Entropic elasticity at the sol-gel transition

The sol-gel transition is studied in two purely entropic models consisting of hard spheres in continuous three-dimensional space, with a fraction p of nearest-neighbor spheres tethered by inextensible bonds. When all the tethers are present (p = 1) the two systems have connectivities of simple cubic and face-centered cubic lattices. For all p above the percolation threshold p_c, the elasticity has a cubic symmetry characterized by two distinct shear moduli. When p approaches P_c, both shear moduli decay as (p - p_c)^f, where f ≃ 2 for each type of connectivity. This result is similar to the behavior of the conductivity in random resistor networks, and is consistent with many experimental studies of gel elasticity. The difference between the shear moduli that measures the deviation from isotropy decays as (p - p_c)^h, with h ≃ 4.