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

5 pages, 3 figures. Expanded text and 3 new SI figures. To appear in PRL

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