Entropic Elasticity of Two-Dimensional Self-Avoiding Percolation Systems

The sol-gel transition is studied on a purely entropic two-dimensional model system consisting of hard spheres (disks) in which a fraction $p$ of neighbors are tethered by inextensible bonds. We use a new method to measure directly the elastic properties of the system. We find that over a broad range of hard sphere diameters $a$ the rigidity threshold is insensitive to $a$ and indistinguishable from the percolation threshold $p_c$. Close to $p_c$, the shear modulus behaves as $({p-p}_c{)}^f$, where the exponent $f\simeq 1.3$ is independent of $a$ and is similar to the conductivity exponent in random resistor networks.