The Geometry behind the Glass Transition and Frictional Jamming in Systems of Two-Dimensional Hard Disks

TL;DR

This work investigates the origin of dynamic arrest in 2D bidisperse hard-disk systems and proposes that a geometric ground state, specifically the floret pentagonal tiling with area fraction $\phi_g = \sqrt{3}\pi/7 \approx 0.777343$, underlies both glassy and frictional jamming transitions. Using event-driven molecular dynamics, radical Voronoi tessellations, and isoperimetric-quotient analysis, the study links structural motifs to dynamics via metrics such as $q = 4\pi A/P^2$, the generalized edge count $n_q$, and the diffusion landscape $D(\phi)$ with an arrest point $\phi_a$ and an inflection $\phi_i$. Across a broad range of size ratios $R$, arrest occurs near $\phi \approx 0.777$, with crystallinity and pentagonal-hole motifs governed by a topological envelope $\bar{n}_q = 5.5$; this supports a unifying geometric mechanism for both glass and jamming, and points to a 3D analogue via square bipyramidal honeycombs that yield random loose packing. The findings provide a topology-guided, geometry-based framework for understanding dense disordered fluids, potentially informing interpretations of random loose packing and the glass/jamming crossover, with testable predictions for 3D systems.

Abstract

The relation between dynamics and structure in systems of Brownian bidisperse 2D hard disks with arrested dynamics is examined using numerical simulations. Surprisingly, the suspensions show dynamic arrest at an area fraction of φ {\approx} 0.777 over a wide range of disk-size ratios. This is in close agreement with the experimental findings of [Nat. Mater. 18, 1118 (2019)] (φ {\approx} 0.776) for a quasi-2D colloidal suspension of spheres with large-to-small size ratio of approximately 1.4. Intriguingly, this also matches a jamming transition (φ {\approx} 0.773 to 0.777) found experimentally in a 2D bidisperse granular packing of disks for a similar aspect ratio [Powders and Grains (2025)]. Adopting a geometric viewpoint allows for the identification of the floret pentagonal tiling (φ {\approx} 0.777343), which is comprised of congruent (elongated) pentagonal tiles. In analogy to foam and tissue models, it is the presence of this congruent reference state that induces a dynamical transition in the disordered fluid of the disks. That is, the change in dynamics observed both in a uniformly compressed granular medium and a colloidal one at finite temperature are implied to be caused by the same topological mechanism. Extending this reasoning to a honeycomb lattice could also explain the experimentally observed onset of caging dynamics (φ {\approx} 0.6) in a similar bidisperse, quasi-2D system of colloidal spheres [Nature 587, 225 (2020)]. An outlook is provided on how this concept may be applied to 3D: removing one in four particles from a face-centered-cubic arrangement leads to a square bipyramidal honeycomb} with associated volume fraction η {\approx} 0.55536. The ideas put forward here thus also provide a possible origin for random loose packing of (frictional) hard spheres in 3D and can potentially shed light onto the nature of the glass transition.

Figures (15)

• Figure 1: Experimental results for the active probe's rotational diffusion. The effective rotational diffusion coefficient $D_{p}$ normalized by the bare rotational diffusion $D_{0,r}$ of a self-propelled probe as a function of the area fraction $\phi$ of the surrounding bidisperse passive spheres (close to a surface, hence quasi-2D). The black dots show the mean value and the black bars the standard deviation, see Ref. lozano2019active. The two red curves are a stretched-exponential (left) and an exponential (right) fit to the data and serve to guide the eyes. The vertical blue dashed line indicates the value $\phi = \sqrt{3}\pi/7 \approx 0.777$, which will be justified later. Data is reproduced with permission from the authors.
• Figure 2: Visualization of topological transitions in a 2D system of disks. (a) Neighbor exchange of particles (red disks) is a T1 transition, as shown using part of the Voronoi lattice for this configuration (black lines). In the central plot, the particles are equidistantly placed and the transition takes place. (b) In a T2 transition, a particle is removed or added. This is illustrated here by shrinking one of the particles. In the middle row, the disk is smaller for every other particle and made increasingly opaque from left to right, starting from the second particle on the left. The black lines indicate the radical Voronoi diagram for this configuration.
• Figure 3: Procedure for computing the isoperimetric quotient using a Voronoi tessellation. (a) Part of a snapshot of a system showing the instantaneous organization of bidisperse disks with size ratio $R^{-1} = 1.4$ at an area fraction of $\phi = 0.777$. The smaller disks are indicated in light red, while the larger disks are in the darker shade of red. The centers of the disks are indicated using black dots and the blue lines surrounding the disks represent the radical Voronoi diagram for the system. Note that none of the disks overlap though this is hard to resolve by eye. (b) An example of a Voronoi cell for a large particle which has $n = 6$ sides, which is the $i$-th cell in this sample. (c) The area $A_{i}$ of this cell (green) together with the perimeter length $P_{i} = \sum_{i=1}^{n} l_{i}$ (purple arrows) are used to compute the isoperimetric quotient $q_{i} = 4 \pi A_{i} / P_{i}^{2}$.
• Figure 4: Structural features of the 2D system characterized by the isoperimetric quotient. (a-c) The probability density function (PDF) $P_{\phi}(q)$ of the isoperimetric quotient $q$ for several values of the area fraction $\phi$ in lin-log representation. The inverse size ratio $R^{-1}$ between the large and small disks in the system is provided in by the label. From red to purple, the area fractions are $\phi = 0.351$, $0.341$, $0.339$ (a-c; red), $0.503$, $0.500$, $0.497$ (a-c; orange), $0.616$, $0.608$, $0.606$ (a-c; green), $0.720$ (cyan), $0.780$ (blue), $0.825$ (purple), respectively. The vertical black lines provide the $q$ values for a regular $n$-gon, with $n = 3,\dots,7$ from left to right; also see visual guide on top. The error bars indicate the standard deviation on the binned value for the interval over which $P_{\phi}(q)$ is sampled, which adheres to counting statistics and is thus always the square root of the number of samples in a bin. (d) The generalized number of edges $n_{\bar{q}}$ belonging to the mean $q$ value --- $\bar{q}$: the first moment of $P_{\phi}(q)$ --- as a function of $\phi$ for all size ratios $R$ considered. From red to purple the value of the size ratio is given by $R^{-1} = 1.00$, $1.05$, $1.10$, $1.15$, $1.17$, $1.20$, $1.22$, $1.23$, $1.24$, $1.25$, $1.30$, $1.35$, $1.40$, $1.45$, $1.50$, $1.55$, $1.60$, $1.65$, and $1.70$, respectively. The inset shows a zoom-in on the range where the systems form crystals, i.e., above the dashed horizontal line provides the value $n_{\bar{q}} = 5.5$. I do not provide error bars here to help improve the representation. The error is about twice the line width, as can be appreciated from the noise that is present on the red curve in the inset.
• Figure 5: Analyzing the dynamics in the system. All the data in this set of graphs was obtained for $R^{-1} = 1.45$. (a) The reduced mean squared displacement (MSD) as a function of the reduced time $t$. From red to blue the value of the area fraction $\phi$ increases from $0.081$, $0.155$, $0.223$, $0.285$, $0.341$, $0.393$, $0.44$, $0.483$, $0.522$, $0.558$, $0.606$, $0.62$, $0.634$, $0.647$, $0.66$, $0.672$, $0.683$, $0.694$, $0.718$, $0.723$, $0.731$, $0.739$, to $0.744$, respectively. The dots show the average and the error bars the standard error of the mean. The dashed black line shows the linear scaling of the MSD for large times. (b) Logarithmic plot of the reduced diffusion coefficient $D$ as a function of $\phi$ for the data in (a). The error is typically much smaller than the symbol size. The red curve shows a power-law fit to the data. The vertical dashed vertical magenta line indicates the inflection point to the change in diffusion. The inset shows the numerical derivative $\partial D/\partial \phi$ as a function of $\phi$. The solid orange curve shows a quintic polynomial fit to this data and the magenta line is the same as in the main panel.