Assignment
The simplest model of rubbers assumes: cross-links were created from polymer chains in equilibrium; polymer sections between all cross links have same number of units, N; affine deformation; no intrachain or interchain interactions. Assume incompressibility throughout.
(i) A rubber is synthesized using a radiation process resulting in very uniformly spaced cross links: the cross-link separation is always b. The relative location of pairs of neighboring cross-links is random and isotropically distributed.
Repeat the simple model calculation for this rubber: calculate the uniaxial stress-strain relation for general stretch factor lambda and determine the shear modulus G. Discuss how the material properties are affected. Is this desirable?
(ii) Another rubber material is synthesized under extruding flow conditions such that the separation of neighboring crosslinks is highly biased in one direction (x direction). Assume that all chain end to end separations lie in the x direction and equal b.
Calculate Young's modulus for uniaxial extension in the x direction, the y direction, and the z direction. Discuss the physical nature of this material.
(iii) Return to the simple model (assume chains crosslinked in equilibrium). In reality there is a molecular weight distribution P(N) of chain segment lengths between cross links. Recalculate G. How does this change things?
(iv) Discuss (don't try to calculate) how including the following might change the final rubber material properties such as G: (1) intrachain energy interactions,
(2) interchain energy interactions and topological constraints (knots).