Polymer Melt Rheology and Flow Birefringence

The present monograph is intended as an introduction into a field which certainly did not receive proper attention in the past. It is one of the aims of this book to verify this suppo­ sition. The author hopes to show that the technique of the measurement of flow birefringence can fulfil an important com­ plementary task in polymer melt rheology. From this point it is expected that the present monograph will attract the atten­ tion of polymer scientists in general, and of rheologists and process engineers in particular. Certainly, the fourth chapter will appeal to the latter group. As a teacher in polymer science and technology the author wants to address also the group of the graduate students. In fact, the standard knowledge acquired during usual university studies in chemistry, physics or engineering does not enable a quick start of research activities in the field of polymer melt rheology. Certainly, in this typically interdisciplinary field everyone can lay emphasis on matters which are familar to hirn because of his preceding education. Significant research activities, however, can only be generated on the basis of a more universal knowledge. In the absence of this knowledge beginners have to rely upon the guidance of their supervisors for an unduly long period. Otherwise they take the risk of losing too much of their costly time. This holds in particular for the experimentalists who cannot be dispensed from being familiar with the necessary theoretical background.

1 Survey of Experimental Results.- 1.1 Preparatory Considerations.- 1.1.1 Introductory Remarks.- 1.1.2 Some Theoretical Basic Concepts.- 1.1.3 Short Review of Mechanical Techniques of Measurements.- 1.1.3.1 Apparatus for the Measurement of Dynamic Shear Moduli.- 1.1.3.2 The Cone-and-Plate Rheometer (Rheogoniometer).- 1.1.3.3 Capillary and Slit Rheometers.- 1.1.3.4 Tensile Rheometers.- 1.1.4 Some Illustrative Results of Mechanical Measurements.- 1.1.4.1 Dynamic Shear Moduli as Functions of the Circular Frequency.- 1.1.4.2 Prescribed Strain Histories.- 1.1.4.3 Prescribed Stress.- 1.1.4.4 Influence of Molecular Parameters.- 1.2 Flow Birefringence Measurements.- 1.2.1 Principles of Measurements.- 1.2.2 A Survey of Flow Birefringence Techniques Suitable for Polymer Melts.- 1.2.2.1 Arrangements for the Measurement of Flow Birefringence in Elongation.- 1.2.2.2 Apparatusses for Birefringence Measurements in the 1, 2-Plane of the Shear Flow.- 1.2.2.3 Apparatusses Derived from Slit and Capillary Rheometers.- 1.2.3 Results of Flow Birefringences Measurements.- 1.2.3.1 The Linear Stress-Optical Rule and its Limits of Validity.- 1.2.3.2 Further Evidence for the Validity of the Linear Stress-Optical Rule.- 1.2.3.3 Zero Shear Properties and Molecular Mass: Evaluation of Cone-and-Plate Measurements.- 1.2.3.4 Evaluation of Slit Rheometer Measurements.- 1.2.3.5 Time-Temperature Superposition and Non-Linear (Equilibrium) Properties.- 1.2.3.6 Further Comments on the Influence of Molecular Mass (Distribution).- 1.2.3.7 Measurement of the Optical Equivalent to the Second Normal Stress Difference.- 1.2.3.8 Transient Flow Birefringence Effects.- References of Chapter 1.- 2 Quasi-Molecular Phenomenological Theories.- 2.1 Rubber Elasticity.- 2.1.1 Results of the Kinetic Theory.- 2.1.1 Behaviour of Real Rubbers.- 2.2 The Linear Stress-Optical Relation.- 2.2.1 Results of the Kinetic Theory of Flexible Cain Molecules.- 2.2.2 Can the Linear Stress-Optical Rule be Understood also without the Assumption of a Temporary Network Structure?.- 2.3 The Most Simple Rubberlike Liquid Model.- 2.3.1 Introduction.- 2.3.2 Formulation of the Theory.- 2.3.2.1 Application to Several Types of Shear Flow.- 2.3.2.2 Application to Elongational Flow.- 2.3.2.3 Problems with Inversion.- 2.3.3 Preliminary Discussion.- 2.3.4 Equivalent Formulations of the Discussed Theory.- 2.3.5 Incorporation of Temperature Changes into the Integral Equation.- 2.3.6 Optical Means to Prove the Validity of the Simple Rubberlike Liquid Theory.- 2.4 More General Discussion of Quasi-Linear Models.- 2.4.1 The Corotational Model.- 2.4.2 Codeformational Models.- 2.4.2.1 Introduction.- 2.4.2.2 Covariant and Contravariant Time Derivatives.- 2.4.2.3 Equations Equivalent to Lodge’s Original Formulation.- 2.4.3 The Independent Alignment Model.- 2.4.3.1 Formal Presentation.- 2.4.3.2 Some Physical Considerations.- 2.4.3.3 Evaluation.- 2.4.3.4 Shear Flow.- 2.4.3.5 Elongational Flow.- 2.4.3.6 Damping Functions.- 2.4.3.7 Postscript.- 2.5 Quasi-Linear Models Containing Adjustable Elements.- 2.5.1 Introduction.- 2.5.2 The n-Measure of Strain.- 2.5.3 The Experimental Damping Function.- 2.5.3.1 Introduction.- 2.5.3.2 Experimental Evidence in Shear.- 2.5.3.3 Non-Linear Shear Creep.- 2.5.3.4 How to Find the Damping Function for an Extension Experiment.- 2.5.3.5 Non-Linear Extensional Creep.- 2.5.4 The Irreversibility of the Disentanglement Process.- 2.5.4.1 Introduction.- 2.5.4.2 The Triple Step-Strain Experiment in Shear.- 2.5.4.3 Description of Recovery Experiments.- 2.5.5 A Differential Type Constitutive Equation Containing Time Dependent Structural Parameters.- 2.5.5.1 Introduction.- 2.5.5.2 Formulation of the Theory.- 2.5.5.3 Evaluation for Steady Homogeneous Shear Flow.- 2.5.5.4 Description of Other Deformations with Given Strain History.- 2.5.5.5 Creep and Recovery.- 2.5.5.6 Free Volume as a Structural Parameter.- 2.5.6 The Non-Affine Displacement of Temporary Network Junctions.- 2.5.6.1 Description of the Network Model.- 2.5.6.2 General Discussion.- 2.5.6.3 Some Comments with Respect to the Usefulness of the Theory.- 2.6 A Theory Derived from a Different Basic Principle.- 2.6.1 Model Considerations.- 2.6.2 Discussion of Results.- 2.6.2.1 Steady State and Relaxation.- 2.6.2.2 Recovery.- 2.6.2.3 Evaluation of Model Parameters.- 2.6.3 Comparison with Experiment.- References of Chapter 2.- 3 Prospects for Predictions on a Moleculat Basis.- 3.1 General Introduction.- 3.2 Relaxion Processes in a Polmer Melt.- 3.2.1 Description of the Model.- 3.2.2 Various Phases of Relaxation.- 3.2.2.1 The Ultra-Fast Phase A of Stress Relaxation.- 3.2.2.2 Equations for the Stress Tensor.- 3.2.2.3 The Second or “Equilibration” Phase (Phase B) of the Stress Relaxation.- 3.2.2.4 The Third or “Disengagement” Phase (Phase C) of the Stress Relaxation.- 3.2.3 Experimental Evidence.- 3.2.3.1 Relaxation after Application of a Step Shear Strain.- 3.2.3.2 Relaxation after Stretching.- 3.3 General Flow Behavior of Polymer Melts.- 3.3.1 The Constitutive Equation.- 3.3.2 The Linear Viscoelastic Properties of the Model.- 3.3.2.1 Monodisperse Samples.- 3.3.2.2 Polydisperse Samples.- 3.3.3 The Non-Linear Properties of the Model.- 3.3.4 A More Elaborate Model.- 3.3.5 Conclusions.- References of Chapter 3.- 4 Industrial Applications.- 4.1 Molecular Orientation.- 4.1.1 The Linear Theory of Strain Recovery after Partial Relaxation.- 4.1.2 Experimental Observations Related to Product Quality.- 4.2 Injection Moulding.- 4.2.1 Two types of Birefringence.- 4.2.2 Heat Transfer During Mould Filling.- 4.2.2.1 A Verbal Description.- 4.2.2.2 Boundary Layer Approach to Mould Filling.- 4.2.2.3 Comparison with Experiment and Discussion.- References of Chapter 4.- Appendix A: Linear Visco-Elasticity.- A.1 Hooke’s Law in Complex Notation.- A.2 Stress Relaxation and Creep.- A.3 Elastico-Viscous Flow and Recovery.- A.4 Some Comments with Respect to the Interconversion of Material Functions.- A.5 Mechanical Spectra.- A.6 Interconversion Formulae Based on the Relaxation Time Spectrum.- A.7 Calculation of the Relaxation Time Spectrum from Experimental Material Functions.- References of Appendix A.- Appendix B: The Time-Temperature Superposition Principle.- B.1 Introduction.- B.2 The WLF-Equation.- B.3 The Free Volume Concept.- B.4 Unified Description of the Temperature Denpendence of the Zero-Shear Viscosity.- References of Appendix B.- Appendix C: The Measurement of Birefringence Effects.- C.1 Matrix Representation of Optical Components.- C.2 Calculations of Intensities of Beams Emerging from the Analyser.- C.3 How to Cope with Imperfections of Windows and Polarizing Prisms.- C.4 Description of the Modulator.- C.5 Problems with Quickly Changing Birefringence Effects.- References of Appendix C.