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Ab initio Alloy Thermodynamics:Recent Progress and Future Directions
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 WalleMark Asta
Materials Science and Engineering Department, Northwestern University
Gerbrand CederMaterials Science and Engineering Department, MIT
Chris WoodwardAir Force Research Laboratory, Wright-Patterson AFB
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• Describe the current capabilities of ab initio thermodynamic calculations
• Illustrate how the Alloy Theoretic Automated Toolkit (ATAT) can help perform such calculations
Goals
ATAT homepage: http://cms.northwestern.edu/atat/
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What can first-principles thermodynamic calculations do for you?
Ducastelle (1991), Fontaine (1994), Zunger (1994,1997), Ozolins et al. (1998),Wolverton et al. (2000), Ceder et al. (2000), Asta et al. (2000,2001)
Composition-temperaturephase diagrams
Thermodynamics of stable
and metastable phases,
Short-range orderin solid solutions
Thermodynamic properties
of planar defectsPrecipitate morphology and
Microstructures
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First-principles thermodynamic data
Quantum Mechanical Calculations
Lattice model &Monte Carlo Simulations
Electronic entropyVibrational entropyEnthalpy
•Large number of atoms•Many configurations
•Small number of atoms•Few configurations
Ab initio thermodynamic calculations
ATAT
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Outline
• Methodology– Modeling configurational disorder– Modeling lattice vibrations
• Applications (Ti-Al and Al-Mo-Ni)– Sample input files– Sample outputs
• Recent innovations
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The Cluster Expansion Formalism
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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
E 1 , , n J
2,,
i
1 1
1 1 i
3,,
i
1 1 1
1 12
12
03
2 3
2
i
i ni, i, i
Same basic form:
“Not in cluster”
Sanchez, Ducastelle and Gratias (1984)Tepesch, Garbulski and Ceder (1995)
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Cluster expansion fit
Which structures andwhich clustersto include in the fit?
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Cross-validation
Example of polynomial fit:
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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)
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Effect of lattice vibrations onphase stability
Ozolins and Asta (2001) (Wolverton and Ozolins (2001))
Stable without vibrations(incorrect)
Stable with vibrations(correct)
How to handle alloy phase diagrams?
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Coupling vibrational and configurational disorder
Need to calculate vibrational free energy for many configurations
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Efficient modeling of lattice vibrations
• Infer the vibrational entropies from bulk moduli(Moruzzi, Janak, and Schwarz, (1988))(Turchi et al. (1991), Sanchez et al. (1991), Asta et al. (1993), Colinet et al. (1994))
• Calculate full lattice dynamics using tractable energy models(Ackland (1994), Althoff et al., (1997), Ravello et al (1998), Marquez et al. (2003))
• Calculate lattice dynamics from first principles in a small set of structures (Tepesch et al. (1996), Ozolins et al. (1998))
• Transferable force constants (Sluiter et al. (1999))
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Bond length vs. Bond stiffness
van de Walle and Ceder (2000,2002)
Relationship holds across different structures
Chemical bond type andbond length: Good predictor of nearest-neighbor force constants (stretching and bending terms)
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Length-Dependent Transferable Force Constants (LDTFC)
van de Walle and Ceder (2000,2002)
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A matter of time…T
ime
Human
Computer
1980 2003
Time needed to complete a given first-principles calculation
The procedure needs to be automated
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Lattice geometry Ab initio code parameters
Effective cluster interactions Ground states
Thermodynamic properties Phase diagrams
MAPS (MIT Ab initioPhase Stability Code)
Cluster expansion construction
Ab initio code(e.g. VASP, Abinit)
Emc2 (Easy Monte Carlo Code)
The Alloy Theoretic Automated Toolkit
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Application to Ti-Al AlloysSimple lattice input file
Simple ab initio code input file
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Effective Cluster Interactions
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Ground States Search
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Ground state search inAl-Mo-Ni system
E
AlNi
Al (fcc)
Mo (bcc)
Ni (fcc)
NiAl (B2)Ni3Al (L12)
Mo
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Monte Carlo output:Free energies
Can be used as input to CALPHAD approach
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Short-range order calculations
Calculated diffuse X-ray scattering in Ti-Al hcp solid-solution
Energy cost of creating a diffuse anti-phase boundary in a Ti-Al short-range ordered alloy by sliding k dislocations
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Calculated Ti-Al Phase Diagram
Assessed Phase Diagram:I. Ohnuma et al., Acta Mater.
48, 3113 (2000)
1st-Principles Calculations:van de Walle and Asta
Temperature Scale off by ~150 K
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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
DH
(kJ
/mo
le)
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
DG
(kJ
/mo
le)
EMF Measurements(Samokhval et al., 1971)
Monte-CarloCalculations
Gibbs Free Energies (T=960 K)Heats of Formation
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Recent Additions to ATAT
• Generation of multicomponent
Special Quasirandom Structures (SQS)
• General lattice dynamics calculations
• Support for GULP and Abinit
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Multicomponent SQS GenerationSQS: Periodic structures of a given size that best approximate a random solid solution. (Zunger, Wei, Ferreira, Bernard (1990))
fcc SQS-12 ABC
bcc SQS-16 ABC2
fcc SQS-16 ABC2
hcp SQS-16 ABC2(2x2x2 supercells shown)
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Automated lattice dynamics calculations
Thermal expansion of Nb
• Automatic determination of • supercell size• minimum number of perturbations (symmetry)
• Implements quasi-harmonic approximation
• crystal structure• force constant range
Input:• Phonon DOS• Free energy, entropy• Thermal expansion
Output:
Phonon DOS of disordered Ti3Al (SQS-16)
Examples:
Features:
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Conclusion• Essential tools for ab initio alloy thermodynamics:
– The cluster expansion (configurational entropy)– Transferable length-dependent force constants (vibrational entropy)
• Automated tools are essential• Thermodynamic properties can now be calculated
with a precision comparable to calorimetric measurements
• Future directions:– Automated Monte Carlo code for general multicomponent
systems.
ATAT homepage: http://cms.northwestern.edu/atat/