Multi-scale simulation methods - Fritz Haber Institute of the Max … · 2018-08-06 · Mie Andersen | Multi-scale simulation methods 12 Rare event dynamics E IS TS FS Brute force

Mie Andersen

Theoretical Chemistry, Technische Universität München

Multi-scale simulation methods

Hands-on DFT and beyond:

Frontiers of advanced electronic structure

and molecular dynamics methods

Peking University

August 6th, 2018

2Mie Andersen | Multi-scale simulation methods

Outline

• Multi-scale modeling for catalysis, crystal growth and particle diffusion

• Ingredients of a multi-scale model

• Getting the processes and rate constants

• Kinetic Monte Carlo simulations

• Examples, challenges and current frontiers

o Sensitivity analysis

o Low-barrier problem

o Lateral interactions


Multi-scale modeling

Ab initio atomistic thermodynamics and statistical mechanics of surface properties and functions

K. Reuter, C. Stampfl, and M. Scheffler, in: Handbook of Materials Modeling Vol. 1,

(Ed.) S. Yip, Springer (Berlin, 2005). http://www.fhi-berlin.mpg.de/th/paper.html


Applications of multi-scale modeling

http://www.lce.hut.fi/publications/annual2000/node22.html

Crystal growth

Heterogeneous catalysis

Diffusion in battery materials

Urban et al., npj Computational Materials 2, 16002 (2016)

Reuter, K., in Modelling and Simulation of Heterogeneous Catalytic Reactions:

From the Molecular Process to the Technical System, O. Deutschmann, Editor.

2011, Wiley-VCH: Weinheim. p. 71-112


Ingredients of a multi-scale model

“Active site” model

IS FS

eq.

X

X

Rate constants

Process identification KMC simulation

Quantum Engine

{Ri} → Etot, Fi

Level of theory

x

xx

xx

x x x xxx

x

x

xxx

xx x

pCO (10-9 atm)

Exp.

Theory

0.0 1.0 2.0 3.06

4

2

0

TO

FC

O2

(10

12

mo

l/c

m2

s)

Model validation

I. Active site model


Mind the materials gap !

Supported „real“ catalyst

Model catalysts

Courtesy: G. Rupprechter and Ch. Weiland, NanoToday 2, 20 (2007).


From single facets to nanoparticle models

Support

Nano-

particle

Common choice:

Limit study to one particular

site type

Reality:

• Bifunctional couplings

between different site types?

• Influence of support?

Reuter et al., J. Chem. Phys. 146, 040901 (2017)

Andersen et al., Angew. Chem. Int. Ed. 128, 1 (2016)

Jørgensen, Grönbeck, Angew. Chem. Int. Ed. 57, 5086 (2018)

II. Quantum engine


Current workhorse: Density functional theory

XC functionals:

Practitioner level:

GGAs (metals)

Hybrids (molecules, insulators)

Major problems:

Self-interaction

(range-separated hybrids)

Van der Waals

(dispersion corrected functionals)

Have to expect inaccuracies in

binding energies/activation

barriers of order ~0.3-0.4 eV!

Jacob’s Ladder (Perdew et al.)

III. Getting the rate constants


Rare event dynamics

E

IS

TS

FS

Brute force approach to rate constants:

i) Have accurate potential energy surface (forces)

ii) Run MD trajectory so long, that it establishes

equilibrium, crossing the barrier many, many

times back and forth:

k =no. of crossings IS FS per unit time

fraction of time system has spent in IS

require approximate theories to obtain process rates!

Yet:

- Relevant time step in MD run is fs (vibrations)

- Typical barrier E for surface reactions ~ 1 eV 10-2 - 102 reactions per second (TOF!)

- Requires to run trajectory over about 1015 – 1020 time steps unfeasable…

…and essentially 99,9999% of the time, the system will

just vibrate around IS basin (short time dynamics)


Transition state (activated complex) theory

Assumptions ( Eyring, Evans, Polanyi, ~1935 ):

i) Reaction system passes the barrier only once

(no recrossings)

ii) Energy distribution of reactant DOF is Boltzmann-

like (many collisions compared to reaction events

yield equilibrium between activated complex and IS,

except with respect to the reaction coordinate)

iii) Passage over barrier is the motion of only one DOF,

the reaction coordinate, which is independent of all

other motions of the activated complex (no concerted

motions)

iv) Passage over barrier is a classical event (no tunneling)

IS FS

eq.

X

X

Derivation: see e.g.

K.J. Laidler, Chemical kinetics,

Harper & Row, New York (1987) kTST = [ ( ) e S/k ] e-E/kTkT

hISFS

IS

FS

TSx

x

xx x

xx

x

x

x

F true

F ||spring

x


Transition state search: nudged elastic band - Initialize with several images {Ri} along a

straight-line interpolation

- Minimize

S(R1, …, RN) = i E(Ri) + i k/2 (Ri+1 - Ri )2

- Problem:

- elastic band cuts corners

- images tend to slide down towards

low-energy IS/FS regions, leaving few

images for relevant TS region

- Solution:

- only spring force component parallel

to path (no corner cutting)

- only true force component perpendicular

to path (no down-sliding)

widely applied workhorse

has problems, if energy varies largely along path,

but very little perpendicular to it (kinky PES)

G. Mills and H. Jónsson,

Phys. Rev. Lett. 72, 1124 (1994)

G. Henkelman, et al.,

J. Chem. Phys. 113, 9901 (2000)


Cheap barriers from BEP relations

Andersen et al., J. Chem. Phys. 147, 152705 (2017)Reaction coordinate [Å]

Energ

y [

eV

]

TS (expensive !)

IS (cheap !) FS (cheap !)

DFT-calculated

dissociation of H2O on

terrace site of Rh(211)

Activation e

nerg

y

Reaction energy



Reaction coordinate [Å]

Energ

y [

eV

]

TS (expensive !)


DFT-calculated



Activation e

nerg

y

Reaction energy

Cu PdPtRhRu

Andersen et al., J. Chem. Phys. 147, 152705 (2017)

Brønsted-Evans-

Polanyi (BEP) relation




Energ

y [

eV

]

TS (expensive !)


DFT-calculated



Step site

of Rh(211)

Activation e

nerg

y

Reaction energy

Cu PdPtRhRu


Brønsted-Evans-





Energ

y [

eV

]

TS (expensive !)


DFT-calculated



Brønsted-Evans-


Step site

of Rh(211)

Activation e

nerg

y

Reaction energy

Same BEP

relation obeyed !

Typical for

hydrogenation

Cu PdPtRhRu



More BEP relationsN2 dissociation

Nørskov et al., Chem. Soc. Rev. 37, 2163 (2008)

Offset in β typical for

dissociation of strongly

bonded molecules !

Brønsted-Evans-Polanyi

(BEP) relation:

𝐸act. = 𝛼 ∙ Δ𝐸react. + 𝛽

Useful for estimating trends and limitations for transition metal catalyst activities !

IV. Process identification


Diffusion at metal surfaces: Surprises…

Hopping mechanism

Ag(100) E = 0.45 eV

Au(100) E = 0.83 eV

B.D. Yu and M. Scheffler, Phys. Rev. B 56, R15569 (1997)

Exchange mechanism

Ag(100) E = 0.73 eV

Au(100) E = 0.65 eV


Automatized process identification

Hyperdynamics

Temperature accelerated dynamics

Accelerated molecular dynamics:

Other approaches: - parallel replica dynamics

- dimer method

…

A.F. Voter, F. Montalenti and T.C. Germann,

Annu. Rev. Mater. Res. 32, 321 (2002)

V. Kinetic Monte Carlo simulations


Basics of KMC: Markovian state dynamics

Kinetic Monte Carlo

N

t

B

A

j

jij

j

ijii tPktPkdt

tdP)()(

)(

A

B

Molecular Dynamics

TS

kA→B

kB→A

ΔEA→B ΔEB→A

Reuter, K., in Modelling and Simulation of Heterogeneous Catalytic Reactions:

From the Molecular Process to the Technical System, O. Deutschmann, Editor.

2011, Wiley-VCH: Weinheim. p. 71-112

Chemical Master equation


KMC: essentially „coarse-grained MD“

Molecular Dynamics:

the whole trajectory

Kinetic Monte Carlo:

coarse-grained hops

ab initio MD:

up to 50 ps

ab initio KMC:

up to minutes


Flowchart of a kinetic Monte Carlo simulation

Determine all possible

processes p for given

system configuration and

build a list.

Get all rate constants

kp

Get two random numbers r1 , r2 ]0,1]

Calculate ktot = p kp

and find process “q”:

q q-1

kp r1 ktot kp

p=1 p =1

Execute process number “q”,

i.e. update configuration update clock

t t – ln(r2)/ktot

START

END

0

ktot

r1 ktotq

VI. Simple toy model: Metal adatom diffusion


Au adatom diffusion: only hopping mechanism

Hopping: E = 0.83 eV

kTST = [ ( ) e S/k ] e-E/kT , T=300 KkT

hISFS

Initialize lattice (no particles -> deadlock)

X

simplification





hISFS X

simplification

Initialize lattice with one particle



𝑫 =𝒓 𝒕 − 𝒓𝟎

𝟐

𝟐𝒅𝒕Calculate diffusion coefficient by

tracking mean squared displacements

d is the lattice

dimension (2)

1 trajectory – stochastic noise !

∆Ehop = 0.83 eV

D = 0.32 Å2/s



𝑫 =𝒓 𝒕 − 𝒓𝟎

𝟐

𝟐𝒅𝒕Calculate diffusion coefficient by

tracking mean squared displacements

d is the lattice

dimension (2)

Average over 50 trajectories

using different initialization

and random seed.∆Ehop = 0.83 eV

D = 0.34 Å2/s


Au adatom diffusion: include exchange mechanism



hISFS X

simplification

Exchange: E = 0.65 eV

Initialize lattice with one particle


Au adatom diffusion: include exchange mechanism

∆Ehop = 0.83 eV

D = 0.34 Å2/s

∆Ehop = 0.83 eV, ∆Eexc = 0.65 eV

D = 720 Å2/s

Garbage in – garbage out !

Overlooking the exchange mechanism affects the result by orders of magnitude.


How do DFT errors affect the result ?

Result is only sensitive to barriers for “rate-limiting” processes.

Trivial for 2-process model, but what about more complex (catalysis) models ?

∆Ehop = 0.83 eV

D = 0.34 Å2/s

∆Ehop = 0.83 eV, ∆Eexc = 0.65 eV

D = 720 Å2/s

∆Ehop = 0.73 eV, ∆Eexc = 0.65 eV

D = 760 Å2/s

VII. Sensitivity analysis


More complex catalysis model

RuO

26 elementary processes (site-specific):

- O2 adsorption/desorption (dissociative/associative)

- CO adsorption/desorption (unimolecular)

- O and CO diffusion

- CO + O reaction

K. Reuter and M. Scheffler, Phys. Rev. B 73, 045433 (2006)

O

CO

K. Reuter, Oil&Gas Sci. Technol. 61, 471 (2006)

CO oxidation @ RuO2(110)

Make use of ergodicity of KMC model to replace ensemble average by time average !


KMC reaction rates

𝑅𝛽 =1

𝑡𝑓𝑖𝑛𝑎𝑙

𝑖=1

𝑁𝑘𝑀𝐶

𝒛

𝑘𝛽 𝒛 𝒚𝑖 Δ𝑡𝑖

𝑅𝛽: rate of reaction 𝛽, e.g. CO2 formation

𝑘𝛽: rate constant for CO2 formation

𝒚, 𝒛: initial and final lattice configuration for possible CO2 formation process

OCO

KMC time steps are discrete


Degree of rate control

• Measures how sensitive the reaction rate is

to a change of the rate constant of process i.

• Positive (negative) value means 𝑅𝛽increases (decreases) when 𝑘𝑖

+ increases.

𝑥𝑖𝛽+ =𝑘𝑖+

𝑅𝛽

𝜕𝑅𝛽

𝜕𝑘𝑖+𝑘𝑗≠𝑖+ ,𝑘𝑗

−

H. Meskine et al., Surf, Sci., 603, 1724 (2009)

Which processes are rate-

determining depends on the

reaction conditions !

Beyond finite difference sampling of

derivatives and global sensitivity analysis:

M. J. Hoffmann et al., J. Chem. Phys. 146, 044118 (2017)

S. Döpking et.al., Chem. Phys. Lett. 674, 28 (2017)

C. Stegelmann et al., JACS, 131, 8077 (2009)

VIII. Low barrier (timescale disparity) problem


Timescale disparity problem

∆Ehop = 0.83 eV

D = 0.34 Å2/s

∆Ehop = 0.83 eV, ∆Eexc = 0.65 eV

D = 7.24 nm2/s

The timescales one can reach in a KMC simulation are limited by the fastest

process of the system.


CO methanation over stepped metalsReaction mechanism:

1) CO(g) + * ⇌ CO*

2) H2(g) + 2* ⇌ 2H*

3) CO* + H* ⇌ C* + OH*

4) C* + H* ⇌ CH* + *

5) CH* + H* ⇌ CH2* + *

6) CH2* + H* ⇌ CH3* + *

7) CH3* + H* ⇌ CH4(g) + 2*

8) OH* + H* ⇌ H2O(g) + 2*

9) OH* + OH* ⇌ H2O(g) + O* + *

10) O* + H* ⇌ OH* + *

11) Diffusion of all species

between all sites

← adsorption

← H-assisted CO dissociation

(only step sites are active)

← hydrogenation of C species

← hydrogenation of O species

Special H reservoir site

Metal(211) sites:

CO diffusion barrier ≈ 0.1 eV

CO dissociation barrier ≈ 1-2 eV

(metal dependent)


Acceleration of KMC simulations

Fast, quasi-equilibrated reaction channels are scaled with

equal scaling factors applied to the forward and reverse

processes to preserve thermodynamics (detailed balance):

𝑛+ − 𝑛−𝑛𝑒

< δ Dynamically partition processes into quasi-equilibrated and

non-equilibrated reaction channels:

ki+′ = αiki

+

ki−′ = αiki

−

αi ∈ ]0; 1]

Nf = r2 / r1

Dybeck et al., J. Chem. Theory Comput. 13, 1525 (2017)


Towards KMC-based catalyst screening

Use scaling relations (adsorption energies) and BEP relations (barriers) to express trends

over transition metal series in terms of only the C and O adsorption energies.

Traditional volcano plot computed

in the mean-field approximation

KMC volcano plot enabled by

acceleration algorithm


IX. Lateral interactions


Lateral interactions in KMC simulations

Cluster expansion model for interactions between species j

at site q and nearest neighbor species i at sites s:

can account for attractive / repulsive interactions, ordering and island formation

Straightforward generalization to many-body interactions and distant neighbors

high computational cost for complex interaction models


Interactions at DFT level

H-assisted CO dissociation:

Andersen et al. (in preparation)

Low coverage High coverage

Top view

Side view


Interactions at DFT level

Low


High

Coverage

Cu Pd Pt Re Rh Ru

Low coverage High coverage

Top view

Side view



Interactions at DFT levelLow coverage High coverage

Top view

Side view

Low


High

Coverage

Cu Pd Pt Re Rh Ru



Interactions at KMC level

Standard KMC implementation:

• Every neighbour configuration is treated as a

separate process.

• Cost grows linearly with number of processes

-> exponentially with number of interactions.

Seibt et al. (in preparation)

standardCO oxidation on RuO2(110) surface

Reuter et al., Phys. Rev. B 73, 045433 (2006)

? ?

?

? ?

?



“on-the-fly” implementation:

• Rate constants of processes affected by lateral

interactions are calculated “on-the-fly” instead of

at model initiation.

standard

on-the-fly

CO oxidation on RuO2(110) surface

? ?

?

? ?

?





standard

on-the-fly

10 × 10

lattice

50 × 50


…

? ?

?

? ?

?









standard

on-the-fly

10 × 10

lattice

50 × 50


…

? ?

?

? ?

?

In progress:

Combining the “on-the-fly”

implementation with the

acceleration algorithm.







All models were realized using the “kmos” software package:

Development: https://github.com/mhoffman/kmos

Documentation: http://kmos.readthedocs.io/

Mailing list: https://groups.google.com/group/kmos-users/

Further tutorials (incl. crystal growth models): https://github.com/jmlorenzi/intro2kmos


“kmos” KMC code


Acknowledgements & developers

Karsten Reuter

(TU Munich)

Max Hoffmann

(Stanford Uni.)Sebastian Matera

(Freie Uni., Berlin)Juan Lorenzi

(Siemens)

Mie Andersen

(TU Munich)

Andy Garhammer

(Siemens)

Michael Seibt

(TU Munich)

Documents

Multi-scale simulation methods - Fritz Haber Institute of the Max … · 2018-08-06 · Mie Andersen | Multi-scale simulation methods 12 Rare event dynamics E IS TS FS Brute force