Viscoelastic scaling regimes for marginally-rigid fractal spring networks

Abstract

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 non-trivial scaling regimes were observed, (i) G ≈ G ∝ ω ∆ at low frequencies, consistent with ∆ = 1/2; (ii) G ∝ G ∝ ω ∆ for intermediate frequencies corresponding to fractal structure, consistent with a theoretical prediction ∆ = (ln 3 − ln 2)/(ln 3 + ln 2). The cross-over between these two regimes occurred at lower frequencies for larger fractals in a manner suggesting diffusive-like dispersion. Solid gels generated by introducing internal stresses exhibited similar behaviour above a low-frequency cut-off, indicating the relevance of these findings to real-world applications.