Mathematical Modelling for Polymer Processing

Polymers are substances made of macromolecules formed by thousands of atoms organized in one (homopolymers) or more (copolymers) groups that repeat themselves to form linear or branched chains, or lattice structures. The concept of polymer traces back to the years 1920's and is one of the most significant ideas of last century. It has given great impulse to indus­ try but also to fundamental research, including life sciences. Macromolecules are made of sm all molecules known as monomers. The process that brings monomers into polymers is known as polymerization. A fundamental contri­ bution to the industrial production of polymers, particularly polypropylene and polyethylene, is due to the Nobel prize winners Giulio Natta and Karl Ziegler. The ideas of Ziegler and Natta date back to 1954, and the process has been improved continuously over the years, particularly concerning the design and shaping of the catalysts. Chapter 1 (due to A. Fasano ) is devoted to a review of some results concerning the modelling of the Ziegler- Natta polymerization. The specific ex am pie is the production of polypropilene. The process is extremely complex and all studies with relevant mathematical contents are fairly recent, and several problems are still open.

Contenu

Polymerization.- 1 Mathematical Models for Polymerization Processes of Ziegler-Natta Type.- 1.1 Introduction.- 1.2 Description of the Process.- 1.3 Governing Equations for the Macroscopic Processes (Heat and Mass Transfer).- 1.4 Governing Equation for the Microproblem (Polymerization).- 1.5 Computation of the Coupling Terms. Expansion of the Agglomerate.- 1.6 Rescaling and Simplifications.- 1.7 Summary of the Simplified Model and Sketch of the Proof.- 1.8 Fragmentation.- 1.9 Conclusions and Open Questions.- 1.10 Acknowledgements.- References.- Nucleation.- 2 Classical Kinetic Theory of Nucleation and Coarsening.- 2.1 Introduction.- 2.2 Kinetics of Clusters.- 2.3 Phase Equilibria of a Binary Material.- 2.4 Macroscopic Kinetics.- 2.5 Quasistatic Nuclei.- 2.6 Material and Energy Parameters of Kinetic Theory Determined from Xiao-Haasen Data.- 2.7 Discussion.- 2.8 Acknowledgements.- 2.9 Appendix: BD Kinetics for n < nc.- References.- 3 Multidimensional Theory of Crystal Nucleation.- 3.1 Introduction.- 3.2 One-dimensional Theory of Nucleation.- 3.3 Modifications of the Classical Theory.- 3.4 Multidimensional Theory of Nucleation.- 3.5 Some AppHcations.- 3.6 Closing Remarks.- References.- 4 Kinetic Theory of Nucleation in Polymers.- 4.1 Introduction.- 4.2 Nucleation Theory.- 4.3 Kinetic Equations from Non-Equilibrium Thermodynamics.- 4.4 Isothermal Homogeneous Nucleation.- 4.5 Nucleation in Spatially Inhomogeneous Non-Isothermal Conditions.- 4.6 Homogeneous Nucleation in Spatially Inhomogeneous Systems. Diffusion Regime.- 4.7 Nucleation in a Shear Flow.- 4.8 Kinetic Equations for the Crystallinity.- 4.9 Further Extensions.- 4.10 Discussion and Conclusions.- 4.11 Acknowledgments.- References.- Crystallization.- 5 Mathematical Models for Polymer Crystallization Processes.- 5.1 Stochastic Models of the Crystallization Process.- 5.2 The Causal Cone.- 5.3 The Hazard Function.- 5.4 The Random Tessellation of Space.- 5.5 Estimating the Local Density of Interfaces.- 5.6 Interaction with Latent Heat.- 5.7 Simulation of the Stochastic Model.- 5.8 Stochasticity of the Causal Cone.- 5.9 A Hybrid Model.- 5.10 Scaling Limits.- 5.11 Related Mathematical Problems.- References.- 6 Polymer Crystallization Processes via Many Particle Systems.- 6.1 Introduction.- 6.2 The Nucleation Process.- 6.3 Growth of Crystals.- 6.4 The Mass Distribution of Crystals.- 6.5 The Rates.- 6.6 The Time Evolution of the Processes ? N and Y N.- 6.7 Heuristic Derivation of a Continuum Dynamics.- References.- Manufacturing.- 7 Modelling of Industrial Processes for Polymer Melts: Extrusion and Injection Moulding.- 7.1 Introduction.- 7.2 Flow Instabihties in the Extrusion of Polymer Melts.- 7.3 Injection Moulding.- 7.4 Closing Remarks.- References.- Appendix. Color Plates.