ATAT - A software toolkit for modeling coupled

configurational and vibrationaldisorder in alloy systems

This work was supported by:NSF under program DMR-0080766 and DMR-0076097.DOE under contract no. DE-F502-96ER 45571.AFOSR-MEANS under grant no. F49620-01-1-0529

Axel van de WalleMaterials Science and Engineering Department

A matter of time...Ti

me

Human

Computer

1980 2005

Time needed to complete a given first-principles calculation

The procedure needs to be automated

Thermodynamic data

Quantum Mechanical Calculations

Lattice model &Monte Carlo Simulations

Electronic excitationsLattice vibrations

First-principles Thermodynamic Calculations

http://cms.northwestern.edu/atat

Configurational disorder

Thermodynamic data

Quantum Mechanical Calculations

Lattice model &Monte Carlo Simulations

Electronic excitationsLattice vibrationsConfigurational disorderConfigurational disorder

Coupled SublatticesMulticomponent Cluster Expansion

Example: binary fcc sublattice withternary octahedral sites sublattice

Occupation variables:

1 2

1 1

“Decorated” clusters:

i 0, ,ni 1

1 , ,n

i 0, ,ni 1

E1 , ,n J

2, ,

i

1 11 1 i

3, ,

i

1 1 11 1

2 12

0 32 3

2

i

i ni, i, i

Same basic form:

“Not in cluster”

Sanchez, Ducastelle and Gratias (1984)Tepesch, Garbulski and Ceder (1995)

# components

Automated Cluster Expansion Construction

Van de Walle, Asta and Ceder (2002),Murray (1987) (exp.) Many other examples…

hcp Ti

DO19Ti3Al

Likely source of the discrepancy:Vibrational entropy.

Fultz, Nagel, Antony, et al. (1993-1999)

Ceder, Garbulsky, van de Walle (1994-2002)

de Fontaine, Althoff, Morgan (1997-2000)

Zunger, Ozolins, Wolverton (1998-2001)

Temperature scale problem

Thermodynamic data

Quantum Mechanical Calculations

Lattice model &Monte Carlo Simulations

Electronic excitationsLattice vibrationsConfigurational disorderLattice vibrations

The Cluster Expansion Formalismwith Coarse-Graining

E1, ,n

F1, ,n JT

Coarse-Graining of the Free EnergyGraphically:

Formally: (Ceder (1993), Garbulski and Ceder (1994-1996))

F 1 ln i eEi 1 ln i eEi

1 ln eF

F 1 ln i eEiwhere

kBT1

First-principles lattice dynamics

Least-squares fit toSpring model

First-principles data

ThermodynamicProperties

Phonon density of states

Computationallyintensive!

Direct force constant method(Wei and Chou (1992), Garbuski and Ceder (1994), among many others)

Must be done formany configurations!

Transferable Force Constants

van de Walle and Ceder (2000,2002)

Relationship holds across different structures on the same lattice (here fcc is shown).

Chemical bond type andbond length:Good predictorof nearest-neighbor force constants(stretching and bending terms)

Further tests...Wu, Ceder, van de Walle (2002)

Au-Au bonds

Pd-Pd bonds

Cu-Cu bonds

Cu-Pd bonds

Au-Pd bonds

Au-Cu bonds

Accuracy ~ 0.03 kB

Length-Dependent TransferableForce Constants (LDTFC)

van de Walle and Ceder (2000,2002)

Quasi-harmonic model

F(T,V) = E(V) + FH(T,V)

V*(T) = arg minV F(T,V)

F(T) = F(T,V*(T))

Energy of a relaxed motionless latticewith externally imposed volume V

Vibrational free energy of a harmonic solidat temp. T with externally imposed volume V

Thermal expansion:

“True” free energy:

Ideal for use with length-dependent transferable force constants

Calculated Ti-Al Phase Diagram

Assessed Phase Diagram:I. Ohnuma et al., ActaMater. 48, 3113 (2000)

1st-Principles Calculations:van de Walle and Asta

Ti-Al Thermodynamic Properties1st-Principles Calculations vs.

Measurements

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

0 0.1 0.2 0.3 0.4 0.5Al Concentration

∆H

(kJ/

mol

e)

Calorimetry(Kubaschewski and Dench, 1955)

FLMTO-LDA

VASP-GGA

-30

-25

-20

-15

-10

-5

0

0 0.1 0.2 0.3 0.4

Al Concentration

∆G

(kJ/

mol

e)

α

α2

EMF Measurements(Samokhval et al., 1971)

Monte-CarloCalculations

Gibbs Free Energies (T=960 K)Heats of Formation

Ordering in the Cu-Li system?

Ordered Cu4Li phase also suggested by

Borgsttedt & Gumiński (1996),

Krauss, Mendelson, Gruen, et al. (1986),

Old & Trawena (1981).

Widely used assessments

do not include ordered phases

(Pelton (1986), Saunders (1998)).

Evidence from EMF measurements

Calculated Thermodynamic datafor Cu-Li system

Tsimulation = 950 KTmeasurement = 885 K

2 adjustable parameters:Tsimulation + chemical potential of Li (liq)

bcc

fcc

B32

1st order transition

2nd order transition

Phase diagram Electromotive Force (EMF)

Thermodynamic data

Quantum Mechanical Calculations

Lattice model &Monte Carlo Simulations

Electronic excitationsLattice vibrationsConfigurational disorderElectronic excitations

Electronic Excitations

F1, ,n JT

Electronic DOS

Finite-temperature DFT

Fermi-Dirac Distribution

012345678

-4 -2 0Energy (eV)

Den

sity

of S

tate

s

T-independent DOS and charge density

Electronic Free energy

Advantage: Get Felec “for free”

FelecT EelecT Eelec0 TSelecT

EelecT f,Tgd

SelecT kB f ,T ln f ,T 1 f ,T ln1 f,Tgd

g

f,T

Cluster expansion:

Using ATAT

• Overview of the input/output files

• Syntax of the files

• Sample output

Commands:

File structureABlat 0

5

. . .

str.out energy str_relax.out

str.out energy str_relax.out

0.2_0_0

0.2_0_1

0.2_0.5_0

0.2_0.5_1

str.out

str.out

str.out

str.out

force.out

force.out

force.out

force.out

lat.invasp.wrap

maps -drun ab initio codefitsvsl -d

fitsvsl -f

run ab initio code

slsprings.outstrnames.in

eci.outclusters.outgs.outmaps.log Perturbations

clusterexpand fvibforeachfile str.out svsl -d

fvib

fvib

Trange.infvib.eci

mkteci fvib.eci

teci.out

Files:

Example of input filesSimple lattice input file

Simple ab initio code input file

Clusters and ECI files

10.0000000

20.00000011.000000 1.000000 1.000000

62.93999920.666667 0.333333 0.5000001.000000 0.000000 1.000000

62.94000420.666667 0.333333 0.500000-0.333333 -0.666667 0.500000

etc.

0.229683-0.3502750.0479380.041720

etc.

multiplicitydiameter

# of pointssites

(using coordinatesystem in lat.in)

clusters.out eci.out fvib.eci

0.0826-0.0008-0.0006-0.0005...

0.0793-0.0008-0.0006-0.0006...

T=0K

T=100K

etc.

teci.out

2000 21

0.312346-0.3510770.04733490.0411423...

0.308527-0.3509030.04725380.0410874...

Tmax # division

maps graphical output

Search for new compoundsin the Al-Mo-Ni system

E

AlNi

Al (fcc)

Mo (bcc)

Ni (fcc)

NiAl (B2)Ni3Al (L12)

Mo

plotted with MEDIT, INRIA-Rocquencourt.

Ni

Predicted Compound

Al

MoAlMo2Ni2

(fcc)

(bcc)

plotted with MEDIT, INRIA-Rocquencourt.

fitsvsl graphical output

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

2.8 2.82 2.84 2.86 2.88 2.9

Al-Al

s'f_Al-Al.dat' u 1:2

b'f_Al-Al.dat' u 1:3

-0.5

0

0.5

1

1.5

2

2.84 2.86 2.88 2.9 2.92 2.94

Al-Ti

s'f_Al-Ti.dat' u 1:2

b'f_Al-Ti.dat' u 1:3

0

0.5

1

1.5

2

2.5

3

2.8 2.85 2.9 2.95 3

Ti-Ti

s'f_Ti-Ti.dat' u 1:2

b'f_Ti-Ti.dat' u 1:3

length (Å)

Forc

e co

nsta

nt (e

V/Å

2 )

Forc

e co

nsta

nt (e

V/Å

2 )

length (Å)

length (Å)

emc2 graphical output

200

400

600

800

1000

1200

1400

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

'mc1.out' u 4:1

T

x

1) T: Temperature2) mu: chemical potential3) E-mu*x: Average energy4) x: Average Concentration5) phi: grand canonical potential6) E2: heat capacity7) x2: susceptibility8) E_lte-mu*x_lte: from LTE9) x_lte10) phi_lte11) E_mf-mu*x_mf: from MF12) x_mf13) phi_mf14) E_hte-mu*x_hte: from HTE15) x_hte16) phi_hte17) lro: Long Range Order parameter18-) corr: correlations

Other outputs:

Miscellanea (I)maps : Cluster expansion buildermmaps : Multicomponent version of mapsemc2 : General purpose Monte Carlo codephb : Phase-transition-tracing Monte Carlo codecheckrelax : Excessive relaxation detectorcorrdump : Cluster generator/correlation calculatorclusterexpand : Manual cluster expansion generatorgenstr : Super structure generatorgensqs : Special Quasirandom Structure generatorpdef : Point defect supercells generatorcsfit : Constituent strain calculatorcellcvrt : General crystal structure file format conversion utilitylsfit : Least-squares fitting codefitfc : Phonon calculation with direct force methodfitsvsl : Length-Dependent Transferable Force Constants generatorsvsl : Phonon calculations using LDTFCfelec : Electronic free energy calculator

ATATUtilites

Miscellanea (II)runstruct_vasp : Interface with vasprunstruct_abinit : Interface with abinitrunstruct_pwscf : Interface with pwscfrunstruct_gulp : Interface with gulp

pollmach : A job dispatcher for computer clustersforeachfile : A “loop over directories” utilitystr2xyz : File conversion utility for viewing with rasmol

makelat : Database of crystal structures

getvalue, (just)after, (just)before, (just)between sspp : Text extractors

memc2 : Multicomponent version of emc2 (in development)

ATATUtilites

ab initio

empirical potential

• Run command without any argument to get usage help • Use –h option to get detailed help• http://cms.northwestern.edu/atat/manual/manual.html

Getting Help

References

• A. van de Walle and G. Ceder. The effect of lattice vibrations on substitutional alloy thermodynamics. Rev. Mod. Phys., 74:11, 2002.

• A. van de Walle, M. Asta, and G. Ceder. The alloy theoretic automated toolkit: A user guide. CALPHAD Journal, 26:539, 2002.

• A. van de Walle and G. Ceder. Automating first-principles phase diagram calculations. Journal of Phase Equilibria, 23:348, 2002.

• A. van de Walle and M. Asta. Self-driven lattice-model monte carlosimulations of alloy thermodynamic properties and phase diagrams. Modelling Simul. Mater. Sci. Eng., 10:521, 2002.

• Web site: http://cms.northwestern.edu/atat• This afternoon’s tutorial: http://cms.northwestern.edu/atat/tutorial