The Ideal Glass and the Ideal Disk Packing in Two Dimensions

TL;DR

By constructing ideal jammed packings of polydisperse disks in 2D via radii as a mutable degree of freedom, the paper realizes a zero-configurational-entropy jammed state with a fully triangulated contact network. The package uses radical Delaunay triangulation and a constrained Lagrangian to produce packings at $\varphi \approx 0.910$ that are hyperuniform, mechanically ultrastable, and isotropic despite being amorphous, sharing crystal-like bulk and shear moduli. These ideal packings exhibit zero configurational entropy in the triangulated limit, no crystalline orientational or translational order, and vibrational spectra consistent with Debye scaling without a Boson peak, along with enhanced thermodynamic stability indicated by $T_m$ and $\varphi_m$ values. The work resolves aspects of the Kauzmann paradox in 2D and provides a practical shortcut to generate well-equilibrated glassy states, enabling comprehensive exploration of jammed and glassy physics in two dimensions.

Abstract

The ideal glass, a disordered system of particles with zero configurational entropy, cannot be realized through thermal processes. Nevertheless, we present a method for constructing ideal jammed packings of soft spheres, and thus the zero temperature ideal glass, in two dimensions. In line with the predicted properties, these critically jammed packings have high bulk and shear moduli as well as an anomalously high density. While the absence of pressure scaling in the shear moduli of crystalline materials is often attributed to the ordered nature of the particles, we show for the first time that disordered ideal packings also have this feature. We also find that the density of states avoids the low frequency power law scaling famously found in most amorphous materials, these configurations display hyperuniformity, and they melt at unusually high temperatures as compared to conventional packings. In addition to resolving a long-standing mystery, this methodology represents a valuable shortcut in the generation of well-equilibrated glassy systems. The creation of such an ideal packing makes possible a complete exploration and explanation of two dimensional jammed and glassy systems.

Figures (5)

• Figure 1: a) Triangulated packing, and b) conventionally jammed packing with $N=256$ and an identical set of radii. The contact network is overlaid in black, while face coloring of disks corresponds to the number of contacts contacting neighbors (grey for zero, light green for three, periwinkle for four, blue for five, purple for six, red for seven and teal for eight. c) The orientational correlation function, $C_6$, for triangulated packings (magenta and conventionally jammed packings (tan) of $N=2048$ (+'s) and $4096$ (o's). d) The finite size scaling for the translational order parameter, $\tau$, for triangulated packings (magenta), conventionally jammed packings (tan), and the hexagonal crystal (black).
• Figure 2: a) Bulk modulus per particle $K/N$ are plotted as a function of pressure, $P$ for triangulated packings (magenta points), conventionally jammed packings (tan points), and hexagonal crystals (black line) for $N$ ranging from 64 to 4096 in powers of two (indicated by Ys, squares, pentagons, triangles, stars, +'s and o's respectively). Moduli are averaged over 10 independent packing. Error bars indicate standard error. b) Shear modulus per particle, $G/N$, is presented with the same colors and symbols. c) Density of vibrational states, $D(\omega)$, are presented with the same colors and symbols; only $N=4096$ is shown for clarity.
• Figure 3: Binned averages of spectral density $\tilde{\chi}$, scaled by $\left<r\right>$ to collapse curves from $N=64-8192$; respective markers are, in increasing powers of two, Ys, squares, pentagons, triangles, stars, +'s, o's and hexagons. Color indicates protocol; tan for position-minimized packings and magenta for triangulated packings. Dashed lines with slope of $1/2$(black) and 2/3 donev_linear_2004 (grey) are overlaid as a visual guide. Error bars indicate standard geometric error of mean for each point; points without error bars are unaveraged.
• Figure 4: Measurements of the relaxation time for a sample triangulated packing (magenta) and conventionally jammed packing (tan), each with $N=8192$ using 10 thermal averages for A) thermal soft spheres and B) hard spheres. Solid line fits are to the modified VFT form of Eq. \ref{['eq:VFT']} and to the power-law diffusion form of Eq. \ref{['eq:powerLawDiff']} respectively, and dashed lines show the ideal glass melting point. Insets show a lack of statistically significant finite size scaling for $N>128$.
• Figure A1: The probability distribution of disk diameters, $P(\sigma)/\langle \sigma \rangle)$ averaged over 10 packings of 4096 disks. The starting distribution is at 20% polydispersity (gray dashed line), the distribution after radii-minimization is plotted as gray x's, the distribution after triangulation is plotted as magenta triangles, and for comparison a 26% polydispersity distribution is shown in black.