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 $G^{\ast }(\omega )$ calculated. Two non-trivial scaling regimes were observed, (i)~$G^{\mathrm{\prime }}\approx G^{\mathrm{\prime }\mathrm{\prime }}\propto {\omega }^{\mathrm{\Delta }}$ at low frequencies, consistent with $\mathrm{\Delta }=1\mathrm{/}2$; (ii)~$G^{\prime}\propto G^{\prime\prime}\propto\omega^{\Delta^{\prime}}$ for intermediate frequencies corresponding to fractal structure, consistent with a theoretical prediction ${\mathrm{\Delta }}^{\mathrm{\prime }}=(\ln 3-\ln 2)\mathrm{/}(\ln 3+\ln 2)$. 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.