Distinct viscoelastic scaling for isostatic spring networks of the same fractal dimension

Abstract

Fractal structure emerges spontaneously from the chemical cross\-linking of monomers into hydrogels, and has been directly linked to power law visco\-elasticity at the gel transition, as recently demonstrated for isostatic (marginally--rigid) spring networks based on the Sierpinski triangle. Here we generalize the Sierpinski triangle generation rules to produce 4 fractals, all with the same dimension $d_{\rm f}=\log 3/\log 2$, with the Sierpinski triangle being one case. We show that spring networks derived from these fractals are all isostatic, but exhibit one of two distinct exponents for their power--law viscoelasticity. We conclude that, even for networks with fixed connectivity, power--law viscoelasticity cannot generally be a function of the fractal dimension alone.