Materials Process Design and Control Laboratory Topological characterization of adsorption phenomena...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


Topological characterization of adsorption phenomena using

multi-body potential expansions

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

B. Ganapathysubramanian and Prof. Nicholas Zabaras

http://mpdc.mae.cornell.edu/

http://mpdc.mae.cornell.edu/




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




Alternate means of energy production

Anode:

2H24H++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

“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




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)

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)

Need to store atleast 4 kg of hydrogen for commercial usage of hydrogen1.




Towards designing materials with enhanced adsorption

Top layer

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

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




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




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)




• 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

Multi-body expansionTotal energy 1,2

Total energy is the sum of energies of higher and higher levels of interaction

Symmetric function

Position and species

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)




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




Multi-body expansion: Simple examples

E0 = V0

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

2

3




Multi-body expansion: link to other Hamiltonians

• All potential approximations can be shown to be a All potential approximations can be shown to be a special case of multi-body expansionspecial case of multi-body expansion– Embedded atom potentials Embedded atom potentials




• 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

• 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


Comparison with the Cluster Expansion Method

1. Sanchez and de Fontaine, 19812. Sanchez, et al, Generalized Cluster Description of Multicomponent Systems, Physica A 128 (1984)3. Connolly,Williams, Density-functional theory applied to phase transformations in transition-metal alloys

Phys Rev B, 27 (1983 )





∑= ∑+ ∑+ + …

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?




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




High dimensional surfaces

1 2 3

4As 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

Have to approximate high dimensional surfaces accurately

Cannot utilize a tensor product space!

Come up with intelligent schemes to sample from the hyper-surface

DimensioDimensionn

pointspoints

11 5050

22 25002500

44 6.25e66.25e6

88 3.9e133.9e13

1616 1.52e21.52e277

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

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)




SMOLYAK ALGORITHM

LET OUR BASIC 1D INTERPOLATION SCHEME BE SUMMARIZED AS

IN MULTIPLE DIMENSIONS, THIS CAN BE WRITTEN AS

( ) ( )i

i i

i i

xx X

U f a f x

1 11

1 1

( )( ) ( ) ( , , )d di id

i i i id d

i ii i

x xx X x X

U U f a a f x x

TO REDUCE THE NUMBER OF SUPPORT NODES WHILE MAINTAINING ACCURACY WITHIN A LOGARITHMIC FACTOR, WE USE SMOLYAK METHOD

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

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




SMOLYAK ALGORITHM

Extensively used in statistical mechanics

Provides a way to construct interpolation functions based on minimal number of points

Univariate interpolations to multivariate interpolations

( ) ( )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

ORDER SC FE

3 1581 8000

4 8801 40000

5 41625 100000

D = 10

Accuracy the same as tensor product

Within logarithmic constant

Increasing the order of interpolation increases the number of points sampled




SMOLYAK ALGORITHM: REDUCTION IN POINTS

ORDER SC FE

3 1581 1000

4 8801 10000

5 41625 100000

D = 10

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

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.




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.




MINIMAL 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 geometry

Improving the computational efficiency by reducing the problem dimension

ClusteCluster sizer size

Cluster specifierCluster specifier DimensionalitDimensionalityy

22 RR1212 11

33 RR1212, , RR2323, , RR313133

44 RR1212,,RR2323,,RR3434,,RR4141,,RR42 42 ,,RR313166

MM RR1212,,RR2323,,RR3434,,RR4141…… 3M-63M-6




Constructing the multi-body potentialsE

nerg

y

Position

•Needs the least number of ab initio calculations toconstruct the potential,

•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

•Completely independent of the number of dimensions of the problem.

•Provides a way of constructing fully–transferable ab initio based potentials.

atomsatoms accuracaccuracyy

tensortensor sparsesparse

33 1010-6-6 6604966049 15371537

44 1010-5-5 1.9x101.9x101919 0.6x100.6x1066

55 2x102x10-5-5 5.4x105.4x103333 20x1020x1066




• 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.

Abinitio computation of the energies




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?

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)

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

correct energy

Energies (En) calculated from an n-body

expansion




Computation of MBE energy filters

Weighted MBE

+

+

+ ..




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.




Platinum clusters

+

+

Depth of interpolation

4 120

4 560

4 1820

Number of isolated cluster calculations

Actual energy

Weighted MBE 4th order

Energy minimaLattice parameter

• Coefficients obtained using an 8 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




MULTIBODY EXPANSION ALGORITHM

Select the max. number of terms in expansion

Generate atom positions in interpolant space

Transform

to re

al space

Create ab-initio energy database

Build database of interpolants

Transform to interpolant spaceGiven a phase structure

Compute E from Interpolation function

Interpolation algorithm

Decompose to two-atom, three atom etc. positions

Multibody energy summation

Energy of phase structure

I. Database step

II. Computation step

Inte

rpol

atio

n al

gorit

hm




LINKING THE MULTIBODY EXPANSION TO OTHER SOFTWARE

http://lammps.sandia.gov/pictures.html#twin

The multibody expansion software written in C++

Two parts: potential generation & energy computation

Energy computation part is the Hamiltonian

Molecular dynamics- LAMMPS

Multi Body Expansio (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 ensembels

Monte Carlo- MCCCS Towhee




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) FCC (110) FCC (111)







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

TOPTOP FCCFCC BRIDGEBRIDGE HCPHCP

-0.410 -0.455-0.455 -0.404-0.404 -0.420-0.420

1 G.Kallen,G.Wahnstrom, Quantum treatment of H on a Pt(111) surface, Phys Rev B, 65 (2001)


Compare’s extremely well with the abinitio based results of Kallen et.al1




The atomic potential energy surface (APES) computed from ab-initio techniques

First step towards efficient , quick computation of the PES

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)

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)

Confined to fcc-hcp-fcc valleys


FCC site

Computational cost

MBE: ~ 10 minutes

DFT: ~ days

From ref 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 Hallow Top

-49.5522 Ry -49.81611 Ry -49.22788 Ry





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




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

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

Pt(111) (2x1) H adatom configuration




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

1 Q. Wang, J. K. Johnson, Optimization of Carbon Nanotube Arrays for Hydrogen Adsorption, J Phys. CHem B

Determine best surface

Determine suitable components

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 surfacesSurface 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

PDF is usually assumed to be a Gaussian





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?




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

Temperature = 300K, Pressure = 10 bar

Minima reached in 5 iterations

Each Monte Carlo simulation=100000 steps

Time taken for one iteration= 8 hours





Atoms beneath the first layer: leads to embrittlement

Normalized distance

Ato

mdi

stri

butio

npr

ofile

0 0.25 0.5 0.75 10

0.5

1

1.5

2

2.5

3

3.5

4

For this simple case the wavelength of the optimal surface is 0.71 μm

Convergence history of topological optimization PDF of the adsorbate distribution




A LOOK AHEAD ….

Determine best surface

Determine suitable components

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

- Can optimize surface and chemistry to inhibit one type of material and enhance another (prevent CO poisoning )




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

Materials Process Design and Control Laboratory Topological characterization of adsorption phenomena...

Documents

Transcript of Materials Process Design and Control Laboratory Topological characterization of adsorption phenomena...