Viscoelastic scaling regimes for marginally-rigid fractal spring networks

A family of marginally rigid (isostatic) spring networks with fractal structure up to a controllable length was devised, and the viscoelastic spectra G∗(ω) calculated. Two nontrivial scaling regimes were observed, (i) G′≈G′′∝ωΔ at low frequencies, consistent with Δ=1/2, and (ii) G′∝G′′∝ωΔ′ for intermediate frequencies corresponding to fractal structure, consistent with a theoretical prediction Δ′=(ln3−ln2)/(ln3+ln2). The crossover between these two regimes occurred at lower frequencies for larger fractals in a manner suggesting diffusivelike dispersion. Solid gels generated by introducing internal stresses exhibited similar behavior above a low-frequency cutoff, indicating the relevance of these findings to real-world applications.