Page 1
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
Page 2
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
Page 3
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
Page 4
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
Page 5
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 )]
Page 6
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
Page 7
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
Page 8
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
Page 9
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:
Page 10
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
Page 11
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
Page 12
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
Page 13
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)
Page 14
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
Page 15
• 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
Page 16
• 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
Page 17
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
Page 18
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
Page 19
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
Page 20
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
Page 21
),;(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
Page 22
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