Vibrational Properties of Hard and Soft Spheres Are Unified at Jamming

The unconventional thermal properties of jammed amorphous solids are directly related to their density of vibrational states. While the vibrational spectrum of jammed soft sphere solids has been fully described, the vibrational spectrum of hard spheres, a model glass former often related to physical colloidal glasses, is still unknown due to the difficulty of treating the nonanalytic interaction potential. We bypass this difficulty using the recently described effective interaction potential for the free energy of thermal hard spheres. By minimizing this effective free energy, we mimic the rapid compression of hard spheres and produce typical configurations of the thermal system. We measure the resulting vibrational spectrum and characterize its evolution toward the jamming point where configurations of hard and soft spheres are trivially unified. For densities approaching jamming from below, we observe low-frequency modes which agree with those found in numerical simulations of jammed soft spheres. Our measurements of the vibrational structure demonstrate that the jamming universality extends away from jamming: hard sphere thermal systems below jamming exhibit the same vibrational spectra as thermal and athermal soft sphere systems above the transition.

Figure 1

Minimization of the effective logarithmic potential in $d=2$ with packing fraction $\phi =0.55$. Left: packing after harmonic minimization. Right: final configuration of the same packing after the logarithmic potential minimization.

Figure 2

Gap distribution of hard sphere packings in $d=3$. The distance from jamming increases from left to right: data from decompressions (blue squares) $\mathrm{\Delta }\phi =1.1\times {10}^{-7},1.3\times {10}^{-6},1.5\times {10}^{-5}$, data from compressions (green diamonds) $\mathrm{\Delta }\phi =2.3\times {10}^{-5},2.1\times {10}^{-4},3\times {10}^{-3},2\times {10}^{-2}$. The distributions peak around the value of the typical nearest neighbor gap and then decay following a power-law scaling (black line) consistent with the mean field prediction $h^{-\gamma }$ with $\gamma =0.41296\dots$ [5]. Gaps are cutoff at $h=1$ to avoid showing next nearest neighbor behavior.

Figure 3

(a) Participation ratio (PR) and vibrational density of states for a packing of $N=8192$ particles in $d=3$ at $\mathrm{\Delta }\phi =3\times {10}^{-2}$. (b) Real space representation of the eigenvectors for a packing of $N=8192$ particles in $d=2$ with distance from jamming $\mathrm{\Delta }\phi =3\times {10}^{-2}$. Left: phonon with characteristic plane wave modulation. Center: quasilocalized mode with localized excitations distributed over the whole system. Right: extended anomalous mode which correlates a large portion of the system with random excitations.

Figure 4

Evolution of participation ratio (PR) and vibrational density of states along the compression as a function of $\mathrm{\Delta }\phi$ in $d=2$ (left) and $d=3$ (right). Data from compressions are shown in green and that from decompressions in $d=3$ in blue. Each scatter plot of PR shows data from 10 samples while the density of state curves are averaged over the same number of samples. The distance from jamming increases from left to right. In $d=3$ $\mathrm{\Delta }\phi =1.1\times {10}^{-7}$, $\mathrm{\Delta }\phi =2.3\times {10}^{-5},5\times {10}^{-4},3\times {10}^{-2}$. In $d=2$ $\mathrm{\Delta }\phi =2.7\times {10}^{-6},3\times {10}^{-4},3\times {10}^{-2}$. The low-frequency decay of the density of states in $d=2$ follows ${\omega }^2$ for every value of $\mathrm{\Delta }\phi$, while in $d=3$ it follows ${\omega }^4$ sufficiently far from jamming.

Figure 5

Scaling of ${\omega }^{*}$ as a function of $\mathrm{\Delta }\phi$ for different system sizes from decompressions (blue squares $N=4096$) and compressions (green circles $N=1024$, upward triangles $N=2048$, downward triangles $N=4096$, diamonds $N=8192$). Data are consistent with the critical scaling ${\omega }^{*}\sim \mathrm{\Delta }{\phi }^{1/2}$ observed for soft spheres.