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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
Using multi-body energy expansions from ab-initio
calculations for computation of alloy phase structures
Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes HallCornell University
Ithaca, NY 14853-3801
Email: [email protected]: http://mpdc.mae.cornell.edu
V. Sundararaghavan and Prof. Nicholas Zabaras
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
PREDICTION OF STABLE STRUCTURES
Computational techniques -Exhaustive or heuristic search aided by DFT calculations-Cluster expansion
CuCa
hP6oP12
Stable Pt clusters (Doye
and Wales, New J. Chem., 1998)
Stable configurations of adsorbed 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
• Only configurational degrees of freedom• Relaxed calculation required but only a few calculations required • Periodic lattices, Explores superstructures of parent lattice
• Configurational and positional degrees of freedom• Relaxed DFT calculations are not required• Periodicity is not required • Requires a large number of cluster energy evaluations• Convergence issues
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
Comparison with CE
Cluster 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
Hybrid cluster expansions
• Allow positional degrees of freedom in cluster expansions
• For periodic lattices
Cluster expansion for the fixed lattice
Pair potentials for local relaxations
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
Geng, Sluiter et al, Phys Rev B 2006
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
JW Martin - Journal of Physics C, 1975, Empirical potentials (3 body): Murrell-Mottram (Mol. Phys 1990)
∑= ∑+ ∑+ + …
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 multibody potentialsExample of calculation of multibody potentials
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
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 potentialsInversion of potentials
EL is found from ab-initio energy database, L << M
Calculation of energiesCalculation of energies
Drautz, Fahnle, Sanchez, J Phys: Condensed matter, 2004
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
Fitting energy surfaces
To calculate the energy of a 3 body structure (E3), we need to identify E2 and E1, values.
• Two body energy E2(X1,X2) is the energy of an isolated cluster of 2 atoms at positions X1 and X2.
• The database may not contain this energy since the energy values have only been obtained for atoms at locations (xi,yi) that are different from (X1,X2)
• We use interpolation methods for retrieving energy at (X1,X2) from the database of energies at (xi,yi) . For example, we can use a polynomial interpolation of the form:
2 3 4 5 6 70
0.5
1
1.5
2
2.5
3
3.5x 10
6
Order of expansion
Num
ber o
f add
ition
al c
alcu
latio
ns
Interpolation allows us to compute a large number of energies from a well-sampled
database
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
Smolyak algorithm
Extensively used in statistical mechanics
Provides a way to construct interpolation functions based on minimal number of points
( ) ( )i
i i
i i
xx X
U f a f x
1
0 11
, 1,
0, ,
( ) ( ) ( )( )d
i i id
iiq d q d
i q
U U U i i i
A f A f f
Uni-variate interpolation
Multi-variate interpolation
Smolyak interpolation
Accuracy the same as tensor product
Within logarithmic constant
Increasing the order of interpolation increases the number of points sampled
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
Smolyak algorithm: reduction in points
For 2D interpolation using Chebyshev nodes
Left: Full tensor product interpolation uses 256 points
Right: Sparse grid collocation used 45 points to generate interpolant with comparable accuracy
For multi-atom systems, sample all combinations of atoms (eg. E(A-A-A), E(A-A-B), E(A-B-B),E(B-B-B) and construct interpolants.
Results in multiple orders of magnitude reduction in the number of points to sample
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
CLUSTER REPRESENTATION
Specification of clusters of various order by position variables
1 2 3
4 5
5
1 2 3
4
a
bba
• Convex hull technique to represent all atoms in the positive z-direction
• Use independent coordinates to represent the cluster geometryA point in 6 dimensional
space
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
CLUSTER ENERGY COMPUTATIONS
• Executables Executables –Cluster coordinatesCluster coordinates–Energy interpolationEnergy interpolation–Batch input for PWSCFBatch input for PWSCF–Read energies from Read energies from
PWSCFPWSCF–Energy calculationEnergy calculation
• Plane-wave electronic density functional program ‘quantum espresso’ (http://www.pwscf.org) calculations are used to compute energies given the atomic coordinates and lattice parameters. •These calculations employ LDA and use ultra-soft pseudopotentials. • Single k-point calculations were used for isolated clusters, the cell size was selected so that the effect of periodic neighbors are negligible.•For multi-component systems, a constant energy cutoff equal to cutoff for the hardest atomic potential (e.g. B in B-Fe-Y-Zr) is used. MP smearing (ismear=1, sigma=0.2) is used for the metallic systems.
Computations were performed in parallel on a 64 node quad-processor LINUX cluster
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
LINKING THE MULTIBODY EXPANSION TO OTHER SOFTWARE
The multibody expansion software written in C++
Two parts: potential generation & energy computation
Energy computation part is the Hamiltonian
Molecular dynamics- LAMMPS
Multi Body Expansion (MBE)
Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is a classical molecular dynamics (MD) code developed by S. Plimpton et. al (Sandia national lab)
Directly linked energy computation part in LAMMPS with MBE
Useful for molecular dynamics and energy minimization
Monte Carlo for Complex Chemical Systems (MCCCS) developed by M. G. Martin, J. I. Siepmann et. al. Available at http://towhee.sourceforge.net/
Fortran based code. Linked Towhee and MBE using a library
Performs a variety of calculations in all ensembles
Monte Carlo- MCCCS Towhee
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
3 4 5 6 7 8 9 10 11 12-104.8
-104.7
-104.6
-104.5
-104.4
-104.3
-104.2
Interatomic distance (Bohr)
En
erg
y (
Ry
d)
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
ENERGY SURFACES FOR ISOLATED CLUSTERS
b
a
X
Y
a = (1.5*X+0.5)*7.5 Bohr
b = (1.5*Y+0.5)*7.5 Bohr
4 5 6 7 8 9 10 11
4
5
6
7
8
9
10
11
-157
-156.9
-156.8
-156.7
-156.6
-156.5
-156.4
-156.3
Platinum isolated cluster energies computed using multi-body potentials
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
COMPUTATION OF CLUSTER ENERGIES
4 5 6 7 8 9 10 11
4
5
6
7
8
9
10
11
-157
-156.9
-156.8
-156.7
-156.6
-156.5
-156.4
-156.3
4 5 6 7 8 9 10 11
4
5
6
7
8
9
10
11
Distance between atoms 1-2 (Bohr)
Dis
tan
ce b
etw
ee
n a
tom
s 1
-3 (
Bo
hr)
Distance between atoms 1-2 (Bohr)
Dis
tan
ce b
etw
ee
n a
tom
s 1
-3 (
Bo
hr)
(a) (b)
The complete potential surface for a 3 Pt cluster. Figure (a) shows computed Platinum three-atom cluster energies, while (b) shows extension of energies using pair potential terms beyond the cutoff.
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
Convergence results for different energy functions
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30
10
20
30
40
50
60
70
80
90
Cont
ribut
ion
to to
tal e
nerg
y in
%
value of n in n-body interation term
convergence result
Energy of the system scales as n2 where n is the number of atoms
Order of interactions necessary for full convergence: 2
3 body term contribution
=0
1 1.5 2 2.5 3 3.5 4 4.5 50
50
100
150
200
250
Con
tribu
tion
to to
tal e
nerg
y in
%
value of n in n-body interation term
convergence result
Energy of the system scales as n1/2 where n is the number of atoms
Order of interactions necessary for full convergence: 4
5 body term contribution =0
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
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30
20
40
60
80
100
120
Co
ntr
ibu
tio
n t
o t
ota
l e
ne
rgy
in
%
value of n in n-body interation term
convergence result
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
Convergence results for different energy functions
Using pair potentials: Lennard Jones for Helium atoms
Order of interactions necessary for full convergence is 2 as expected (since it is a pair potential)
3 body term contribution =0
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
Oscillations in MB energy for complex energy functionals
-Energies oscillate around the true energy
-Approach: Low pass filtering (convolution operation) that cuts off high frequency oscillations.
-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
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
Computation of MBE energy filters
Weighted MBE
+
+
+ ..
Is the total energy correlated with
structural energies of clusters ?
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
Weighted MB energy
11 2( , ,.., )M ME X X X
21 2( , ,.., )M ME X X X
31 2( , ,.., )M ME X X X
a1
a2
a31 2 2 1 2
3 1 2 4 1 2
( , ,.., ) 0.1484 ( , ,.., )
0.5721 ( , ,.., ) 0.2794 ( , ,.., ).M M M
M M
E X X X E X X X
E X X X E X X X
6 6.5 7 7.5 8 8.5 9-8.2
-8
-7.8
-7.6
-7.4
-7.2
-7
-6.8
Lattice Parameter (Bohr)
Co
he
siv
e E
ne
rgy
(R
yd
)
MBE energyTrue energy
2 3 4
-20
-15
-10
-5
0
Truncation order
Co
he
siv
e E
ne
rgy
(R
yd
)
True energy
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
4 4.5 5 5.5 6 6.5 7-10
0
10
20
30
40
50
60
70
80
Lattice Parameter (Bohr)
Co
he
siv
e E
ne
rgy
(R
yd
)
MBE energyTrue energy
Extrapolatory tests on weighted MBE
True energy
MBE 4th order
1 2 2 1 2
3 1 2 4 1 2
( , ,.., ) 0.1484 ( , ,.., )
0.5721 ( , ,.., ) 0.2794 ( , ,.., ).M M M
M M
E X X X E X X X
E X X X E X X X
Weighted MBE energies once built for a small set of configurations provide accurate energy fit for various different inter-atomic distances within that configuration.
16 atom Au-Cu FCC cluster
4 unit cell, 4 at/cell
AuCu3
6 6.5 7 7.5 8 8.5 9-8.2
-8
-7.8
-7.6
-7.4
-7.2
-7
-6.8
Lattice Parameter (Bohr)
Coh
esiv
e E
nerg
y (R
yd)
MBE energyTrue energy
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
Selection of order of expansion
Weighted 2nd order MBE
Weighted 3rd order MBE
Weighted 4th order MBE
True energies
True energies
True energies
Weighted MBE expansion coefficients are fitted using 12 atom cluster energies and the results are presented for a 16 atom cluster.
Energies may differ but the weighted MBE captures the energy minima within 4th order expansion.
Coh
esi
ve e
ne
rgy
(Ryd
)
Coh
esi
ve e
ne
rgy
(Ryd
)
Coh
esi
ve e
ne
rgy
(Ryd
)
Test various MBE orders in extrapolatory modes
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
Platinum clusters
+
+
Depth of interpolation
4 120
4 560
4 1820
Number of isolated cluster calculations
• Coefficients obtained using an 12 atom cluster energies at different lattice parameters
16 atom FCC cluster
1 2 2 1 2
3 1 2 4 1 2
( , ,.., ) 0.5884 ( , ,.., )
0.3014 ( , ,.., ) 0.0353 ( , ,.., ).M M M
M M
E X X X E X X X
E X X X E X X X
Actual energy
Weighted MBE 4th order
Energy minima
6 6.5 7 7.5 8 8.5 9-6
-5
-4
-3
-2
-1
0
Lattice parameter (Bohr)
Coh
esi
ve e
ne
rgy
(Ryd
)
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
+
+
Depth of interpolation
4 276
4 2024
4 10626
Number of isolated cluster calculations
Actual energy
24 atom FCC cluster
1 2 2 1 2
3 1 2 4 1 2
( , ,.., ) 0.5884 ( , ,.., )
0.3014 ( , ,.., ) 0.0353 ( , ,.., ).M M M
M M
E X X X E X X X
E X X X E X X X
Platinum clusters
Weighted MBE 4th order
Energy minima
6 6.5 7 7.5 8 8.5 9-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
Lattice parameter (Bohr)
Coh
esi
ve e
ne
rgy
(Ryd
)
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
+
+
Depth of interpolation
4 276
4 2024
4 10626
Number of isolated cluster calculations
Actual energy
Weighted MBE 4th order
1 2 2 1 2
3 1 2 4 1 2
( , ,.., ) 0.5884 ( , ,.., )
0.3014 ( , ,.., ) 0.0353 ( , ,.., ).M M M
M M
E X X X E X X X
E X X X E X X X
A random 24 atom configuration
Platinum clusters
6 6.5 7 7.5 8 8.5 9-9
-8
-7
-6
-5
-4
-3
-2
-1
Lattice parameter (bohr)
Coh
esi
ve e
ne
rgy
(Ryd
)
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
Stable phase structures of Au-Cu alloy
Super-cell approach
For computing stable structures of periodic lattices, a 4x4x4 supercell (216 atoms) is used as an approximation.
Weighted MBE is several orders of magnitude faster than a relaxed DFT calculation.
Useful for amorphous structures
Small cluster calculations are used to compute the weights in the weighted MBE expansion
FCC structures are considered here for Au-Cu.
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
6 6.5 7 7.5 8 8.5 9-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Lattice parameter (A)
Coh
esiv
e en
ergy
(Ryd
/ato
m)
6 6.5 7 7.5 8 8.5 9-0.46
-0.44
-0.42
-0.4
-0.38
-0.36
-0.34
-0.32
-0.3
-0.28
-0.26
Lattice parameter (Bohr)
Co
he
siv
e e
ne
rgy
(R
yd
/ato
m)
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
Stable phase structures of Au-Cu alloy
AuCu3 cell relaxation
3x3x3 supercell
a = 6.62 bohr = 3.50 A
Au3Cu cell relaxation
a = 7.3 bohr = 3.86 A
1 2 2 1 2
3 1 2 4 1 2
( , ,.., ) 0.1484 ( , ,.., )
0.5721 ( , ,.., ) 0.2794 ( , ,.., ).M M M
M M
E X X X E X X X
E X X X E X X X
AuCu3 lattice parameter: 3.76 A Au3Cu lattice parameter: 4.04 A
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
6 6.5 7 7.5 8 8.5 9-0.19
-0.18
-0.17
-0.16
-0.15
-0.14
-0.13
-0.12
-0.11
Lattice parameter (Bohr)
Coh
esiv
e en
ergy
(R
y/at
om)
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
Stable phase structures of Au-Cu alloy
AuCu3 cell relaxation
4x4x4 supercell
a = 6.71 bohr = 3.55 A
Au3Cu cell relaxation
6 6.5 7 7.5 8 8.5 9-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Lattice parameter (Bohr)
Coh
esiv
e E
nerg
y (R
yd/a
t)
a = 7.4 bohr = 3.92 A
AuCu3 lattice parameter: 3.76 A Au3Cu lattice parameter: 4.04 A
1 2 2 1 2
3 1 2 4 1 2
( , ,.., ) 0.1484 ( , ,.., )
0.5721 ( , ,.., ) 0.2794 ( , ,.., ).M M M
M M
E X X X E X X X
E X X X E X X X
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
G.Kallen,G.Wahnstrom, Quantum treatment of H on a Pt(111) surface, Phys Rev B, 65 (2001)
Minimum energy surface of h on Pt(111)
Plot of minimum energy in z direction for the primitive cell
Highly anharmonic potential energy surface
FCC->HCP (55 mev), FCC->TOP (160 mev)
H confined to FCC-HCP-FCC valleys
APPLICATION TO SURFACE PHENOMENA
FCC site
(Baskar and Zabaras, 2007)
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 structures.
• Ab-initio database of cluster energies are created and interpolation for various cluster positions are generated using efficient sparse grid interpolation algorithms.
•Weighted MBE is fast and captures the energy minima within a small order of expansion.
• Technique is applicable to study stability of amorphous systems, molecules and clusters.