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

Figure 1

(a) Schematic of network generation for system size $2^n\times 2^n$ and maximum fractal length $2^m$. Upward (dark) and downward (light) oriented triangles spanning a periodic box were subdivided into four subtriangles (left). Each subtriangle was then recursively subdivided into three triangles each over $m=3$ further iterations, as per standard Sierpinksi triangles (right). (b) Examples of networks for $n=4$ and $1\le m\le 4$. Dots denote nodes, and straight lines denote springs. The maximum fractal extent is indicated by the shaded triangles, and the origin has been shifted so all nodes are clearly visible. Arrows indicate the two nodes for $m=n$ with $z=2$. Images for $n=6$ are provided in Ref. [34].

Figure 2

Viscoelastic spectra for networks of $2^n\times 2^n$ nodes with $n=10$, with fractal structure up to a length $\propto 2^m$ with $m$ given in the legend. Solid lines denote $G^{\prime }(\omega )$ scaled to the spring constant $k$, and dashed lines denote the network contribution (i.e., less the solvent contribution $\eta \omega$ for viscosity $\eta$) to $G^{\prime \prime }(\omega )$. Thin horizontal lines give the affine prediction, $G_{\mathrm{aff}}^{\prime }=(\sqrt{3}/2)k(3^m-2^m/4^m)$. The same data is replotted against $\omega /{\omega }_f$ with ${\omega }_f\zeta /k=(2^m{)}^{-2}$ in the lower panel, with fits to the intermediate scaling regime shown. In both panels, thick line segments have the annotated slope.

Figure 3

(a) Viscoelastic spectra for fractal length $\propto 2^m$ with $m=7$, varying the magnitude of internal stress $\mathrm{\Lambda }$. The upper set of curves (solid) are $G^{\prime }$, and the lower (dashed) are the network contribution to $G^{\prime \prime }$. Selected values of $\mathrm{\Lambda }$ are given in the legend; curves for other $\mathrm{\Lambda }$ interpolate these. (b) The same data replotted with $G^{*}$ scaled by $G_0=G^{\prime }(\omega =0)$ and $\omega$ scaled by ${\omega }_{\mathrm{\Lambda }}$. (c) and (d) give the corresponding values of $G_0$ and ${\omega }_{\mathrm{\Lambda }}$ as functions of $\mathrm{\Lambda }$, increasing incrementally from $m=3$ (top curves) to $m=8$ (bottom curves). For all figures, the solid line segments have the annotated slopes.