Simulation

We lay the trifunctional network down on an hexagonal lattice in a two-dimensional box. The initial configuration is an equilibrium, isotropic one in which the ellipses are all circles. For this case, the spring equilibrium lengths are all $r_0$, and the equilibrium angles are those of the hexagon with $\theta_0=120^{\circ}$.

This corresponds to the physical case of an annealed nematic elastomer for which any orientational preference has been removed. A deformation is applied to the top and bottom boundaries of the box with the goal being to see how the whole network responds to the imposed deformation. Using the assumed potentials, a Metropolis Monte Carlo scheme is implemented in order to find this response for various degress of extension. Except for the cross-link points on the upper and lower boundaries, which are constrained to remain there, all other points are free to move in the domain. The angular orientations of springs (and, hence, ellipses) are kept track of by a binning procedure using thirty-six equally spaced bins for the interval 0 to $\pi$. After each sweep, the angular deviation from the horizontal for each spring is calculated and binned accordingly, so that a distribution of orientations is obtained. Also, average asphericity and average width (perpendicular to direction of stretch) of the domain are computed after each sweep. These quantities are monitored to determine when the system has equilibrated. After equilibration, sufficient additional steps are run to acquire adequate statistics.

A characterstic of rubber materials is their tendency to deform in a volume-preserving (or, in two dimensions, area-preserving) fashion (DeSimone and Dolzmann 2000). Rather than imposing such a deformation to the network a priori, we decided instead to see whether the system responded to the imposed elongation in an area-preserving manner on its own. Since the new height of the box is known from the imposed extension, monitoring the average width of the box directly leads to knowledge of the new domain area. Thus, this serves as a check on how well our model represents reality.