Table of Contents

Part I. Constitutive Relations 1

1 Elastic Solid 5
Christopher W. Macosko

1.1 Introduction 5

1.2 The Stress Tensor 8

1.2.1 Notation 11

1.2.2 Symmetry 16

1.2.3 Pressure 18

1.3 Principal Stresses and Invariants 20

1.4 Finite Deformation Tensors 24

1.4.1 Finger Tensor 29

1.4.2 Strain Tensor 32

1.4.3 Inverse Deformation Tensors 32

1.4.4 Principal Strains 34

1.5 Neo-Hookean Solid 37

1.5.1 Uniaxial Extension 38

1.5.2 Simple Shear 40

1.6 General Elastic Solid 40

1.6.1 Strain‒Energy Function 42

1.6.2 Anisotropy 44

1.6.3 Rubber-like Liquids 45

1.7 Equations of Motion 45

1.7.1 Mass Balance 45

1.7.2 Momentum Balance 47

1.8 Boundary Conditions 52

1.9 Summary 58

1.10 Exercises 59

References 62

2 Viscous Liquid 65
Christopher W. Macosko

2.1 Introduction 65

2.2 Velocity Gradient 68

2.2.1 Rate of Deformation Tensor 72

2.3 Newtonian Fluid 77

2.3.1 Uniaxial Extension 79

2.4 General Viscous Fluid 83

2.4.1 Power Law 84

2.4.2 Cross Model 86

2.4.3 Other Viscous Models 86

2.4.4 The Importance of II2D 89

2.4.5 Extensional Thickening Models 91

2.5 Plastic Behavior 92

2.5.1 Other Viscoplastic Models 95

2.6 Balance Equations 98

2.6.1 Equations of Motion 99

2.6.2 Boundary Conditions 99

2.6.3 Energy Equation 100

2.6.4 Temperature and Pressure Dependence Viscosity 100

2.7 Summary 104

2.8 Exercises 105

References 106

3 Linear Viscoelasticity 109
Christopher W. Macosko

3.1 Introduction 109

3.2 General Linear Viscoelastic Model 111

3.2.1 Relaxation Spectrum 115

3.2.2 Linear Viscoelasticity in Three Dimensions 115

3.2.3 Differential Form 115

3.3 Small Strain Material Functions 117

3.3.1 Stress Relaxation 118

3.3.2 Creep 119

3.3.3 Sinusoidal Oscillations 121

3.4 Exctciscs 126

Appendix 3A 127

Robert B. Secor

Curve Fitting of Relaxation Modulus 127

Approximating Form 127

Error Measure 128

Search Procedures 129

References 133

4 Nonlinear Viscoelasticity 135
Ronald G. Larson

4.1 Introduction 135

4.2 Nonlinear Phenomena 138

4.2.1 Normal Stress Difference in Shear 138

4.2.2 Shear Thinning 139

4.2.3 Interrelations Between Shear Functions 140

4.2.4 Extensional Thickening 142

4.3 Simple Nonlinear Constitutive Equations 146

4.3.1 Second-Order Fluid 146

4.3.2 Upper-Converted Maxwell Equation 149

4.3.3 Lodge Integral Equation 153

4.4 More Accurate Constitutive Equations 158

4.4.1 Integral Constitutive Equations 158

4.4.2 Maxwell-Type Differential Constitutive Equations 166

4.5 Summary 170

4.6 Exercises 171

References 172

Part II Measurements: Rheometry 175

5 Shear Rheometry: Drag Flows 181
Christopher W. Macosko

5.1 Introduction 181

5.2 Sliding Plates, Falling Ball 184

5.2.1 Falling Cylinder 185

5.2.1 Falling Ball 187

5.2.3 Rolling Ball 187

5.3 Concentric Cylinder Rheometer 188

5.3.1 Shear Stress 190

5.3.2 Shear Strain and Rate 191

5.3.3 Normal Stresses in Couette Flow 195

5.3.4 Rod Climbing 198

5.3.5 End Effects 200

5.3.6 Secondary Flows 202

5.3.7 Shear Healing in Couette Flow 203

5.4 Cone and Plate Rheometer 205

5.4.1 Shear Stress 206

5.4.2 Shear Strain Rate 207

5.4.3 Normal Stresses 208

5.4.4 Inertia and Secondary Flow 209

5.4.5 Edge Effects with Cone and Plate 213

5 4.6 Shear Heating 216

5.4.7 Summary 216

5.5 Parallel Disks 217

5.5.1 Normal Stresses 221

5.6 Drag Flow Indexers 222

5.6.1 Rotating Disk in a Sea of Fluid 223

5.6.2 Rotating Vane 224

5.6.3 Helical Screw Rheometer 224

5.6.4 Instrumented Mixers 225

5.7 Eccentric Rotating Geometries 226

5.7.1 Rotating Cantiliver Rod 227

5.7.2 Eccentric Rotating Disks 227

5.7.3 Other Eccentric Geometries 231

References 231

6 Shear Rheometry: Pressure‒Driven Flows 237
Christopher W. Macosko

6.1 Introduction 237

6.2 Capillary Rheometer 238

6.2.1 Shear Rate 240

6.2.2 Wall Slip. Melt Fracture 244

6.2.3 True Shear Stress 247

6.2.4 Shear Heating 252

6.2.5 Extrudate Swell 254

6.2.6 Melt Index 256

6.3 Slit Rheometry 257

6.3.1 Normal Stresses 260

6.3.2 Exit Pressure 261

6.3.3 Pressure Hole 262

6.4 Other Pressure Rheometers 266

6.4.1 Axial Annular Flow 266

6.4.2 Tangential Annular Flow 267

6.4.3 Tilted Open Channel 268

6.4.4 Squeezing Flow 270

6.5 Comparison of Shear Methods 275

6.6 Summary 277

References 280

7 Extensional Rheometry 285
Christopher W. Macosko

7.1 Introduction 285

7.2 Simple Extension 288

7.2.1 End Chimps 291

7.2.2 Rotating Clamps 292

7.2.3 Buoyancy Baths 294

7.2.4 Spinning Drop 296

7.3 Lubricated Compression 297

7.3.1 Planar Squeezing 303

7.4 Sheet Stretching, Multiaxial Extension 303

7.4.1 Rotating Clamps 304

7.4.2 Inflation Methods 306

7.5 Fiber Spinning 308

7.5.7 Tubeless Siphon 315

7.6 Bubble Collapse 317

7.7 Stagnation Flows 320

7.7.1 Lubricated Dies 322

7.7.2 Unlubricated Dies 322

7.7.3 Opposed Nozzles 323

7.8 Entrance Flows 326

7.9 Summary 332

References 333

8 Rheometer Design 337
Christopher W. Macosko

8.1 Introduction 337

8.2 Drag Flow Rheometers 338

8.2.1 Controlled Strain 339

8.2.2 Torque Measurement 342

8.2.3 Normal Stresses 345

8.2.4 Alignment 347

8.2.5 Controlled Stress 349

8.2.6 Environmental Control 352

8.3 Data Analysis 357

8.3.1 Sinusoidal Oscillations 359

8.3.2 Transient 363

8.4 Pressure-Driven Rheometers 364

8.5 Extensional Rheometers 368

8.6 Process Line Rheometers 370

8.7 Summary 373

References 374

9 Rheo-Optics: Flow Birefringence 379
Timothy P. Lodge

9.1 Introduction 379

9.2 Review of Optical Phenomena 381

9.2.1 Absorption and Emission Spectroscopies 382

9.2.2 Scattering Techniques 382

9.2.3 Birefringence and Dichroism 384

9.3 Polarized Light 386

9.3.1 Transmission Through a Series of Optical Elements 390

9.4 Flow Birefringence: Principles and Practice 393

9.4.1 The Stress–Optical Relation 393

9.4.2 Range of Applicability of the Stress–Optical Relation 397

9.4.3 Geometries for Measuring Flow Birefringence 400

9.4.4 Birefringence in Steady and Transient Couette Flow 403

9.4.5 Birefringence in Oscillatory Shear Flow 405

9.4.6 Experimental Considerations 407

9.5 Flow Birefringence: Applications 408

9.5.1 Stress Field Visualization 408

9.5.2 Extensional Flow 409

9.5.3 Dynamics of Isolated, Flexible Homopolymers 409

9.5.4 Dynamics of Isolated Block Copolymers 412

9.5.5 Dynamics of Block Copolymer Melts 415

9.5.6 Dynamics of a Binary Blend 415

9.5.7 Birefringence in Transient Flows 416

9.5.8 Rheo-Optics of Suspensions 416

9.5.9 Rotational Dynamics of Rigid Rods 417

9.6 Summary 419

References 419

Part III. Applications 423

10 Suspension Rheology 425
Jan Mewis and Christopher W. Macosko

10.1 Introduction 425

10.2 Dilute Suspensions of Spheres 428

10.2.1 Hard Spheres 428

10.2.2 Particle Migration 430

10.2.3 Emulsions 434

10.2.4 Deformable Spheres 437

10.3 Particle–Fluid Interactions: Dilute Spheroids 439

10.3.1 Orientation Distribution 440

10.3.2 Constitutive Relations for Spheroids 443

10.4 Particle‒Particle Interactions 449

10.4.1 Dispersion Forces 450

10.4.2 Electrostatic Forces 451

10.4.3 Polymeric (Steric) Forces 452

10.4.4 Scaling 454

10.5 Brownian Hard Particles 455

10.5.1 Monodisperse Hard Spheres 455

10.5.2 Particle Size Distribution 458

10.5.3 Nonspherical Particles 459

10.5.4 Non-Newtonian Media 460

10.5.5 Extensional Flow of Ellipsoids 460

10.6 Stable Colloidal Suspensions 461

10.6.1 Electrostatic Stabilization 462

10 6.2 Polymeric (Steric) Stabilization 464

10.7 Flocculated Systems 465

10.7.1 Structure in Flocculated Dispersions 465

10.7.2 Static Properties 467

10.7.3 Flow Behavior 468

10.8 Summary 470

References 471

11 Rheology of Polymeric Liquids 475
Matthew Tirrell

11.1 Introduction 475

11.2 Polymer Chain Conformation 476

11.3 Zero Shear Viscosity 479

11.3.1 Dilute Solution 479

11.3.2 Nondilute Polymeric Liquids 489

11.3.3 Coil Overlap 482

11.4 Rheology of Dilute Polymer Solutions 487

11.4.1 Elastic Dumbbell 487

11 4.2 Rouse and Other Multihead Models 495

11.5 Concentrated Solutions and Melts 497

11.5.1 Entanglements 497

11.5.2 Reptation Model 502

11.5.3 Effects of Long Chain Branching 505

11.5.4 Effect of Molecular Weight Distribution 506

11.6 Temperature Dependence 510

11.7 Summary 512

References 512

Appendix Solutions to Exercises

Chapter 1 515

Chapter 2 521

Chapter 3 527

Chapter 4 531

Index 535

About the Author

Christopher W. Macosko is the author of Rheology: Principles, Measurements, and Applications, published by Wiley.