Jamming as a Multicritical Point

The discontinuous jump in the bulk modulus $B$ at the jamming transition is a consequence of the formation of a critical contact network of spheres that resists compression. We introduce lattice models with underlying undercoordinated compression-resistant spring lattices to which next-nearest-neighbor springs can be added. In these models, the jamming transition emerges as a kind of multicritical point terminating a line of rigidity-percolation transitions. Replacing the undercoordinated lattices with the critical network at jamming yields a faithful description of jamming and its relation to rigidity percolation.

Figure 1

(a) Three-sublattice model showing NN (solid lines) and NNN (dashed and dotted lines) bonds, the latter of which connect sites in either of the triangular sublattices containing the first (black) or second (open) sites of the honeycomb lattice. (b) 3D phase diagram showing the surfaces $S_{FRabc}$ (blue), $S_{RbRabc}$ (green), $S_{RcRabc}$ (khaki), $S_{FRb}$ (dark green), and $S_{FRc}$ (dark khaki) and the jamming line $L_J$ (red); (c)–(e) 2D slices of the 3D diagram at (c) constant $2/3<p_a<1$, (d) constant $0<p_a<2/3$, and (e) both $p_c=1/6$ at $p_b=p_c$, showing the $F$, $R_b$, $R_c$, and $R_{abc}$ phases. $U{U}^{\prime }$ is the jamming line when $p_a=1$ and an RP line when $p_a<1$. $X{X}^{\prime }$, $XY$, $X^{\prime }{Y}^{\prime }$, $JX$, and $JY$ are RP lines, and $XZ$ marks the transition from the $R_b$ to the $R_{abc}$ phase. In (e), the region to the right of $JX$ is the $R_{abc}$ phase when $p_a=p_b$, and the line $JX$ is not a part of the figure when $p_c=1/6$. CJD is a jamming path with a discontinuous jump in $B$.

Figure 2

Left: Simulations (points) and EMT solutions (surfaces) for $B$ (yellow) and $G$ (blue) as a function of $p_a$ and $p_b$ for $p_c=0$. Red and green lines correspond to $JX$ and $XY$ in Fig. 1, respectively. Right: $B$ and $G$ (inset) as a function of $p_b$ from EMT (lines) and simulations (circles) for $p_c=0$ and $p_a=1$ (filled circles) and $p_a=0.7$ (open circles).

Figure 3

(a) $k_b/|\mathrm{\Delta }\tilde{p}|$ as a function of $\gamma \omega /|\mathrm{\Delta }\tilde{p}{|}^2$ in the low-frequency limit $w=i\gamma \omega$. Blue (red) circles: Numerical solutions to full EMT equations for approach to jamming in the rigid (floppy) phase; black dashed line: asymptotic solutions [Eq. (9)] near jamming critical point; hollow circles: $\mathrm{Re}{k}_b/|\mathrm{\Delta }\tilde{p}|$; filled circles: $-\mathrm{Im}{k}_b/|\mathrm{\Delta }\tilde{p}|$, which is independent of the sign of $|\mathrm{\Delta }\tilde{p}|$. Inset: $k_a$ as a function of $\gamma \omega /(\mathrm{\Delta }{p}_a^J|\mathrm{\Delta }\tilde{p}|)$. (b) Density of states $\rho (\omega )$ for $p_a=1$ and $\mathrm{\Delta }\tilde{p}=\mathrm{\Delta }{p}_b^J={10}^{-2}$ (solid lines), ${10}^{-3}$ (dashed line), ${10}^{-4}$ (dot-dashed line), and ${10}^{-5}$ (dotted line). Inset: Linear behavior of crossover frequency ${\omega }^{*}(\mathrm{\Delta }{p}_b^J)$.