Experimental Evidence of the Gardner Phase in a Granular Glass

Analyzing the dynamics of a vibrated bidimensional packing of bidisperse granular disks below jamming, we provide evidence of a Gardner phase deep into the glass phase. To do so, we perform several compression cycles within a given realization of the same glass and show that the particles select different average vibrational positions at each cycle, while the neighborhood structure remains unchanged. The separation between the cages obtained for different compression cycles plateaus with an increasing packing fraction, while the mean square displacement steadily decreases. This phenomenology is strikingly similar to that reported in recent numerical observations when entering the Gardner phase, for a mean-field model of glass as well as for hard spheres in finite dimension. We also characterize the distribution of the cage order parameters. Here we note several differences from the numerical results, which could be attributed to activated processes and cage heterogeneities.

Figure 1

Gardner transition. Left: A glass metabasin breaks into a hierarchy of marginally stable subbasins (inspired from Ref. [22]). In the glass phase $({\varphi }_g<\varphi <{\varphi }_G)$, the particle (in blue) is caged by its neighbors; those of which establish contact at jamming $({\varphi }_J)$ are not selected yet. In the Gardner phase $(\varphi >{\varphi }_G)$, each of the subbasins eventually corresponds to one structure of a contact network (red neighbors). Right top: Experimental realization of this scenario in a bidisperse system of disks for two independent compressions up to ${\varphi }_J$ within the same glass state. Right bottom: In the Gardner phase, below jamming $({\varphi }_G<\varphi <{\varphi }_J)$, successive compressions starting from the same glass state, where particles vibrate in a large (gray) cage, lead to a different caging location in a smaller cage (blue, red, and yellow). While $\mathrm{\Delta }(\varphi )$, the cage size, decreases with $\varphi$, ${\mathrm{\Delta }}_{AB}$, the typical distance between the cages obtained for different compressions, plateaus at the cage size of the transitional packing fraction, ${\varphi }_G$.

Figure 2

Cage order parameters. Mean square displacement $\mathrm{\Delta }(t_0)$ and mean square distance between cages ${\mathrm{\Delta }}_{AB}$ averaged over 200 time steps and ten compression cycles from ${\varphi }_0=0.8185$ to $\varphi$ as a function of $\varphi$. Inset: Aging. Mean square displacement $\mathrm{\Delta }(t_0,t_w)$ as a function of $t_w$ for $\varphi =0.8196$, 0.8212, 0.8222, 0.8233. The color codes are for the packing fraction.

Figure 3

Statistics of $\mathrm{\Delta }$ and ${\mathrm{\Delta }}_{AB}$. Probability distribution of $\mathrm{\Delta }/⟨\mathrm{\Delta }⟩$ sampled (a) over time and compression cycles and (b) over time only [inset: dynamical susceptibility associated to the fluctuations of ${\mathrm{\Delta }}^k(t_0,t_w)$ over time]. Color code as in Fig. 2. Temporal evolution of ${\mathrm{\Delta }}^{k,k^{\prime }}(t)$ for ten different pairs of compression cycles (color coded) for (c) $\varphi =0.8196$ and (d) $\varphi =0.8228$, as compared to the temporal evolution of ${\mathrm{\Delta }}^k(t,{\tau }_0)$ (in black). Colors indicate different pairs of compression cycles. (e) Probability distribution function of ${\mathrm{\Delta }}_{AB}$ sampled across compression cycles. Color code as in Fig. 2.