Robust Quantum-Based Interatomic Potentials

for Transition Metals

John A. MoriartyLawrence Livermore National Laboratory

Atomistic Simulations for Industrial Needs Workshop

NIST Gaithersburg, MDJuly 27, 2010

Collaborators: J. N. Glosli, R. Q. Hood, D. A. Orlikowski, P. Söderlind,

M. Tang , C. Wu and L. H. Yang (LLNL)

L. Burakovsky (LANL), A. Belonoshko (RIT, Sweden)

This work was performed under the auspices of the U. S. Department of Energy by

Lawrence Livermore National Laboratory under contract No. DE-AC52-07NA27344

LLNL-PRES-443673

Outline

• Quantum-based interatomic potentials

linking first-principles quantum mechanics to large-scale

atomistic simulation via GPT method

simplified model GPT or MGPT for central transition metals

• Selected MGPT applications in Ta, Mo and V prototypes

high-pressure phase transitions

multiphase equation of state, melt and polymorphism

dislocations and multiscale modeling of yield strength

• Beyond the standard theory: advanced MGPT capabilities

matrix MGPT: f-electrons, non-canonical bands, fast algorithms

sp-d hybridization: series-end transition metals, e.g., Ni

inclusion of electron temperature: T-dependent potentials

Bridging the gap from quantum mechanics to

large-scale atomistic simulation

“ab initio”

Exact

quantum

mechanics

“empirical”

Total

empirical

description

Number of atoms

1 - 102 10 - 103 102 - 109 103 - 1010

Quantum simulation Atomistic simulation

Material-dependent gap

Coarse-grained …… No electronic

electronic structure structure

Many-electron …… Self-consistent

states mean field

Electronic: electron + ion motion Atomic: ion motion only

Correlated

electron

theory

QMC

DMFT

…

Density

Functional

Theory

(DFT)

QMD

PP

FP-LMTO

…

Quantum

based

potentials

GPT

MGPT

BOP

…

Empirical

potentials

EAM

FS

…

with

BG/L

Generalized Pseudopotential Theory (GPT)

sp pseudopotential:

d-d tight-binding:

sp-d hydridization:

k q | w | k

d

i | | d

j & d

i | d

j

k | | d & k |d

• Mixed basis: (DFT quantum mechanics)

expansions in weak

matrix elements

self-consistent screening

| k , |d

Etot (R1, , RN ) NEvol () 1

2

' v2(ij;)i, j

1

6

' v3(ijk;)i,j,k

1

24

' v4

i, j,k,l

(ijkl;)

volume radial forces angular forces

• Total-energy functional: (bulk formulation: atomic volume )

structure-independent potentials: rigorous transferability

atomistic simulation: MS, MD, MC

ab initio GPT: simple & series-end transition metals: Mg, Cu, …

binary and ternary alloys: TMxAl1-x …

model GPT: central transition metals: Mo, Ta, …

canonical d bands; analytic v3 and v4

GPT

MGPT

Simplified MGPT for central d-transition metals

• Systematic approximations in GPT:

neglect sp-d hybridization beyond Evol and fold d-state non-orthogonality into v2

introduce canonical d bands:

Etot (R1, , RN ) NEvol () 1

2

' v2(ij;)i, j

1

6

' v3(ijk;)i,j,k

1

24

' v4

i, j,k,l

(ijkl;)

d d (Rij ) d | | d m(RWS / Rij )5 m(RWS / Rij )

p p ~ 4-5 0: 1: 2

bcc metals 6: -4: 1

f (r) (1.8RWS / r)p

L, P, M universal

angular functions

(d symmetry)

v2(r) v2

sp(r) v2

hc (r) va[ f (r)]4 vb[ f (r)]2

v3(r1,r2 , r3 ) vc f (r1) f (r2 ) f (r3)L(1,2 ,3) vd{[ f (r1) f (r2 )]2 P(3)

[ f (r2) f (r3 )]2 P(1) [ f (r3 ) f (r1)]2 P(2 )}

v4(r1,r2 , r3 ,r4 , r5, r6) ve[ f (r1) f (r2 ) f (r3 ) f (r4 )M(1,2 ,3,4 ,5,6 )

f (r3) f (r2 ) f (r6) f (r5 )M(7,8,9 ,10 ,5 ,12)

f (r1) f (r6) f (r4) f (r3)M(11,12 ,5,6 ,3 ,4 )]

Advanced-generation MGPT potentials: Ta, Mo, V

• Pressure ranges treated:

Ta to 1000 GPa

Mo to 400 GPa

V to 230 GPa

Evol: Ecoh va: C44 vb: E0vac compressibility sum rule reduces

vd: B ve: C vc:D independent parameters to 5

• Parameter constraints: bcc DFT and/or experimental data as a function of

Two-ion Three-ion Four-ion

-30

-20

-10

0

10

20

30

1.4 1.6 1.8 2.0 2.2 2.4 2.6

Tw

o-i

on

pote

ntia

l (m

Ry)

Relative separation r /RWS

MGPT: v2

Mo

V

Ta

-2

0

2

4

6

8

40 80 120 160

Th

ree

-ion p

ote

ntial (m

Ry)

Angle (deg)

MGPT: v3

d

d

d = 1.8 RWS

V

MoTa

-8

-4

0

4

8

12

16

40 60 80 100 120 140

Fo

ur-

ion

po

ten

tia

l (m

Ry)

Angle (deg)

MGPT: v4

dd

d

d

d = 1.8 R

WS

V

MoTa

Volume-dependent MGPT potentials

available over wide pressure ranges

Transition-metal MGPT potentials have been widely

applied to thermodynamic and mechanical properties

• Structural and thermo-

dynamic properties:

phase transitions

phonons

high-pressure melting

multiphase EOS

rapid solidification

thermoelasticity

• Defects and mechanical

properties:

high-pressure elastic

moduli

vacancy, self-interstitial

formation and migration

grain boundary structure

dislocation structure and

mobility

multiscale modeling of

plasticity and strength

Reviews: JPCM 14, 2825 (2002); JMR 21, 563 (2006); Dislocations in Solids 16, 1 (2010)

Dislocation

structure

and

mobility

Multi-

phase

EOS

0

50

100

150

200

250

300

70 80 90 100 110 120

Pre

ssure

(G

Pa

)

Atomic volume (a.u.)

Present EOS

Ta

Shock data

DAC data

Hugoniot

300 K isotherm

High-P,T

elastic

moduli

0

200

400

600

800

1000

0 20 40 60 80 100 120

Ela

stic m

odu

li (G

Pa

)

C44

MGPT

DAC: SAX

Ultrasonic

Ta

C11

C12

Pressure (GPa)

DAC: ISLS

Grain

boundary

atomic

structure

High-P

melt

0

4000

8000

12000

16000

0 100 200 300 400 500

Tem

pe

ratu

re (

K)

Pressure (GPa)

Ta

Hugoniot

liquid

melt

solid (bcc)

observed

shockmelting

expt

Rapid

solid-

ification

Structural phase stability and high-pressure phase

transitions in central d-transition metals

Ti V Cr (Mn)

Zr Nb Mo Tc

Hf Ta W Re

IVB VB VIB VIIB

hcp bcc bcc hcp

• Primary trends

structure controlled by d-band

filling: hcp – bcc – hcp sequence

sp d electron transfer under

pressure

high-P transition to structure on

immediate right:

IVB metals bcc

VB metals remain stable in bcc

VIB metals hcp

• Secondary trends

driven by details of electronic structure

IVB metals: intermediate phase, so high-P sequence is hcp bcc

bcc metals: competitive A15 structure, especially in Ta and W at low P

possibly stable phase in Ta at high P,T

VB metals: Fermi-surface driven elastic anomalies: bcc rhom bcc in V

Structural phase stability in Mo

• MGPT structural energies and high-pressure trends:

good description without constraint: v4 essential to correct physics

bcc hcp predicted beyond 400 GPa: sign of v4 changes

systematic improvement possible: beyond canonical bands and/or beyond v4

0

10

20

30

40

50

60 70 80 90 100 110

Re

lative

en

erg

y (

mR

y)

MGPT

Atomic volume (au)

fcc-bcc

A15-bcc

hcp-bccMo:

0

10

20

30

40

50

60 70 80 90 100 110

Re

lative

en

erg

y (

mR

y)

FP-LMTO

Atomic volume (au)

fcc-bcc

A15-bcc

hcp-bcc

Mo:

Elastic anomalies and bcc rhom transition in V

• bcc rhom transition seen in DAC at 69 GPa: Ding et al., PRL 98, 085502 (2007)

• MGPT potentials capture this behavior through elastic moduli Cij :

softening of C44 is precursor to transition

transition onset near 65 GPa: T2[110] zone-boundary phonon becomes imaginary

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

-0.02 0.00 0.02 0.04 0.06R

ela

tive

en

erg

y (

mR

y)

Strain

V: rhom - bcc

MGPT

bcc

65 GPa

130 GPa

152 GPa

-50

0

50

100

150

200

250

0 50 100 150 200 250

Ela

stic c

on

sta

nt (G

Pa

)

C44

C'

V

MGPT

FP-LMTO

Pressure (GPa)

Expt

Multiphase equation of state and melt: Ta prototype

• Cold and electron-thermal components: FP-LMTO coupled to MGPT

T = 0 properties:

E0 , P0 ; Cij , …

constraints for

MGPT potentials

finite temperature:

Ael , Pel , Eel , Sel

high-T structure

from MD/MGPT

, ( 15, , ),bcc A liquid 0( , ) ( ) ( , ) ( , )ion elA T E A T A T

free energy: cold ion-thermal electron-thermal

• Ion-thermal components: MGPT

lattice/liquid thermal properties: Aion , Pion , Eion , Sion

MD simulation in high-T solid and liquid

• Extension to thermoelasticity, and recently polymorphism, in high-P,T solid

0

50

100

150

200

250

300

70 80 90 100 110 120

Pre

ssure

(G

Pa

)

Atomic volume (a.u.)

MGPT

Ta

Shock data

DAC data

Hugoniot

300 K isotherm

High-pressure Ta melt curve and shock melting

Predicted melt curve agrees with shock and isobaric data, and there

is improved agreement with new DAC data: PRL 104, 255701 (2010)

2

3

4

5

6

7

8

0 100 200 300 400 500

So

un

d v

elo

city (

km

/s)

Pressure (GPa)

Ta: vs

liquid: vs

B

bcc: vs

L

Expt

MGPT

melt

0

2000

4000

6000

8000

10000

12000

14000

0 100 200 300 400

Te

mp

era

ture

(K

)

Pressure (GPa)

Ta

Hugoniot

liquid

solid (bcc)

shockmelt

old DAC

isobaric newDAC

Structural disorder of bcc in Ta under shear loading

Cold bcc Hot bcc Plastic Flow Hot Liquid

{111}

planes

{112}

planes

-

{110}

planes

-

Planar projections of Ta atomic coordinates (P = 30 GPa)

g(r

)

Wu et al., Nature Materials 8, 223 (2009)

MD/MGPT simulations of

partial disorder of bcc

structure at high T

Dislocation-free plastic flow

on {110} planes matches

original DAC “melt”

Possible link to newly

discovered polymorphism:

Burakovsky et al., PRL 104,

255702 (2010)

Accurate atomistic simulations of dislocation

properties in bcc transition metals

• Core structure and its pressure

dependence

a/2<111>

screw

dislocation

• Peierls stress P

and orientation dependence

• Kink and kink-pair energetics,

including stress-dependent

activation enthalpy H ()

• Pressure scaling of P and H

• Dynamic simulations of

structure and mobility

P = 0 P = 1000 GPa

non-degenerate,

isotropic

degenerate,

polarized

Current consensus

view near ambient

But… pressure

sensitive core

P

bcc Ta

• Screw dislocations move via thermally

activated kinks

vscrew( ) exp[H() / kBT]

• Stress-dependent activation enthalpy H

controls dislocation mobility and is

key input into DD simulations of

yield stress

orientation dependence and pressure

scaling through H (0) and P

Extended to high pressure for

DD yield stress simulations

Linking atomistics to microscale dislocation

dynamics (DD) simulations

Left kink of

a/2<111>

kink pair

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

Ta

Atomistic simulation

Empirical:Tang et al.(yield stress)

Relative stress /P

Ene

rgy (

eV

)

Kink-pair activation enthalpy H

• Atomistic calculations of kink-pair

enthalpy fitted to analytic form

needed for DD:

Atomistically informed DD simulations of

single-crystal yield stress in bcc metals

qp

PHH ])/(1)[0()(

for each pressure considered

• Additional needed quantities

P , D , b etc. are also imported

from atomistic calculations

• In Ta, P is scaled down by about

a factor of two to account for

ambient-pressure overestimate

• Similar DD simulations have also

been done for Mo, but without

the need for P scalingYang et al., Dislocations in Solids 16, 1 (2010)

0

100

200

300

400

500

0 100 200 300 400 500

Re

solv

ed

yie

ld s

tre

ss (

MP

a)

Temperature (K)

Ta

ambient

51 GPa

204 GPa

DD

Expt

Matrix MGPT for f electrons, non-canonical bands

and high speed

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 30 60 90 120 150 180

Th

ree

-io

n a

ng

ula

r fu

nction

Angle (deg)

d

d

d bandsf bands

L(): canonical d and f bands

0.0

0.2

0.4

0.6

0.8

1.0

0 30 60 90 120 150 180

Th

ree-i

on

an

gu

lar

fun

ctio

n

Angle (deg)

d

d

d bands

f bands

P(): canonical d and f bands

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 30 60 90 120 150 180

Fo

ur-

ion a

ng

ula

r fu

nctio

n

Angle (deg)

dd

d

d

M(): canonical d and f bands

d bandsf bands

• Canonical bands: d f extension for actinide metals

d states: = 2 0: 1: 2

p = 2+ 1 = 5 6: -4: 1

f states: = 3 0: 1: 2:3

p = 2+ 1 = 7 20: -15: 6: -1

• On-the-fly matrix multiplication of angular functions P, L, M :

)ˆˆˆˆ( kiikjiij HHHHTrP )ˆˆˆ( kijkij HHHTrL )ˆˆˆˆ( likljkij HHHHTrM

• Non-canonical bands: p and m become variable input parameters

• Fast algorithms: analytic forces with 10-fold speed increase over standard MGPT

Non-canonical bands for improved accuracy

• Non-canonical bands permit

an improved description of the

underlying electronic structure

at no additional computational

cost

• For d bands, there are two

additional MGPT parameters:

200 /c211 /c

which can be volume dependent

• For a given electron temperature

a single set of parameters

improve all phonons frequencies

at all volumes

10 ,cc

H-point bcc phonon:

0

5

10

15

70 80 90 100 110 120 130

Fre

qu

en

cy (

TH

z)

Atomic volume (a.u.)

Mo H:L,T

Tel = 5000 K

MGPT canonical

MGPT non-canonical

DFT

Inclusion of sp-d hybridization in MGPT formalism

is important for series-end transition metals

2 2 2 2 2

3 3 3

4 4 4

hc sp sp d d

sp d d

sp d d

v v v v v

v v v

v v v

In full GPT:

MGPT pair potentials for

Ni with sp-d hybridization

In MGPT:

model vnd via canonical or non-

canonical d bands in usual way

introduce screened effective

hybridization potential:

2 2 3 4

2

sp d sp d sp d sp d

eff

sp d

scr

v v v v

v f

potential range reduced by factor of two

successfully applied to Ni at v2 level

Ni generalized stacking fault (GSF) energies

at ambient pressure

• MGPT closely matches

DFT over the entire

{111}<112> boundary

• Intrinsic stacking fault

energy isf calculated

in experimental range

• In contrast, short-range

EAM potentials under-

estimate isf

• Ni MD/MGPT simulations

of dynamic fracture in

progress

),;(24

1),;(

6

1),;(

2

1),(),,(

,,,

4

'

,,

3

'

,

2

'

1 TijklvTijkvTijvTNARRAlkjikjiji

volNtot

volume radial forces angular forces

Temperature-dependent MGPT potentials for

strong-coupling transition metals: Mo prototype

• Use free-energy functional at finite electron temperature: Tel = Tion = T

subsumes electron thermal Ael and provides T-dependent forces for MD

• Density of states (DOS), structural and elastic properties sensitive to Tel

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

80 100 120 140

Sh

ear

mo

dulu

s (

Mbar)

Atomic volume (a.u.)

bcc Mo: C'Tel = 0 K

5000 K

DFT

MGPT

10000 K

7500 K

Melting in Mo requires temperature-dependent

MGPT potentials

0

2000

4000

6000

8000

10000

12000

14000

0 100 200 300 400

Te

mp

era

ture

(K

)

Pressure (GPa)

Mo

shock

T = 0 MGPT

T-dep MGPT

QMD: LLNL

QMD: LANL

isobaric

liquid

solid (bcc)

• T = 0 MGPT potentials

overestimate melt Tm by

factor of two

• T-dep MGPT melt in good

agreement with isobaric

and shock data as well as

with first-principles QMD

simulations

• Polymorphism in high-P,T

solid currently being studied

with T-dependent potentials