Understanding Modeling and Simulation of Polymerization Reactions

Esmaiel Jabbari, PhD is Professor of Chemical and Biomedical Engineering at the University of South Carolina. His research focuses on the application of resorbable polymeric biomaterials in medicine for drug delivery and regenerative medicine. He was elected a Fellow of the American Institute of Medical and Biological Engineering in 2013.

• Preface xviiAcknowledgment xxiAbout the Companion Website xxii1 Mechanisms and Methods of Polymerization 11.1 Introduction 11.2 Mechanisms of Polymerization 21.2.1 Addition Polymerization 21.2.1.1 Conventional Radical Polymerization 21.2.1.2 Anionic Polymerization 31.2.1.3 Cationic Polymerization 41.2.1.4 Reversible-deactivation Radical Polymerization 71.2.1.5 Ring-opening Polymerization 111.2.1.6 Group Transfer Polymerization 131.2.2 Step-growth Polymerization 141.2.3 Step Transfer-addition and Radical Termination Polymerization 151.3 Living Polymerization 171.4 Copolymerization 211.5 Architecture of Polymer Chains 221.6 Summary 231.7 List of Symbols and Abbreviations 241.8 Practice Problems 27References 282 Elementary Reactions in Polymerization 432.1 Introduction 432.2 Initiation Reaction 442.2.1 Radical Initiation 442.2.1.1 Irreversible Radical Initiation 442.2.1.2 Reversible Radical Initiation 492.2.2 Anionic Initiation 512.2.3 Cationic Initiation 522.3 Propagation Reaction 522.4 Termination Reactions 522.5 Chain Transfer Reactions to Small Molecules 542.6 Chain Backbiting Reaction 572.7 Chain Combination Reactions 582.7.1 Termination by Combination 592.7.2 Radical Transfer to Polymer Reaction 592.7.3 Radical Transfer to Terminal Double Bond of Polymer Reaction 612.7.4 Radical Transfer to Pendant Double Bond of Polymer Reaction 612.7.5 Condensation Reaction in Step-growth Polymerization 622.8 Chain Dissociation Reactions 632.9 Summary 642.10 List of Symbols and Abbreviations 652.11 Practice Problems 69References 703 Functions with Distributed Variables 853.1 Introduction 853.2 Moments of a Distribution 863.3 Relating Molar-based Distribution of Polymer Chains to Weight-based Distribution 883.4 Relating Molar-based Moments of Polymer Chains to Weight-based Moments 893.4.1 Example Problem 1 903.5 Relating Number-average, Weight-average, and z-average Degree of Polymerization of Polymer Chains to Molar-based Moments of Chain Distribution 923.5.1 Example Problem 2 943.6 Relating Viscosity-average Degree of Polymerization of Polymer Chains to Molar-based Moments of Chain Distribution 943.6.1 Example Problem 3 953.7 Variance and Polydispersity of a Polymer Chain Distribution 973.7.1 Example Problem 4 983.8 Skewness and Kurtosis of a Polymer Chain Distribution 983.8.1 Example Problem 5 1003.9 Standard Distribution Functions Used in Polymerization Reactions 1013.9.1 Poisson Distribution Function 1013.9.2 Flory–Schulz Distribution Function 1023.9.3 Exponential Distribution Function 1043.9.4 Schulz–Zimm Distribution Function 1053.9.5 Lognormal Distribution Function 1063.9.6 Weibull Distribution Function 1073.9.7 Example Problem 6 1083.10 Summary 1113.11 List of Symbols and Abbreviations 1113.12 Practice Problems 113References 1164 Z-transform in Polymerization Reactions 1214.1 Introduction 1214.2 Definition of Z-transform 1224.3 Moments of a Distributed Function in Z Domain 1224.4 Properties of a Function in the Z Domain 1244.5 Inverse Z-transform 1264.5.1 Example Problem 1 1274.6 Application of Z-transform in Addition Polymerization 1284.6.1 Z-transform in Termination by Combination 1284.7 Application of Z-transform in Step-growth Polymerization 1304.7.1 Z-transform in Condensation of Bifunctional Monomers A—R—B 1304.7.1.1 Conservation Equation for n-mer Chains 1314.7.1.2 Overall Conservation Equation 1324.7.1.3 Conservation Equation for n-mer Chains in Terms of Conversion 1334.7.1.4 Z-transform of Conservation Equation for n-mers 1344.7.1.5 Z-transform of Initial Reaction Condition 1344.7.1.6 Inverse Z-transform of Chain Distribution Function 1354.7.2 Z-transform in Condensation Polymerization of Bifunctional Monomer A—R—B and Monofunctional Monomer A—x 1364.7.2.1 Conservation Equation for Bifunctional n-mer Chains 1384.7.2.2 Conservation Equation for Monofunctional n-mer Chains 1404.8 Summary 1414.9 List of Symbols and Abbreviations 1424.10 Practice Problems 145References 1485 Conservation of Moments of Distribution of Chains in Polymerization 1535.1 Introduction 1535.2 Elementary Reactions in Polymerization in a Differential Time Interval 1535.3 Mixing Theory of Polymer Chains 1545.3.1 Example Problem 1 1575.4 Conservation of kth Order Moment of Polymer Chains in a Reaction 1585.4.1 Conservation of kth Order Moment of Terminated Polymer Chains in the Absence of Convective or Diffusive Mass Transfer 1605.4.2 Conservation of kth Order Moment of Dormant Polymer Chains in the Absence of Convective or Diffusive Mass Transfer 1615.4.3 Conservation of kth Order Moment of Propagating Polymer Chains in the Absence of Convective or Diffusive Mass Transfer 1615.5 Derivation of the General Conservation Equation for kth Order Moment of Polymer Chains in a Reaction 1625.5.1 Moment Conservation Equations Applied to Batch Stirred Vessel Reactor 1675.5.2 Moment Conservation Equations Applied to Continuous Stirred Tank Reactor 1685.5.3 Moment Conservation Equations Applied to Plug-flow Tubular Reactor 1695.5.4 Moment Conservation Equations Applied to Batch Bulk, Solution, and Suspension Polymerizations 1715.5.5 Moment Conservation Equations Applied to Batch Emulsion Polymerization 1725.6 Formation of Reactive Species in Conventional Radical Polymerization and the Quasi-steady State Assumption 1745.6.1 Quasi-steady State in Dilute Polymerization Regime 1755.6.2 Quasi-steady State in Concentrated Polymerization Regime with Reaction Temperature Above Glass Transition Temperature 1755.6.3 Quasi-steady State in Concentrated Polymerization Regime with Reaction Temperature Below Glass Transition Temperature 1765.7 Initiation and Formation of Reactive Species in Polymerization 1765.7.1 Conventional Radical Polymerization with Equal Reactivity of Radical Species 1765.7.2 Reversible-deactivation Radical Polymerization with Unequal Reactivity of Reactive Species 1785.7.3 Anionic Polymerization with Free, Stable, and Reactive Anionic Species 1815.7.4 Cationic Polymerization with Free and Ion-pair, Reactive Cationic Species 1825.8 Summary 1845.9 List of Symbols and Abbreviations 1855.10 Practice Problems 187References 1896 Moments of Elementary Reactions in Addition Polymerization 2056.1 Introduction 2056.2 Instantaneous kth Order Moments of Propagating Chains After Propagation in Terms of Propagating Chains Prior to Propagation 2056.3 Instantaneous kth Order Moments of Chains Produced by Termination in Terms of kth Order Moments of Propagating Chains 2076.3.1 Termination by Disproportionation 2076.3.2 Termination by Combination 2076.4 Instantaneous kth Order Moments of Chains Produced by Transfer Reactions in Terms of kth Order Moments of Propagating Chains 2086.4.1 Instantaneous kth Order Moments of Terminated Chains Produced by Transfer to Small Molecules and Transfer Agent 2086.4.2 Instantaneous kth Order Moments of Dormant Chains Produced by Reaction with Transfer Agent with Reversible Deactivation 2096.4.3 Instantaneous kth Order Moments of Terminated Chains Produced by Transfer to Polymer Chains 2096.4.3.1 Instantaneous kth Order Moments of Terminated Chains Produced by Transfer to Polymer Chains via Hydrogen Abstraction 2096.4.3.2 Instantaneous kth Order Moments of Terminated Chains Produced by Chain Transfer to Polymer by Reaction with Terminal Double Bond 2106.4.3.3 Instantaneous kth Order Moments of Terminated Chains Produced by Chain Transfer to Polymer by Reaction with Double Bond Along Polymer Chain 2126.4.4 Instantaneous kth Order Moments of Terminated Chains Produced by Intramolecular Hydrogen Abstraction or Backbiting 2136.5 Instantaneous kth Order Moments of Chains Produced by Chain Scission 2136.5.1 Instantaneous kth Order Moments of Chains Produced by β 2 Scission 2136.5.2 Instantaneous kth Order Moments of Chains Produced by β 3 Scission 2156.6 Summary 2186.7 List of Symbols and Abbreviations 2186.8 Practice Problems 221References 2237 Conservation of Moments Applied to Polymerization Reactions 2297.1 Introduction 2297.2 General Conservation Equations for kth Order Moment of Propagating, Dormant, and Terminated Chains 2297.3 Modeling Radical Polymerization with Termination by Disproportionation in the Absence of Chain Transfer 2307.3.1 Modeling Radical Polymerization with Termination by Disproportionation in a Batch Stirred Vessel Reactor (BSVR) 2317.3.1.1 Time Dependence of Concentration of Propagating Chains in Radical Polymerization with Termination by Disproportionation 2327.3.1.2 Time Dependence of Monomer Concentration in Radical Polymerization with Termination by Disproportionation 2337.3.1.3 First-order Moment of Propagating Chains in Radical Polymerization with Termination by Disproportionation 2347.3.1.4 Second-order Moment of Propagating Chains in Radical Polymerization with Termination by Disproportionation 2357.3.1.5 Concentration of Terminated Polymer Chains in Radical Polymerization with Termination by Disproportionation 2377.3.1.6 First-order Moment of Terminated Polymer Chains in Radical Polymerization with Termination by Disproportionation 2387.3.1.7 Second-order Moment of Terminated Polymer Chains in Radical Polymerization with Termination by Disproportionation 2397.3.1.8 (1+a)-order Moment of the Terminated Chains in Radical Polymerization with Termination by Disproportionation 2407.3.1.9 Example Problem 1 2417.3.2 Modeling Radical Polymerization with Termination by Disproportionation in a Continuous Stirred Tank Reactor (CSTR) 2467.3.3 Modeling Radical Polymerization with Termination by Disproportionation in a Plug Flow Tubular Reactor (PFTR) 2507.4 Modeling Radical Polymerization with Chain Transfer Agent 2517.4.1 Concentration of Propagating Chains in Radical Polymerization with Chain Transfer Agent 2527.4.2 Concentration of Chain Transfer Agent in Radical Polymerization with Chain Transfer Agent 2537.4.3 Monomer Concentration in Radical Polymerization with Chain Transfer Agent 2537.4.4 kth Order Moments of Propagating Chains in Radical Polymerization with Chain Transfer Agent 2547.4.5 Concentration of Terminated Polymer Chains in Radical Polymerization with Chain Transfer Agent 2547.4.6 Higher Moments of Terminated Chains in Radical Polymerization with Chain Transfer Agent 2557.5 Modeling Radical Polymerization with Chain Transfer to the Terminated Polymer Chains 2567.5.1 Higher Moments of Propagating Chains in Radical Polymerization with Chain Transfer to Terminated Chains 2577.5.2 Higher Moments of Terminated Polymer Chains in Radical Polymerization with Chain Transfer to Terminated Chains 2587.5.3 Long Branching Density in Radical Polymerization with Chain Transfer to Terminated Chains 2587.6 Modeling Radical Polymerization with β 2 Chain Scission 2597.6.1 Higher Moments of Propagating Chains in Radical Polymerization with β 2 Chain Scission 2607.6.2 Concentration of Terminated Polymer Chains in Radical Polymerization with β 2 Chain Scission 2617.6.3 Higher Moments of Terminated Polymer Chains in Radical Polymerization with β 2Chain Scission 2617.6.4 Density of Terminal Double Bonds in Terminated Polymer Chains in Radical Polymerization with β 2 Chain Scission 2627.7 Modeling Radical Polymerization with Photoinitiation 2637.7.1 Absorbed Radiation in Radical Polymerization with Photoinitiation 2647.7.2 Concentration of Primary Radicals and Propagating Chains in Radical Polymerization with Photoinitiation 2677.7.3 Monomer Concentration in Radical Polymerization with Photoinitiation 2687.7.4 Higher Moments of Propagating Chains in Radical Polymerization with Photoinitiation 2697.7.5 Concentration of Terminated Polymer Chains in Radical Polymerization with Photoinitiation 2707.7.6 Higher Moments and Average Degree of Polymerization of Terminated Polymer Chains in Radical Polymerization with Photoinitiation 2717.8 Modeling Reversible-deactivation Radical Polymerization by Atom transfer Radical Polymerization 2717.8.1 Atom Transfer Radical Polymerization in the Absence of Chain Termination 2717.8.1.1 Conversion of Primary Radicals to Propagating Chains in ATRP in the Absence of Chain Termination 2727.8.1.2 Monomer Concentration in ATRP in the Absence of Chain Termination 2737.8.1.3 Higher Moments of the Propagating Chains in ATRP in Absence of Chain Termination 2737.8.1.4 Higher Moments of Dormant Chains in ATRP in the Absence of Chain Termination 2757.8.2 Atom Transfer Radical Polymerization in the Presence of Chain Termination 2767.8.2.1 Concentration of Propagating Chains in ATRP in the Presence of Termination Reaction 2777.8.2.2 Higher Moments of Propagating Chains in ATRP in the Presence of Termination Reaction 2787.8.2.3 Concentration and Higher Moments of Dormant Chains in ATRP in the Presence of Termination Reaction 2797.8.2.4 Concentration and Higher Moments of Terminated Chains in ATRP in the Presence of Termination Reaction 2797.9 Modeling Ionic Polymerization 2817.9.1 Modeling Anionic Polymerization with Butyl Lithium Initiation 2817.9.2 Modeling Cationic Polymerization with HI/ZnI 2 Initiation 2827.9.2.1 Modeling Cationic Polymerization with HI/ZnI 2 Initiation in the Absence of nBu 4 NCl 2847.9.2.2 Modeling Cationic Polymerization with HI/ZnI 2 Initiation in the Presence of nBu 4 NCl 2867.10 Summary 2877.11 List of Symbols and Abbreviations 2887.12 Practice Problems 294References 2978 Diffusion Controlled Polymerization Reactions 3078.1 Introduction 3078.2 Relating Diffusion Coefficient and Viscosity to Molecular Weight 3088.2.1 Dilute Polymerization Regime Below the Onset of Chain Entanglement 3088.2.2 Semi-dilute or Concentrated Regime Below Threshold Molecular Weight for Reptation 3128.2.3 Concentrated Regime Above Threshold Molecular Weight for Reptation 3128.3 Role of Diffusion in Bulk Polymerization of Methyl Methacrylate 3138.3.1 Diffusion Time of Monomer and Other Small Molecules 3138.3.2 Diffusion Time of Propagating and Terminated Polymer Chains 3148.3.2.1 Diffusion Time of PMMA Propagating Chains in Dilute Regime 3188.3.2.2 Diffusion Time of PMMA Propagating Chains in Semi-dilute Regime 3188.3.2.3 Diffusion Time of PMMA Propagating Chains in Reptation Regime 3198.3.2.4 Diffusion Time of PMMA Propagating Chains in Glassy Regime 3198.4 Termination Rate Constant in Diffusion-limited Polymerization Reactions 3218.4.1 Mass Conservation Equation for Propagating Chains in Diffusion-limited Polymerization Reactions 3218.4.2 Diffusive Flux of Propagating Chains in Diffusion-limited Polymerization Reactions 3248.4.3 Transfer Rate of Reactive Species on Propagating Chains in Diffusion-limited Polymerization Reactions 3248.4.4 Local Termination Rate of the Reactive Species in Diffusion-limited Polymerization Reactions 3258.4.5 Local Concentration of Propagating Chains in Diffusion-limited Polymerization Reactions 3268.4.6 Apparent Termination Rate Constant in Diffusion-limited Polymerization Reactions 3268.4.7 Time Constant for Termination Reaction in Diffusion-limited Polymerization Reactions 3278.5 Quasi Steady State in Diffusion-limited Polymerization Reactions 3288.5.1 Mass Conservation Equation for Primary Reactive Species in Diffusion-limited Polymerization 3298.6 Modeling Diffusion-limited Radical Polymerization with Termination by Disproportionation and Combination and Chain Transfer to Monomer 3308.6.1 Time dependence of Reaction Volume in Diffusion-limited Polymerization 3318.6.2 Time Dependence of Initiator Concentration in Diffusion-limited Polymerization 3328.6.3 Time Dependence of Monomer Concentration in Diffusion-limited Polymerization 3328.6.4 Conservation Equation for kth Order Molar Concentration Moments of Propagating Chains in Diffusion-limited Polymerization 3338.6.4.1 Concentration of Propagating Chains in Diffusion-limited Polymerization 3358.6.4.2 Kinetic Chain Length of Propagating Chains in Diffusion-limited Polymerization 3358.6.5 Conservation Equation for kth Order Molar Concentration Moments of Terminated Chains in Diffusion-limited Polymerization 3368.6.5.1 Concentration of Terminated Chains in Diffusion-limited Polymerization 3378.6.5.2 Number-average Degree of Polymerization of Terminated Chains in Diffusion-limited Polymerization 3378.6.5.3 Weight-average Degree of Polymerization of Terminated Chains in Diffusion-limited Polymerization 3388.7 Modeling Diffusion-limited Bulk Radical Polymerization of Methyl Methacrylate 3388.7.1 Dependence of Polymer Diffusion Coefficient on Conversion in Bulk MMA Polymerization 3418.7.2 Dependence of Apparent Initiation Efficiency and Rate Constants on Conversion in Bulk MMA Polymerization 3428.7.3 Time Dependence of Conversion in Bulk MMA Polymerization 3438.7.4 Time Dependence of kth Order Concentration Moments of Propagating Chains in Bulk MMA Polymerization 3438.7.5 Dependence of kth Order Concentration Moments of Terminated Chains on Conversion in Bulk MMA Polymerization 3458.8 Summary 3478.9 List of Symbols and Abbreviations 3488.10 Practice Problems 356References 3589 Markov Chain Modeling of Polymerization Reactions 3679.1 Introduction 3679.2 Definition of a Markov Chain 3679.3 State Space of a Markov Chain 3689.4 Initial Probability Vector 3689.5 One-step Uniform Transition Probability Matrix 3689.6 n-step Uniform Transition Probability Matrix 3699.7 n-step Absolute Probability Vector 3709.8 Types of States in State Space of a Markov Chain 3709.9 n-step Probability of Reaching an Absorbing State 3719.10 Matrix Operations 3729.10.1 Multiplication of a Matrix by a Scalar 3729.10.2 Addition and Subtraction of Matrices 3739.10.3 Multiplication of Matrices 3739.10.4 Determinant of a Matrix 3739.10.5 Matrix of Cofactors 3749.10.6 Transpose of a Matrix 3749.10.7 Adjugate Matrix 3759.10.8 Inverse of a Matrix 3759.11 Application of Markov Chains to Radical Polymerization 3759.11.1 Radical Polymerization with Termination by Disproportionation 3759.11.2 Radical Polymerization with Exclusion of Monomers from Chain Distribution 3789.11.3 Radical Polymerization with Termination by Disproportionation and Combination and Exclusion of Monomers from Chain Distribution 3809.12 Application of Markov Chain to Condensation Polymerization 3839.12.1 Condensation Polymerization of ARB Monomers 3839.12.2 Condensation Polymerization of AR 1 A, AR 2 , BR 3 B, and BR 4 Monomers 3859.12.3 Condensation Polymerization of AR 1 A and BR 3 B Monomers with Unequal Number of A and B Groups 3879.13 Summary 3899.14 List of Symbols and Abbreviations 3909.15 Practice Problems 393References 39510 Markov Chain Modeling of Copolymerization Reactions 39910.1 Introduction 39910.2 Number Representation of Copolymer Sequences 39910.3 Order of Copolymerization 40010.4 State Space of Copolymerization Reaction 40210.5 Conditional Probability Matrix for Copolymerization Reaction 40310.6 Composition of Copolymer 40310.7 Copolymer Sequence Distribution 40410.7.1 Probability of Finding n-mer or Longer Sequences of a Monomer in Copolymer 40510.7.2 Probability of Finding n-mer Sequences of a Monomer in Copolymer 40510.7.3 Fraction of n-mer Sequences of a Monomer in Copolymer 40610.7.4 Number-average of n-mer Sequences of a Monomer in Copolymer 40710.7.5 Deviation Parameter from Random Copolymer 40710.8 Modeling Terminal Copolymerization of Two Monomers 40910.8.1 Copolymer Composition in Terminal Copolymerization 41010.8.1.1 Random Terminal Copolymer 41110.8.1.2 Ideal Terminal Copolymer 41110.8.1.3 Alternating Terminal Copolymer 41110.8.1.4 Block Terminal Copolymer 41110.8.2 Sequence Distributions in Terminal Copolymerization and Sequence Averages 41310.8.2.1 Probability of Finding n-mer Sequences of a Monomer 41310.8.2.2 Mole Fraction of n-mer Sequences and Number-average of Sequences of M 0 n-mers 41310.8.2.3 Weight Fraction of n-mer Sequences and Weight-average of Sequences of M 0 n-mers 41310.8.2.4 Deviation Parameter from Random Copolymer 41410.8.2.5 Determination of Ratio of Reaction Rate Constants 41410.8.2.6 Testing Terminal Copolymerization Model 41710.8.2.7 Limitations of Terminal Copolymerization Model 41810.9 Modeling Terminal Terpolymerization of Three Monomers 41810.9.1 Conditional Probabilities in Terminal Terpolymerization 41910.9.2 Copolymer Composition in Terminal Terpolymerization 42010.9.2.1 Random Terminal Terpolymer 42110.9.2.2 Ideal Terminal Terpolymer 42110.9.2.3 Alternating Terminal Terpolymer 42210.9.2.4 Block Terminal Terpolymer 42210.9.3 Terpolymer Sequence Distributions and Sequence Averages 42210.9.3.1 Probability of Finding an n-mer Sequence of a Monomer 42310.9.3.2 Mole Fraction of n-mer Sequences and Number-average of Sequences of M 0 n-mers 42310.9.3.3 Weight Fraction of n-mer Sequences and Weight-average of Sequences of M 0 n-mers 42410.9.3.4 Polydispersity Index for Sequences of M 0 n-mers 42510.9.4 Deviation Parameter from Random Terpolymer 42510.9.5 Determination of Ratio of Reaction Rate Constants 42610.9.6 Testing Terminal Terpolymerization Model 42610.10 Summary 42710.11 List of Symbols and Abbreviations 42810.12 Practice Problems 430References 43411 Monte Carlo Simulation of Polymerization Reactions 43911.1 Introduction 43911.2 Theory of Monte Carlo Simulation 43911.3 Estimation of an Integral Function with Monte Carlo Method 44011.4 Probability Distributions of Random Numbers in Monte Carlo Simulation 44111.4.1 Rectangular Probability Distribution 44211.4.2 Normal Probability Distribution 44211.4.3 Poisson Probability Distribution 44211.4.4 χ 2 Probability Distribution 44211.5 Error Estimation in Monte Carlo Simulation 44311.6 Testing Uniformity of Random Numbers 44511.6.1 Test of Uniformity 44511.6.2 Example Problem 1 44611.6.3 Test of Interval Correlation 44711.6.4 Example Problem 2 44811.6.5 Test of Serial Correlation 44811.6.6 Example Problem 3 44911.6.7 Example Problem 4 44911.7 Monte Carlo Simulation of Initiator Dissociation 45111.8 Monte Carlo Simulation of Methyl Methacrylate Polymerization 45211.9 Monte Carlo Simulation of Branching in Butadiene Emulsion Polymerization 45711.9.1 Elementary Reactions 45811.9.2 Reaction Conditions and Rate Constants 45911.9.3 Simulation Assumptions 46011.9.4 Simulation Method 46111.9.4.1 Aqueous Phase Reactions 46111.9.4.2 Micelle Phase Reactions 46211.9.4.3 Determination of Reaction Times of Elementary Reactions 46511.9.4.4 Simulation Results 46511.10 Summary 46711.11 List of Symbols and Abbreviations 46811.12 Practice Problems 472References 474Index 481