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

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 $d_{\mathrm{f}}=\log 3\mathrm{/}\log 2$, 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.