A Gaussian Slip-Link Model for Polymer Single and Double Networks

In this study, we developed Schieber's slip-link model for lightly cross-linked polymers assuming the equilibration of deformed Gaussian chains. Our simulation consists of two steps: preparation and deformation. In the preparation step, cross-links and slip-links are assumed to be distributed randomly along the chain, but with independent statistical parameters: the average number of Kuhn steps between entanglements, $N_{e}$, and the average number of Kuhn steps between cross-links, $N_{c}$. In the second step, the cross-links and slip-links are deformed affinely, but since the chain can slide through the slip-links, its deformation is non-affine. The stress tensor can be determined as a function of deformation using Brownian dynamics as a sort of Monte Carlo algorithm. The Mooney plot of our simulation result has good agreement with most experimental data for uniaxial elongation deformation for cross-linked NR, PDMS, and PBd. The model is used to predict values for the Mooney plot parameters ($C_{1}$ and $C_{2}$) as a function of $N_{e}$ and the $N_{c}/N_{e}$ ratio. The $C_{2}/C_{1}$ ratio is found to be strongly dependent on $ N_{c}/N_{e}$, but weakly dependent on $N_{e}$. This observation provides a new way of predicting the cross-link density and separating it from the entanglement density and for systems of known $N_{e}$ and $N_{c}$, the model requires no adjustable parameters. We are also developing our model for double network polymers in order to demonstrate different applications for the model.