Scaling of Entropic Shear Rigidity

The scaling of shear modulus near the gelation-vulcanization transition is explored heuristically and analytically. It is found that in a dense melt the effective chains of the infinite cluster have sizes that scale sublinearly with their contour length. Consequently, each chain contributes $k_{\mathrm{B}}T$ to the rigidity, which leads to a shear-modulus exponent $d\nu$. In contrast, in phantom elastic networks the scaling is linear in the contour length, yielding an exponent identical to that of the random resistor network conductivity, as predicted by de Gennes. For nondense systems, the exponent should cross over to $d\nu$ when the percolation correlation length is much larger than the density-fluctuation length.

Figure 1

The infinite cluster is a network of effective chains: (a) In a dense melt each effective chain is one single thermal blob with typical size ${\xi }_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}}$, and contributes $k_{\mathrm{B}}T$ to the shear rigidity. (b) In a phantom network each chain is sequence of “tension blobs” (dashed circles), each of size ${\xi }_{\mathrm{t}\mathrm{e}\mathrm{n}}$. Each tension blob contributes $k_{\mathrm{B}}T$ to the shear rigidity.

Figure 2

Shear deformation (full vs dotted lines) is applied to measurement but not preparation replicas. After the coordinate transformation, the original boundaries are recovered at the cost of introducing a nontrivial metric tensor.