Scaling Theory for the Frictionless Unjamming Transition

We develop a scaling theory of the unjamming transition of soft frictionless disks in two dimensions by defining local areas, which can be uniquely assigned to each contact. These serve to define local order parameters, whose distribution exhibits divergences as the unjamming transition is approached. We derive scaling forms for these divergences from a mean-field approach that treats the local areas as noninteracting entities, and demonstrate that these results agree remarkably well with numerical simulations. We find that the asymptotic behavior of the scaling functions arises from the geometrical structure of the packing while the overall scaling with the compression energy depends on the force law. We use the scaling forms of the distributions to determine the scaling of the total grain area $A_G$ and the total number of contacts $N_C$.

Figure 1

A section of a jammed configuration of soft frictionless disks. The centers of the grains with radii $\{{\sigma }_g\}$ are located at positions $\{{\overrightarrow{r}}_g\}$. The contact points between grains are located at positions $\{{\overrightarrow{r}}_c\}$, with contact vectors ${\overrightarrow{r}}_{g,c}={\overrightarrow{r}}_c-{\overrightarrow{r}}_g$. The distance vectors ${\overrightarrow{r}}_{g,g^{\prime }}={\overrightarrow{r}}_{g^{\prime }}-{\overrightarrow{r}}_g$ form a network of faces (minimum cycles) with $z_v$ sides each. The polygonal tiling associated with the packing partitions the space into areas occupied by grains (white) and areas occupied by voids (blue). The triangle formed by the points $({\overrightarrow{r}}_g,{\overrightarrow{r}}_c,{\overrightarrow{r}}_{c^{\prime }})$ (shaded area) is uniquely assigned to the contact $c$ and has an associated area $a\equiv a_{g,c}$, with a normalized area ${\alpha }_c=a_{g,c}/{\sigma }_g^2$.

Figure 2

Scaling collapse of the distribution of areas $p(\alpha ,>3)$ of the $z_v>3$ cycles at different energies. The plot shows distributions for $N_G=4096$ disks interacting via harmonic potentials ($\mu =2$). $x\to 0$ corresponds to disks with contact angles close to $\pi /2$. The scaling is consistent with Eq. (2). The limiting behaviors of the distribution are provided in Eq. (3). Inset: Comparison between the distributions obtained from the theory (bold lines) and numerical simulations demonstrating very good agreement.

Figure 3

Scaling of global quantities with energy for the same system as Fig. 2: (i) the excess number of contacts in the $z_v=3$ cycles [$\mathrm{\Delta }{n}_3=n_3(E_G)-n_3(0)]$ and (ii) the excess normalized area per contact of the $z_v>3$ cycles [$\mathrm{\Delta }⟨\alpha {⟩}_{>3}=⟨\alpha {⟩}_{>3}(E_G)-⟨\alpha {⟩}_{>3}(0)$]. The scaling is consistent with predictions in Eqs. (18) and (19). Inset: Scaling of excess grain area $\mathrm{\Delta }{A}_G=A_G(E_G)-A_G(0)$ displaying a scaling consistent with $\mathrm{\Delta }{A}_G\sim \mathrm{\Delta }{N}_C$ and Eq. (5).