## Elasticity

Paper: "Strain-Dependent Localization, Microscopic Deformations, and Macroscopic Normal Tensions in Model Polymer Networks" Physical Review Letters 93, 257801 (2004). Authors: Carsten Svaneborg, Gary S. Grest, and Ralf Everaers.

We use molecular dynamics simulations to investigate the microscopic and macroscopic response of model polymer networks to uniaxial elongations. By studying networks with strand lengths ranging from Ns 20 to 200 we cover the full crossover from cross-link to entanglement dominated behavior. Our results support a recent version of the tube model which accounts for the different strain dependence of chain localization due to chemical cross-links and entanglements.

The left plot shows the effective square deformation (vertical axis) vs. chemical distance along the polymer (horizontal axis) for several macroscopic deformations (elongation factor 1.5, 2, 3, 4). At large length scales the observed deformation matches that imposed at macroscopic scales which is consistent with affine deformation response, at very short length scales the chain displays random walk statistics independent of the macroscopic deformation. This is a result of the large conformational degree of freedom of polymer chains. The cross-over length scale between these two regimes is given by the (deformation dependent) tube diameter. The estimated tube diameters are shown in the right plot. The red symbols denote the tube confinement due to cross-links, while the green symbols denote the tube due to entanglements. Finally the blue symbols denote the effective tube diameter. For network with long strands the effective tube is dominated by the entanglement contribution, while for progressively stronger cross-linked systems the cross-link contribution becomes more and more important. The entanglement tube is also seen to depend on cross-linking suggesting an influence of strands that are trapped by cross-links.

Paper: "Disorder effects on the strain response of model polymer networks" Polymer 46, 4283 (2005)

Molecular dynamics simulations are used to investigate polymer networks made by either end-linking or randomly crosslinking a melt of linear precursor chains. The resulting network structures are very different, since end-linking leads to nearly ideal monodisperse networks, while random crosslinking leads to polydisperse networks, characterized by an exponential strand length distribution. Networks with average strand length 20 and 100 were generated. These networks were used to study the effects of disorder in the network connectivity on observables averaged either over the entire network or selected sub-structures. Heterogeneities in the randomly crosslinked networks cause significant differences in the localization of monomers, however, neither the localization of crosslinks nor the microscopic strain response are significantly affected. Compared to end-linked networks, randomly crosslinked networks have a slightly increased tube diameter, and as a result a slightly decreased shear modulus, but otherwise identical stress–strain behavior. For the investigated systems, we conclude that the microscopic strain response, tube diameter, and stress–strain relation are all insensitive to the heterogeneities due to the linking process by which the network were made.

The figure shows a visualization of the thermal fluctuations of three chains in a network (top left), and the fluctuations of the same three chains in the network when deformed lambda=2 along the x axis (bottom). Finally the affine back transformed representation of the bottom graph is shown (top right)