Echoes of the Glass Transition in Athermal Soft Spheres

Recent theoretical advances have led to the creation of a unified phase diagram for the thermal glass and athermal jamming transitions. This diagram makes clear that, while related, the mode-coupling—or dynamic—glass transition is distinct from the jamming transition, occurring at a finite temperature and significantly lower density than the jamming transition. Nonetheless, we demonstrate a prejamming transition in athermal frictionless spheres which occurs at the same density as the mode-coupling transition and is marked by percolating clusters of locally rigid particles. At this density in both the thermal and athermal systems, individual motions of an extensive number of particles become constrained, such that only collective motion is possible. This transition, which is well below jamming, exactly matches the definition of collective behavior at the dynamical transition of glasses. Thus, we reveal that the genesis of rigidity in both thermal and athermal systems is governed by the same underlying topological transition in their shared configuration space.

Figure 1

Definition of local rigidity: the blue particles are considered locally rigid if all of the red particles marked with an $\times$ are constrained. This represents the largest locally rigid set of the system.

Figure 2

The fraction of locally rigid particles as a function of scaled packing fraction. (a) For dimensions 3 (black), 4 (red), 5 (blue), and 6 (green) each averaged over 8 systems of $N=16384$ particles. The inset shows the fraction of systems with a percolating locally rigid set. (b) Finite size effects in $d=4$ seen for system sizes ranging from $N=256$ (light gray) to 16 384 (black) in powers of 2 and the largest system 32 768 in red. We average over 512 systems at $N=256$, 256 at $N=512$, 128 at $N=1024$, 64 at $N=2048$, 32 at $N=4096$, 16 at $N=8192$, 8 at $N=16384$, and 16 at $N=32768$. The inset shows the change of ${\varphi }_{\mathrm{LR}}$ as a function of system size. (c) Differences in precision in $d=4$ as measured by $-{\log }_{10}{E}_{\mathrm{cut}}=10$, 20, 30, 40, 50 (light gray to black) and the highest precision of 60 in red. This data is averaged over 16 systems of $N=8192$ particles. The inset shows how ${\varphi }_{\mathrm{LR}}$ changes with precision.

Figure 3

A simplified illustration of the allowed configuration space shared by thermal hard spheres and energy minimized athermal soft spheres. White areas represent allowed configurations and black areas represent forbidden configurations. Energy minimized soft sphere athermal systems will generically sit on the boundaries between allowed and forbidden states, while thermal hard sphere systems will explore allowed states. As packing fraction is increased, the volume and dimensionality of allowed configurations decrease, undergoing two distinct transitions at ${\varphi }_d\simeq {\varphi }_{\mathrm{LR}}$ and ${\varphi }_J$.

Figure 4

Values of jamming point ${\varphi }_J$ [19], the dynamical glass transition point ${\varphi }_d$ in $d=3$ [9] and $d=4–6$ [10], and local rigidity onset point ${\varphi }_{\mathrm{LR}}$ in systems with 16 384 particles. ${\varphi }_{\mathrm{LR}}$ values have their error in the least significant digit, which is quoted in parentheses.