Topological characterization of
adsorption phenomena using multibody potential expansions
B. Ganapathysubramanian and Prof. Nicholas Zabaras
Materials Process Design and Control Laboratory
Sibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes Hall
Cornell University
Ithaca, NY 14853-3801
Email: zabaras@cornell.edu
URL: http://mpdc.mae.cornell.edu
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OVERVIEW
1. Problem statement
2. Multibody expansions: Representing the PES
3. Constructing the Multi body expansions: Large dimensions,
interpolation and the Smolyak algorithm
4. Simple problems in adsorption
5. Coupling MBE with a Grand Canonical simulator
6. Towards topological design
7. Conclusions
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Alternate means of energy production
“Catching up energy production
with energy demand (is) one of
the top 10 problems for the next
50 years” – Prof Smalley
Among the most promising
means is through fuel cells.
Chemical reaction or
combustion produces heat and
electricity with high efficiency
Anode:
2H24H++4e-
Cathode:
O2 +4H+ +4e-2H2O
Cell: 2H2(g) + O2(g) 2H2O(l)
Advantages:
 High efficiency
 Fuel can be obtained from sources other than petroleum
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Alternate means of energy production for mobile applications
Chemical reaction or
combustion produces heat and
electricity with high efficiency
Major issue is the onboard
storage of the fuel (hydrogen)
Need to store atleast 4 kg of hydrogen for
commercial usage of hydrogen1.
Many techniques investigated:
Most promising is the
physisorbtion of hydrogen onto
metallic and metallic-hydride
surfaces
L. Schlapbach, A. Züttel, Hydrogen-storage materials for mobile applications, NATURE 414
(2001)
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Towards designing materials with enhanced adsorption
Platinum based surfaces have large potential to
adsorb hydrogen
Recent developments have shown that alloying
platinum with metals like Bi and Rb produce cheaper
surfaces with similar properties
This is the first aspect of designing materials for
enhanced adsorption behavior
Top layer
Adsorption is essentially a surface phenomena
Can the surface be designed to enhance
adsorption?
Research shows that certain surfaces and
topological characteristics improve coordination of
hydrogen
Q. Wang, J. K. Johnson, Optimization of Carbon Nanotube Arrays for Hydrogen Adsorption, J Phys. CHem B
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Issues with modeling adsorption on metallic surfaces
Hydrogen molecule or hydrogen atom?
Hydrogen molecule, trajectory and velocity
of approach is important for chemisorption.
Recently shown that scattering of H2 is
electronically adiabatic 1.
Accurate potential energy surface to find
adsorption sites
Quantum delocalization effects: hydrogen
appears to be smeared out on the
surface
Medium range effects due to smearing
1. P.Nieto, et. al, Reactive and Nonreactive Scattering of H2 from a Metal Surface Is Electronically Adiabatic,
Science (2006) 312. 86 - 89
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Modeling adsorption on metallic surfaces
To take into account the quantum effects
need an essentially ab-initio approach.
Various studies have been performed that
investigate the adsorbtion of hydrogen on
metallic (specifically Pt) surfaces in a
quantum mechanical framework
In the context of designing topological
features one needs to necessarily model
larger scale structures (~O(μm))
Need a abinitio level accurate strategy
that can model large structures in a
computationally tractable way
1.
Watson G et. al, A comparision of the adsorption and diffusion of hydrogen on the {111} surfaces of Ni, Pd,
and Pt from density functional theory calculations, Journal of Physical Chemistry 105, 4889-4894 (2001)
2.
G. Källen, G. Wahnström, Quantum treatment of H adsorbed on a Pt(111) surface, Phys Rev B 65 (2001)
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Multi-body expansion
Total energy 1,2
Position and species
Symmetric function
Total energy is the sum of energies of higher and higher levels of interaction
•
•
•
•
•
All degrees of freedom included
No relaxations needed
Needs a database of calculations, regression schemes required
Periodicity is not required (large cell, one k-point calculation)
Can predict energies over several different lattices
1.
R Drautz, M Fahnle, J M Sanchez, General relations between many-body potentials and cluster
expansions in multicomponent systems, J. Phys.: Condens. Matter 16 (2004) 3843–3852
2.
J W Martin, Many-body forces in metals and the Brugger elastic constants, J. Phys. C, 8 (1975)
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Multi-body expansion
Need to find a
representation for
these functions
Inversion of potentials: Going from energies to potentials,
Mobius transformation
EL is found from ab-initio
energy database
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Multi-body expansion: Simple examples
E0 = V 0
E1(X1) = V (1)(X1) + V0
E2(X1,X2) = V (2)(X1,X2) + V (1)(X1) + V (1)(X2) + V0
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) – E0)
1
3
2
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Multi-body expansion: link to other Hamiltonians
• All potential approximations can be shown to be a
special case of multi-body expansion
– Embedded atom potentials
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Comparison with the Cluster Expansion Method
•
•
•
•
Only chemical degrees of freedom
Relaxed calculation required but only a few calculations required
Periodic lattice only
Results are obtained from superstructures of parent lattice
Multi-body expansion
•
•
•
•
•
1.
2.
3.
All degrees of freedom included
No relaxations needed
Needs a database of calculations, regression schemes required
Periodicity is not required (large cell, one k-point calculation)
Can predict energies over several different lattices
Sanchez and de Fontaine, 1981
Sanchez, et al, Generalized Cluster Description of Multicomponent Systems, Physica A 128 (1984)
Connolly,Williams, Density-functional theory applied to phase transformations in transition-metal alloys
Phys Rev B, 27 (1983 )
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Multi-body expansion
= ∑
+ ∑
+∑
+…
Total energy represented as hierarchical sum of isolated clusters of
atoms
- No periodicity
- Fully transferable
- No relaxation necessary
Two issues to be taken care of:
1)
How to construct each of these multi body potentials?
2)
When to stop the expansion?
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Constructing the multi-body potentials
Approximate the n-body potential as a polynomial in the corresponding dimension
Use the theory of interpolation to find these polynomials
Compute energies of a finite number of n-atom isolated clusters using ab-initio
methods and fit the polynomials to these energies
Well established theory to find the ‘best approximating polynomial’: again two
issues: which polynomial to choose and which points to sample at?
Very simple for two-body interactions
Enforcing symmetry and reducing the dimensions, this
becomes a one dimensional function
Just sample at roots of the chebyshev polynomial
Have rigorous bounds on the quality of the interpolant
generated
Becomes more complicated for higher body potentials
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High dimensional surfaces
4
As the number of atoms in the n-body potential increases, the
dimensionality of the n-body potential increases.
‘Curse of dimensions’ comes into play very quickly
1
2
3
Have to approximate high dimensional surfaces accurately
Cannot utilize a tensor product space!
Come up with intelligent schemes to sample from the hypersurface
Dimension points
Multi body expansions not a new theory.
One of the standing mathematical problems in representation
potential energy surfaces- Roszak & Balasubramanian J. Math
Chem (1994)
Techniques devised for representing the PES: but specific to
dimension and could not be generalized to higher body
interaction
1
50
2
2500
4
6.25e6
8
3.9e13
16
1.52e27
Murrell & Varandas, Molecular Physics (1986), Salazar, Chem Phys Let (2002), Wu et.al
PCCP (1999), Aquilanti et.al, PCCP (2000), Ischtwan & Collins, J. Chem Phys (1993),
Schatz, Rev. Mod. Phy (1989), Becker & Karplus, J. Chem Phys (1997)
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SMOLYAK ALGORITHM
LET OUR BASIC 1D INTERPOLATION SCHEME BE SUMMARIZED AS
Ui( f ) 
a
xi  X i
x
i
f ( xi )
IN MULTIPLE DIMENSIONS, THIS CAN BE WRITTEN AS
(U i1 
 U id )( f ) 

xi1 X i1

xid X id
(axi1 
 axid ) f (x i1 ,
, x id )
TO REDUCE THE NUMBER OF SUPPORT NODES WHILE MAINTAINING
ACCURACY WITHIN A LOGARITHMIC FACTOR, WE USE SMOLYAK METHOD
U 0  0, i  U i  U i 1 ,
i  i1 
Aq ,d ( f )  Aq 1,d ( f )   (i1 
 id
id )( f )
i q
IDEA IS TO CONSTRUCT AN EXPANDING SUBSPACE OF COLLOCATION
POINTS THAT CAN REPRESENT PROGRESSIVELY HIGHER ORDER
POLYNOMIALS IN MULTIPLE DIMENSIONS
A FEW FAMOUS SPARSE QUADRATURE SCHEMES ARE AS FOLLOWS:
CLENSHAW CURTIS SCHEME, MAXIMUM-NORM BASED SPARSE GRID AND
CHEBYSHEV-GAUSS SCHEME
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SMOLYAK ALGORITHM
Extensively used in statistical mechanics
Uni-variate interpolation
Provides a way to construct interpolation
functions based on minimal number of
points
Ui( f ) 
a
xi  X i
x
i
f ( xi )
Multi-variate interpolation
(U i1 
Univariate interpolations to multivariate
(U i1   U id )( f )  
interpolations
xi1 X i1
 U id )( f ) 

xid X id
(axi1 


(axi1 
 axid ) f (x ,
, x id )
xi1 X i1
xid iX id
1
Smolyak interpolation
U 0  0, i  U i  U i 1 ,
Accuracy the same as tensor
product
i  i1 
Aq ,d ( f )  Aq 1,d ( f )   (i1 
 id
id )( f )
i q
D = 10
Within logarithmic constant
Increasing the order of interpolation increases
the number of points sampled
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ORDER
SC
FE
3
1581
8000
4
8801
40000
5
41625
100000
Materials Process Design and Control Laboratory

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
Results in multiple orders of magnitude
reduction in the number of points to sample
D = 10
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.
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ORDER
SC
FE
3
1581
1000
4
8801
10000
5
41625
100000
Materials Process Design and Control Laboratory
ADAPTIVE SPARSE GRID COLLOCATION
The conventional sparse grid method treats every dimension equally.
Functions may have widely varying characteristics in different directions (discontinuities,
steep gradients) or the function may have some special structure (additive, nearly-additive,
multiplicative).
The basis proposition of the adaptive sparse grid collocation is to detect these
structures/behaviors and treat different dimensions differently to accelerate convergence.
Must use some heuristics to select the sampling points.
Such heuristics have been developed by Gerstner and Griebel
Have to come up with a way to make the Smolyak algorithm treat different dimensions
differently.
Generalized Sparse Grids:
Convention sparse grids imposes a strict admissibility condition on the indices. By relaxing
this to allow other indices, adaptivity can be enforced.
Admissibility criterion for a set of indices S.
where ej is the unit vector in the j-th direction
1. T. Gerstner, M. Griebel, Numerical integration using sparse grids, Numerical Algorithms, 18 (1998) 209–232.
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MINIMAL CLUSTER REPRESENTATION
Specification of clusters of various order by position variables
Cluster
size
Cluster specifier
Dimensionality
2
R12
1
3
R12, R23, R31
3
4
R12,R23,R34,R41,R42 ,R31
6
M
R12,R23,R34,R41…
3M-6
4
4
5
a
1
2
3
b
b
1
2
3
5
a
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Improving the
computational
efficiency by
reducing the
problem dimension
• Convex hull
technique to
represent all atoms in
the positive z-direction
• Use independent
coordinates to
represent the cluster
geometry
Materials Process Design and Control Laboratory
Constructing the multi-body potentials
•Needs the least number of ab initio
calculations toconstruct the
potential,
Energy
•Provides capabilities to
hierarchically improve the quality of
interpolation using the previous
interpolant,
•Can be made to adaptively sample
the different dimensions to further
reduce the computational
requirements
Position
atoms
accuracy
tensor
sparse
3
10-6
66049
1537
4
10-5
1.9x1019
0.6x106
5
2x10-5
5.4x1033
20x106
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U N I V E R S I T Y
•Completely independent of the
number of dimensions of the
problem.
•Provides a way of constructing
fully–transferable ab initio based
potentials.
Materials Process Design and Control Laboratory
Abinitio computation of the energies
• Executables
– Cluster coordinates
– Energy interpolation
– Batch input for PWSCF
– Read energies from
PWSCF
– Energy calculation
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U N I V E R S I T Y
• 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.
Materials Process Design and Control Laboratory
Selection of order of expansion
= ∑
+ ∑
+∑
+…
Two issues to be taken care of:
1)
How to construct each of these multi body potentials?
2)
When to stop the expansion?
Work of B.Paulus 1,2 show that the computed
energy oscillates between even and odd
number of expansion terms, asymptotically
converging to the exact energy
Stop the expansion when energy is accurate
enough
Energies (En) calculated
from an n-body
expansion
correct energy
1.
B. Paulus et. al, The convergence of the ab-initio many-body expansion for the cohesive energy of solid mercury Phys. Rev. B
70, 165106 (2004)
2.
B. Paulus, The method of increments -- a wavefunction-based ab-initio correlation method for solids, Phys Rep 428 (2006)
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Computation of MBE energy filters
+
+
+ ..
Weighted MBE
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Selection of order of expansion
Weighted 4th order
MBE
Weighted 2nd order
MBE
True energies
True energies
Weighted 3rd order
MBE
Weighted MBE
expansion coefficients
are fitted using 12
atom cluster energies
and the results are
presented for a 16
atom cluster.
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True energies
Materials Process Design and Control Laboratory
Platinum clusters
16 atom FCC
cluster
Weighted MBE
4th order
EM ( X 1 , X 2 ,.., X M )  0.5884 E2 ( X 1 , X 2 ,.., X M ) 
0.3014 E3 ( X 1 , X 2 ,.., X M )  0.0353E4 ( X 1 , X 2 ,.., X M ).
4
Number of
isolated cluster
calculations
120
4
560
4
1820
Depth of
interpolation
Actual
energy
+
+
Lattice parameter
Energy minima
• Coefficients obtained using an 8 atom
cluster energies at different lattice parameters
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MULTIBODY EXPANSION ALGORITHM
Interpolation
algorithm
Generate atom positions in
interpolant space
II. Computation step
Given a phase structure
Create ab-initio energy
database
Interpolation
algorithm
Select the max. number of
terms in expansion
I. Database step
Build database of
interpolants
Compute E from Interpolation function
Transform to interpolant space
Multibody energy summation
Decompose to two-atom, three
atom etc. positions
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Energy of phase structure
Materials Process Design and Control Laboratory
LINKING THE MULTIBODY EXPANSION TO OTHER SOFTWARE
Multi Body Expansio (MBE)
The multibody expansion software written in C++
Two parts: potential generation & energy
computation
Energy computation part is the Hamiltonian
Molecular dynamics- LAMMPS
Large-scale Atomic/Molecular Massively Parallel Simulator
(LAMMPS) is a classical molecular dynamics (MD) code
developed by S. Plimpton et. al (Sandia national lab)
http://lammps.sandia.gov/pictures.html#twin
Directly linked energy computation part in LAMMPS with MBE
Useful for molecular dynamics and energy minimization
Monte Carlo- MCCCS Towhee
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 ensembels
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APPLICATION TO SURFACE PHENOMENA: Ex 1
Predict the most stable adsorption site of hydrogen on metallic
surfaces
Test for FCC Platinum
Depending on the surface there are multiple adsorption sites
Many investigations performed using EAM and other semi-emperical
models
These predict the binding sites fairly accurately
Try to predict favorable binding sites and energies using MBE
FCC (100)
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FCC (110)
FCC (111)
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APPLICATION TO SURFACE PHENOMENA: Ex 1
Test for FCC(111)
Generate a 5x5x5 atom cell of Pt
Coordination number is 9
Position of hydrogen atom varied along the first
primitive cell
The potential energy surface is constructed
Standing problems in surface chemistry
Compare’s extremely well with the abinitio
based results of Kallen et.al1
TOP
FCC
BRIDGE
HCP
-0.410
-0.455
-0.404
-0.420
1 G.Kallen,G.Wahnstrom, Quantum treatment of H on a Pt(111) surface, Phys Rev B, 65 (2001)
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APPLICATION TO SURFACE PHENOMENA: Ex 1
The atomic potential energy surface (APES) computed from ab-initio techniques
First step towards efficient , quick computation of the PES
Computational cost
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)
MBE: ~ 10 minutes
DFT: ~ days
FCC site
Confined to fcc-hcp-fcc valleys
From ref 1
1.
G.Kallen,G.Wahnstrom, Quantum treatment of H on a Pt(111) surface, Phys Rev B, 65 (2001)
2.
S.C.Badescu et al, Energetics and Vibrational states for Hydrogen on Pt(111), PRL 88 (2002)
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APPLICATION TO SURFACE PHENOMENA: Ex 1
Predict the most stable adsorption site of hydrogen on metallic surfaces
Test for FCC Platinum= (1 0 0) surface
Ab initio studies reveal Hallow > Bridge > Top sites
Bridge
-49.5522 Ry
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U N I V E R S I T Y
Hallow
-49.81611 Ry
Top
-49.22788 Ry
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SEARCHING FOR GROUND STATE CONFIGURATIONS
Design of nanostructures: multiple applications like design of
memories for data storage
Adatoms on surfaces
Need to establish which configuration of adatoms can stabilize
the surface the most
Previously done using abinitio calculations (problems of
periodicity and long range interactions)
Recently done using a modified cluster expansion method
Apply multibody expansion to this problems
Take FCC(111) surface.
Stable configuration should be a Pt(111) (2x1) H adatom
configuration
1.
Drutz, Singer, Fahnle, PHYSICAL REVIEW B 67 (2003) 035418
2.
Sluiter, Kawazoe, PHYSICAL REVIEW B 68 (2003) 085410
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SEARCHING FOR GROUND STATE CONFIGURATIONS
Finding the stable structure:
1. Consider a super cell (3x3x3 cell)
2. Place n number of hydrogen atoms on the surface
3. Apply periodic boundary conditions
4. Displace hydrogen atoms to get minima
Pt(111) (2x1) H adatom configuration
Can link to various other software
Minimization found using LAMMPS (Large-scale
Atomic/Molecular Massively Parallel Simulator)
The Multi body expansion converted into a library
Library included into the makefile of LAMMPS
Can directly run a variety of Molecular dynamics and
minimization scenarios
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TOPOLOGICAL OPTIMIZATION: A SIMPLE EXAMPLE
The adsorption behavior of the atom not only depends on the
chemistry of the local surface but also depends on the topology of the
global surface
The availability of an efficient,
computationally tractable method of finding
the interaction energy between a large set of
atoms paves the way for topological design
of surfaces
Surface characterization:
Roughness: Small scale perturbations to the surface
Representing roughness:
Roughness represented by two components: PDF of
a point above a datum z and the correlation between
two points (ACF)
ACF depends on the processing methodology, ex
shot peening, sand blasting and milling
Determine
suitable
components
Determine
best surface
PDF is usually assumed to be a Gaussian
1 Q. Wang, J. K. Johnson, Optimization of Carbon Nanotube Arrays for Hydrogen Adsorption, J Phys. CHem B
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TOPOLOGICAL OPTIMIZATION: A SIMPLE EXAMPLE
Can consider topological optimization in a functional framework
Define a cost functional. Here taken to be the fraction of available sites
occupied
This cost functional is defined in terms of
the topology. Simplest case of roughness
is a sinusoidal wave
Start from a random configuration,
compute the cost functional and minimize
the cost functional
The cost functional here depends on the
frequency of oscillations of the surface
How to compute the fraction of available
sites occupied?
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TOPOLOGICAL OPTIMIZATION: A SIMPLE EXAMPLE
Compute the fraction of available sites occupied using Monte Carlo methods
The Multibody expansion (MBE) provides a accurate PES of the adsorbate
Coupling this energy descriptor with a validated Monte Carlo simulator
Monte Carlo for Complex Chemical Systems (MCCCS) Towhee is very suitable
for such a task
Coupled the library of MBE to towhee software
Platinum (111) surface
Surface dimensions= 0.28 μm x 5.6 nm
Total number of Pt atoms in the simulation=
9600
Perform Grand canonical ensemble Monte Carlo to
model adsorption
Minima reached in 5 iterations
Each Monte Carlo simulation=100000
steps
Time taken for one iteration= 8 hours
Temperature = 300K, Pressure = 10 bar
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TOPOLOGICAL OPTIMIZATION: A SIMPLE EXAMPLE
4
Atom distribution profile
3.5
3
2.5
2
1.5
1
0.5
0
Convergence history of topological optimization
0
0.25
0.5
0.75
Normalized distance
1
PDF of the adsorbate distribution
Atoms beneath the
first layer: leads to
embrittlement
For this simple case the wavelength of the optimal surface is 0.71 μm
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A LOOK AHEAD ….
The next step is to utilize more realistic
representations for the surface:
-Spline representations and Bezier curves
- Larger number of atoms
- Must relax the surface atoms also, can
analyze the effects of the adsorbate as well
as embrittlement effects
Analysis/Design of multiple component
surfaces:
- Platinum + Bismuth predicted to have
good adsorption behavior
Determine
suitable
components
Determine
best surface
- Can optimize surface and chemistry to
inhibit one type of material and enhance
another (prevent CO poisoning )
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AND A LOOK BEHIND
1)
Represented the energy of a set of atoms as a hierarchical sum of isolated
clusters of atoms: The multi body expansion (MBE)
2)
Provided a methodology to compute these high dimensional surfaces using
sparse grid techniques: Smolyak theorem, adaptive sparse grid methods
3)
Coupled the multibody potential framework to several publicly available
molecular dynamics and Monte Carlo software
4)
Computed the atomic potential energy surface of H adsorption on Pt to high
accuracy
5)
Applicability of the MBE to finding the ground state stable configurations
6)
Laid the groundwork for functional topological optimization of surfaces
towards enhancing adsorption with a simple example
B. Ganapathysubramanian and N. Zabaras, "Topological characterization of adsorption
phenomena using multi-body potential expansions", in preparation.
V. Sundararaghavan and N. Zabaras, "Many-body expansions for computing stable
structures of multi-atom systems", Physical Reviews B, submitted
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