1 Introduction to Chain Molecules .................................................................1
1.1 Introduction................................................................................................................... 1
1.2 How Big is Big? ............................................................................................................ 3
1.2.1 Molecular Weight ..............................................................................................3
1.2.2 Spatial Extent.....................................................................................................4
1.3 Linear and Branched Polymers, Homopolymers, and Copolymers ............................. 7
1.3.1 Branched Structures ...........................................................................................7
1.3.2 Copolymers ........................................................................................................9
1.4 Addition, Condensation, and Natural Polymers.......................................................... 11
1.4.1 Addition and Condensation Polymers .............................................................11
1.4.2 Natural Polymers .............................................................................................13
1.5 Polymer Nomenclature ................................................................................................ 18
1.6 Structural Isomerism.................................................................................................... 20
1.6.1 Positional Isomerization ..................................................................................20
1.6.2 Stereo Isomerization ........................................................................................21
1.6.3 Geometrical Isomerization...............................................................................22
1.7 Molecular Weights and Molecular Weight Averages................................................. 24
1.7.1 Number-, Weight-, and z-Average Molecular Weights ..................................25
1.7.2 Polydispersity Index and Standard Deviation .................................................26
1.7.3 Examples of Distributions ...............................................................................28
1.8 Measurement of Molecular Weight............................................................................. 31
1.8.1 General Considerations....................................................................................31
1.8.2 End Group Analysis.........................................................................................32
1.8.3 MALDI Mass Spectrometry ............................................................................35
1.9 Preview of Things to Come......................................................................................... 37
1.10 Chapter Summary ........................................................................................................ 38
References............................................................................................................................... 41
Further Readings...................................................................................................................... 41
2 Step-Growth Polymerization......................................................................43
2.1 Introduction................................................................................................................... 43
2.2 Condensation Polymers: One Step at a Time ............................................................... 43
2.2.1 Classes of Step-Growth Polymers .....................................................................43
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2.2.2 First Look at the Distribution of Products ........................................................44
2.2.3 A First Look at Reactivity and Reaction Rates ................................................46
2.3 Kinetics of Step-Growth Polymerization ...................................................................... 49
2.3.1 Catalyzed Step-Growth Reactions.....................................................................50
2.3.2 How Should Experimental Data Be Compared with
Theoretical Rate Laws? .....................................................................................52
2.3.3 Uncatalyzed Step-Growth Reactions.................................................................53
2.4 Distribution of Molecular Sizes .................................................................................... 56
2.4.1 Mole Fractions of Species .................................................................................56
2.4.2 Weight Fractions of Species..............................................................................58
2.5 Polyesters ....................................................................................................................... 60
2.6 Polyamides ..................................................................................................................... 64
2.7 Stoichiometric Imbalance .............................................................................................. 67
2.8 Chapter Summary .......................................................................................................... 71
Problems ................................................................................................................................. 71
References............................................................................................................................... 76
Further Reading ....................................................................................................................... 76
3 Chain-Growth Polymerization....................................................................77
3.1 Introduction .................................................................................................................... 77
3.2 Chain-Growth and Step-Growth Polymerizations: Some Comparisons....................... 77
3.3 Initiation ......................................................................................................................... 79
3.3.1 Initiation Reactions ............................................................................................80
3.3.2 Fate of Free Radicals .........................................................................................81
3.3.3 Kinetics of Initiation ..........................................................................................82
3.3.4 Photochemical Initiation ....................................................................................84
3.3.5 Temperature Dependence of Initiation Rates....................................................85
3.4 Termination.................................................................................................................... 86
3.4.1 Combination and Disproportionation ................................................................86
3.4.2 Effect of Termination on Conversion to Polymer ............................................88
3.4.3 Stationary-State Radical Concentration ............................................................89
3.5 Propagation .................................................................................................................... 90
3.5.1 Rate Laws for Propagation ................................................................................91
3.5.2 Temperature Dependence of Propagation Rates ...............................................92
3.5.3 Kinetic Chain Length.........................................................................................94
3.6 Radical Lifetime ............................................................................................................ 96
3.7 Distribution of Molecular Weights................................................................................ 99
3.7.1 Distribution of i-mers: Termination by Disproportionation .............................99
3.7.2 Distribution of i-mers: Termination by Combination.....................................102
3.8 Chain Transfer ............................................................................................................. 104
3.8.1 Chain Transfer Reactions ................................................................................105
3.8.2 Evaluation of Chain Transfer Constants .........................................................106
3.8.3 Chain Transfer to Polymer ..............................................................................108
3.8.4 Suppressing Polymerization ............................................................................109
3.9 Chapter Summary ........................................................................................................ 110
Problems ............................................................................................................................... 110
References............................................................................................................................. 114
Further Reading ..................................................................................................................... 115
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4 Controlled Polymerization .......................................................................117
4.1 Introduction................................................................................................................ 117
4.2 Poisson Distribution for an Ideal Living Polymerization ......................................... 118
4.2.1 Kinetic Scheme..............................................................................................119
4.2.2 Breadth of the Poisson Distribution ..............................................................122
4.3 Anionic Polymerization ............................................................................................. 126
4.4 Block Copolymers, End-Functional Polymers, and Branched
Polymers by Anionic Polymerization........................................................................ 129
4.4.1 Block Copolymers .........................................................................................129
4.4.2 End-Functional Polymers ..............................................................................133
4.4.3 Regular Branched Architectures....................................................................135
4.5 Cationic Polymerization ............................................................................................ 137
4.5.1 Aspects of Cationic Polymerization ..............................................................138
4.5.2 Living Cationic Polymerization ....................................................................140
4.6 Controlled Radical Polymerization ........................................................................... 142
4.6.1 General Principles of Controlled Radical Polymerization............................142
4.6.2 Particular Realizations of Controlled Radical Polymerization .....................144
4.6.2.1 Atom Transfer Radical Polymerization.........................................144
4.6.2.2 Stable Free-Radical Polymerization (SFRP).................................145
4.6.2.3 Reversible Addition-Fragmentation Transfer (RAFT)
Polymerization ...............................................................................145
4.7 Polymerization Equilibrium....................................................................................... 147
4.8 Ring-Opening Polymerization ................................................................................... 150
4.8.1 General Aspects .............................................................................................150
4.8.2 Specific Examples of Living Ring-Opening Polymerizations......................152
4.8.2.1 Poly(Ethylene Oxide) ....................................................................152
4.8.2.2 Polylactide......................................................................................153
4.8.2.3 Poly(dimethyl siloxane).................................................................154
4.8.2.4 Ring-Opening Metathesis Polymerization (ROMP) .....................155
4.9 Dendrimers................................................................................................................. 156
4.10 Chapter Summary ...................................................................................................... 160
Problems ............................................................................................................................... 161
References............................................................................................................................. 163
Further Reading ..................................................................................................................... 163
5 Copolymers, Microstructure, and Stereoregularity................................165
5.1 Introduction................................................................................................................ 165
5.2 Copolymer Composition............................................................................................ 166
5.2.1 Rate Laws ......................................................................................................166
5.2.2 Composition versus Feedstock ......................................................................168
5.3 Reactivity Ratios........................................................................................................ 170
5.3.1 Effects of r Values.........................................................................................171
5.3.2 Relation of Reactivity Ratios to Chemical Structure....................................173
5.4 Resonance and Reactivity.......................................................................................... 175
5.5 A Closer Look at Microstructure .............................................................................. 179
5.5.1 Sequence Distributions ..................................................................................180
5.5.2 Terminal and Penultimate Models ................................................................183
5.6 Copolymer Composition and Microstructure: Experimental Aspects ...................... 185
5.6.1 Evaluating Reactivity Ratios from Composition Data .................................186
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5.6.2 Spectroscopic Techniques..............................................................................188
5.6.3 Sequence Distribution: Experimental Determination ...................................190
5.7 Characterizing Stereoregularity ................................................................................. 193
5.8 A Statistical Description of Stereoregularity ............................................................ 196
5.9 Assessing Stereoregularity by Nuclear Magnetic Resonance................................... 200
5.10 Ziegler¿Natta Catalysts ............................................................................................. 205
5.11 Single-Site Catalysts .................................................................................................. 208
5.12 Chapter Summary ...................................................................................................... 211
Problems ............................................................................................................................... 212
References............................................................................................................................. 216
Bibliography .......................................................................................................................... 216
6 Polymer Conformations ...........................................................................217
6.1 Conformations, Bond Rotation, and Polymer Size..................................................... 217
6.2 Average End-to-End Distance for Model Chains ....................................................... 219
Case 6.2.1 The Freely Jointed Chain .......................................................................220
Case 6.2.2 The Freely Rotating Chain .....................................................................221
Case 6.2.3 Hindered Rotation Chain ........................................................................222
6.3 Characteristic Ratio and Statistical Segment Length.................................................. 223
6.4 Semiflexible Chains and the Persistence Length........................................................ 225
6.4.1 Persistence Length of Flexible Chains............................................................227
6.4.2 Worm-Like Chains ..........................................................................................228
6.5 Radius of Gyration....................................................................................................... 229
6.6 Spheres, Rods, and Coils ............................................................................................. 234
6.7 Distributions for End-to-End Distance and Segment Density .................................... 235
6.7.1 Distribution of the End-to-End Vector............................................................236
6.7.2 Distribution of the End-to-End Distance.........................................................239
6.7.3 Distribution about the Center of Mass ............................................................240
6.8 Self-Avoiding Chains: A First Look ........................................................................... 241
6.9 Chapter Summary ........................................................................................................ 242
Problems ............................................................................................................................... 242
References............................................................................................................................. 244
Further Reading ..................................................................................................................... 245
7 Thermodynamics of Polymer Solutions .................................................247
7.1 Review of Thermodynamic and Statistical Thermodynamic Concepts ..................... 247
7.2 Regular Solution Theory ............................................................................................. 249
7.2.1 Regular Solution Theory: Entropy of Mixing.................................................249
7.2.2 Regular Solution Theory: Enthalpy of Mixing ...............................................251
7.3 Flory¿Huggins Theory................................................................................................. 254
7.3.1 Flory¿Huggins Theory: Entropy of Mixing by a Quick Route ......................255
7.3.2 Flory¿Huggins Theory: Entropy of Mixing by a Longer Route ....................255
7.3.3 Flory¿Huggins Theory: Enthalpy of Mixing ..................................................257
7.3.4 Flory¿Huggins Theory: Summary of Assumptions ........................................258
7.4 Osmotic Pressure ......................................................................................................... 258
7.4.1 Osmotic Pressure: General Case .....................................................................259
7.4.1.1 Number-Average Molecular Weight ...............................................261
7.4.2 Osmotic Pressure: Flory¿Huggins Theory ......................................................263
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7.5 Phase Behavior of Polymer Solutions......................................................................... 264
7.5.1 Overview of the Phase Diagram......................................................................265
7.5.2 Finding the Binodal .........................................................................................268
7.5.3 Finding the Spinodal........................................................................................269
7.5.4 Finding the Critical Point ................................................................................270
7.5.5 Phase Diagram from Flory¿Huggins Theory ..................................................271
7.6 What is in x?................................................................................................................ 275
7.6.1 x from Regular Solution Theory .....................................................................275
7.6.2 x from Experiment...........................................................................................276
7.6.3 Further Approaches to x..................................................................................278
7.7 Excluded Volume and Chains in a Good Solvent ...................................................... 280
7.8 Chapter Summary ........................................................................................................ 283
Problems ............................................................................................................................... 284
References............................................................................................................................. 287
Bibliography .......................................................................................................................... 288
8 Light Scattering by Polymer Solutions...................................................289
8.1 Introduction: Light Waves........................................................................................... 289
8.2 Basic Concepts of Scattering....................................................................................... 291
8.2.1 Scattering from Randomly Placed Objects .....................................................292
8.2.2 Scattering from a Perfect Crystal ....................................................................292
8.2.3 Origins of Incoherent and Coherent Scattering ..............................................293
8.2.4 Bragg¿s Law and the Scattering Vector ..........................................................294
8.3 Scattering by an Isolated Small Molecule .................................................................. 296
8.4 Scattering from a Dilute Polymer Solution................................................................. 298
8.5 The Form Factor and the Zimm Equation .................................................................. 304
8.5.1 Mathematical Expression for the Form Factor ...............................................305
8.5.2 Form Factor for Isotropic Solutions ................................................................306
8.5.3 Form Factor as qRg P 1 ..................................................................................307
8.5.4 Zimm Equation ................................................................................................307
8.5.5 Zimm Plot ........................................................................................................308
8.6 Scattering Regimes and Particular Form Factors........................................................ 312
8.7 Experimental Aspects of Light Scattering .................................................................. 314
8.7.1 Instrumentation ................................................................................................316
8.7.2 Calibration........................................................................................................317
8.7.3 Samples and Solutions.....................................................................................319
8.7.4 Refractive Index Increment .............................................................................319
8.8 Chapter Summary ........................................................................................................ 320
Problems ............................................................................................................................... 321
Bibliography .......................................................................................................................... 325
Further Reading ..................................................................................................................... 325
9 Dynamics of Dilute Polymer Solutions ...................................................327
9.1 Introduction: Friction and Viscosity............................................................................ 327
9.2 Stokes¿ Law and Einstein¿s Law................................................................................. 330
9.2.1 Viscous Forces on Rigid Spheres....................................................................331
9.2.2 Suspension of Spheres .....................................................................................332
9.3 Intrinsic Viscosity ........................................................................................................ 334
9.3.1 General Considerations....................................................................................334
9.3.2 Mark¿Houwink Equation ................................................................................336
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9.4 Measurement of Viscosity ........................................................................................ 341
9.4.1 Poiseuille Equation and Capillary Viscometers ...........................................341
9.4.2 Concentric Cylinder Viscometers .................................................................345
9.5 Diffusion Coefficient and Friction Factor ................................................................ 346
9.5.1 Tracer Diffusion and Hydrodynamic Radius................................................347
9.5.2 Mutual Diffusion and Fick¿s Laws...............................................................348
9.6 Dynamic Light Scattering ......................................................................................... 354
9.7 Hydrodynamic Interactions and Draining ................................................................ 357
9.8 Size Exclusion Chromatography............................................................................... 360
9.8.1 Basic Separation Process ..............................................................................361
9.8.2 Separation Mechanism ..................................................................................365
9.8.3 Two Calibration Strategies ...........................................................................367
9.8.3.1 Limitations of Calibration by Standards.......................................367
9.8.3.2 Universal Calibration ....................................................................368
9.8.4 Size Exclusion Chromatography Detectors ..................................................369
9.8.4.1 RI Detector ....................................................................................369
9.8.4.2 uv¿vis Detector .............................................................................370
9.8.4.3 Light Scattering Detector..............................................................371
9.8.4.4 Viscometer.....................................................................................372
9.9 Chapter Summary...................................................................................................... 372
Problems............................................................................................................................. 373
References ........................................................................................................................... 378
Further Reading................................................................................................................... 379
10 Networks, Gels, and Rubber Elasticity .................................................381
10.1 Formation of Networks by Random Cross-Linking............................................... 381
10.1.1 Definitions .................................................................................................381
10.1.2 Gel Point....................................................................................................383
10.2 Polymerization with Multifunctional Monomers ................................................... 386
10.2.1 Calculation of the Branching Coefficient.................................................387
10.2.2 Gel Point....................................................................................................388
10.2.3 Molecular-Weight Averages .....................................................................389
10.3 Elastic Deformation ................................................................................................ 392
10.4 Thermodynamics of Elasticity ................................................................................ 394
10.4.1 Equation of State.......................................................................................394
10.4.2 Ideal Elastomers ........................................................................................396
10.4.3 Some Experiments on Real Rubbers ........................................................397
10.5 Statistical Mechanical Theory of Rubber Elasticity: Ideal Case ........................... 398
10.5.1 Force to Extend a Gaussian Chain ...........................................................400
10.5.2 Network of Gaussian Strands....................................................................402
10.5.3 Modulus of the Gaussian Network ...........................................................403
10.6 Further Developments in Rubber Elasticity ........................................................... 406
10.6.1 Non-Gaussian Force Law..........................................................................406
10.6.2 Front Factor ...............................................................................................407
10.6.3 Network Defects........................................................................................408
10.6.4 Mooney¿Rivlin Equation ..........................................................................409
10.7 Swelling of Gels...................................................................................................... 410
10.7.1 Modulus of a Swollen Rubber ..................................................................411
10.7.2 Swelling Equilibrium ................................................................................412
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10.8 Chapter Summary.................................................................................................... 414
Problems............................................................................................................................. 416
References ........................................................................................................................... 418
Further Reading................................................................................................................... 418
11 Linear Viscoelasticity .............................................................................419
11.1 Basic Concepts ........................................................................................................ 419
11.1.1 Stress and Strain........................................................................................421
11.1.2 Viscosity, Modulus, and Compliance .......................................................421
11.1.3 Viscous and Elastic Responses .................................................................422
11.2 Response of the Maxwell and Voigt Elements ...................................................... 423
11.2.1 Transient Response: Stress Relaxation .....................................................423
11.2.2 Transient Response: Creep........................................................................426
11.2.3 Dynamic Response: Loss and Storage Moduli.........................................427
11.2.4 Dynamic Response: Complex Modulus and Complex Viscosity ............430
11.3 Boltzmann Superposition Principle ........................................................................ 431
11.4 Bead¿Spring Model................................................................................................. 433
11.4.1 Ingredients of the Bead¿Spring Model.....................................................433
11.4.2 Predictions of the Bead¿Spring Model.....................................................434
11.5 Zimm Model for Dilute Solutions, Rouse Model for
Unentangled Melts .................................................................................................. 440
11.6 Phenomenology of Entanglement ........................................................................... 445
11.6.1 Rubbery Plateau ........................................................................................445
11.6.2 Dependence of Me on Molecular Structure ..............................................448
11.7 Reptation Model...................................................................................................... 451
11.7.1 Reptation Model: Longest Relaxation Time and Diffusivity...................452
11.7.2 Reptation Model: Viscoelastic Properties ................................................455
11.7.3 Reptation Model: Additional Relaxation Processes .................................457
11.8 Aspects of Experimental Rheometry ...................................................................... 459
11.8.1 Shear Sandwich and Cone and Plate Rheometers....................................459
11.8.2 Further Comments about Rheometry........................................................461
11.9 Chapter Summary .................................................................................................... 462
Problems............................................................................................................................. 462
References ........................................................................................................................... 465
Further Reading................................................................................................................... 465
12 Glass Transition......................................................................................467
12.1 Introduction ............................................................................................................. 467
12.1.1 Definition of a Glass .................................................................................467
12.1.2 Glass and Melting Transitions ..................................................................468
12.2 Thermodynamic Aspects of the Glass Transition .................................................. 470
12.2.1 First-Order and Second-Order Phase Transitions.....................................471
12.2.2 Kauzmann Temperature ............................................................................473
12.2.3 Theory of Gibbs and DiMarzio ................................................................474
12.3 Locating the Glass Transition Temperature ........................................................... 476
12.3.1 Dilatometry................................................................................................476
12.3.2 Calorimetry................................................................................................478
12.3.3 Dynamic Mechanical Analysis .................................................................480
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12.4 Free Volume Description of the Glass Transition.................................................. 481
12.4.1 Temperature Dependence of the Free Volume ........................................482
12.4.2 Free Volume Changes Inferred from the Viscosity .................................483
12.4.3 Williams¿Landel¿Ferry Equation.............................................................485
12.5 Time¿Temperature Superposition........................................................................... 488
12.6 Factors That Affect the Glass Transition Temperature.......................................... 493
12.6.1 Dependence on Chemical Structure..........................................................493
12.6.2 Dependence on Molecular Weight ...........................................................494
12.6.3 Dependence on Composition ....................................................................494
12.7 Mechanical Properties of Glassy Polymers ............................................................ 498
12.7.1 Basic Concepts ..........................................................................................498
12.7.2 Crazing, Yielding, and the Brittle-to-Ductile Transition .........................500
12.7.3 Role of Chain Stiffness and Entanglements .............................................503
12.8 Chapter Summary.................................................................................................... 506
Problems............................................................................................................................. 507
Bibliography....................................................................................................................... 510
Further Reading................................................................................................................... 510
13 Crystalline Polymers...............................................................................513
13.1 Introduction and Overview ..................................................................................... 513
13.2 Structure and Characterization of Unit Cells ......................................................... 515
13.2.1 Classes of Crystals ....................................................................................515
13.2.2 X-Ray Diffraction .....................................................................................517
13.2.3 Examples of Unit Cells .............................................................................520
13.3 Thermodynamics of Crystallization: Relation of Melting Temperature
to Molecular Structure ............................................................................................ 523
13.4 Structure and Melting of Lamellae......................................................................... 528
13.4.1 Surface Contributions to Phase Transitions..............................................528
13.4.2 Dependence of Tm on Lamellar Thickness...............................................529
13.4.3 Dependence of Tm on Molecular Weight .................................................532
13.4.4 Experimental Characterization of Lamellar Structure .............................534
13.5 Kinetics of Nucleation and Growth ........................................................................ 538
13.5.1 Primary Nucleation ...................................................................................539
13.5.2 Crystal Growth ..........................................................................................541
13.6 Morphology of Semicrystalline Polymers .............................................................. 547
13.6.1 Spherulites .................................................................................................547
13.6.2 Nonspherulitic Morphologies....................................................................550
13.7 Kinetics of Bulk Crystallization ............................................................................. 554
13.7.1 Avrami Equation .......................................................................................554
13.7.2 Kinetics of Crystallization: Experimental Aspects ..................................559
13.8 Chapter Summary.................................................................................................... 564
Problems............................................................................................................................. 565
References ........................................................................................................................... 568
Further Reading................................................................................................................... 568
Appendix .........................................................................................................569
A.1 Series Expansions ...............................................................................................................569
A.2 Summation Formulae..........................................................................................................570
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A.3 Transformation to Spherical Coordinates...........................................................................571
A.4 Some Integrals of Gaussian Functions ...............................................................................572
A.5 Complex Numbers ..............................................................................................................574