Reaction Route Graphs – An Effective Tool In Studying Complex Kinetic Mechanisms

Reaction Route Graphs – An Effective Tool In Studying

Complex Kinetic Mechanisms

Worcester Polytechnic InstituteWorcester, MA

November 11, 2004

Ilie Fishtik, Caitlin A. Callaghan, Ravindra Datta

Motivation• There is a tremendous interest in general network theory

(e.g., small world networks) in various areas of science

• Detailed and complex kinetic mechanisms are increasingly available

• Graph theoretical methods were proved to be a powerful tool in chemical kinetics

• Little is known about the topology of kinetic mechanisms

• Reduction, simplification and comprehension of complex kinetic mechanisms is a necessity

• Place the species at the nodes of the graph

• Branches represent the connectivity of the species according to the stoichiometry of elementary reactions

• Useful in studying topological characteristics of chemical reaction networks

• Leaves open kinetic issues such as reduction and simplification

The Conventional Graph Theoretical Approach

Reaction Route Graphs • The branches are elementary reactions

• The nodes represent connectivity of the elementary reactions and satisfy the quasi-steady state for intermediates and terminal species

• Any walk between two terminal nodes is a full reaction route

• Any walk between two intermediate nodes is an empty route or cycle

• RR graphs are easily converted into electrical networks – Elementary reactions are associated with the resistances– Overall reaction is associated with a power source– Kirchhoff’s laws are applicable

RR Graphs and Kinetics

A RR graph may be viewed as hikes through a mountain range:– Valleys are the energy levels of reactants and products– Elementary reaction is hike from one valley to adjacent

valley– Trek over a mountain pass represents overcoming the

energy barrier

Notation

ρs

ρ

ρs

:

ρs :

l

k

n

iiρikρk βα

1 1

0TI ( ρ = 1, 2, …, p)

OR: 0TTT 2211 nnν...νν

npplpp

nl

nl

β...ββα...αα

........................

β...ββα...αα

β...ββα...αα

222121

2222122221

1121111211

ν

Elementary Reaction:

Overall Reaction:

Stoichiometric Matrix:

Graph Topological Characteristics of the RR Graphs

• Full Routes (FRs) – a linear combination of the elementary reactions that cancels all of the intermediates and produce the desired OR

• Direct FR - a FR that involves a minimal number of elementary reactions

:),,...,,(121 qq iiii ssssFR OR

sα...αα

sα...αα

...............

sα...αα

sα...αα

qqqq

qq

i,qi,i,i

i,qi,i,i

i,qi,i,i

i,qi,i,i

1111

11

2222

1111

21

21

21

21

FRg: ORsσp

ρρgρ

1


• Empty Routes (ERs or cycles) – a linear combination of the elementary reactions that cancels all of the intermediates and terminal species and produce a “zero” OR

• Direct ER - an ER that involves a minimal number of elementary reactions

:),,,...,,(2121 qqq jjjjj sssssER 0

22222

11111

22222

11111

121

121

121

121

121

qqqqq

qqqqq

qqqqq

j,j,qj,j,j

j,j,qj,j,j

j,j,qj,j,j

j,j,qj,j,j

j,j,qj,j,j

sβα...αα

sβα...αα

sβα...αα

..................

sβα...αα

sβα...αα

ERg: 01

p

ρρgρ sσ


• Intermediate Nodes (INs) - a node including ONLY the elementary reaction steps and satisfying the quasi-steady state conditions for the intermediates

• Direct IN – an IN that involves a minimal number of elementary reactions

:),...,,(121I qphhh sssn

k

kq

kq

kq

h

qp

k

q,h,ql,ql,ql

,h,l,l,l

,h,l,l,l

s

αα...αα

...............

αα...αα

αα...αα

1

1

2222

1111

121

121

121

ra rb

rcrd

sa sb

scsd


• Terminal Nodes (TNs) - a node including the OR in addition to the elementary reaction steps

• Direct TN – a TN that involves a minimum number of elementary reactions

:)(21T qphhh ,...,s,ssn

ORs

βα...αα

βα...αα

...............

βα...αα

βα...αα

ν k

kkkk

qqqq

h

qp

k

,hq,h,h,h

,lq,l,l,l

,lq,l,l,l

,lq,l,l,l

1

121

121

121

121

1

2222

1111

Δ

1

rOR rb

rcrd

sOR sb

scsd

Electrical Circuit Analogy• Kirchhoff’s Current Law

– Analogous to conservation of mass

• Kirchhoff’s Voltage Law– Analogous to thermodynamic consistency

• Ohm’s Law– Viewed in terms of the De Donder Relation

ab

c

d

ea b c d e 0r r r r r

f g h i 0 A +A A Af g

i h

Rr

A=

Minimal, Non-Minimal and Direct RR Graphs

• Minimal RR Graph – a RR graph that involves each elementary reaction only once

• Non-Minimal RR Graph – a RR graph that involves an elementary reaction twice, thrice, etc.

• Direct RR Graph – a RR graph that involves only direct FRs

Electrochemical Hydrogen Evolution and

Oxidation Reactions

Electrocatalytic Reaction

sT: H2 + 2M 2HM

sV: H2O + HM M + H3O+ + e –

sH: H2O + H2 + M HM + H3O+ + e –

OR: H2 + 2H2O 2H3O+ + 2e-

electrochemical hydrogen oxidation and evolution reactions electrochemical hydrogen oxidation and evolution reactions

sT: 2HM 2M + H2

sV: M + H2O + e- HM + OH-

sH: HM + H2O + e- M + H2 + OH-

OR: 2H2O + 2e- H2 + 2OH-

HHYDROGEN YDROGEN OOXIDATION XIDATION RREACTIONS EACTIONS

HHYDROGEN YDROGEN EEVOLUTION VOLUTION RREACTIONS EACTIONS

2HT

2HTT )1( θkθkr

�

RT

FEβθk

RT

FEβθkr

)1(exp)exp(1 V

HVV

HVV

�

RT

FEβθk

RT

FEβθkr

)1()exp1(exp H

HHH

HHH

�

Topological Characteristics of the RR Graph


ORRVH: sV + sH = OR

ORRVT: 2sV + sT = OR

ORRHT: 2sH – sT = OR

TN1: OR - sH - sT

TN2: OR - sV + sT

TN3: 2OR - sV - sH

IN: -sV + sH + 2sT

ERR: sV - sH + sT = 0

OVERALL REACTION ROUTESOVERALL REACTION ROUTES EMPTY REACTION ROUTESEMPTY REACTION ROUTES

INTERMEDIATE NODESINTERMEDIATE NODES TERMINAL NODESTERMINAL NODES

Constructing the RR Graph

(a)

(b)

intermediate nodes

terminal nodes

peripheral nodes

sV

sV

sH

sH

sT sT

sV

sV

sH

sH

sT sT

sV

sV

sH

sH

OR

OR


The RR NetworkOR

OR

RC

RB

RA

RV

RH

+ -

+ -

OR

OR

RH

RV

RT

RV

RH

RT

+ -

+ -

OR

OR

RH

RV

RV

RH

RT/2

+ -

+ -

- Transformation


C

COR RRRR

RRRRRR

BHV

HBVA

))((

/2)( THV

HVA RRR

RRR

/2)(

/2)(

THV

THB RRR

RRR

/2)(

/2)(

THV

TVC RRR

RRR

Resistances

2HT

2HT

2HT

2HT

TT

)1(

)1(ln

θkθk

θk

θk

rR

T

��

A

RT

FEβθk

RT

FEβθk

RT

FEβθk

RT

FEβθk

rR

)1(exp)exp(1

)1(exp

)exp(1

ln

VHV

VHV

VHV

VHV

V

VV �

�

A

RT

FEβθk

RT

FEβθk

RT

FEβθk

RT

FEβθk

rR

)1()exp1(exp

)1()exp1(

exp

ln

HHH

HHH

HHH

HHH

H

HH �

�

A


Numerical Simulations

-1.5

-1.2

-0.9

-6.0 -4.0 -2.0 0.0

log(i /A cm-2)

E(V

)

Eq. (38)

Eq. (39)

I

II

I

III II

Eq. (45)Eq. (46)

Polarization curves

15.0

20.0

25.0

30.0

-1.45 -1.3 -1.15 -1 -0.85E (V )

lnR

RV

RH

RT

15.0

20.0

25.0

30.0

-1.45 -1.3 -1.15 -1 -0.85E (V)

lnR

RVH

RVT

RHT


CBHV

CHBVA

HV

))(()(

2

RRRR

RRRRR

FFri OR

AA

38

HV rrFi 39

22

)(22

TV

TV

/RR

FFri OR

AA45

HV

HV )(22

RR

FFri OR

AA46

Limiting Cases

OR

RV RH

+ - OR

RV

RT

RV RT

-1.1-1.5 -0.9E (V)

OR

OR

RH

RV

RT

RV

RH

RT

+ -

+ -

+ -


Conclusions

• The classical theory of direct RRs has been extended by defining direct ERs, INs and TNs.

• The extension of the RR theory leads to a new type of reaction networks, i.e., RR graphs.

• The RR graphs may be converted into electrical networks.

• The analogy between a reaction network and electrical network is an effective tool in reducing, simplifying and rationalizing complex kinetic mechanisms.

Reaction Route Graphs – An Effective Tool In Studying Complex Kinetic Mechanisms

Documents

Transcript of Reaction Route Graphs – An Effective Tool In Studying Complex Kinetic Mechanisms