Chapter 9: Cross-linked polymers and rubber elasticity

1. Chapter 9: Cross-linked polymers and rubber elasticity

6. Natural rubber: cis-isoprene Before crosslinking (as obtained from the tree) - Linear , - Mn ~ 106 g/mol - amorphous (no crystallinity), Glass transition Tg–70 oC  not elastic as such (no recovery after streching) The rubber elastic state Cross-linking with sulphur: Now the chains cannot flow past each others upon stretching  Elastic recovery and it does not melt, flow or dissolve in solvents Also physical cross-linking… Crystalline domains, glassy domains (SBS), entanglements…

7. Extremly high extensibility – up to 10 times the original length • Complete recovery after mechanical deformation General properties of rubbers: • Under the constant load the stretched length decreases on heating and increases on cooling • Rubber warms on streching and cools on being allowed to contract

8. Definitions: Stress-Strain curves

9. typical for crystalline solids – example steel wire can be streched reversibly about 1% extension. Near the equlibrium point the potential Elastic force: Stress is Hooke’s law Energy driven elasticity: Due to strong enthalpic contribution – Entropy is a minor factor

10. Stretched: Entropy S low Coil: Entropy S high Entropy driven elasticity - Rubbers Individual polymer chains tends to retract into the coiled conformation after streching – Entropy driven springs. If the chains are connected (cross-linked) the elastic behavior manifests also to the bulk material. Stress Mc is the molecular weight between the crosslinks

11. Thermoelastic behavior and thermodynamics: energetic and entropic elastic forces

12. Make now a thermodynamic model for rubber. We need to consider the free energies. A suitable free energy depends on the surrounding conditions. Now it is feasible to assumme that pressure p and T are constant.  Gibbs free energy where H is Enthalphy and S is Entropy. E is the internal energy, p pressure and V volume Force Volume: V0 = A0L0 V= AL Ideal rubber - constant volume: V0=V  A = (L0/L)A0 = A0/l

13. Energetic and Entropic components of the elastic force

16. Statistical mechanical theory of rubber elasticity

23. Affine vs. Phantom network model Affine network model: Stress-Stain relation Phantom network model: Stress-Stain relation Both theories however fail to predict stress-strain behavior at large strains (e.g. l > 4 for natural rubber)

24. Realistic materials and corrections to the ideal model Network structure and defects: a) Elastically aqctice chain, b) loop, and c) dangling chain end. Mc = molecular weight between the cross-links M = total molecular weight before the cross-link formation Ideal network no ”loose ends”

26. Non-Gaussian chains (large stretching) Ideal rubber – Gaussian chains Deviation at large strains due to non-gaussian effects and due to strain induced crystallization

30. Example 1 First law of Thermodynamics

31. Example 2 Specimen of example 1: initial lenght is L0 = 10 cm, what is the force required to double its lenght

32. Example 3