Reaction Route Graphs – An Effective Tool In Studying Complex Kinetic Mechanisms Worcester Polytechnic Institute Worcester, 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 : 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

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

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 rara rbrb rcrc rdrd sasa sbsb scsc sdsd

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 r OR rbrb rcrc rdrd s OR sbsb scsc sdsd

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 a b c d e fg ih

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:sT: H 2 + 2M  2HM sV:sV: H 2 O + HM  M + H 3 O + + e – sH:sH: H 2 O + H 2 + M  HM + H 3 O + + e – OR: H 2 + 2H 2 O  2H 3 O + + 2e - electrochemical hydrogen oxidation and evolution reactions sT:sT: 2HM  2M + H 2 sV:sV: M + H 2 O + e -  HM + OH - sH:sH: HM + H 2 O + e -  M + H 2 + OH - OR: 2H 2 O + 2e -  H 2 + 2OH - HYDROGEN OXIDATION REACTIONS HYDROGEN EVOLUTION REACTIONS

Topological Characteristics of the RR Graph electrochemical hydrogen oxidation and evolution reactions ORR VH : s V + s H = OR ORR VT : 2s V + s T = OR ORR HT : 2s H – s T = OR TN 1 :OR - s H - s T TN 2 :OR - s V + s T TN 3 :2OR - s V - s H IN:-s V + s H + 2s T ERR: s V - s H + s T = 0 OVERALL REACTION ROUTES EMPTY REACTION ROUTES INTERMEDIATE NODES TERMINAL NODES

Constructing the RR Graph (a) (b) intermediate nodes terminal nodes peripheral nodes sVsV sVsV sHsH sHsH sTsT sTsT sVsV sVsV sHsH sHsH sTsT sTsT sVsV sVsV sHsH sHsH OR electrochemical hydrogen oxidation and evolution reactions

The RR Network OROR OROR RCRC RBRB RARA RVRV RHRH OROR OROR RHRH RVRV RTRT RVRV RHRH RTRT OROR OROR RHRH RVRV RVRV RHRH R T /  -  Transformation electrochemical hydrogen oxidation and evolution reactions

Resistances electrochemical hydrogen oxidation and evolution reactions

Numerical Simulations electrochemical hydrogen oxidation and evolution reactions

Limiting Cases OROR RVRV RHRH +- OROR RVRV RTRT RVRV RTRT E (V) OROR OROR RHRH RVRV RTRT RVRV RHRH RTRT 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.