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

1. 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

2. 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

3. The Conventional Graph Theoretical Approach • 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

4. 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

5. 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 Stop Start

6. Notation Elementary Reaction: Overall Reaction: Stoichiometric Matrix: :

7. 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

8. 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

9. 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 sa ra sb rb sd sc rd rc

10. 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 rb sb rOR sOR rc sc rd sd

11. a b e c d f g i h 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

12. 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

13. Electrochemical Hydrogen Evolution and Oxidation Reactions

14. Electrocatalytic Reaction HYDROGEN OXIDATION REACTIONS HYDROGEN EVOLUTION REACTIONS electrochemical hydrogen oxidation and evolution reactions

15. Topological Characteristics of the RR Graph OVERALL REACTION ROUTES EMPTY REACTION ROUTES ORRVH: sV + sH = OR ORRVT: 2sV + sT = OR ORRHT: 2sH – sT = OR ERR: sV - sH + sT = 0 INTERMEDIATE NODES TERMINAL NODES IN: -sV + sH + 2sT TN1: OR - sH - sT TN2: OR - sV + sT TN3: 2OR - sV - sH electrochemical hydrogen oxidation and evolution reactions

16. sH sV sH sV sT sT sV sV sH sH OR sH sV sT sT peripheral nodes sV sH terminal nodes intermediate nodes OR Constructing the RR Graph (a) (b) electrochemical hydrogen oxidation and evolution reactions

17. OR OR + - + - + OR - RC RV RH RV RH RV RA RT RT RT/2 RV RV RB RH RH RH OR OR + - + - + OR - The RR Network - Transformation electrochemical hydrogen oxidation and evolution reactions

18. Resistances electrochemical hydrogen oxidation and evolution reactions

19. 38 39 45 46 Numerical Simulations electrochemical hydrogen oxidation and evolution reactions

20. Limiting Cases + - OR RH RV RT RT RV RH + - OR RT RV RH RV RV RT + OR - + - OR -1.1 -1.5 -0.9 E (V) electrochemical hydrogen oxidation and evolution reactions

21. 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.