reaction route graphs – an effective tool in studying complex kinetic mechanisms
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Reaction Route Graphs – An Effective Tool In Studying Complex Kinetic Mechanisms. Ilie Fishtik, Caitlin A. Callaghan, Ravindra Datta. Worcester Polytechnic Institute Worcester, MA November 11, 2004. Motivation. - PowerPoint PPT PresentationTRANSCRIPT
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
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
Graph Topological Characteristics of the RR Graphs
• 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
Graph Topological Characteristics of the RR Graphs
• 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
Graph Topological Characteristics of the RR Graphs
• 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
electrochemical hydrogen oxidation and evolution reactions electrochemical hydrogen oxidation and evolution reactions
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
electrochemical hydrogen oxidation and evolution reactions electrochemical hydrogen oxidation and evolution reactions
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
electrochemical hydrogen oxidation and evolution reactions electrochemical hydrogen oxidation and evolution reactions
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
electrochemical hydrogen oxidation and evolution reactions electrochemical hydrogen oxidation and evolution reactions
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
electrochemical hydrogen oxidation and evolution reactions electrochemical hydrogen oxidation and evolution reactions
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
+ -
+ -
+ -
electrochemical hydrogen oxidation and evolution reactions electrochemical hydrogen oxidation and evolution reactions
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.