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Transcript of Materials Process Design and Control Laboratory
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
ON THE DEVELOPMENT OF WEIGHTED MANY-BODY EXPANSIONS USING AB-INITIO
CALCULATIONS FOR PREDICTING STABLE CRYSTAL STRUCTURES
1Department of Aerospace Engineering,University of Michigan, Ann Arbor
2Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
Cornell University
Email: [email protected]: http://mpdc.mae.cornell.edu
Veera Sundararaghavan1 and Nicholas Zabaras2
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
The crystal structure prediction The crystal structure prediction problemproblem
Predict the stable low-temperature phases of an alloy comprising of
elements X,Y.
Decreasing form
ation energies
Composition - XY2
Trial structures
True phase structure ?
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Ab Initio Structure PredictionAb Initio Structure Prediction
• Identify optimum structure through Identify optimum structure through Monte carlo/GA optimization using Monte carlo/GA optimization using relaxed ab initio energy calculations relaxed ab initio energy calculations
(vs)(vs)
• Identify optimum structures using Identify optimum structures using simplified Hamiltonianssimplified Hamiltonians – potentials, – potentials, cluster expansion, multi-body expansion cluster expansion, multi-body expansion (fitting challenges/ transferability/ (fitting challenges/ transferability/ accuracy issues)accuracy issues)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Ortho-normal and complete set of basis functions are introduced. is the configuration variable (+/- 1 for binary systems)
Basis for M lattice sites is given as:
Energy of the lattice (M sites) is given as:
For all cluster sizes For all clusters with number of atoms =K
Average of energies of all configurations projected onto the basis function
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Sanchez and de Fontaine, 1981, Sanchez et al, 1984 Physica A
Cluster expansion
-1
+1
+1
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Cluster expansionCluster expansion
...
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Cluster expansion fitCluster expansion fit
• The cluster expansion is able to represent any function E() of configuration by an appropriate selection of the values of J. • Converges rapidly using relatively compact structures (e.g. short-range pairs or small triplets). • Unknown parameters of the cluster expansion is determined by fitting first-principles energies as shown.
Connolly-Williams method, Phys Rev B, 1983
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
• Only configurational degrees of freedomOnly configurational degrees of freedom• Relaxed calculation required but only a few calculations Relaxed calculation required but only a few calculations
required required • Periodic lattices, Explores superstructures of parent latticePeriodic lattices, Explores superstructures of parent lattice
• Configurational and positional degrees of freedomConfigurational and positional degrees of freedom• Relaxed DFT calculations are not requiredRelaxed DFT calculations are not required• Periodicity is not required Periodicity is not required • Requires a large number of cluster energy evaluations*Requires a large number of cluster energy evaluations*• Convergence issues*Convergence issues*
Multi-body expansionMulti-body expansion
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Comparison with CEComparison with CE
Cluster expansionCluster expansion
*V. Sundararaghavan and N. Zabaras, "Many-body expansions for computing stable structures", Physical Review B, in review.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Multi-body expansion
Total energy
Symmetric function
Position and species
∑= ∑+ ∑+ + …
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Multi-body expansion
Example of calculation of multi-body Example of calculation of multi-body potentialspotentials
E1(X1) = V (1)(X1)
E2(X1,X2) = V (2)(X1,X2) + V
(1)(X1) + V (1)(X2)
Inversion of potentials
Evaluate (ab-initio) energy of several two atom structures to arrive at a
functional form of E2(X1,X2) V (2)(X1,X2) = E2(X1,X2) - (E1(X1) + E1(X2) )
E1(X2) = V (1)(X2)
Drautz, Fahnle, Sanchez, J Phys: Condensed matter, 2004
= Increment in energy due to pair interactions
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
= Increment in energy due to pair interaction
= Increment in energy due to trimer interaction
Multi-body expansion
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Multi-body expansion
Inversion of potentials (Mobius formula)Inversion of potentials (Mobius formula)
EL is found from ab-initio energy database, L << M
Calculation of energiesCalculation of energies
Drautz, Fahnle, Sanchez, J Phys: Condensed matter, 2004
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
• All potential approximations can be shown to be a special case of multi-body expansion– Embedded atom potentials
Multi-body expansion
Drautz, Fahnle, Sanchez, J Phys: Condensed matter, 2004
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Specification of clusters of various order by position variables
Cluster specifiers
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Cluster configurational spaces
Space of all possible three atom clusters of interest
Geometric constraints
Symmetry constraints
Corresponds to 9 planes forming a convex hull
El Er
Fourth order space (6D)
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Locating a cluster in the configurational space
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
User imposed cut offs
Lower cutoff- unstable
configurations
Upper cutoff- weak
interaction
3-atom cluster energy surface
2-atom cluster energy surface
Approximated using lower order (pair)
interactions
Upper cutoff
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Issues with larger orders of expansion
Explosion in number of clusters needed to calculate energies
Increase in configurational spaces required for an N-
atom cluster
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
-Energies oscillate around the true energy
-Approach: Weight MBE terms.
-Compute the energy at the minima using self consistent field calculation
correct energy
Energies (En) calculated from an n-body expansion
EAM potentials: Platinum system
Weighted Multi-body expansion
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Weighted MBE fit
• The multi body expansion is able to represent energy E of configuration of N atoms by an appropriate selection of the values of coefficients. • Converges rapidly using relatively compact structures (e.g. short-range pairs or small triplets). • Unknown parameters of the expansion is determined by fitting first-principle energies as shown.
Cluster Energies
Structures
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Weighted Multi-body expansion (Pt)
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Convergence test for extrapolatory cases
16 atom Pt Cluster with perturbed atoms
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Interpolated ab-initio MBE for Pt
Calculation of Pt lattice parameter
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
MBE for alloys
Multi-body expansion for -Alumina (Al2O3) system using cluster energies computed using the Streitz-Mintmire (SM) model. -Alumina
has a rhombohedral primitive unit cell and is describedin space group R-3c (no.167).
Converges at fourth order.
-Alumina (Al2O3) system
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Ab-initio MBE for alloys – Au-Cu systemCu-Cu-Au Cu-Au-Au
Structure optimization to find the lattice constants for FCC CuAu3 system (space group no. 221) using interpolated energies of clusters computed from first principles DFT calculations.
For computing stable structures of periodic lattices, a 6x6x6 supercell (864 atoms) is used.
Weighted MBE is several orders of magnitude faster than a relaxed DFT calculation.
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Sum over all k-atom clusters
Fix the k-atom cluster, and find the energy of system for various types of all other atoms
Link between MBE and CE
CE Global projection operation
Drautz et al, J Phys: Condensed matter, 2004
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Freeze the k-atom cluster Sum the N body potential for various positions and types of all other atoms
Contribution of N-body potential to K-atom cluster coefficients
Sum over all orders of expansion > K
N = 3, K = 2
Link between MBE and CE
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Advantages
• Because in any calculation only a finite number of clusters can be involved in the CE, there is always an open question of which of the various possible clusters are the most essential ones.– Can be identified using MBE
• Allows one to calculate the energetic contribution of relaxations to the cluster expansion coefficients on a lattice from the many-body potential expansion.– Allows consideration of the effect of relaxation, vibrational dofs
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Conclusions
• MB expansion provides atom position dependent potentials that are used to identify stable phase structures.
• Ab-initio database of cluster energies are created and interpolation for various cluster positions are generated using efficient finite element interpolation.
•Weighted MBE is fast and captures the energy minima within a small order of expansion.
Publication
V. Sundararaghavan and N. Zabaras, "Many-body expansions for computing stable structures", Physical Review B, in review.
Preprint available for download at http://mpdc.mae.cornell.edu