Force percolation transition of jammed granular systems

The mechanical and transport properties of jammed materials originate from an underlying percolating network of contact forces between the grains. Using extensive simulations we investigate the force-percolation transition of this network, where two particles are considered as linked if their interparticle force overcomes a threshold. We show that this transition belongs to the random percolation universality class, thus ruling out the existence of long-range correlations between the forces. Through a combined size and pressure scaling for the percolative quantities, we show that the continuous force percolation transition evolves into the discontinuous jamming transition in the zero pressure limit, as the size of the critical region scales with the pressure.

Figure 1

A percolative analysis of jammed configurations is introduced by considering particles as connected if interacting with a force whose magnitude is greater than a threshold $f_t$. This schematic phase diagram illustrates the existence of a continuous percolation line that ends at the jamming volume fraction ${\mathrm{\Phi }}_J$, where the percolation transition becomes discontinuous. The panels illustrate the percolative analysis of a $N={10}^4$ particle system across the transition. Lines connect particles belonging to the same cluster. For clarity, not all of clusters are shown.

Figure 2

Critical exponents of the force percolation transition: (a) Percolation probability as a function of force threshold $f_t$, fitted to sigmoidal function $f\left(x\right)$ (see text), (b) Inverse percolation transition width ${\mathrm{\Delta }}^{-1}\left(L\right)$ scales as $L^{1/\nu }$, [(c) and (d)] largest cluster size $C_1$ and mean cluster size $S$ at critical transition $f_c$ scale as $L^{-\beta /\nu }$ and $L^{\gamma /\nu }$, respectively. Performing linear fits for the largest system sizes ($L>100$), we estimate $\nu =1.32\left(2\right), \beta =0.147\left(7\right)$, and $\gamma =2.32\left(6\right)$. Data are for $p=5\times {10}^{-3}$.

Figure 3

Random percolation universality class of force percolation transition. Panels (a)–(d) show the finite-size scaling of the fraction of particles in the largest cluster $C_1$ and mean cluster size $S$ using Eq. (1), and exponents determined in Fig. 2. Panels (e) and (f) show the normalized cluster size distribution $n\left(s\right)$, and the pair connected correlation function $g_{\text{pc}}\left(r\right)$, respectively, at the percolation threshold for the $N=4\times {10}^4$ system. Exponents are determined to be $\tau =2.07\left(4\right)$, and $\eta =0.208\left(6\right)$. Data are for $p=5\times {10}^{-3}$.

Figure 4

Pressure dependence of the force percolation threshold, $f_c$, and of the average force, $\langle f\rangle$. These two forces are proportional, and scale as $f_c={\alpha }_c{p}^q$ and as $\langle f\rangle =\alpha p^q$, with $q\approx 0.98\left(1\right), {\alpha }_c=1.00\left(4\right)$, and $\alpha =0.67\left(3\right)$.

Figure 5

Dependence of the size of the largest cluster, $C_1$, on $f-f_c$, for different values of pressure (a), and pressure dependence of its derivative at the inflection point (b). The derivative diverges as $p^{-q}$ in the $p\to 0$ limit. Panel (c) clarifies that $C_1$ scales as $N^{-\beta /d\nu }$, for all values of the pressure, ruling out the presence of a discontinuous transition in the thermodynamic limit, for $p\to 0$. Panel (d) shows that the $C_1$ data of (a) collapse when plotted versus $(f-f_c)/p^q$, thus clarifying that the size of the critical region scales as $p^q$. In panels (a), (b), and (d), $N=2\times {10}^4$.

Figure 6

Combined pressure and size scaling of the largest cluster size $C_1$ (left panel) and of the mean cluster size $S$ (right panel). The presented data are for three system size $N=4\times {10}^3,1\times {10}^4,\text{and}2\times {10}^4$, each corresponding to three pressure values $p=5\times {10}^{-4},5\times {10}^{-5},\text{and}5\times {10}^{-6}$. We have fix $q\simeq 0.98$.

Figure 7

Same as Fig. 3, but for Hertzian potential ($\alpha =2.5$). The value of the pressure is $p=0.005$.

Figure 8

Same as Fig. 3, for a system of Harmonic spheres in three dimensions. The value of the pressure is $p=0.005$.