Challenges for Quantum-Mechanics-BasedMultiscale Modeling

Emily A. Carter Princeton University

• Stressed Al:Predict onset of plasticity during nanoindentation via on-the-fly coupling of orbital-free density functional theory to the local quasicontinuum method

(Hayes, Fago, Ortiz, and EAC, PRB, 2004; MMS, submitted)

• Shocked Fe: Predict microstructure as a function of applied compression and shear via quantum-mechanically-informed sequential lamination model

(Caspersen, Lew, Ortiz, and EAC, PRL, 2004)

Funding: DoD-MURI, ARO, & DoE-ASCI

• Ductile Fracture of Al under Shock:Predict kinetics of vacancy coalescence, void formation, and spall viaorbital-free DFT rates coupled to kinetic Monte Carlo simulations

(Ong, Serebrinsky, Caspersen, Ortiz, and EAC)

• Stressed and Embrittled Fe: Predict fracture behavior as a function of applied tension and impurity concentration via quantum-mechanically-informed cohesive zonemodel (Serebrinsky, Jiang, Hayes, Ortiz, and EAC, PRB, 2004; Acta Materialia, 2004; JMPS, 2004)

Quantum Mechanics and Local Quasicontinuum Coupling

Quantum-derived stress tensor for representative unit cells stress on nodes in each finite element

energy +stress tensor

continuummechanics

(~billions atoms/FEM)

quantummechanics(few atoms/cell) Deformation

gradientHayes, Fago, Ortiz, EAC, PRB 04

Density Functional Theory

)]([)]([)]([)]([)]([ rErErJrTrE extxcs ρρρρρ +++=

Kohn-Sham• Uses orbitals for density ⇒ O[Nkn3]

• Exact Kinetic Energy (Ts) is known

• Can use orbital dependent non-localpseudopotentials

Orbital-Free• Based only on the density ⇒ O[n*ln(n)]

• Kinetic Energy DensityFunctional only known in certain limits

• Requires localpseudopotentials

Only OFDFT can efficiently model large 3D periodic metallic materials from “First Principles”

Application of OFDFTApplication of OFDFT--LQC to LQC to NanoindentationNanoindentation: : Probe Dislocation NucleationProbe Dislocation Nucleation

750 nm indentation in 1.0 µm Al film

Acta Mater. 2002, 50, 23.

Experiments detect dislocation formation by jumps in the load-displacement curveBut …• many dislocations form before a jump occurs

• spatial location and character of initial dislocation unknown1

• Characterize elastic instability by:

0)(),(min =+=Λ ljijkiijkl kkwwCkw τElastic constants

matrixCauchy stress

tensor

Dislocation on plane normal k with

Burgers vector w

2

[1] Phys. Rev. B 55 (1997) 16057; [2] Nature 418 (2002) 307.

Al (111) IndentationAl (111) IndentationOFDFTOFDFT EAMEAM

OFOF--DFTDFT KSKS--DFTDFT EAMEAM

ΛΛminmin [GPa][GPa] 0.00.0 --6.36.3 --4.34.3

slip plane slip plane normalnormal

--1 1 11 1 1 --1 1 11 1 1 --1 1 11 1 1

slip slip directiondirection

0 0 --1 11 1 0 0 --1 11 1 0 0 --1 11 1

• OFOF--DFT and EAM predict DFT and EAM predict different numbers and locations different numbers and locations of dislocationsof dislocations

•• EAM and KSEAM and KS--DFT agree with the DFT agree with the character and stability of the character and stability of the first OFfirst OF--DFT dislocationDFT dislocation

OFOF--DFT & EAM Validation Against KSDFT & EAM Validation Against KS--DFTDFTStacking Fault EnergyStacking Fault Energy

Unstable Unstable Stacking Stacking

Fault Fault Energy Energy [mJ/m[mJ/m22]]

OFOF--DFTDFT 8585

KSKS--DFT DFT (local)(local) 93

93

KSKS--DFT DFT (non(non--local)local)11 213

213

EAMEAM22 130130

[1] Phys Rev. B, 58 (1998) 2487. [2] Europhys Lett. 26 (1994) 583.

• OFOF--DFT has lower energy barriers DFT has lower energy barriers than EAMthan EAM

•• Small error in OFDFT Small error in OFDFT vsvs KSDFT (local) KSDFT (local) due to kinetic energy functionaldue to kinetic energy functional

•• Local potential Local potential →→ largest error largest error (can be improved via Zhou’s method) (can be improved via Zhou’s method)

-0.3 -0.2 -0.1 0.1 0.2 0.3

5

10

15

Resolved Shear Stress Resolved Shear Stress along the directionalong the direction

εyzσ y

z[G

Pa]

EAM

OF-DFT

KS-DFT (local)

[-1-12]

[111]

[-110]

LoadLoad--Displacement CurvesDisplacement Curves

Displacement [µm]

0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.00025

0.0005

0.00075

0.001

0.00125

0.0015

0.00175

ODFDT

EAM

Load

[N

]

(110) surface(110) surface

*

0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.00025

0.0005

0.00075

0.001

0.00125

0.0015

0.00175

(111) surface(111) surface

Displacement [µm]

ODFDT

EAM

Load

[N

]

First dislocation

forms*

* *

• Larger load and displacement needed to create firstLarger load and displacement needed to create firstdislocation in OFDFT compared to EAM, despite low SFEdislocation in OFDFT compared to EAM, despite low SFE

•• (111) surface forms dislocations more easily than (110) (111) surface forms dislocations more easily than (110) surface, consistent with experimentsurface, consistent with experiment11

[1] Acta mater. 48 (2000) 2277.

Advantages and Limitations of Advantages and Limitations of OFDFTOFDFT--LQCLQC

•• Can have comparable accuracy to KSCan have comparable accuracy to KS--DFT, DFT, but orders of magnitude faster so that DFTbut orders of magnitude faster so that DFT--LQC can be performed at allLQC can be performed at all

•• Pinpoints spatial location and character Pinpoints spatial location and character of first dislocation, but cannot atomically of first dislocation, but cannot atomically resolve dislocation and continue probing resolve dislocation and continue probing mechanical response => need mechanical response => need aperiodicaperiodicOFDFT (currently formulated in real and OFDFT (currently formulated in real and reciprocal space)reciprocal space)

•• Limited to nearly freeLimited to nearly free--electronelectron--like materials like materials until better kinetic energy until better kinetic energy functionalsfunctionals and and local local pseudopotentialspseudopotentials are developed…are developed…

0.65 2.58 4.52 6.46 GPa Indentation = 0.05 micron

-0.3 -0.2 -0.1 0.1 0.2 0.3

5

10

15

εyz

σ yz[G

Pa]

Local Pseudopotentials (PP) (valence electron interaction with nuclei +core electrons)

Orbital-Free DFT can use only local PP, vloc(r)since Ene[ρ] =

(more accurate, nonlocal PP involves projections onto orbitals)

Challenge: to develop vloc(r) as accurate as vnonloc(r) for all elements in the periodic table

5. Acid test: vloc(r) in OF-DFT; compare to vnonloc(r) in KSDFT

Strategy: Exploit the first Hohenberg-Kohn theorem

“The external potential v(r) is determined, within a trivial additive constant, by the electron density ρ(r).”1. Use accurate vnonloc(r) to obtain target ρ(r) (from KS-DFT) for atoms and bulk

crystals 2. Invert KS equations to determine total KS potential, veff,KS([ρ],r) from target ρ(r).

N.b. no unique path exists. Figure of merit: veff,KS([ρ],r) must reproduce target ρ(r).

3. Extract vloc(r) = veff,KS([ρ],r) - vHartree - vXC4. Compare vloc(r) to vnonloc(r) in KS (orbital-based) DFT

(Zhou, Wang, Carter, PRB 2004)c.f. S. Watson et al. Europhys. Lett. 41, 37 (1998).

Towards OFDFT for Covalent Materials: Local Towards OFDFT for Covalent Materials: Local PseudopotentialsPseudopotentials for for SiSi

The LPSs become much more repulsive near core (r< 0.5 Bohr) in order to force higher angular momentum electrons out of core region.

BLPS (derived from bulk crystal) significantly more repulsive than ALPS (derived from atom) in core region.

Si

• Real space PS for Si: the NLPS (green), ALPS(red), and the BLPS (blue), and Coulombicpotential (cyan).

Zhou, Wang, and EAC, PRB 04

Testing LPS within KSDFT: Equilibrium Energy Relative to CD Testing LPS within KSDFT: Equilibrium Energy Relative to CD SiSi

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

HD cbcc β-tin bct5 sc hcp sbcc fcc

∆E

min (e

V/at

om)

NLPSBLPSALPS

cbcc: complex bcc, bct5: body-centered tetragonal

Equilibrium energy relative to CD Si: ∆Emin=Emin-Emin,CD

Energy gap (~0.15-0.4 eV/atom) between semiconducting and metallic phasesseverely underestimated by ALPS but well reproduced by BLPS

• Phase ordering with BLPS correct for almost all phases

• Overall agreement much better with BLPS than with ALPS

OFDFT Equilibrium Energies Relative to CD OFDFT Equilibrium Energies Relative to CD SiSi

Equilibrium energy relative to CD Si: ∆Emin=Emin-Emin,CD

-0.1

0

0.1

0.2

0.3

0.4

0.5

HD cbcc β-tin bct5 sc hcp sbcc fcc

∆E m

in (e

V/at

om)

NLPS–KSALPS–OFBLPS–OF

ALPS-OF(WGC-KEDF): wrongly predicts hcp Si to be most stable BLPS-OF (WGC-KEDF): correctly predicts CD Si to be most stableSubstantial errors in relative equilibrium energies and wrong phase ordering.Remaining error mainly due to WGC KEDF. Next: new parameterization for semiconducting phases…

Linear-Response-Based Kinetic Energy Density Functionals

For example, WGC functional with a density-dependent kernel (PRB 1999)

Analytical form of wα,β is unknown. Evaluate in reciprocal space as solution to ODE

Nanoindentation into (111) AlSpherical indenter,

Radius = 12 nm

1 µm Al on Si substrate1

grains as thick as film1

5 nm Al2O3

• 1st nucleation occurs when τmax = τtheory, τ = shear stress• Symmetric nucleation of glide loops on {111} planes

[1] Acta mater. 48 (2000) 2277. [2] Nature 418 (2002) 307.

Non-Periodic Orbital-Free Density Functional Theory

Challenge:To develop an accurate O(N) method in real space to investigate:

• The properties of non-periodic defects, vacancies, and impurities (i.e., vacancy diffusion/void aggregation)

• The effects of single dislocations that CANNOT be made periodic (i.e., a screw dislocation)

• The properties of nanoscale metallic structures, such as nanowires and clusters, on surfaces or in vacuum

Current Work:Developing a code that:

• Applies the Multigrid method to OFDFT

• Is truly O(N) (Eliminates Fast Fourier Transforms)

• Is easily parallelizable• Can be married to QC to create a

fully nonlocal quantum-based multiscale method

Outcomes:• A better understanding of the effect

and evolution of defects in metals• A new ability to investigate within

OFDFT:

• Materials with regions of low electron density

• Systems where image effects are important

Ho, Brandt, and EAC

In EAM-NLQC, can use clusters of atoms to resolve dislocations. Can’t do this in quantum mechanics! In Absence of Realistic Boundary Conditions:

Vacuum Vacuum

Instead use realistic boundary conditions (density matching conditions):

Aperiodic Region:R-S Multigrid OF-DFT

Periodic Regions:Finite Element, QC

Ho, Brandt, E, Ortiz and EAC

Conclusions• OFDFT-LQC provides an efficient on-the-fly coupling

between quantum and continuum mechanics

• Current Challenges: – generality and accuracy of OFDFT– Nature of quantum wave functions => difficult to extend LQC

to nonlocal QC

• Current Approaches:– Bulk-derived LPSs for all elements– KEDFs based on real material (interacting electron)

response functions– Multigrid real space OFDFT with density matching boundary

conditions…=> fully nonlocal QC for any material someday…

Thanks to:

Michael Ortiz

Matt Fago

Achi BrandtCarter Group 2003-04: Robin Hayes, Baojing Zhou, Alex Wang,Greg Ho, Vincent Ligneres for this work… Weinan E

Quantum Mechanics and Local Quasicontinuum CouplingDensity Functional TheoryApplication of OFDFT-LQC to Nanoindentation: Probe Dislocation NucleationAl (111) IndentationOF-DFT & EAM Validation Against KS-DFTLoad-Displacement CurvesAdvantages and Limitations of OFDFT-LQCTowards OFDFT for Covalent Materials: Local Pseudopotentials for SiTesting LPS within KSDFT: Equilibrium Energy Relative to CD SiOFDFT Equilibrium Energies Relative to CD SiConclusionsThanks to: