and Applied MathematicsVol. 4, No. 2, pp. 359–389

PREDICTION OF DISLOCATION NUCLEATION DURINGNANOINDENTATION BY THE ORBITAL-FREE DENSITY

FUNCTIONAL THEORY LOCAL QUASI-CONTINUUM METHOD∗

ROBIN L. HAYES† , MATT FAGO‡ , MICHAEL ORTIZ‡ , AND EMILY A. CARTER§

Abstract. We introduce the orbital-free density functional theory local quasi-continuum(OFDFT-LQC) method: a first-principles-based multiscale material model that embeds OFDFTunit cells at the subgrid level of a finite element computation. Although this method cannot addressintermediate length scales such as grain boundary evolution or microtexture, it is well suited to studymaterial phenomena such as continuum level prediction of dislocation nucleation and the effects ofvarying alloy composition. The model is illustrated with the simulation of dislocation nucleation dur-ing indentation into the (111) and (110) surfaces of aluminum and compared against results obtainedusing an embedded atom method interatomic potential. None of the traditional dislocation nucleationcriteria (Hertzian principal shear stress, actual principal shear stress, von Mises strain, or resolvedshear stress) correlates with a previously proposed local elastic stability criterion, Λ. Discrepancies indislocation nucleation predictions between OFDFT-LQC and other simulations highlight the need foraccurate, atomistic constitutive models and the use of realistically sized indenters in the simulations.

Key words. multiscale modeling, indentation, dislocation nucleation, embedded atom method,density functional theory

AMS subject classifications. 74C15, 74E05, 74E10, 74E15, 74E40, 74S05, 81V55, 92E10

DOI. 10.1137/040615869

1. Introduction. Multiscale modeling is an important tool, both in the devel-opment of high-fidelity material models and in direct application to problems thatinherently exhibit multiple scale behavior. Material response is often strongly in-fluenced by the coupling of processes occurring over a wide range of length scales.The electronic structure at the Angstrom level, dislocation interactions at the micronscale, and long-range structures over centimeters or meters all may matter. Whilesimple models such as Hooke’s law provide a useful—albeit coarse—description atlarge scales, and first-principles calculations can be quite accurate at the smallestscales, multiscale methods are necessary both at intermediate scales and when higheraccuracy is required for describing macroscopic systems.

For reviews of multiscale modeling as it pertains to solid mechanics, see, forexample, [16, 24, 49, 51, 56, 64, 73]. One view of this large field is to categorizemultiscale modeling into two classes: embedded models and hierarchical models. Ex-amples of embedded models include macroatomistic ab initio dynamics by Abrahamet al. [1, 2, 10], Rudd and Broughton’s coarse-grained molecular dynamics [63, 64],

∗Received by the editors September 27, 2004; accepted for publication (in revised form) Janu-ary 25, 2005; published electronically June 27, 2005. This work was supported by DOD-MURI andDOD-ASCI programs.

http://www.siam.org/journals/mms/4-2/61586.html†Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095-

1569 ([email protected]). This author was supported by a DOD-MURI grant and a NationalDefense Science and Engineering Graduate Fellowship.

‡Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125([email protected], [email protected]). The second author was supported by the DOE andKrell Institute through a Computational Science Graduate Fellowship.

§Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095-1569 and Department of Mechanical and Aerospace Engineering, D404A Engineering Quadrangle,Princeton, NJ 08544 ([email protected]).

359

360 R. L. HAYES, M. FAGO, M. ORTIZ, AND E. A. CARTER

and the nonlocal quasi-continuum method [37, 66, 72]. All of these approaches spa-tially embed smaller length scale models in localized regions—for example at a cracktip—with suitable handshaking often necessary to feed information between separatesimulations. The usefulness of such approaches is generally limited to specific prob-lems, but they can be a valuable tool in the study of important material processes[29, 46, 51].

Alternatively, the hierarchical approach consists of identifying the importantphysical processes and modeling each at the appropriate scale. The results at thesmaller scales are then fed upwards, either manually or concurrently, into the larger-scale simulations, finally resulting in an effective model. An example of the formermethod is the use of molecular dynamics results to construct interaction rules usedin dislocation dynamics [11], the results of which can then be used to construct con-tinuum plasticity models [83]. Concurrent hierarchical simulations include subgridmethods that are the basis for the algorithm discussed here. In these algorithms,such as sequential lamination [4], the local quasi-continuum method [72], and the in-teratomic potential finite element method (IPFEM) [75], the smaller-scale models aredirectly embedded at the integration points of the finite element model.

In developing multiscale algorithms, care must be taken in the selection of val-idation problems. The simulation of indentation is especially suitable due to thewealth of past results, the simplicity of the geometry and boundary conditions, andthe rich diversity of observed behaviors. Indentation experiments probe the initialfailure mechanisms of materials by pinpointing the minimum indenter load requiredto nucleate specific dislocation types. Both experiments and simulations are criticalcomponents of this endeavor. Experiments easily extend to macroscopic sample sizes,but real-time identification of the spatial location and character of the first dislocationis not possible [15, 23]. Minor et al. [47] used in situ transmission electron microscopyto obtain real-time images of large-scale dislocation structures formed during Al in-dentation, but they could not identify dislocation nucleation sites because the dislo-cations moved too quickly. In addition, they found that the dislocation structuresanalyzed in postmortem studies differ significantly from the dislocation structuresactually present under indenters due to dislocation movement, casting doubt on thevalidity of the interpretations of most experimental dislocation structure studies. Al-though theoretical simulations easily locate the dislocations, computational expenserequires either small indenter tip sizes with accurate material representations [61] orexperimental tip and sample sizes with less accurate material models [8, 60].

Conventional wisdom identifies the onset of dislocation activity with the first jumpin the load vs. displacement curve during indentation. The underlying assumptionsare that all dislocations produce a measurable drop in the load and that the theoreticalload vs. displacement curves generated from small indenter tips, which do show thesedrops, can be extrapolated to the larger tips used experimentally. However, Knapand Ortiz [38] recently demonstrated that the load-displacement curves are not agood indication of the onset of plasticity. Their results showed that at a typicalexperimental tip radius (70 nm), dislocations formed in (001) Au before the load-displacement curve exhibited a drop, whereas identical calculations using an indentertip radius typical of simulations (7 nm) produced drops in the load-displacement curve.Likewise, Tymiak et al. [74] studied indentation into (001) Al with spherical indentertips ranging in size from 470–18100 nm. Under equally applied loads, smaller radiusindenter tips penetrated deeper and yielded larger values of hardness than the largertips, indicating that more dislocation structures were formed with the smaller tips.To enable direct comparisons between experiments and theory, simulations should use

ORBITAL-FREE DENSITY FUNCTIONAL THEORY LQC METHOD 361

similarly sized indenters to those actually employed in experiments.The orbital-free density functional theory local quasi-continuum (OFDFT-LQC)

method captures the effects of the nanoscale on macroscopic level phenomena. Al-though this method cannot treat intermediate length scales such as grain boundaryevolution or microtexture, it is well suited to study material phenomena such as ini-tial dislocation formation or phase transformations under experimentally attainableindenter sizes. The purpose of this paper is twofold. First, we would like to detail thedevelopment of a novel technique combining the accuracy and flexibility of real-timefirst-principles calculation with the power of the LQC method [19]. Second, we wouldlike to validate this method by studying indentation-induced homogeneous disloca-tion nucleation in fcc Al with the LQC method, where the nanoscale information isobtained either from OFDFT or from an empirically based embedded atom method(EAM) potential for comparison.

2. Problem description. In the exact case, material properties depend on thecombined wavefunction of all of the particles in a solid—a completely intractableproblem. The Born–Oppenheimer approximation allows the nuclei to be handledclassically, but one is still left with a many-body electron problem. Density functionaltheory (DFT) is a popular first-principles method that is capable of handling tensto thousands of atoms (depending on species and algorithm), but such problem sizesare still of limited applicability to general material science problems. Molecular dy-namics has been used to study hundreds of millions of atoms, but it requires massivecomputer resources and simplified interatomic potentials, and such simulations stilldo not represent a significant material volume. It is thus necessary to combine theideas of such methods with more traditional solid mechanics techniques to developmore robust material models.

The LQC finite element method has been highly successful in treating engineering-sized systems [71, 75], but its application is limited by the accuracy of the underlyingatomistic level calculations. Previous implementations relied upon empirical analyticinteraction potentials for treating atomic interactions. Changing to a first-principlesmethod at the atomistic scale is highly advantageous because it in principle improvesthe trustworthiness of the results, especially under complicated strain or compositionconditions, and it eliminates the need to fit analytic potentials. Empirical poten-tials can give good results but usually only when the local environment matchesthe conditions to which they were fit; if we add different atoms to form an alloy orhighly deform the structure, then the empirical method may very well give incorrectanswers. Therefore, our strategy is to keep the reliable LQC method for the macro-scopic calculations but replace the empirical atomic-scale calculations with the fastand inexpensive first-principles orbital-free density functional theory (OFDFT).

3. Methodology.

3.1. LQC method. The local formulation of the QC theory [72] begins witha finite element construction, where, instead of using material constitutive relationsderived from macroscopic properties, these relations are extracted from underlyingatomistic calculations. The IPFEM method by Van Vliet et al. [75] is an independentimplementation of the LQC method. Many excellent reviews [5, 9, 32, 86, 87, 88] ofthe finite element method exist, so we will only briefly highlight the parts critical toour method. Uppercase variables refer to the undeformed material reference frame,while lowercase variables pertain to the deformed spatial frame.

In this approach, the deformation field of the body is described within an element


Fig. 3.1. Illustration of the Cauchy–Born hypothesis showing a single element with deformedunit cells at each interior mesh integration point (crosses). The nodes are depicted as black circleson the edges of the element.

by the nodal deformations and the usual finite element shape functions:

x =∑

i∈nodes

Nixi,(3.1)

where Ni is the shape function of node i and xi are the corresponding nodal coor-dinates in the deformed configuration. The present implementation uses ten-nodequadratic tetrahedral elements, as shown in Figure 3.1. In order to allow for largestrains, the energy is required to be a function of the deformation gradient F , whichis the gradient of the deformation mapping (3.1) with respect to the material coordi-nates:

F =∑

i∈nodes

xi∂Ni

∂X,(3.2)

where X are the nodal coordinates in the undeformed reference frame. The globalsolution is obtained by energy minimization, where the total energy E(x) is obtainedby integrating the energy density W (F ) over the entire body. In the finite elementformulation, this integral reduces to performing a sum over all of the integrationqpoints in every element:

E(x) =∑

i∈elements

∑j∈qpoints

vjiWji (F ),(3.3)

where vji are quadrature weights.In the LQC or Cauchy–Born approach, the energy density at each quadrature

point is obtained by applying the local deformation ((3.2)) uniformly to the properlyoriented crystallographic unit cell, Figure 3.1, and then computing the resulting energyand stress tensor, with either EAM, OFDFT, or another suitable model. Given theperiodic boundary conditions of each unit cell, this method approximates the materiallocal to each integration or quadrature point as a perfect infinite crystal undergoingthe specified uniform deformation. Local electronic effects are accounted for in thecase of OFDFT, but each unit cell, apart from interactions through the global energy,is isolated from its neighbors.

3.2. OFDFT. DFT is a first-principles electronic-structure method that deter-mines the ground state total energy of atoms, molecules, and crystals using the elec-tron density, while the ion positions are assumed fixed per the Born–Oppenheimer


approximation. Energy derivatives can then be used to compute the mechanical stress,while relaxation of the ion positions yields an optimized atomic structure. A basicintroduction to DFT is given by Argaman and Makov [3], while Parr and Yang [52]is a standard reference. For a review of DFT and the complexities of its applicationto periodic systems, see Payne et al. [53] and the references therein.

DFT relies upon two basic theorems. In 1964, Hohenberg and Kohn [31] provedthat there was a one-to-one mapping of the electron density to the potential up toan additive constant, and hence the total energy can be expressed solely in terms ofthe density. The energy can be broken up into electronic potential, electronic kinetic,and ionic energy terms:

ETot[ρ(�r); F ] = EHart[ρ(�r); F ] + Exc[ρ(�r); F ] + Ts[ρ(�r); F ] + EExt[ρ(�r); F ] + Eii[�r; F ],

(3.4)

where EHart[ρ(�r); F ] is the Hartree electron-electron classical Coulomb repulsion en-ergy, Exc[ρ(�r); F ] is the purely quantum mechanical electron exchange-correlationenergy, Ts[ρ(�r); F ] is the electronic kinetic energy, EExt[ρ(�r); F ] is the external (typ-ically ion-electron) potential energy, and Eii[�r; F ] is the ion-ion energy. Here “ion”could be either the bare nuclei or “ions,” i.e., nuclei screened by their core electrons.Although the exchange-correlation functional is only known in a few limits, the contri-bution is small relative to the total energy. The local density approximation (LDA),exact in the limit of uniform electron density, has proven to be reliable for metals andother nearly free-electron-like materials.

The second theorem uses the variational principle to determine the minimumground state energy:

δETot[ρ(�r); F ]

δρ(�r)=

δEHart[ρ(�r); F ]

δρ(�r)+

δExc[ρ(�r); F ]

δρ(�r)+

δTs[ρ(�r); F ]

δρ(�r)+ VExt[�r; F ] = μ.

(3.5)

Here μ is the chemical potential, and VExt = δEExt

δρ .DFT calculations of periodic systems generally use plane wave basis sets to rep-

resent the density. Near the center of the atom, the electron density has many sharppeaks due to orthogonality constraints on the electronic orbitals. These peaks areexpensive to represent with plane waves but do not contribute to interatomic bond-ing. Therefore, the potential in the inner atomic core region is often replaced bya smoothly varying pseudopotential which decreases the number of required planewaves and allows only the valence electrons to be treated explicitly.

Kohn and Sham [39] suggested the density be expanded into a set of orbitals:

ρ(�r) =N∑i

ϕ∗i (�r)ϕi(�r).(3.6)

This has the advantage of providing a simple formula for the exact electronic kineticenergy for a set of noninteracting electrons whose density is the same as the densityof interacting electrons:

Ts[ρ(�r); F ] =

N∑i

⟨ϕi(�r)

∣∣∣∣−1

2∇2

∣∣∣∣ϕi(�r)

⟩.(3.7)


However, the introduction of orbitals comes at a cost: the required orthogonalizationof the orbitals. Since orthogonalization is an O[N3] operation, large calculationsbecome expensive. Furthermore, orbitals require k-space sampling [53], which canincrease the computational cost by two to five orders of magnitude if accurate stressesin metallic systems are required, as in this study. If only a few Kohn–Sham (KS)DFT calculations were needed, the expense would not be a problem, but duringthe indentation simulation, millions of first-principles calculations will be required.Traditional KSDFT calculations are prohibitively expensive; a faster method must beused.

If the electron density is used directly, the calculations scale as O[N ln(N)] sincethe orthogonalization and k-point sampling steps are eliminated. Unfortunately, theexact form of the kinetic energy functional directly in terms of the density is unknown.Nonetheless, OFDFT has proven to be reliable and efficient for simple metallic sys-tems when a linear-response-based kinetic energy functional [21, 55, 69, 77, 79, 80] isemployed.

Using only the density also impacts the form of the pseudopotential. KS calcula-tions typically utilize nonlocal pseudopotentials, where the ion-electron potential feltdepends on the orbital angular momentum. This angular momentum dependence typ-ically improves the accuracy of the KSDFT calculations but cannot be done in OFDFTbecause there are no orbitals. However, Zhou, Wang, and Carter [84] recently demon-strated that the error in OFDFT calculations can be limited to the kinetic energyfunctional when a new method based on bulk KSDFT densities is used to constructlocal pseudopotentials. This should remove the error due to local pseudopotentialsfrom future work.

The DFT-LQC method requires two physical quantities from the underlyingOFDFT calculation: the total energy and the stress tensor. The analytic expressionsfor the individual terms of the OFDFT stress tensor, originally derived by Jesson [33]except for Tα,β

s , are given in the appendix. Only linear scaling OFDFT reduces thecomputational cost sufficiently to allow nanoindentation simulations of metallic sys-tems from first principles.

3.3. Dislocation emission criteria. There are a variety of failure criteria inuse to predict the onset of plasticity. Some methods calculate the shear stress resolvedon each of the important crystallographic directions and compare these values againstsome critical resolved shear stress [22, 35, 36, 89]. Others investigate the values of thevon Mises stress or principal shear stress compared to the theoretical shear strength ofthe material [15, 26, 71]. Experiments typically combine the maximum load before thefirst drop in the load vs. displacement curve with Hertzian contact mechanics [34] toobtain the maximum value of the Hertzian principal shear stress [26, 47, 75]. Anothermethod which has proven more effective [41, 75, 85] is to look directly at the stabilityof the material tangent stiffness or acoustic tensor. This approach, first used byHill [30] and Rice [58], is a well-known practice in continuum plasticity [7, 40, 81]. Inthe present finite strain formulation, it is more convenient to reexpress the stabilityfactor given by Li et al. [41] as

Λ =3∑

i,J,k,L=1

CiJkLNJNLkikk,(3.8)

where C are the mixed material tangent moduli, N are the normals to the slip planesin the reference frame, and k are the slip directions in the deformed, spatial frame.


Note that the slip direction is equivalent to the Burgers’ vector, except for a constantfactor. Since the tangent moduli are defined as

CiJkL =∂W

∂FiJ∂FkL,(3.9)

where W is the energy density, it thus follows from the Legendre–Hadamard condi-tion [17] that the solution loses ellipticity, and thus may become localized and unsta-ble, whenever Λ ≤ 0. In practice, we calculate CiJkL as a finite difference of the firstPiola–Kirchhoff stress tensor, P = Det[F ]σF−T ,

CiJkL =PiJ [FkL + h] − PiJ [FkL − h]

2h,(3.10)

where σ is the Cauchy stress tensor. h was chosen to be 10−2 for the DFT calcula-tions and 10−3 for the EAM calculations to ensure that differences in the stress werenumerically significant.

Van Vliet et al. [75] perform this calculation in general terms by minimizingthe eigenvalues of the acoustic tensor, CiJkLNJNL, with a zero eigenvalue signify-ing the formation of an instability. They found an exact correspondence betweenthe predicted dislocation and the dislocation that actually formed during moleculardynamics (MD) simulations. Leroy and Ortiz [40] perform a similar calculation byminimizing the determinant of the acoustic tensor and equating a nonpositive valuewith localization.

For simplicity, the present implementation explicitly uses the slip systems of an fcccrystal. Thus Λ is computed at the end of each loadstep per (3.8) for each of the {111}family of slip planes and the 〈110〉 and 〈112〉 family of slip directions, corresponding toperfect and partial dislocations, respectively. The minimum of these 24 scalar values,Λmin, is then chosen to represent the stability of the given unit cell, with a nonpositivevalue indicating the nucleation of a dislocation. Since the simulation follows only theequilibrium path, any dislocations which arise under highly nonequilibrium conditions,such as shock, will be missed.

4. Indentation setup. Al indentation has been well studied both experimen-tally and theoretically, is important technologically, and is easily treated withinOFDFT and with empirical potentials, so it is an excellent initial candidate for theOFDFT-LQC method. Two surfaces were considered: (111) and (110). Al was treatedwith first-principles OFDFT [80] and with empirical EAM potentials [18]. The bulkAl crystal is represented by a fully three-dimensional (3D) 2 μm × 2 μm × 1 μmtetragonal mesh with 210 ten-node tetrahedral elements, each with four quadraturepoints. By exploiting the symmetry of the system, the computational expense can benearly halved by using a half mesh. In each case, the surface normal is oriented alongthe z-axis, and the surface is set to z = 0.

Rather than displacing the nodes in contact with the indenter, the indenter ismodeled by adding an energy penalty to the surface nodes of the form

I =1

2μ|L|2,(4.1)

where μ is a hardness parameter and |L| is the distance that the deformed solidpenetrates the indenter, and then allowing the surface nodes to relax to the minimumenergy configuration. In practice, the energy penalty given by (4.1), summed over


X = [110]

Y = [112]

Z = [111]

(a)

X = [110]

Y = [001]

Z = [110]

(b)

Fig. 4.1. Orientation and symmetry of computational meshes used in example calculations forthe (a) (111) and (b) (110) surfaces. The +X surface is a plane of symmetry, the indenter isapplied on the +Z surface, and the other surfaces are fixed.

all surface nodes, is added to (3.3). A hard (μ ≈ 5W(unit length)2 ) 750 nm spherical

indenter was chosen to fall in the size range of common experimental tips. Unlikefully atomistic calculations, which are limited to tip sizes on the order of 10 nm, ourindenter size can be chosen to match those used in experiments.

Like all fcc crystals, the preferred slip directions are in the {111} planes in the〈110〉 directions. These 〈110〉 directions may be split into Shockley partials of 〈112〉character. For the (111) surface, the x-axis and y-axis are oriented along the [110] and[112] directions, respectively; see Figure 4.1(a). Likewise, the x-axis and y-axis for the(110) surface are oriented along the [110] and [001] directions, respectively; see Figure4.1(b). In the latter configuration, the (111) and (111) slip planes are perpendicularto the surface.

The simulation must be stopped once the first dislocation is predicted to formbecause there is no way to quickly and systematically incorporate the nonlocal effectsthat arise from the local structure around the dislocation. The bottleneck lies in thefast Fourier transforms which make the OFDFT code linear scaling but which require aperiodic crystal. In order to prevent periodic images of the dislocation core interactingwith each other, prohibitively large cells would be required. A linear scaling, real-space OFDFT method [14] would remove this constraint, enabling the simulation ofdislocations during indentation. Research towards this goal is being pursued by theauthors. Previous QC implementations which used EAM for the constitutive law donot suffer from this limitation because the calculation can be formulated entirely inreal space, allowing the mesh to resolve individual atoms.

4.1. EAM calculational details. While looking for data to validate theOFDFT-LQC results, no 3D macroscopic simulations were discovered for Al. Al-though some QC results were available, other than a recent paper by Knap and Or-


Table 4.1

Comparison of lattice spacing, aeq, bulk modulus, B, and elastic moduli, Cij , from OFDFT(this work), EAM [18], and experimental [67] results.

Source aeq (A) B (GPa) C11 (GPa) C12 (GPa) C44 (GPa)OFDFT 4.037 71.1 83.6 65.0 20.1EAM 4.032 80.9 118.1 62.3 36.7Expt. 4.03 79.4 114.3 61.9 31.6

tiz [38], these were primarily in two dimensions. Van Vliet et al. [75] have performedtwo closely related 3D simulations. The first model used MD with the Ercolessi–Adams [18] EAM Al potential to examine the dislocation structures that form whena 6.5 nm radius indenter penetrates the (111) surface. However, as Knap and Ortizshowed, the behavior is highly dependent on the indenter size. The second model theyrefer to, IPFEM, is an independent implementation of the EAM-LQC method we de-scribe here. However, they have studied EAM Cu, not Al. Therefore, it was decidedto repeat the calculations for the well-known Ercolessi–Adams Al potential using theEAM implementation made available to the public by Miller and Tadmor [45].

4.2. OFDFT calculational details and validation. The OFDFT calcula-tions [80] employed the commonly used Goodwin–Needs–Heine (GNH) local pseudo-potential [25], the LDA for the exchange and correlation [12, 54], the Wang–Teterdensity-independent linear response kinetic energy functional [77], and a kinetic en-ergy cutoff of 60 Rydberg. Electron density convergence was achieved using a real-space conjugate gradient minimization scheme. In order to quantify the error in theOFDFT energies and stress due to the kinetic energy functional approximation, wealso performed KSDFT calculations with the CASTEP code [65]. Both the LDA forthe exchange and correlation and the GNH local pseudopotential were used. Elec-tron minimization was carried out with the density mixing scheme in CASTEP. Thelocal pseudopotential requires a higher kinetic energy cutoff of at least 44.1 Rydbergcompared to typical nonlocal pseudopotential calculations. The required accuracy inthe energies and stresses were obtained by utilizing the finite basis set correction pro-vided by CASTEP and by adjusting the k-point grid depending on the application.Whenever OFDFT and local KSDFT calculations are compared, the difference arisessolely from the kinetic energy functional approximation because all other numericalparameters are either identical or converged.

Table 4.1 compares the equilibrium lattice constant, bulk modulus, and elasticconstants for our OFDFT method using the GNH pseudopotential, for publishedEAM calculations, and for experiments extrapolated to 0 K. The OFDFT latticeconstant and bulk modulus compare favorably with experimental values, but OFDFTunderestimates the C11 and C44 elastic constants near the equilibrium position. Itis not surprising that EAM Al matches the experimental elastic constants better,since elastic constants were included in the fitting algorithm for the EAM potential.However, it is less clear that the EAM potential will capture the correct physicalbehavior under highly deformed configurations, a region where it was not explicitlyfit. At the larger strains where dislocations are expected to form, the first-principlesOFDFT results may be superior.

Dislocations are expected to form in slip systems that have low energy barriers forion movement. In fcc metals, the lowest and second-to-lowest energy barriers occurin the (111) slip plane along the 〈112〉 and 〈110〉 directions, respectively. There-fore, one measure of the suitability of a method is the resolved shear stress (σ23)


-0.3 -0.2 -0.1 0 0.1 0.2 0.3

0

5

10

15

Fig. 4.2. Resolved shear stress (σ23) curve along the [112] direction (ε23) in fcc Al usingEAM (dotted line), OFDFT (solid line), KSDFT with the GNH local pseudopotential (stars), andKSDFT with a nonlocal norm-conserving pseudopotential (squares). All four methods agree fairlywell, although EAM overestimates the stress when the atoms in adjacent layers approach each other.The difference between the local and nonlocal KSDFT stress reflects errors due to the choice of localpseudopotential. Stresses are in GPa.

along the (111)[112] direction; here the y- and z-axes are oriented along the [112]and [111] directions, respectively. All of the ions were fixed at their ideal bulk fccpositions. Figure 4.2 shows that EAM, OFDFT, and KSDFT results all agree fairlywell, although EAM significantly overestimates the stress when atoms in adjacentlayers approach each other. However, the on-top configuration is highly unfavorableenergetically and therefore is not likely to play a significant role during this simula-tion. One set of KSDFT calculations (stars) were done using the same GNH localpseudopotential as the OFDFT calculations. A kinetic energy cutoff of 58.8 Rydbergand a Monkhorst–Pack grid of 16 × 11 × 7 for the 6-atom orthogonal cell was usedin KSDFT to converge the stress to 0.5 GPa. The error due to the kinetic energyfunctional is small, as seen by the agreement between the local KSDFT and OFDFTresults, so the approximation is fine. A second set of KSDFT calculations (squares)were done using a nonlocal norm-conserving pseudopotential with a kinetic energycutoff of 22.0 Rydberg. The small difference in stress, especially for the lower bar-rier where the atoms are not directly on top of each other, indicates that the errorintroduced by using a local pseudopotential is not unreasonable. The small peaks inthe OFDFT stress are due to a singularity at specific reciprocal space vectors arisingfrom the Lindhard response function (χLind in (A.22)) that appears in the Wang–Teter kinetic energy term. However, the discontinuity in the Lindhard function isthought to give rise to the physically important Friedel oscillations in the density, soit cannot be omitted. Instead, to minimize spurious contributions to the stress due tothe Wang–Teter kinetic energy functional, those contributions are set to zero when areciprocal space vector nears the singularity.

The unstable stacking fault energy (γus), the intrinsic stacking fault energy (γisf ),the unstable twinning energy (γut), the extrinsic stacking fault energy (γesf ), and thetwinning energy (2γt) provide an alternative measure to gauge whether the variousmethods have captured the material properties correctly. Indeed, these quantities


Table 4.2

Comparison of stacking fault and twinning energies. (ur) signifies that the ions were fixed totheir ideal bulk positions. (r) signifies that the ions were allowed to relax and the crystal to expandin the [111] direction. All values are in mJ/m2. References are in [ ].

Source γus γisf γut γesf 2γtγisf

γus

γusγut

OFDFT (r) 73 35 94 34 15 0.48 0.77OFDFT (ur) 86 41 107 36 16 0.47 0.81

KSDFT(local) (ur)

89 47 120 70 26 0.53 0.74

KSDFT(nonlocal) (r)

224 [44] 164 [44] 138 [28] 122 [28] 0.73

EAM (r) [70] 128 106 169 117 116 0.83 0.76

Expt.135 [68],166 [48]

180 [68] 150 [68]

have been published using a variety of theoretical methods. Rice [59] and Tadmor andHai [70] have used ratios of these materials properties (γisf/γus, γus/γut) to predictwhether a material will emit a partial dislocation or twin from a crack tip. Conceptu-ally, these quantities can be calculated by the rigid displacement of two (111) crystalsurfaces along the 〈112〉 direction. In order to avoid introducing surfaces and to re-duce the total number of layers needed, we use the configuration detailed by Bernsteinand Tadmor to calculate these quantities [6]. γisf , γus, and 2γt are obtained froma 20-layer 3D periodic cell whose final configuration contains two intrinsic stackingfaults and two twin boundaries, while γut and γesf are extracted from a 22-layer 3Dperiodic cell whose final configuration contains two twin boundaries and two extrinsicstacking faults. In our case, each [111] layer contains two atoms with in-plane period-icity along the [110] and [112] directions. Table 4.2 compares stacking fault energiesfor OFDFT, KSDFT, EAM, and experiment. Both the OFDFT and KSDFT (lo-cal) calculations were done with the GNH local pseudopotential. The KSDFT (local)calculations utilized a kinetic energy cutoff of 44.1 Rydberg and a Monkhorst–Packk-point grid of 19 × 11 × 1 (maximum spacing between k points ≈ 0.02 A−1). Thetotal energy was converged to 1 meV/atom with respect to kinetic energy cutoff andk-point sampling, resulting in an uncertainty in the stacking fault and twinning en-ergies of ±8 mJ/m2. Experimentally, the ions relax, and the unit cell expands in the[110] direction relative to the bulk equilibrium configuration. However, this adds sig-nificantly to the computational expense of the calculation, especially with the KSDFTmethod. Two series of OFDFT calculations, an unrelaxed and relaxed case, were doneto estimate the error that arises from constraining the ions and lattice vectors to theirideal values. The difference in energies is minimal, because Al is a nearly free-electronmetal that is known to exhibit little ionic relaxation at surfaces [50]. Hence the addi-tional computational expense to allow relaxation in the KSDFT (local) calculationswas deemed unnecessary.

The close agreement between the OFDFT and KSDFT (local) stacking fault andtwinning energies demonstrate that the majority of the error is attributable to thelocal pseudopotential. Fortunately, it should be possible to drastically improve theagreement between OFDFT and KSDFT nonlocal pseudopotentials calculations whenthe recently developed bulk local pseudopotentials of Zhou, Wang, and Carter [84]become available.

EAM predicts larger stacking fault energies than OFDFT, indicating that dislo-cations should form more readily in the OFDFT rather than the EAM-based LQC


simulations. Since the EAM value is closer to experimental values, this suggests thatthe EAM model might possess more predictive capabilities than OFDFT for Al. How-ever, if the dislocations form more easily during the EAM-LQC simulation than inthe OFDFT-LQC simulation, other factors beyond the resolved shear stress in the[112] direction must be critical to initial dislocation formation. As we shall see, thisis indeed the case.

Experiments and theory disagree about the relative ordering of γisf , the ener-getic cost to form a stacking fault; γesf , the energetic cost to form a two layer thickmicrotwin; and 2γt, the energetic cost to form a full twin. The only experiment [68]which measures all three quantities finds γisf < 2γt < γesf . Using the more commonlycited value, γisf = 166 mJ/m2 switches the order of γisf and 2γt. By contrast, mostKSDFT theoretical studies [6] find either 2γt < γesf < γisf or 2γt = γesf < γisf ,regardless of whether or not the ions are allowed to relax in the [111] direction. Sincethe experimental measurement of γesf is difficult to measure and may contain errorsand since we are using a method that approximates KSDFT, the KSDFT (nonlocal)results are the appropriate comparison. Even though the magnitude of the stackingfault and twinning energies are too low with the OFDFT method, the relative order-ing (2γt < γesf < γisf ) is correct—a feature the EAM potential fails to capture. Arecently revised Al EAM potential [43] yields an intrinsic stacking fault energy closerto experimental values than other EAM potentials, but whether it also yields thecorrect relative ordering awaits testing.

Although the current OFDFT model does not provide a superior materials de-scription for single crystal Al, it does provide an excellent initial test case with whichto assess the feasibility of the OFDFT-LQC method. The power of the OFDFT-LQCmethod lies in more complicated systems than Al. Complex metallic materials likealloys or those with multiple possible phases under a localized strain field can becalculated within OFDFT without additional modifications, while fitting an EAMpotential to study such systems becomes a formidable and tedious task.

5. Results.

5.1. Load-displacement curve. The load vs. displacement curve provides alink between indentation experiments and theoretical simulations. Figure 5.1 com-pares the load vs. displacement curves for indentation into the (a) (111) and (b) (110)surfaces using the OFDFT-LQC (solid) and EAM-LQC (dashed) models. The dis-placement where the first dislocation is predicted to form is denoted by a “∗”. For the(111) surface, the OFDFT-LQC model predicts that the first dislocation will form ata displacement of 50 nm and a load of 0.98 mN, while the EAM-LQC model predictsa smaller displacement of 35 nm and a nearly equal load of 1.04 mN. By contrast,the OFDFT-LQC model predicts a larger displacement of 70 nm and a larger load of1.80 mN, while the EAM-LQC model remains somewhat closer to the (111) resultswith a displacement of 45 nm and a load of 1.46 mN for the (110) surface. TheOFDFT-LQC model predicts that the first dislocation will form at a larger displace-ment than the EAM-LQC model, regardless of the surface, but at a smaller load forthe (111) surface and a larger load for the (110) surface. Since OFDFT has a lowerstacking fault energy than EAM, we might expect dislocations to form more read-ily during the OFDFT-LQC than in the EAM-LQC simulation, but in practice theEAM dislocations were predicted to form at either smaller or equivalent loads to theOFDFT dislocations, indicating the stacking fault energy is not the critical factordetermining dislocation nucleation.


0 20 40 60Displacement [nm]

0

0.5

1

1.5

2

Forc

e [m

N]

0 20 40 60Displacement [nm]

(a) (b)

***

*

Fig. 5.1. The load vs. displacement curve for indentation into the (a) (111) and (b) (110)surfaces. The solid (dashed) line represents the OFDFT-LQC (EAM-LQC) model. The “ ∗” denoteswhere the first dislocation is predicted to form. Both the OFDFT-LQC and the EAM-LQC modelspredict that the (110) surface forms dislocations less readily than the (111) surface, in agreementwith experiment.

The OFDFT-LQC model correctly predicts that the (111) surface forms disloca-tions more readily than the (110) surface, in agreement with experiment [26]. TheEAM-LQC model predicts the same trend, but the effect is less pronounced. Bycontrast, a 3D MD indentation simulation with the same Al EAM potential claimedthat the (110) surface was easier to indent than the (111) surface [82], based onthe depth of indenter penetration into the surface at a fixed load. Experimentallymeasured elastic-plastic boundaries provide an upper bound to the location of thefirst dislocation. After indenting a 500 nm radius tip 92 nm into the (100) surfaceof single crystal Al, Tymiak et al. [74] used atomic force microscopy to measure anelastic-plastic boundary radius of 950 nm. The first dislocation in both our EAM andOFDFT-LQC simulations lie within this upper bound.

Several experiments estimate the load and indenter depth at which the first dis-location forms in single crystal fcc Al from the first discontinuity in the load vs. dis-placement curve. Gouldstone et al. [26] used a Berkovish indenter, where the end ofthe tip can be approximated as a spherical indenter with a radius of 50 nm, to indentboth the (111) and (110) surfaces of single crystal Al. The first displacement burstoccurred when the load reached 8 μN and 30 μN at an indenter depth of ≈ 9 nm and≈ 14 nm for the (111) and (110) surfaces, respectively. A later experimental studyby Van Vliet et al. [75] also indented the (111) surface of single crystal Al but used a150 nm radius spherical tip instead. The first displacement burst occurred when theload was 10.80 μN at an indenter depth of ≈ 13 nm. As expected, a larger indenter tipproduces dislocations at larger loads and indentation depths when applied to crystalswith identical surface orientations. One final experiment by Minor et al. [47] foundthat dislocations started to nucleate at 10 μN when a 50–75 nm radius tip indented≈ 7 nm into an unspecified single crystal Al surface. Since there is no inherent lengthscale in our LQC simulation, we can scale our results to match the experimental in-denter size as long as the quadrature points are sufficiently isolated from each other.


Based on periodic surface and stacking fault energy calculations, we estimate that theminimum distance between quadrature points should be at least a few angstroms. Ifwe scale the length in our simulation so that the indenter has a radius of 75 nm, anindenter radius that satisfies the above constraint, the previously listed loads wouldbe scaled by 1/100 and the indenter depths by 1/10, resulting in predictions that areof the same order of magnitude as the experimental results. Quantitative results arenot expected because many dislocations will likely have already formed when the loadvs. displacement curve exhibits its first discontinuity.

The only other 3D simulation that studied indentation of Al was also conductedby Li et al. [41] and Van Vliet et al. [75]. They used MD to follow the penetration of arigid 6.5 nm radius spherical indenter into the (111) surface of Ercolessi–Adams EAMAl. The first dislocation formed at an indenter depth of ≈ 6 A and a load of 0.02 μN(estimated from Figure 8(a) in [75]). If we scale their results such that the indenter hasa radius of 750 nm, the resulting load is nearly four times smaller than our (111) EAMload, once again highlighting the need for simulations with experimentally relevantindenter sizes. We speculate that the difference arises from the atomic nature of theindenter tip and surface. For instance, if the tip of a nanosized indenter first contactsthe top of a surface atom, the load response may be dramatically different than ifthe first contact was the side of an atom. For mesoscale and macroscale indentersconsidered here, the tip is flat enough that the various types of tip-surface contactsare averaged out. Since our OFDFT-LQC simulation is based on infinite crystals, itmost closely resembles the mesoscale indentation scenario.

5.2. Dislocations from indentation into the (111) surface. Figures 5.2(a)and 5.2(b) show the localization criterion, Λmin, in the x = 0 plane, for the indentationstep where dislocations are first predicted to form during indentation into the (111)surface for OFDFT and EAM, respectively. The location of the dislocations, denotedby white circles, are projected onto the x = 0 plane. Table 5.1 lists the spatiallocation, slip plane normal, N , and slip direction, k, of the first dislocations that arepredicted to form, using the instability criterion given in (3.8). Although both OFDFTand EAM predict one dislocation that slips in the [011] direction, the character ofeach is quite different. The dislocation in the OFDFT-LQC simulation forms on the(111) slip plane, 0.56 z/a below the initial surface (a is the radius of the sphericalindenter at the surface), slightly off-axis, and 104 nm from the spherical indentersurface. By contrast, the dislocation in the EAM-LQC simulations forms on the (111)slip plane, 0.16 z/a below the surface, off-axis, and only 11 nm from the sphericalindenter surface. Due to finite spacing of the LQC grid, this is essentially on theindenter surface. The lack of surfaces in the underlying constitutive calculationscast doubt on the veracity of surface nucleation. Although some simulations indicatesurface nucleation of dislocations when sharp indenter edges are present [71], surfacenucleation is probably unphysical in the spherical indenter case.

Although the calculation should be stopped after the first dislocation is formedbecause our current reciprocal-space formulation of the LQC method does not allowatomic dislocation structures to be incorporated in the underlying atomistic calcula-tions, it is instructive to look at the next couple of load steps to get an estimate of theoverall stress state of the material. Since the stress is not allowed to relax by form-ing dislocations, highly stressed regions should remain localized between load steps.Phase transformations can sometimes be observed in the LQC method by monitoringthe mesh. However, this requires that the resolution of the mesh relative to the inden-ter size be sufficiently small and that the underlying mesh have the correct structure


-1 0 1-1

0

1.32 7.24 13.16Λ min (GPa)μ m [112]

[111]

(a)

-1 0 1-1

0

2.80 15.34 27.8Λ min (GPa)μ m

[111]

[112]

(b)

Fig. 5.2. Contour plot of the instability criterion, Λmin, in the x = 0 plane for indentationinto the (111) surface of fcc Al using the (a) OFDFT and (b) EAM-based constitutive models. Thefirst dislocation occurs at an indentation depth of 50 nm for the OFDFT-LQC model and 35 nmfor the EAM-LQC model. The white circle designates the location of the first dislocation, projectedonto the x = 0 plane, as predicted by (3.8). The inset shows the local atomic configuration of theunstable point and the slip plane normal (magenta arrow) and slip direction (black arrow) of thepredicted dislocation. The x- and z-axes are shown by the blue and red arrows, respectively. Thepredicted EAM-LQC dislocation is on the indenter surface, while the OFDFT-LQC dislocation is104 nm away from the indenter surface.

to highlight the relevant phase transformation. In our case, the mesh is too coarse todisplay a phase transformation during our simulation. Therefore, we are limited toobserving how the Λ criterion behaves during subsequent load steps. Indeed, in all butone indentation simulation, once a dislocation is predicted to form, its character andgeneral location do not change in subsequent load steps. The exception occurs duringindentation into the (111) surface using the EAM-LQC model. The first dislocationis barely predicted to form (Λmin = 0 GPa) at a load step of 35 nm and then disap-pears in all future load steps. Since the dislocation appears essentially on the indentersurface, its appearance and disappearance may be due to errors that arise from the


Table 5.1

The OFDFT-LQC and EAM-LQC models predict different characters (for Al(111)) and spatiallocations (for Al(111) and Al(110)) of the initial dislocations that nucleate during indentation intofcc Al. The indentation depth when the first dislocation forms is listed in ( ). N is the slip planeand k the slip direction of the dislocation. z/a, the ratio of the distance below the initial undeformedsurface where the dislocation forms divided by the radius of the indenter in contact with the surface,is given for comparison to Hertzian analysis. There are also initial dislocations at the corresponding+x locations due to the underlying symmetry of the mesh.

Point N k x [nm] y [nm] z/a

OFDFT

(50 nm)1 (111) [011] −58.5 −38.6 0.56

(111) (35 nm) 1 (111) [011] −104.4 −52.5 0.16

EAM 1 (111) [011] −122.2 −77.9 0.16

(45 nm) 2 (111) [011] −97.3 −103.8 0.26

1 (111) [121] −195.5 26.9 0.50

OFDFT 2 (111) [121] −195.5 −26.9 0.50

(70 nm) 3 (111) [121] −168.5 52.8 0.59

4 (111) [121] −168.5 −52.8 0.59(110)

1 (111) [121] −123.8 77.9 0.16

EAM 2 (111) [121] −123.8 −77.9 0.16

(45 nm) 3 (111) [121] −125.1 38.9 0.32

4 (111) [121] −125.1 −38.9 0.32

Table 5.2

When the instability criterion, Λmin, first becomes nonpositive, a dislocation is predicted toform in the corresponding distorted fcc cell. The table lists the value of the instability criterion,slip plane, and slip direction for the first predicted dislocation during the OFDFT-LQC indentationsimulation. As a comparison, the instability analysis is repeated with both KSDFT (local) andEAM using the predicted unstable OFDFT cell geometry. All three methods predict the same type ofdislocation will spontaneously form using the distorted fcc geometry taken from the (111) indentationsimulation. By contrast, the first unstable points during the OFDFT-LQC (110) indentation arepredicted to be stable by both KSDFT and EAM.

(110) (111)

Λmin N k Λmin N k Λmin N k

[GPa] [GPa] [GPa]

Point #1 #2 #1

OFDFT 0 (111) [121] 0 (111) [121] 0 (111) [011]

KSDFT 13 (111) [101] 13 (111) [101] −6 (111) [011]

EAM 5 (111) [121] 5 (111) [121] −4 (111) [011]

Point #3 #4

OFDFT −1 (111) [121] −1 (111) [121]

KSDFT 16 (111) [121] 16 (111) [121]

EAM 4 (111) [121] 4 (111) [121]

simple indenter model. Furthermore, the next dislocations form at an indenter depthof 45 nm and load of 1.53 mN, nearly identical to the indenter depth and load in the(110) indentation. These two new dislocations persist at larger indentation depthsand share the same dislocation characters as the OFDFT-LQC simulation, castingdoubt on the validity of the first predicted EAM-LQC dislocation.

The most interesting configuration is the one where the first dislocation is pre-dicted to form, and hence a comparison of the three atomistic models at this pointwill highlight the critical differences in the underlying models. Table 5.2 compares


Λmin for OFDFT, KSDFT, and EAM methods at the atomic configurations that theOFDFT-LQC model predicted would first form dislocations. The KSDFT calculationsused a kinetic energy cutoff of 44.1 Rydberg and a k-point density of 0.010 A−1 inorder to converge the stress to 0.04 GPa for a 4-atom cell. In the case of indentationinto the (111) surface, all three methods agree that a dislocation of the same charactershould form. By contrast, for indentation into the (110) surface, only the OFDFTmethod predicts these four configurations will spontaneously form dislocations. How-ever, EAM and OFDFT both agree on which slip system and slip direction is mostlikely to become unstable for all four configurations, while KSDFT agrees only intwo of the cases. Clearly, dislocation prediction is very sensitive to the quality of theunderlying atomistic model.

The above-mentioned 3D EAM MD simulation by Li et al. [41] and Van Vlietet al. [75] agrees only partially with our EAM-LQC results. They too find that thefirst dislocation nucleates off-axis on one of the {111}〈110〉 planes but at a depthof z = 0.51a compared to our prediction of z = 0.16a. The discrepancy may arisefrom either the different indenter sizes or the implementation of the underlying LQCcode. Tadmor et al. [71] used a two-dimensional (2D) QC simulation, which resolvesdown to individual atoms in regions of high stress, to study indentation into the (111)surface of EAM Al using a 25 A rectangular indenter. By contrast to our EAM-LQC simulation, the first dislocations occur on the (111) slip planes with a [112] slipdirection. The difference is attributed to either the 2D constraint or the difference inindenter shape.

Although there are three equivalent {111} planes underneath the indenter, we ob-serve nucleation on only two of the planes. The second dislocation on the (111)[110]slip system at +x originates from the imposed symmetry of the LQC mesh. This asym-metry may arise from the underlying LQC mesh not being equivalent in the three{111} planes. However, several other indentation simulations for Au [35], Cu [85],and Al [71] have also observed asymmetry during dislocation nucleation, which theyascribe to a combination of numerical noise akin to thermal fluctuations during exper-iments and to the minimum deformation required to accommodate the imposed strainbeing satisfied by activation of only two out of the three equivalent {111} planes.

5.3. Dislocations from indentation into the (110) surface. Figures 5.3(a)and 5.3(b) show the localization criterion, Λmin, in the x = 0 plane for indentationinto the (110) surface using OFDFT and the EAM, respectively. The location of thefirst dislocations, projected onto the x = 0 plane, are denoted by white circles. Thespatial location and character of the dislocations are listed in Table 5.1. Both theEAM and the OFDFT predict that four dislocations will form simultaneously: twoon the (111) slip plane in the [121] direction and two on the (111) slip plane in the[121] direction. However, each atomistic method predicts a significant difference inthe spatial distribution of the dislocations. During the OFDFT-LQC simulation, thefirst dislocations appear at an indenter depth of 70 nm, 0.50 z/a (#1 and #2) or0.59 z/a below the surface, off-axis, 113 nm (#1 and #2) or 133 nm (#3 and #4)from the spherical indenter surface, and on adjacent quadrature points. By contrast,the EAM-LQC simulation predicts that dislocations will form at an indenter depth of45 nm, 0.16 z/a (#1 and #2) or 0.32 z/a (#3 and #4) below the surface, off-axis, andonly 10 nm (#1 and #2) or 48 nm (#3 and #4) from the spherical indenter surface.Due to the finite spacing between quadrature points, the OFDFT dislocations are inadjacent regions, while one set of EAM dislocations is on the indenter surface and theother is well below the surface. Furthermore, only four of the six most probable slip


(a)

-1 0 1-1

0

2.78 15.27 27.76Λmin (GPa)μm [001]

[110]

(b)

Fig. 5.3. Contour plot of the instability criterion, Λmin, on the x = 0 plane for indentationinto the (110) surface of fcc Al using the (a) OFDFT and (b) EAM-based constitutive models. Thefirst dislocations occur at an indentation depth of 70 nm for the OFDFT-LQC model and 45 nmfor the EAM-LQC model. The white circles designate the location of the first dislocation, projectedonto the x = 0 plane, as predicted in (3.8). The inset shows the local atomic configuration of themost unstable point and the slip plane normal (magenta arrow) and slip direction (black arrow)of the predicted dislocation. The x-, y-, and z-axes are shown by the blue, green, and red arrows,respectively. One set of EAM-LQC dislocations is on the indenter surface, while the other set isbelow the surface. Both sets of OFDFT-LQC dislocations are below the surface and adjacent to eachother.

system exhibit dislocations initially. (Two are listed in Table 5.1, and two more arisefrom the reflection symmetry across the x = 0 plane.) However, in this case, not allsix planes are equivalent, so the asymmetry is not unexpected.

Tadmor et al. [71] used a 2D QC simulation to study indentation into the (110)surface of EAM Al using a cylindrical indenter of radius 11.64 A. Like our 3D EAM-LQC simulation, the first dislocations appear off-axis, on the indenter surface, and of〈112〉 character, but the 2D EAM simulation predicts that only two Shockley partial


dislocations will form on the (111) slip plane with ± 16 [211] Burgers vectors. The

difference in the slip system likely arises from the 2D constraint.

5.4. Dislocation criteria. Traditionally, Hertzian analysis [34] has been usedin both experiments and large numerical simulations to extract the location and max-imum value of the principal shear stress where the first dislocation should form whenan elastic material is indented by a spherical indenter. Typical measured and calcu-lated Al elastic constants yield a Poisson ratio, ν, between 0.32 and 0.44. Hertziananalysis then predicts that the first dislocation will form between 0.49 and 0.53 z/adirectly below the indenter. Several recent simulations [35, 41, 71, 75] have shownthat the first dislocations actually nucleate off-axis due to the asymmetries that arisefrom the elastic constants and the underlying crystal structure. Therefore, it is notsurprising that our dislocations are also located off-axis. However, the depth beneaththe surface varies significantly between the OFDFT and EAM models; from 0.51 to0.59 z/a for OFDFT and from 0.16 to 0.32 z/a for EAM. By comparison, the AlEAM-based 3D MD simulation of Van Vliet et al. [75] predicted that the first dislo-cation would be at a depth of 0.51 z/a, indicating the indenter size may significantlyalter dislocation nucleation.

Although Li et al. [41] and Van Vliet et al. [75] have recently shown in simulationsthat the actual maximum value of the principal shear stress does not coincide with thespatial location of the first dislocation or the instability criterion given in (3.8), theyand others [15, 26, 71] have observed that when the maximum value of the Hertzianprincipal shear stress reaches the ideal shear strength of the material, dislocationsstart to form. The maximum value of the Hertzian principal shear stress [34] is givenby

τHertzmax = f [ν]

(6PE∗2

π3R2

)1/3

,(5.1)

where f [ν] is a function that depends on ν but is about 0.31 for Al, P is the maximumload, and R is the indenter radius. For the case of a rigid indenter, E∗, the reducedYoung’s modulus is given by

E∗ =Ehkl

1 − ν2.(5.2)

Ehkl, the Young’s modulus in the [hkl] crystallographic direction, and ν are calculatedfrom the elastic constants in Table 4.1 using

1

Ehkl=

C11 + C12

(C11 − C12)(C11 + 2C12)− 2

[1

C11 − C12− 1

2C44

](l21l

22 + l21l

23 + l22l

23)

and ν = C12/(C11 − C12), where li are the direction cosines between [hkl] and the[100] axis.

Using (5.1) and (5.2), we calculate the maximum Hertzian principal shear stress(τHertz

max ) to be between 3.3 and 5.2 GPa. The individual values are listed in Table 5.3.By comparison, the Al ideal shear strength has been calculated using KSDFT withnonlocal pseudopotentials [62] to be 1.85 GPa and 3.4 GPa with and without elasticrelaxation, respectively, and is experimentally estimated [75] from G/2π, where Gis the resolved shear modulus, to be 4.6 GPa and 4.4 GPa for the (111) and (110)surfaces, respectively. Experiments have provided values of the maximum Hertzianprincipal shear stress (τHertz

expt ), based on the first drop in the load vs. displacement


Table 5.3

Comparison of dislocation criteria proposed in the literature. τHertzmax is the maximum value of

the Hertzian principal shear stress given by (5.1). The coordinates of the maximum value are givenas (x, y, z/a), where a is the contact radius of the indenter. The subscript Λmin denotes the valueof the given criterion measured at the point where the instability criterion predicts dislocations willform, according to (3.8). For the (110) surface, the maximum value among the four unstable pointsis reported. (x, y, z/a) based on the Λmin criterion are given in Table 5.1 and are marked with whitedots in Figures 5.4, 5.5, and 5.6. τmax is the maximum value of the principal shear stress. τRSS

max isthe maximum resolved shear stress on any of the 24 active fcc slip systems given by (5.3). εMises

max isthe maximum von Mises strain given by (5.4). None of the criteria match the dislocation predictionsgiven by Λmin, although εMises

max matches for (111) OFDFT and (110) EAM. Stresses are in GPa;distances are in nm.

Criterion OFDFT EAM

[111] [110] [111] [110]

τHertzmax 3.3 3.4 4.8 5.2

(x, y, z/a) (0, 0, 0.53) (0, 0, 0.53) (0, 0, 0.50) (0, 0, 0.50)

τΛmin3.6 3.4 4.6 7.6

τmax 4.5 5.7 7.2 8.5

(x, y, z/a) (−98,−105, 0.26) (−37,−65, 0.42) (−87,−26, 0.57) (−24, 53, 0.44)

τRSSΛmin

0.01 −2.2 0.04 −2.5

Slip system (111)[011] (111)[121] (111)[011] (111)[121]

τRSSmax −3.7 4.4 5.4 −6.6

Slip system (111)[211] (111)[112] (111)[211] (111)[112]

(x, y, z/a) (−101,−65, 0.41) (−57, 38, 0.54) (−99,−64, 0.43) (−57,−38, 0.58)

εMisesΛmin

0.19 0.16 0.09 0.16

εMisesmax 0.19 0.17 0.12 0.16

(x, y, z/a) (−59,−39, 0.56) (−87,−26, 0.57) (−99, 64, 0.43) (−124,−78, 0.16)

curve, of 4.7 GPa (50 nm radius tip) and 2.38 GPa (150 nm radius tip) for the (111)surface and 7.2 GPa (50 nm radius tip) for the (110) surface [75]. Although thetheoretical shear strength, experimentallyextracted τHertz

expt , and our calculated τHertzmax

are qualitatively all the same order of magnitude, quantitative dislocation predictionsare not possible. The experimentally estimated shear strength predicts that the (111)surface will require a larger Hertzian principal shear stress to nucleate dislocationsthan the (110) surface, in contradiction to what is observed in both experiments andour work. Zhu et al. [85] have already noted that the ideal shear strength is highlydependent on the local stress state. Hence, there is no universal material-dependentshear strength that τHertz

max must exceed in order to nucleate a dislocation.

In our LQC calculations, the maximum value of the principal shear stress (τmax)(4.5 to 8.5 GPa), calculated directly from the Cauchy stress tensor, bears little re-semblance in either magnitude or spatial distribution to τHertz

max . In addition, not onlyare the positions of the dislocations predicted by the Λmin criterion far away fromthose predicted by τmax, but the value of the principal shear stress at those positions(τΛmin

) are significantly lower. Figure 5.4 shows the principal shear stress on thex = 0 plane for the OFDFT-LQC and EAM-LQC simulations. The spatial locationof the dislocations predicted by Λmin are marked with white circles (τΛmin

), while themaximum values of the principal shear stress are marked with purple circles (τmax).Since the maximum values do not occur on the x = 0 plane, the positions are pro-jected onto the x = 0 plane. Clearly, the maximum value of the principal shear stressis not an adequate criterion.

Several other researchers prefer the critical resolved shear stress (CRSS) to predict


-1 0 1-1

0

0.35 1.93 3.50

-1 0 1-1

0

0.54 2.94 5.35

(a) (b)

-1 0 1-1

0

0.49 2.71 4.93-1 0 1

-1

0

0.74 4.06 7.38

(c) (d)

Fig. 5.4. The principal shear stress in GPa along the x = 0 plane for the DFT and EAM-basedconstitutive models. The various models are (a) OFDFT-LQC and (b) EAM-LQC for indentationinto the (111) surface and (c) OFDFT-LQC and (d) EAM-LQC for indentation into the (110)surface. The position of the maximum value is denoted with a purple circle, while the white circlesmark the positions that are predicted to be unstable according to the Λmin criterion given in (3.8).Since all the maxima occur off the x = 0 plane, the positions are projected onto this plane for clarity.The two criteria do not predict the same location for dislocation nucleation.

the onset of dislocation nucleation. The maximum resolved shear stress (RSS) is

τRSSmax = max

α{Det(F )FRKα · σNα(FR)−1},(5.3)

where α refers to the specific slip system, F is the deformation tensor, K is the slipdirection in the reference system, N is the slip plane normal in the reference system,R is the rotation matrix that specifies the axis orientation, and σ is the Cauchy stresstensor. Traditionally, the CRSS denotes the RSS value that must be exceeded on anyslip system α before a dislocation will form.

The fcc Cu phase is expected to display similar dislocation behavior to Al, soCu studies may also be a useful comparison. Kiely and Houston [36] experimentallyindented Cu (111), (110), and (001) surfaces with 70 nm and 175 nm spherical tips.Based on a modified Hertzian analysis, they claim plastic deformation occurs onlywhen the RSS reaches 1.8 GPa (CRSS) on all the active {111} planes, not just on thefirst plane to exceed the critical value. A companion theoretical paper by Gannepalliand Mallapragada [22] used MD to study indentation into (001) Cu with a truncatedpyramidal indenter (15 A× 15 A base). They conclude that the RSS must exceed themaximum value (CRSS) of the derivative of the generalized stacking fault energy withrespect to the displacement along a given slip system before a dislocation will form.The basic idea is that the imposed stress along the slip direction must exceed the stressneeded to move two layers of atoms past each other. Since the generalized stacking


-1 0 1-1

0

0.24 1.31 2.38-1 0 1

-1

0

0.40 2.20 3.99

(a) (b)

-1 0 1-1

0

0.38 2.08 3.79-1 0 1

-1

0

0.56 3.09 5.62

(c) (d)

Fig. 5.5. The maximum resolved shear stress in GPa along the x = 0 plane for the DFT andEAM-based constitutive models. The various models are (a) OFDFT-LQC and (b) EAM-LQC forindentation into the (111) surface and (c) OFDFT-LQC and (d) EAM-LQC for indentation intothe (110) surface. The position of the largest value is denoted with a purple circle, while the whitecircles mark the positions that are predicted to be unstable according to the Λmin criterion given in(3.8). Since all the maxima occur off the x = 0 plane, the positions are projected onto this planefor clarity. The two criteria do not predict the same location for dislocation nucleation. Dislocationnucleation during indentation into the (111) surface is controlled by local softening of the tangentshear modulus, not the RSS.

fault energy (the curve used to calculate the values in Table 4.2) depends on the choiceof slip system, a different CRSS must be defined for each slip system. Consequently,a simple comparison of the RSS to the lowest value of all the possible CRSS valuesmay predict the wrong dislocation. Likewise, Zimmerman et al. [89] deduced froman MD simulation of (111) Cu indentation using a 40 A spherical indenter that theRSS must exceed the unstable stacking fault energy before a dislocation will form.However, other theoretical studies have found that the RSS may not be a reliableindicator of dislocation activity. Tadmor et al. [71] noted in their 2D QC study thatthe critical RSS depended on the indenter size, with smaller indenters having largervalues. Similarly, Zhu et al. [85] found, for indentation into the (111) surface ofCu with a 50 nm radius spherical indenter using their IPFEM code with an EAM-based potential, that the critical resolved shear stress is highly dependent on the localdeformation and resulting local stress state; in their case, this resulted in a lowerbound on the theoretical shear strength two times larger than the elastically relaxedvalue. These studies strongly suggest that the stress barrier that must be overcomefor dislocation nucleation is highly dependent on the local deformation state, andtherefore a single CRSS does not exist for a given metal.

For comparison, we plot the RSS on the x = 0 plane in Figure 5.5 using (5.3),where we have picked the maximum value at each point from the 24 expected fcc α


slip systems. The position of the global maximum RSS and the unstable points aspredicted by Λmin are projected onto the x = 0 plane and marked with purple andwhite circles, respectively. Unlike previous studies, the RSS does not even correctlypredict the region where dislocations will form. The discrepancy is most strikingfor the case of indentation into the (111) surface. Both the OFDFT and the EAMconstitutive models predict that the maximum RSS is nearly 0 GPa in the slip systemwhere the dislocation is predicted to form with the Λmin criterion, while the globalmaximum RSS occurs about 60 nm away on another slip system. This implies thatthere is a softening of the tangent shear modulus along this slip system for thesespecific local deformation gradients. Recently, Zhu et al. [42] showed with an EAMsimulation that certain phonon modes in fcc Al can become soft when strained underlow temperature conditions. A simple screening for dislocations based solely on theRSS would completely miss this effect.

Stress-based dislocation criteria seem fraught with local deformation dependen-cies. Tadmor et al. [71] briefly mentioned that a nucleation criterion based on theresolved shear strain may provide a more universal dislocation nucleation criterion.In a similar vein, we find that the instability criterion (Λmin) predictions most closelymatch the global maximum of the von Mises strain, defined as

εMises =

√(εxx − εyy)2 + (εyy − εzz)2 + (εzz − εxx)2 + 6(ε2

xy + ε2yz + ε2

xz)

2,(5.4)

where ε ≡ (1/2) log(C) is the logarithmic strain, defined in terms of the right Cauchy–Green deformation tensor C = F TF . For deformations, the logarithmic strain canbe approximated as ε ≈ (1/2)(C−I), which is the Lagrangian strain. We plot εMises

along the x = 0 plane in Figure 5.6 and denote the dislocation nucleation predictions,projected onto the x = 0 plane, for εMises and Λmin with purple and white circles,respectively. The two dislocation nucleation criteria predict the same location forOFDFT-LQC (111) indentation and EAM-LQC (110) indentation. However, theyexhibit significant differences in the other two cases. Direct verification of dislocationnucleation is not possible because our system is too large to treat with MD. However,we have verified that the energy-based Λmin criterion predicts fundamentally differentinitial dislocations than other common stress and strain-based dislocation criteria.

5.5. Twinning. Controversy remains as to whether deformation twinning playsa significant role in Al. Bulk Al has long been thought not to exhibit deformationtwinning [76], but thin film crack experiments have observed deformation twinning inAl under specific orientations and loading configurations [13, 57]. In each case, needle-like twins form by identical dislocations travelling on adjacent planes. In Pond andGarcia-Garcia’s thin film aluminum experiment [57], the observed twins formed bypartial dislocations with the Burgers vector [121] travelling on adjacent (111) planes.Hai and Tadmor [27] recently verified these experimentally observed deformation twinsusing 2D QC calculations, and Farkas et al. [20] found using MD that atomisticallysharp Al cracks grew by forming twins which subsequently cracked. Nanoindentationstudies are split on the issue. The 2D QC rectangular indentation study by Tadmoret al. [71] found that pure edge partial dislocations with the Burgers vectors 1

6 [112]and line directions [110] form on adjacent (111) planes to create a twin in the [112]twinning direction. However, the 3D MD nanoindentation simulation [41] with a13 nm spherical indenter did not form deformation twins. Deformation twinning mayoccur in Al only under limited conditions: when 2D constraints are imposed through


-1 0 1-1

0

0.02 0.09 0.16-1 0 1

-1

0

0.01 0.06 0.11

(a) (b)

-1 0 1-1

0

0.02 0.08 0.15-1 0 1

-1

0

0.01 0.06 0.11

(c) (d)

Fig. 5.6. The von Mises strain along the x = 0 plane for the DFT and EAM-based constitutivemodels. The various models are (a) OFDFT-LQC and (b) EAM-LQC for indentation into the(111) surface and (c) OFDFT-LQC and (d) EAM-LQC for indentation into the (110) surface. Theposition of the largest value is denoted with a purple circle, while the white circles mark the positionsthat are predicted to be unstable according to the Λmin criterion given in (3.8). Since all the maximaoccur off the x = 0 plane, the positions are projected onto this plane for clarity. The two criteriagive the same prediction for (111) OFDFT-LQC indentation (a) and for one of the unstable areaspredicted for (110) EAM-LQC indentation (d), but they disagree for the other cases.

either a thin film or dislocation pinning and when a sufficiently large resolved shearstress coincides with the twinning system.

The assumption that each of the quadrature points are well separated from eachother precludes a mesh that could resolve the onset of the observed needle-like twins.This awaits a real-space implementation of the OFDFT code. However, the characterand spatial location of the initial instabilities indicate whether twinning is likely.Indentation into the (110) surface produced two sets of spatially adjacent partialdislocations, [121] on (111) planes and [121] on (111) planes. The former is identical tothe dislocations leading to twinning in the Pond and Garcia-Garcia experiment [57].Although not conclusive, this hints that twinning may be favored for indentationinto the (110) surface. No such evidence of twinning exists from our simulations forindentation into the (111) surface.

6. Summary and conclusions. We have shown that the LQC method can becoupled to first-principles OFDFT to yield dislocation nucleation predictions undernonuniform loading conditions such as indentation. Various dislocation criteria de-rived from energy (Λmin), stress (τHertzian

max , τmax, τRSS), and strain (εMises) consider-ations were compared to each other. None of the other dislocation criteria consistentlymatch the predictions of the localization criterion, Λmin, which has been shown previ-ously during MD simulations to accurately predict the spatial location and character


of the dislocations that actually form. For the first time, the localization criterionpredicted that a dislocation will nucleate due to a softening of the phonon modesand a subsequent lowering of the local shear strength rather than an increase in thestress to a value exceeding the typical theoretical shear strength of the material. Asin previous studies, we find that dislocations nucleate off-axis and only on some of theequivalent available slip planes, when constitutive relations beyond linear elasticityare employed to describe the underlying material.

One recurring theme throughout this research is the high sensitivity of the disloca-tion nucleation predictions to the quality of the underlying constitutive laws, pointingout the need for the development and thorough testing of both pseudopotentials usedin first-principles methods and fast atomistic techniques. A second theme is thatsize matters: nanosized indentation results cannot be scaled to experimental indentersizes. This theme further emphasizes the need for multiscale models that can handlelarge systems under complex loading conditions. In the future, OFDFT-LQC andother first-principles methods promise to provide reliable material descriptions thatwill enable the study of alloys and other multicomponent systems without the needto create empirical potentials for each individual system. Open questions relating tothe twinning behavior of Al, the initial motion of dislocations, grain boundaries, andother atomic-scale effects in large systems under nonuniform loading conditions awaitthe development of a real-space OFDFT code coupled with a full QC method.

Appendix. We calculate the Cauchy stress tensor with the following formula:

σαβ =1

Ω

∑υ

∂E

∂hαυhβυ,(A.1)

where Ω is the periodic unit cell volume, E is the total OFDFT energy, and h arethe lattice vectors. α, β, and υ are the spatial coordinates, {x, y, z}. Since the totalOFDFT energy is a linear combination of ionic and electronic terms, we can calculatethe stress contribution of each term separately.

Stress due to the ion-ion (Ewald) interaction is

σEwaldαβ = − 1

2Ω

∑�L

′∑I,J

ZIZJ

{2η√π

exp(−η2|�x|2) +erfc(η|�x|)

|�x|

}xαxβ

|�x|2

∣∣∣∣�x=�L+�RI−�RJ

+2π

Ω2

∑g �=0

exp(−g2/4η2)

g2

∣∣∣∣∣∑I

ZI exp(i�g · �RI)

∣∣∣∣∣2 {

2gαgβg2

(1 +

g2

4η2

)− δαβ

}

+π

2Ω2η2

(∑I

ZI

)2

δαβ ,(A.2)

where Z is the ionic charge, �RI are the atomic positions, g is the reciprocal spacelattice vector, and η is the parameter that controls convergence of the summation inreal and reciprocal space. I and J represent atoms within the periodic cell, while Lis the vector pointing toward the origin of adjacent unit cells. The prime over thesummation signifies that the I = J term is excluded if L = 0.

Stress due to the ion-electron interaction is

σExtαβ = −

∑g �=0

∑I

ρ(�g)gαgβ|�g|

∂V locIe,I(g)

∂gexp(i�g · �RI) + δαβ

EExt

Ω,(A.3)


where

EExt = Ω∑g �=0

ρ(�g)V locExt(−�g) + Ne

∑I

V ncI (�g = 0)(A.4)

and

V locExt(�g) =

∑I

V locIe,I(|�g|) exp(−i�g · �RI),(A.5)

where ρ(g) is the density in reciprocal space, V locIe,I(�g) is the spherically symmetric

local pseudopotential in reciprocal space for ion I, V ncI (�g = 0) is the non-Coulombic

contribution to the pseudopotential for ion I at g = 0, and Ne is the number of valenceelectrons.

Stress due to the Hartree electron-electron classical Coulomb repulsion is

σHartαβ =

∑g �=0

2π

g2

(2gαgβg2

)ρ(�g)ρ(−�g) − δαβ

EHart

Ω,(A.6)

where

EHart =Ω

2

∑g �=0

4π

g2ρ(�g)ρ(−�g).(A.7)

Stress due to the LDA electron exchange correlation is

σxcαβ =

1

Ω

(Exc −

∫ρ(�r)Vxc(�r)dr

)δαβ ,(A.8)

where

Exc =

∫ρ(�g){εx[ρ(�r)] + εc[ρ(�r)]}d�r(A.9)

and

Vxc(�r) =δExc

δρ(�r),(A.10)

where εx is the exact exchange energy for a uniform electron gas at density ρ andεc is the Perdew and Zunger [54] parameterization of the Ceperley and Alder [12]correlation energy for a uniform electron gas.

Stress due to the Thomas–Fermi kinetic energy functional is

σTFαβ = −2TTF

3Ωδαβ ,(A.11)

where

TTF = CF

∫ρ(�r)

53 d�r(A.12)

and

CF =3

10(3π2)

23 .(A.13)


Stress due to the von Weizsacker kinetic energy functional is

σTWαβ = − 1

4Ω

∫ (∂ρ(�r)

∂rα

)(∂ρ(�r)

∂rβ

)1

ρ(�r)d�r.(A.14)

Stress due to the linear response kinetic energy functional is

σTα′β′lr

αβ = −2Tα′β′

lr

3Ωδαβ

+π2

2α′β′ρα′+β′−2

o kF

∑g �=0

Δρα′(�g)Δρβ

′(−�g)

×(gαgβg2

− 1

3δαβ

)[q

fL(q)2dfL(q)

dq+ 6q2

].(A.15)

The values of α′ and β′ correspond to different linear response kinetic energyfunctionals. α′ = β′ = 1 is the Perrot functional [55], α′ = β′ = 1/2 is theSmargiassi–Madden functional [69], α′ = β′ = 5/6 is the Wang–Teter functional [77],

and α′ = 5+√

56 , β′ = 5−

√5

6 is the Wang–Govind–Carter functional with the density-independent, Wang–Teter-type kernel [78]. In this work, α′ = β′ = 5/6 is used.

The linear response kinetic energy functional is

Tα′β′

lr = −Ω2∑g �=0

Δρ(α′)(�g)Δρ(β′)(−�g) π2

2α′β′ρα′+β′−2

o kFΩ

{(1 + 3q2) − 1

fL(q)

},

(A.16)

where

Δρ(α′)(�r) = ρ(�r)α′ − ρα

′

o .(A.17)

The reference bulk density is

ρo = Ne/Ω.(A.18)

The Fermi wavevector is

kF = (3π2ρo)13 .(A.19)

The reduced momentum is

q = g/2kF .(A.20)

The Lindhard function is

fL(q) =1

2+

1 − q2

4qln

∣∣∣∣1 + q

1 − q

∣∣∣∣.(A.21)

Note that the full Lindhard susceptibility function is

χLind(q) = −kFπ2

fL(q).(A.22)


Acknowledgments. We are grateful to the U.S. Department of Defense forsupport through Brown University’s MURI Center for the “Design and Testing ofMaterials by Computation: A Multi-Scale Approach,” the U.S. Department of Energythrough Caltech’s ASCI/ASAP Center for the Simulation of the Dynamic Response ofSolids, Ron Miller and Ellad Tadmor [45] for providing the EAM code, and Accelrys,Inc. for providing the CASTEP software.

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