John M. Dealy, Ronald G. Larson
Structure and Rheology of Molten Polymers
From Structure to Flow Behavior and Back Again
Preface
6
Contents
9
1 Introduction
16
1.1 Melt Structure and its Effect on Rheology
16
1.2 Overview of this Book
16
1.3 Applications of the Information Presented
18
1.4 Supplementary Sources of Information
19
2 Structure of Polymers
22
2.1 Molecular Size
22
2.1.1 The Freely-Jointed Chain
22
2.1.2 The Gaussian Size Distribution
23
2.1.3 The Dilute Solution and the Theta State
29
2.1.4 Polymer Molecules in the Melt
31
2.2 Molecular Weight Distribution
32
2.2.1 Monodisperse Polymers
32
2.2.2 Average Molecular Weights – Moments of the Distribution
33
2.2.3 Continuous Molecular Weight Distribution
35
2.2.4 Distribution Functions
37
2.2.5 Narrow Distribution Samples
41
2.2.6 Bimodality
42
2.3 Tacticity
42
2.4 Branching
43
2.5 Intrinsic Viscosity
45
2.5.1 Introduction
45
2.5.2 Rigid Sphere Models
46
2.5.3 The Free-Draining Molecule
48
2.5.4 Non-Theta Conditions and the Mark-Houwink-Sakurada Equation
48
2.5.5 Effect of Polydispersity
50
2.5.6 Effect of Long-chain Branching
51
2.5.7 Effects of Short-Chain Branching
52
2.5.8 Determination of the Intrinsic Viscosity – Extrapolation Methods
54
2.5.9 Effect of Shear Rate
54
2.6 Other Structure Characterization Methods
55
2.6.1 Membrane Osmometry
55
2.6.2 Light Scattering
56
2.6.3 Gel Permeation Chromatography
57
2.6.4 Mass Spectrometry (MALDI-TOF)
63
2.6.5 Nuclear Magnetic Resonance
63
2.6.6 TREF and CRYSTAF
64
2.6.7 Molecular Structure from Rheology
66
2.7 Summary
66
3 Polymerization Reactions and Processes
72
3.1 Introduction
72
3.2 Classifications of Polymers and Polymerization Reactions
73
3.3 Structural Characteristics of Polymers
75
3.3.1 Introduction
75
3.3.2 Chemical Composition –Role of Backbone Bonds in Chain Flexibility
75
3.3.3 Chemical Composition – Copolymers
75
3.3.4 Tacticity
76
3.3.5 Branching
76
3.4 Living Polymers Having Prescribed Structures
77
3.4.1 Anionic Polymerization
78
3.4.2 Living Free Radical Polymerization
80
3.4.3 Analogs of Polyethylene for Research
80
3.5 Industrial Polymerization Processes
81
3.6 Free-Radical Polymerization of Low-Density Polyethylene (LDPE)
82
3.7 High-Density Polyethylene
83
3.7.1 Catalyst Systems
83
3.7.2 Branching in HDPE
84
3.7.3 Ultrahigh Molecular Weight Polyethylene
85
3.8 Linear Low-Density Polyethylene
85
3.9 Single-Site (Metallocene) Catalysts
86
3.9.1 Catalyst System
86
3.9.2 Long-Chain Branching in Metallocene Polyethylenes
88
3.10 Polypropylene
94
3.11 Reactors for Polyolefins
96
3.12 Polystyrene
97
3.13 Summary
98
4 Linear Viscoelasticity – Fundamentals
106
4.1 Stress Relaxation and the Relaxation Modulus
106
4.1.1 The Boltzmann Superposition Principle
106
4.1.2 The Maxwell Model for the Relaxation Modulus
110
4.1.3 The Generalized Maxwell Model and the Discrete Relaxation Spectrum
113
4.1.4 The Continuous Relaxation Spectrum
114
4.2 The Creep Compliance and the Retardation Spectrum
115
4.3 Experimental Characterization of Linear Viscoelastic Behavior
119
4.3.1 Oscillatory Shear
120
4.3.2 Experimental Determination of the Storage and Loss Moduli
124
4.3.3 Creep Measurements
127
4.3.4 Other Methods for Monitoring Relaxation Processes
128
4.4 Calculation of a Spectrum from Experimental Data
129
4.5 Moments of the Relaxation Spectrum as Indicators of Molecular Structure
133
4.6 Time-Temperature Superposition
135
4.7 Time-Pressure Superposition
141
4.8 Summary
141
5 Linear Viscoelasticity – Behavior of Molten Polymers
146
5.1 Introduction
146
5.2 The Zero-Shear Viscosity
146
5.2.1 Effect of Molecular Weight
147
5.2.2 Effect of Polydispersity
150
5.3 Relaxation Modulus
152
5.3.1 General Features
152
5.3.2 How a Melt Can Act Like a Rubber
155
5.4 The Storage and Loss Moduli
155
5.5 The Creep and Recoverable Compliances
159
5.6 The Steady-State Compliance
161
5.7 The Storage and Loss Compliances
163
5.8 Determination of the Plateau Modulus
165
5.9 The Molecular Weight Between Entanglements, Me
167
5.9.1 Definitions of Me
168
5.9.2 Effects of Molecular Structure on GN0 and Me
170
5.9.3 Molecular Weight Between Entanglements (Me) Based on Molecular
5.9.3 Molecular Weight Between Entanglements (Me) Based on Molecular
170
170
5.10 Rheological Behavior of Copolymers
174
5.11 Effect of Long-Chain Branching on Linear Viscoelastic
5.11 Effect of Long-Chain Branching on Linear Viscoelastic
175
175
5.11.1 Introduction
175
5.11.2 Ideal Branched Polymers
176
5.11.3 Storage and Loss Moduli of Model Branched Systems
181
5.11.4 Randomly Branched Polymers
184
5.11.5 Low-Density Polyethylene
186
5.12 Use of Linear Viscoelastic Data to Determine Branching Level
188
5.12.1 Introduction
188
5.12.2 Correlations Based on the Zero-Shear Viscosity
189
5.13 The Cole-Cole Function and Cole-Cole Plots
191
5.13.1 The Complex Dielectric Constant and the Cole-Cole Function
191
5.13.2 Cole-Cole Plots for Characterizing Linear Viscoelastic Behavior
192
5.13.3 Van Gurp-Palmen Plot of Loss Angle Versus Complex Modulus
197
5.14 Summary
198
6 Tube Models for Linear Polymers – Fundamentals
208
6.1 Introduction
208
6.2 The Rouse-Bueche Model for Unentangled Polymers
209
6.2.1 Introduction
209
6.2.2 The Rouse Model for the Viscoelasticity of a Dilute Polymer Solution
210
6.2.3 Bueche’s Modification for an Unentangled Melt
212
6.3 Entanglements and the Tube Model
218
6.3.1 The Critical Molecular Weight for Entanglement MC
219
6.3.2 The Plateau Modulus GN0
221
6.3.3 The Molecular Weight Between Entanglements Me
222
6.3.4 The Tube Diameter a
223
6.3.5 The Equilibration Time te
226
6.4 Modes of Relaxation
227
6.4.1 Reptation
227
6.4.2 Primitive Path Fluctuations
229
6.4.3 Reptation Combined with Primitive Path Fluctuations
230
6.4.4 Constraint Release – Double Reptation
233
6.4.5 Rouse Relaxation Within the Tube
243
6.5 Summary
244
7 Tube Models for Linear Polymers – Advanced Topics
248
7.1 Introduction
248
7.2 Limitations of Double Reptation Theory
248
7.3 Constraint-Release Rouse Relaxation
251
7.3.1 Non-Self-Entangled Long Chains in a Short-Chain Matrix
251
7.3.2 Self-Entangled Long Chains in a Short-Chain Matrix
255
7.3.3 Polydisperse Chains
256
7.4 Tube Dilation or “Dynamic Dilution”
257
7.5 Input Parameters for Tube Models
261
8 Determination of Molecular Weight Distribution Using Rheology
274
8.1 Introduction
274
8.2 Viscosity Methods
274
8.3 Empirical Correlations Based on the Elastic Modulus
281
8.4 Methods Based on Double Reptation
282
8.5 Generalization of Double-Reptation
286
8.6 Dealing with the Rouse Modes
287
8.7 Models that Account for Additional Relaxation Processes
287
8.8 Prediction of Polydispersity Indexes
290
8.9 Summary
290
9 Tube Models for Branched Polymers
294
9.1 Introduction
294
9.2 General Effect of LCB on Rheology
295
9.3 Star Polymers
300
9.3.1 Deep Primitive Path Fluctuations
300
9.3.2 Dynamic Dilution
302
9.3.3 Comparison of Milner-McLeish Theory to Linear Viscoelastic Data
305
9.4 Multiply Branched Polymers
313
9.4.1 Branch-Point Motion
313
9.4.2 Backbone Relaxation
315
9.4.3 Dynamic Dilution for Polymers with Backbones
316
9.4.4 Predictions for Molecules with Moving Branch Points:
9.4.4 Predictions for Molecules with Moving Branch Points:
318
318
9.5 Theories and Algorithms for Polydisperse Branched Polymers
322
9.5.1 Hierarchical Dynamic Dilution Model
323
9.5.2 Slip Link Simulations
329
9.6 Dilution and Combinatorial Rheology
336
9.7 Summary
339
10 Nonlinear Viscoelasticity
344
10.1 Introduction
344
10.2 Nonlinear Phenomena – A Tube Model Interpretation
344
10.2.1 Large Scale Orientation – The Need for a Finite Strain Tensor
345
10.2.2 Chain Retraction and the Damping Function
345
10.2.3 Convective Constraint Release and Shear Thinning
347
10.3 Constitutive Equations
347
10.3.1 Boltzmann Revisited
349
10.3.2 The Rubberlike Liquid
351
10.3.3 Wagner’s Equation
352
10.3.4 Other Integral Constitutive Equations
353
10.3.5 Differential Constitutive Equations
355
10.4 Nonlinear Stress Relaxation
356
10.4.1 Doi and Edwards Predictions of the Damping Function
356
10.4.2 Estimating the Rouse Time of an Entangled Chain
358
10.4.3 Damping Functions of Typical Polymers
359
10.4.4 Normal Stress Relaxation
363
10.4.5 Double-Step Strain
366
10.5 Dimensionless Groups Used to Plot Rheological Data
366
10.5.1 The Deborah Number
366
10.5.2 The Weissenberg Number
367
10.6 Transient Shear Tests at Finite Rates
368
10.6.1 Stress Growth and Relaxation in Steady Shear
368
10.6.2 Nonlinear Creep
371
10.6.3 Large-Amplitude Oscillatory Shear
371
10.6.4 Exponential Shear
372
10.7 The Viscometric Functions
373
10.7.1 Dependence of Viscosity on Shear Rate
373
10.7.2 Normal Stress Differences in Steady Simple Shear
379
10.8 Experimental Methods for Shear Measurements
383
10.8.1 Optical Methods
383
10.8.2 Generating Step Strain
383
10.8.3 Rotational Rheometers
384
10.8.4 Measurement of the Second Normal Stress Difference
387
10.8.5 Capillary and Slit Rheometers
388
10.8.6 The Cox-Merz Rule
390
10.8.7 Sliding Plate Rheometers
392
10.9 Extensional Flow Behavior – Introduction
392
10.10 Extensional Flow Behavior of Melts and Concentrated Solutions
402
10.10.1 Linear, Monodisperse Polymers
402
10.10.2 Effect of Polydispersity
402
10.10.3 Linear Low-Density Polyethylene
403
10.10.4 Model Branched Systems
404
10.10.5 Long-Chain Branched Metallocene Polyethylenes
404
10.10.6 Randomly Branched Polymers and LDPE
405
10.11 Experimental Methods for Extensional Flows
407
10.11.1 Introduction
407
10.11.2 Rheometers for Uniaxial Extension
407
10.11.3 Uniaxial Extension – Approximate Methods
412
10.11.4 Rheometers for Biaxial and Planar Extension
413
10.11.5 Extensional Rheometers – Summary
414
10.12 Shear Modification
414
10.13 Summary
415
11 Tube Models for Nonlinear Viscoelasticity of Linear and Branched Polymers
430
11.1 Introduction
430
11.2 Relaxation Processes Unique to the Nonlinear Regime
431
11.2.1 Retraction
431
11.2.2 Convective Constraint Release
432
11.3 Monodisperse Linear Polymers
433
11.3.1 No Chain Stretch: the Doi-Edwards Equation
433
11.3.2 Chain Stretch: the Doi-Edwards-Marrucci-Grizzuti (DEMG) Theory
436
11.3.3 Convective Constraint Release (CCR).
440
11.3.4 Differential Constitutive Equations Containing CCR
445
11.4 Polydisperse Linear Polymers
448
11.5 Comparison of Theory with Data for Linear Polymers
451
11.5.1 Shearing Flows
451
11.5.2 Extensional Flows
455
11.5.3 Processing Flows
462
11.5.4 Constitutive Instabilities and Slip
462
11.6 Polymers with Long-Chain Branching
463
11.6.1 The Pom-Pom Model
11.6.1 The Pom-Pom Model
11.6.2 Revisions to the Pom-Pom Model
474
11.6.3 Empirical Multi-Mode Pom-Pom Equations for Commercial Melts
476
11.7 Summary
479
12 State of the Art and Challenges for the Future
488
12.1 State of the Art
488
12.2 Progress and Remaining Challenges
491
Appendix
496
A Structural and Rheological Parameters for Several Polymers
496
B Some Tensors Useful in Rheology
498
Nomenclature
502
Author Index
508
Subject Index
520
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