Claude Franklin Goldsmith Kinetic Modeling in Heterogeneous Catalysis 120127

8/12/2019 Claude Franklin Goldsmith Kinetic Modeling in Heterogeneous Catalysis 120127

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Kinetic modeling inheterogeneous catalysis.

C. Franklin Goldsmith

Modern Methods in Heterogeneous Catalysis ResearchFriday, January 27th, 2012



Microkinetic modeling is avaluable tool in catalysis.

2

• Disentangle complex phenomena

• Provide insight into what’s going on at the atomic level

• Make predictions for new conditions

• Save time and money

• Optimize reactor design



Review: Power Law

3

Typically a differential reactor is used:

• low loading, low concentration

• isothermal

• high flow rate.Reaction order na, nb are determined experimentally.

r = k A[ ]na

B[ ]nb



Review: Langmuir-Hinshelwood

4

A simple, global mechanism is assumed:

• adsorbates are equilibrated with gas phase

• surface reaction is rate determining step

Apparent reaction order varies between -1 and 1:

r = k K AK B A[ ] B[ ]1+ K

A A[ ]+ K

B B[ ]( )

2

na !

" ln r

" ln A[ ] = 1#

2K A

A[ ]1+ K

A A[ ]+ K B B[ ]



Typically the experimentally observedreaction order is used to interpret aLangmuir-Hinshelwood mechanism.

5

This approach suffers two flaws:

• Experimental na, nb are only valid overa narrow range of conditions.

! it cannot be used to predict anything

• The LH mechanism is over simplistic.! it cannot be used to explain anything



We need a modeling approachthat is predictive, not postdictive.

6

Elementary reaction mechanism:

•attempt to describe real chemistry• is valid over a broad range of conditions

An elementary reaction occurs in a single step, i.e.• it can be described by a single reaction coordinate

• or it passes through a single transition state





Scale is not the only challenge in catalysis.

8

(b) ∆Ei = 41 kcal/mol (c) ∆Ei = 23 kcal/mol(a) ∆Ei = 54 kcal/mol

Intrinsic heterogeneity in

adsorbatedistribution

catalyst sites

Many-body effects

Multiple phases

Catalyst dynamics

reconstruction

isomerization

edge corner

support(111)

(100)

Missing row reconstruction on (110) plane

No single modeling approach caninclude all effects at all scales.

Salciccioni et al. Chem. Eng. Sci. 66 (2011) 4319-4355



Which modeling method you chose dependsupon the length scale you intend to model.

9

Kinetic Monte Carlo

Mean field theory

Transition state theory



Outline:

10

1. Transition State Theory

2. Kinetic Monte Carlo3. Mean Field Theory

4. Making sense of complexity



Kinetic Monte Carlo

Mean field theory


Part I: Transition-State Theory



Part I: Transition-State Theory

12

Transition-state theory (TST):

• an approximation to dynamic theory (classical or quantum).• evaluates the reactive flux through a dividing plane on a

potential energy surface.

There are many flavors of TST.

For simplicity we will focus on canonical TST.



TST Assumptions

13

1. The Born-Oppenheimer approximation is valid.

2. A dynamic bottleneck between reactants and products

can be identified.

3. Reactant molecules are distributed in a Maxwell-Boltzmann distribution.



For n internal coordinates, thepotential energy surface (PES) is an

(n+1)-dimensional hypersurface.

14

A B + C

A + B

+ C

A+BC

PES

dAB

dBC



The reaction coordinate is thelowest energy path between

reactants and products.

15

dAB

dBC



The transition state is the maximumalong the reaction coordinate.

16

AB+C

A...

B...

C

A+BC

PES

Reaction Coordinate



TST assumes that the reactant(s)and transition state are equilibrated.

17

R K †

! "!!# !!! TS

k †

! " ! P

d P[ ]dt

= k †TS[ ] = k †K

†

k TST

!R[ ]



The rate constant is proportional to thefrequency of passes over the barrier timesthe transition state equilibrium constant.

18

k TST

T ( ) =k BT

hk †

!

QTS

†

QR

e! E 0 / RT

K †

" #$ %$

Note the functional similarity to the Arrhenius equation:

k Arr

T ( ) = Ae! E a / RT



The activation energy and barrier height arecorrelated but not equivalent:

19

E a ! " R # lnr#1 T

= RT 2 #

#T ln k BT

h

QTS

†T ( )

QR T ( )

$

% &

'

( ) + E

0

Equivalence can be obtained if we assume:

1. classical oscillators

2. change in all other modes is negligible

A !

k BT

h

QTS

†

QR

! " n O 1013 s

-1( )



Transition states are categorized asloose or tight.

20

Loose transition states:1. higher in entropy than reactants2. more energy levels to be occupied

3. 1013 < A < 1017 s-1

Tight transition states:1. lower in entropy than reactants2. fewer energy levels to be occupied3. 109 < A < 1013 s-1



Calculating kTST from first principles requireselectronic structure calculations (e.g. DFT) andstatistical mechanics.

21

E0 is the difference in the zero-point corrected electronicenergy.

• Typical DFT error is 20 - 30 kJ/mol.

• Need error less than 5 kJ/mol for chemical accuracy.

The partition function requires physical properties:

• vibrational frequencies

• reduced moments of inertia for weakly bound rotors

Q = QelecQtransQrot Qvib





We can combine methods toestimate the kinetic parameters.

DFT Compute atomic binding energies

Scaling relationsUBI-QEP Estimate adsorbate binding energies

BEP

UBI-QEP

Estimate activation energies

Reactiontype

Estimate pre-exponential factors

23



Two methods are commonly usedto estimate energies.

1. Linear scaling relations (Nørskov)

2. Unity Bond Index - QuadraticExponential Potential (UBI-QEP)

24



The binding energy of a molecule islinearly proportional to the binding

energy of the central atom.

25

! E AH

x= " ! E

A+ #

" = xmax $ x

xmax

xmax

= 4 for A = C

= 3 for A = N

= 2 for A = O, S

Abild-Pederson et al. Phys. Rev. Let. 99 (2007)

( )



Brønsted-Evans-Polanyi (BEP) canbe used to estimate the activation

energy.

26 Ferrin et al. JACS. 131, (2009)

!," can berefined based

upon type ofbond broken,surface, etc.

E a = ! " E + #

L l l d



Linear scaling relations reduce acomplex mechanism down to a

minimum set of descriptors.

27 Grabow et al. Angew. Chem. 50, (2011)| l

Nørskov et al. PNAS. 108, 3, (2011)

CO + 3H2 # CH4 + H2O CH4 + NH3 # HCN + 3H2

UBI QEP l h d f



UBI-QEP is a semi-empirical method forcoverage-dependent activation energies.

28

Requires atomic binding energies and gas-phase moleculardissociation energies.

1. Two-body interactions are described by a quadraticpotential, exponential in distance (Morse).

2. Total energy of many-body system is the sum oftwo-body interactions.

Fast, easy, popular -- but can lead to big errors for largermolecules.

Maestri and Reuter. Angew. Chem. 50 (2011)Shustorovich and Sellers. Surf. Sci. Rep. 31 (1998)



T



! H rxn

= E a, f " E a,r

!S rxn

=

Rln A f

Ar

Thermodynamic consistency constrainsthe Arrhenius parameters:

30

Most mechanisms are not consistent!Typically entropy is not conserved;equilibrium constants are off by orders of magnitude.

Th h f i



There are two approaches to enforcingthermodynamic consistency:

1. Constrain all Ea,rev and Arev according to a basis set*.

• Requires accurate kadsorption, kdesorption for all intermediates

2. Compute kr directly from the equilibrium constant.

• Requires accurate H(T), S(T) for all intermediates

31 *Mhadeshwar, Wang, & Vlachos. J. Phys. Chem. B., 107, 2003

P II K M C l



32

Kinetic Monte Carlo

Mean field theory


Part II: Kinetic Monte Carlo

P II K M C l



Part II: Kinetic Monte Carlo

33

KMC is a method for simulating state-to-statedynamics of a rare event system.

• Can span a large range of time scales by neglectingunimportant ultrafast phenomena.

• Explicitly accounts for spatial heterogeneity in competingprocesses:

- adsorption/desorption

- surface diffusion

- surface reactions

C i ti l ll t



Coarse-grain time scales allows us tomodel chemical reactions.

34

Molecular Dynamics:

the whole trajectory

Kinetic Monte Carlo:

coarse-grained hops

Molecular Dynamics wastes timemodeling the 109 thermal vibrationsbetween diffusion events.

Reuter, K. Wiley (2009)



35

startwith

complete list of

processes att=0 Execute randomprocess from list

Advance timeclock

Update processes

finish

kMC ld l f d l



36

kMC can yield accurate results for model systems.

cus site bridge sitebridge

sitescussites

[ 0 0 1 ]

3 . 1

2 Å

[110]_

6.43 Å

110-20 10-10

log(TOF)

pO (atm)2

p C O

( a t m )

110-20 10-10

10-20

10-10

COCObrbr/CO/COcuscus

COCObrbr//--

-- // -- OObrbr/O/Ocuscus

1

350K

x

x xx x x xxxx

x

x

x

x

x

x

x

x

x

x

xx x

xx

x

x

xx

x

xx x

2

pco = 10-10 atm pO2 = 10-10 atm

pCO (10-9 atm)

0.0 1.0 2.0 3.0

pO2 (10-9 atm)

0.0 1.0 2.0 3.0

3

2

1

0

T O F C O 2 (

1 0 1 2 c m - 2 s -

1 )

6

5

4

3

2

1

0

Reuter, K. Wiley (2009)

A k d t f kMC i th ti l



37

A key advantage of kMC is the spatialresolution and the ability to model

adsorbate-adsorbate interactions.

Neighboring molecules can affect the stabilityof a species or transition state.

! H i = ! H i ,0 + c j " j ,nn j =1

N species

# + c j ,k " j ,nn" k ,nnk =1

N species

# j =1

N species

# + ...

E a = E a,0 + $ j " j ,nn j =1

N species

# + $ j ,k " j ,nn" k ,nnk =1

N species

# j =1

N species

# + ...

Jansen, APJ. chembond.catalysis.nl/kMC

S



Summary:Kinetic Monte Carlo

38

Pros:

- detailed chemistry over large time scale.

- accurate representation surface heterogeneity.

Cons:

- limited in length scale.- difficult to couple with continuum

(i.e. no transport limitations).

- “home cooked” code.

P III M fi ld h



39

Kinetic Monte Carlo


Part III: Mean field theory

Mean field theory

P III M Fi ld Th



Part III: Mean Field Theory

40

MFT assumes a uniform distribution ofadsorbates and catalyst sites.

• Can span a large range of length and time scales.

• Computationally efficient

• Only real option for process modeling.

• Lots of software available (CHEMKIN, CANTERA)

MFT is a poor approximation for



MFT is a poor approximation forheterogeneous processes.

Mean FieldApproximationprobably not bad probably not good

41

MFT ll i t



MFT generally is more accurateat higher temperatures.

42

Disordered surfaces are higher in entropy.

• desorption increases with temperature, yielding more

empty sites.• repulsive lateral interactions increase homogeneity.

!G = ! H " T !S

One advantage of MFT is the ability to



One advantage of MFT is the ability tocouple detailed chemistry with fluid

mechanics.

43

: .

.

Quinceno et al. Cat. Today, 119 (2007)

Conservation of mass energy and



Conservation of mass, energy, andmomentum share a common formalism.

44 Quinceno et al. Cat. Today, 119 (2007)

accumulation = bulk transport

+ molecular transport

+ chemical reaction

Mass

Energy

Momentum

!

"

##

#

$

%

&&

&in

Mass

Energy

Momentum

!

"

##

#

$

%

&&

&out

Simple systems can be modeled with



Simple systems can be modeled withideal reactors.

45

Batch reactor:perfect mixing

concentration versus time

Perfectly stirred reactor:perfect mixing

lower conversion per unit volume (generally)

concentration versus time

Plug flow reactor:no radial gradients

higher conversion per unit volume

concentration versus position

Networks of ideal reactors can model



Networks of ideal reactors can modelmore complex phenomena.

46

Two dimensional reactor models



Two-dimensional reactor modelsinclude mass transport limitations.

47

CO*

Pt*O*

O2(z,r)

More complex geometries or problems



More complex geometries or problemsrequire computational fluid dynamics.

48

Most CFD codes are developed for fluidmechanics (chemistry is an afterthought).

• CFD code spends >95% computer time on chemistry,not transport

• Simplified, lumped models must be used

!

New computational paradigms are needed

Summary:



Summary:Mean Field Theory

49

Pros:

- covers a broad range of time and length scales.

- Allows for easy modeling of transport limitations.- easily coupled with reactor models and CFD.

- standard software available.

Cons:

- mean field is a poor approximation for inherentlyheterogeneous phenomena.

Part IV: Understanding the results



Part IV: Understanding the results

50

So you’ve successfully modeled a system with

100’s of species and 1000’s of reactions.

Now what?

Sensitivity analysis:



Sensitivity analysis:determine which rates are most important.

51

xi =

! ln TOFCO2( )

! ln k i( )

si =

! ln C2

H4[ ]( )

! ln k i( )

Flux path analysis:



52 Blaylock et al. JPCC, 113 (2009)

Flux path analysis:determine which intermediates

are most im ortant.



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Claude Franklin Goldsmith Kinetic Modeling in Heterogeneous Catalysis 120127