Modeling and Simulation in Polymer Reaction Engineering A Modular Approach

Preface ix

1 Introduction 1

1.1 Special Features of Polymers 1

1.2 Structures in Polymers and Their Influence on Processing and Application Properties 3

1.2.1 Chain Length, Molecular Mass, Moments, and Mean Values 3

1.2.2 Rheological Properties 6

1.2.3 Constitutional Isomers 7

1.2.4 Architectural Isomers 9

1.2.5 Copolymers 11

1.3 Some Analytical Methods for Model Validation 13

1.4 Description of Polymer Properties 15

1.4.1 Chemical Master Equations 17

1.4.2 Approaches to Polymer Properties 21

1.4.3 Stochastic and Deterministic Simulation 22

2 Polymer Reactions 25

2.1 Module Concept 25

2.2 Rate Coefficients in Polymerization Reactions 26

2.3 Building Macromolecules 28

2.4 Only Chain-Forming Reactions Take Place, Step-Growth Polymerization 30

2.4.1 Only One Type of End Group: The A − A Case, A Reacting with A 31

2.4.2 Two Types of Functional Groups A and B at One Molecule; A Reacts with B 40

2.4.3 Introducing Monofunctional Molecules to Control Degree of Polymerization 41

2.4.4 Addition of a Second Bifunctional Monomer, Two Functional Groups on Two Different Molecular Species 43

2.4.5 Reversible Reactions 49

2.5 Chain-Growth Polymerization – Initiation Required 58

2.5.1 Living Polymerization – Only Initiation and Propagation (Chain Growth) Take Place 59

2.5.1.1 Moment Equations 65

2.5.2 Living Polymerization Together with Chain Depropagation 68

2.5.3 Initiation and Chain Growth with Transfer Reactions 75

2.5.4 Initiation and Chain Growth with Deactivation 76

2.5.5 Initiation and Chain Growth with Chain Termination 77

2.5.5.1 Termination by Disproportionation 84

2.5.5.2 Termination by Combination 84

2.5.5.3 Transfer to Monomer or Transfer Agent 86

2.5.5.4 Transfer to Polymer 87

2.5.5.5 Propagation with Change of Characteristics 89

2.5.5.6 𝛽-Scission 90

2.6 Copolymerization 91

2.6.1 Conventional Description of Copolymer Composition 91

2.6.2 Characteristic Values for the Characterization of Copolymers 94

2.6.3 Modules for the Description of Copolymerization 97

2.6.4 Extended Description of a Copolymer 101

2.6.5 Distributed Counters 104

2.7 Nonlinear Polymerization 106

2.7.1 Branching; Graft Polymers via Copolymerization, (Grafting through) 106

2.7.2 Cross-Linking via Copolymerization 107

2.7.3 Nonlinear Structures by Polymerization from an Existing Chain, Grafting from 109

2.7.4 Cross-Linking of Preformed Linear Macromolecules by Low-Molecular-Mass Compounds 111

2.7.5 Nonlinear Step Growth 111

2.7.6 Higher Dimensional Models 113

2.8 List of Modules 114

2.8.1 Elemental Kinetic 115

2.8.2 Combination (P, Q, T, A) 116

2.8.3 Statistical Degradation (P, Q, T, A, B) 119

2.8.4 Change of Characteristics (P, Q, A, B) 120

2.8.5 Intermolecular Transfer (P, Q, T, R, A) 121

2.8.6 Cross Transfer (P, Q, T, R, A) 123

2.8.7 Initiation (P, A, B, C, m) 124

2.8.8 Propagation (P, Q, M, A, m) 125

2.8.9 Depropagation (P, Q, M, A, B, m) 127

2.8.10 Transfer (P, Q, T, M) 128

2.8.11 Disproportionation (P, Q, R, T, A) 129

2.8.12 Transfer to Polymer (P, Q, T, R, A) 131

2.8.13 Scission (P, Q, T, A, B) 132

2.8.14 Cross-Linking (P, Q, T, A) 133

2.8.15 Flow (A1,A2) 134

2.8.16 Phase Transfer (A1,A2) 135

2.8.17 Example System 135

3 Reactors for Polymerization Processes 139

3.1 Introduction 140

3.2 Well-Mixed (Ideal) Batch Reactor (BR) 141

3.2.1 Aspects of the Overall Mass Balance 143

3.2.2 Heat Balance in a Batch Reactor 144

3.2.3 Polymer Properties in Batch Reactors 148

3.3 Semi-Batch Reactor (Semi-BR) 149

3.4 The Continuous Stirred Tank Reactor (CSTR) 151

3.4.1 Homogeneous Continuous Stirred Tank Reactor (HCSTR) 151

3.4.2 Cascade of HCSTR 156

3.4.3 Segregated Continuous Stirred Tank Reactor (SCSTR) 157

3.5 Tubular Reactors 158

3.5.1 Plug Flow Reactor (PFR) 158

3.5.2 Laminar Tubular Reactor 159

3.6 Nonideal Reactor Models with Partial Backmixing 159

3.7 Comparison of Reactors 161

4 Phases and Phase Transitions 163

4.1 Treatment of Volumes and Concentrations 164

4.2 Phase Transfer Modules 165

4.2.1 Two-Film Theory 166

4.2.2 Examples for Phase Transfer Steps 169

4.2.2.1 Evaporation of a Pure Volatile Compound 169

4.2.2.2 Vapor–Liquid Equilibrium of Volatile Compounds 170

4.2.2.3 Adsorption of Gaseous Compounds 170

4.2.2.4 Vapor Pressure Above a Polymer Solution 172

4.2.2.5 Demixing in Polymer Solutions 174

4.2.3 Example: Phase Transfer During Polymerization; Living Anionic Polymerization of Butadiene 175

4.2.4 Summarizing Remarks to the Phase Change Module 178

4.3 Multiphase Polymerization Systems 179

4.3.1 Suspension Polymerization 179

4.3.2 Precipitation/Dispersion Polymerization 180

4.3.3 Emulsion Polymerization 181

5 Numerical Methods 193

5.1 Introduction 193

5.2 Ordinary Differential Equations 195

5.2.1 Consistency and Convergence 195

5.2.2 Stability 197

5.2.3 Error Control 200

5.2.4 A Practical Guide to ODE Solvers 205

5.2.4.1 List of Explicit Methods and Solvers for Non-Stiff ODEs 206

5.2.4.2 List of Implicit Methods and Solvers for Stiff ODEs and Differential Algebraic Equations (DAEs) 206

5.3 Countable Systems of Ordinary Differential Equations – CODEs 208

5.3.1 Theoretical Aspects 208

5.3.2 The Chain-Length Range 209

5.3.3 Initialization of Polymer Distributions 211

5.3.4 Approximation Schemes 212

5.4 Estimating the Numerical Error 217

5.5 Monte Carlo Methods 220

5.6 The Modeling Cycle: Dealing with Different Errors 223

6 Parameter Estimation 227

6.1 Introduction: Forward and Inverse Problems 227

6.2 General Theory 230

6.2.1 Introduction 230

6.2.2 The Minimization Problem 232

6.2.3 Sensitivity Analysis 235

6.3 Correlated Parameters 236

6.3.1 Damping 237

6.3.2 Essential Directions 238

6.4 Example: Parameter Dependencies and Condition 240

7 Styrene Butadiene Copolymers 251

7.1 Model Description 251

7.2 Components of the Model 251

7.2.1 Low-Molecular-Weight Compounds 251

7.2.2 Polymer Distributions 252

7.2.3 Sequence Distributions 253

7.2.4 Counters 253

7.2.5 Computation of Characteristic Values for Copolymers from Counters 254

7.3 Reaction Modules 254

7.3.1 Chain Initiation 254

7.3.2 Chain Propagation 255

7.3.3 LiH-Elimination 256

7.3.4 Chain Transfer 256

7.3.5 Re-Initiation by 1-Phenyl-1-Lithium Ethane 257

7.3.6 Balance Steps 257

7.4 Exemplary Simulations 258

7.5 Exemplification of the Modeling Cycle for the Styrene–Butadiene Example 266

References 269

Appendix 277

Index 283

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