Universal scaling for disordered viscoelastic matter near the onset of rigidity

The onset of rigidity in interacting liquids, as they undergo a transition to a disordered solid, is associated with a rearrangement of the low-frequency vibrational spectrum. In this Letter, we derive scaling forms for the singular dynamical response of disordered viscoelastic networks near both jamming and rigidity percolation. Using effective-medium theory, we extract critical exponents, invariant scaling combinations, and analytical formulas for universal scaling functions near these transitions. Our scaling forms describe the behavior in space and time near the various onsets of rigidity, for rigid and floppy phases and the crossover region, including diverging length scales and timescales at the transitions.

Figure 1

(a) Jammed disk packing, underlying contact network (B bonds in blue), and randomly added next-nearest-neighbor N bonds (red). (b) HTL model with nearest- and next-nearest-neighbor bonds (solid blue and red lines) connecting sites of a honeycomb lattice. Faint blue and red lines show underlying honeycomb and triangular lattices, respectively. (c) Phase diagram of the HTL model in terms of occupation probabilities for B and N bonds. The yellow region corresponds to the rigid state, and is separated from the floppy state by an RP line ending at a jamming point $J$ (red disk). (d) Conjecture for a crossover flow diagram projected into ${\delta }_{\text{RP}}\times {\delta }_{\text{J}}$ space. $J$ (red disk) and RP (black disk) represent fixed points of a putative renormalization-group scheme. The blue, black, and gray lines represent the unstable manifold, the critical line, and a sample trajectory, respectively.

Figure 2

Scaling collapse plots showing the universal behavior of the longitudinal response as a function of rescaled frequency near jamming (first row) and RP (second row), for overdamped dynamics. Blue disks and red triangles are full solutions of the EMT equations for the real and imaginary parts of $|{\delta }_{\text{RP}}{|}^{\gamma }{\chi }_{\text{L}}/{\chi }_0$, respectively. Solid and dashed curves are the universal scaling predictions of Eqs. (2) and (3). We consider points approaching jamming and RP along the paths indicated in the inset graphs of each panel. We use $q/|{\delta }_{\text{RP}}{|}^{\nu }=0.1$ (solid symbols) and 1 (open symbols) in all panels, and ${\delta }_{\text{J}}/{\left|{\delta }_{\text{RP}}\right|}^{\phi }$ equal to $\sqrt{5}/4$ from the rigid side (a), and equal to 2 from the floppy side (b). Full solutions run at $|{\delta }_{\text{RP}}|={10}^{-2}$, ${10}^{-3}$, and ${10}^{-4}$ for RP and a range $|{\delta }_{\text{RP}}|\in [5\times {10}^{-2},5\times {10}^{-6}]$ for jamming show convergence to our universal asymptotic predictions.

Figure 3

Overdamped asymptotic exponents for universal longitudinal response. Diagram in the $u\left(\text{rescaled}\text{wave}\text{vector}\right)\times v\left(\text{rescaled}\text{frequency}\right)$ plane, showing regions of distinct power-law behavior of the jamming (first row) and RP (second row) universal scaling functions for overdamped dynamics in both the rigid and floppy phases. The first and second (third and fourth) columns correspond to the real (imaginary) parts of $\mathcal{L}$ and $\overline{\mathcal{L}}$. We use $w=1$ for jamming.

Figure 4

(a) 3D plot of the universal scaling function for the correlation function $\overline{\mathcal{S}}(u,v)$, for undamped fluids near RP. The blue dashed line corresponds to the rescaled frequency $v^{*}$ (the boson peak) at which $\overline{\mathcal{S}}(u,v)$ is maximum for fixed rescaled wave vector $u$. (b) $u\times v$ diagram showing the boson peak (blue dashed line) and power-law regions for which $\mathcal{S}(u,v)\propto u^{\alpha }{v}^{\beta }$, with $(\alpha ,\beta )$ close to their asymptotic values $(0,-2)$ (red) and $(4,-4)$ (yellow). In the blue region the condition $\overline{\mathcal{S}}(u,v)>\overline{\mathcal{S}}(u,v^{*})/2$ is satisfied.