Two-Scale Scenario of Rigidity Percolation of Sticky Particles

In the presence of attraction, the jamming transition of packings of frictionless particles corresponds to the rigidity percolation. When the range of attraction is long, the distribution of the size of rigid clusters, $P(s)$, is continuous and shows a power-law decay. For systems with short-range attractions, however, $P(s)$ appears discontinuous. There is a power-law decay for small cluster sizes, followed by a low probability gap and a peak near the system size. We find that this appearing “discontinuity” does not mean that the transition is discontinuous. In fact, it signifies the coexistence of two distinct length scales, associated with the largest cluster and smaller ones, respectively. The comparison between the largest and second largest clusters indicates that their growth rates with system size are rather different. However, both cluster sizes tend to diverge in the large system size limit, suggesting that the jamming transition of systems with short-range attractions is still continuous. In the framework of the two-scale scenario, we also derive a generalized hyperscaling relation. With robust evidence, our work challenges the former single-scale view of the rigidity percolation.

Figure 1

Finite size scaling of the probability of finding jammed states with a percolating rigid cluster, $f_p$, as a function of packing fraction $\varphi$. For $\mu ={10}^{-1}$ and ${10}^{-3}$, the scaling exponents $\nu$ used here are 2.30 and 1.08, respectively. ${\varphi }_c$ is the critical packing fraction of the jamming transition in the thermodynamic limit.

Figure 2

System size evolution of the distribution of the size of rigid clusters, $P(s)$, calculated at $f_p\approx 0.5$. The inset of (b) shows the distributions with the largest cluster being removed. The solid and dashed lines have a slope of $-\tau$ and $-{\tau }_m$, respectively. In (a), we plot the lines with $\tau ={\tau }_m=2.05$. In (b), $\tau =4.7$ and ${\tau }_m=2.1$.

Figure 3

(a) Gyration radius $R_g$ versus cluster size $s$. The top and bottom branches are for $\mu ={10}^{-1}$ and ${10}^{-3}$, respectively. To distinguish the two branches, we show $R_g/2$ for $\mu ={10}^{-3}$. The solid and dashed lines have a slope of $1/d_f$ with $d_f=1.781$ and 1.953, respectively. (b) System size dependence of the distribution of the gyration radius $P(R_g)$. The legend in (a) applies to both (a) and (b). (c) Average gyration radius $⟨R_g⟩$ of the largest (circles) and second largest (squares) clusters versus the side length of the system $L\sim N^{1/2}$. The empty and solid symbols are for $\mu ={10}^{-1}$ and ${10}^{-3}$, respectively. The dashed lines show the predicted scaling relation $R_g\sim L^{\kappa }$, as discussed in the text, with $\kappa$ being calculated from the generalized hyperscaling relation using values of ${\tau }_m$, $\tau$, and $d_f$. The dashed lines shown here are with ${\kappa }_1=1.07$ and ${\kappa }_2=1.07$ for the largest and second largest clusters when $\mu ={10}^{-1}$, and ${\kappa }_1=0.93$ and ${\kappa }_2=0.28$ when $\mu ={10}^{-3}$. (d) Dependence of $\tau$ on the range of attraction $\mu$. The horizontal dashed line is $\tau =2.05$, showing the one-scale limit. The dot-dashed line is a logarithmic fit to $\tau (\mu )$ for small values of $\mu$.