🔬 condensed matter

Universal Jamming Criticality and Self-Organizing Principles from Disorder to the Limit of Perfect Crystalline Order

This paper demonstrates that the mechanical properties of jammed elastic packings, ranging from disordered to nearly crystalline structures, are universally governed by jamming criticality and self-organizing principles, revealing a fundamental decoupling between jamming physics and the glass transition.

Original authors: Jianhua Zhang, Jiaqi Si, Ning Xu, Hua Tong

Published 2026-07-21

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CC BY 4.0

Original authors: Jianhua Zhang, Jiaqi Si, Ning Xu, Hua Tong

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world made of tiny, bouncy balls. If you arrange them in a perfect, repeating grid, you get a crystal—think of a diamond or a snowflake. Scientists have known for a long time how to describe these orderly structures because they follow strict rules, like soldiers marching in formation. But what happens when those balls are thrown into a pile with no order at all? This is the world of "amorphous solids," a category that includes things like window glass, sand, and even the foam in your coffee. These materials are messy, diverse, and don't follow the neat marching orders of crystals. For decades, scientists have struggled to find a single rule that explains how these messy piles become solid and rigid.

The key to this puzzle lies in a concept called "jamming." Imagine a traffic jam where cars are so packed together they can't move; that's jamming. In physics, when you squeeze a pile of particles (like sand or glass beads) tight enough, they suddenly lock together and gain the ability to hold their shape, even though they aren't frozen in a crystal pattern. This transition from a loose, flowing pile to a rigid, solid block is called the "jamming transition." Scientists care deeply about this because it might be the secret code that explains why so many different messy materials behave the way they do. But there's a big question hanging over the field: Does this "jamming" rule only work for messy piles, or does it also apply to materials that are almost perfectly ordered, like crystals that have just a tiny bit of disorder?

In this study, researchers Jianhua Zhang, Jiaqi Si, Ning Xu, and Hua Tong decided to test the limits of this idea. They created a digital playground where they could start with a perfect crystal and slowly, carefully introduce disorder, turning it into a messy pile. They wanted to see if the rules of "jamming" would take over the physics of the material, even if the material still looked mostly like a crystal.

Here is what they found: The "jamming" rules are incredibly powerful. They discovered that as long as there is any amount of disorder in the system—even a tiny, almost invisible amount—the material's behavior is eventually dominated by the laws of jamming. It's as if the messy, chaotic nature of the contacts between the particles acts like a master switch. Once the pressure drops below a certain critical point (which depends on how much disorder there is), the material stops acting like a crystal and starts acting like a jammed solid, no matter how ordered it looks on the surface.

This finding is a big deal because it challenges an old idea. Some scientists thought there might be a hard line, a "disorder threshold," that separates the world of crystals from the world of jammed solids. You might have thought, "If it's 99% crystal, it acts like a crystal; if it's 99% messy, it acts like a jam." But this paper suggests that's not true. There is no such wall. Instead, the "jamming" physics stretches all the way to the very edge of perfect order. This implies that the physics of jamming and the physics of glass formation (which happens when liquids cool down too fast to crystallize) might be two different things entirely. The paper suggests that jamming is driven by the chaos at the contact points between particles, not by the overall shape of the particles themselves.

The researchers also uncovered some hidden patterns that appear in these jammed states, regardless of whether the material started as a crystal or a mess. They found that the way particles connect to their neighbors follows a universal distribution. It's like finding that in a crowded room, whether the people are standing in neat rows or scattered randomly, the number of people each person is touching follows a specific, predictable pattern. They also found that the "contacts" between particles are arranged in a way that suppresses randomness, a property they call "hyperuniformity." It's as if the particles are playing a game of "keep your distance" that creates a hidden order within the chaos.

One of the most surprising results is that the density at which these materials jam can get incredibly close to the density of a perfect crystal. This pushes the boundaries of what we thought was possible, suggesting that the "J-line" (the boundary where jamming happens) can go much further than previous theories predicted. The authors note that these results come from computer simulations, so while the patterns are very strong and consistent, they are observing these behaviors in a virtual world first.

In short, this paper suggests that the secret to rigidity in messy solids isn't just about how messy they are, but about how they self-organize at the point where they touch. Whether you start with a perfect crystal or a chaotic pile, if you squeeze them just right, they all seem to follow the same universal playbook of jamming. This helps scientists understand that the transition from a flowing pile to a solid block is a fundamental feature of matter that doesn't care much about whether the underlying structure is a crystal or a mess. It's a reminder that sometimes, the most rigid structures are built on the most chaotic foundations.

Technical Summary: Universal Jamming Criticality and Self-Organizing Principles from Disorder to the Limit of Perfect Crystalline Order

Problem Statement
The nature of amorphous solids remains elusive due to their disordered, diverse, and nonequilibrium structures, lacking the unified structural basis (translational periodicity) that defines crystals. While jamming physics has successfully described random close packings, its exact range of applicability regarding highly ordered solids is unclear. Specifically, it is unknown whether jamming criticality—a phenomenon characterized by specific scaling behaviors near the transition to rigidity—persists as structural order increases toward perfect crystalline states, or if a finite disorder threshold separates crystal physics from jamming physics. Furthermore, the relationship between jamming criticality and the glass transition, particularly regarding the maximum jamming packing fraction ( $\phi_{GCP}$ ), requires re-evaluation in light of strong steric constraints in low dimensions.

Methodology
The authors investigate jammed elastic packings by systematically tuning structural order from perfect crystals to fully disordered states.

System Generation: Jammed packings are generated by quasistatically increasing polydispersity ( $\eta$ ) starting from perfect crystals (hexagonal in 2D, face-centered cubic in 3D). $\eta$ controls the width of the particle-size distribution, modulating structural order continuously while maintaining overall homogeneity (pure phases rather than mixtures).
Interaction Models: Particles interact via harmonic or Hertzian repulsion. The distance to the jamming transition is controlled by pressure ( $p$ ) at fixed $\eta$ .
Simulation Techniques: To overcome slowing down in the weak-disorder limit, a hybrid FIRE algorithm with self-activation and swaps is employed to generate truly stable configurations.
Analysis: The study analyzes mechanical properties (excess coordination number $\Delta Z$ , shear modulus $G$ , bulk modulus $B$ ), nonaffine elastic responses, coordination-number distributions, and spatial fluctuations (hyperuniformity) across a wide range of $\eta$ (from $\eta \approx 10^{-3}$ to $0.6$) and pressures.

Key Contributions and Results

Universal Jamming Criticality Across the Order-Disorder Spectrum:
The study demonstrates that jamming criticality universally governs mechanical properties below a critical pressure $p_j$ for any finite polydispersity $\eta$ .
- Scaling Regimes: Three regimes are identified: (i) a crystal regime at high pressures with constant properties; (ii) a "disordered crystal" regime with intermediate scaling ( $\Delta Z \sim p^{0.83}$ , $G/B \sim p^{0.7}$ ); and (iii) a jamming critical regime below $p_j$ exhibiting universal jamming scalings ( $\Delta Z \sim p^{0.5}$ , $G/B \sim p^{0.5}$ ).
- Critical Pressure Scaling: The crossover pressure scales as $p_j \sim \eta^\delta$ , with $\delta = 2.5$ in 2D and $\delta = 2$ in 3D. This scaling implies there is no finite-disorder threshold separating crystal and jamming behaviors; instead, the domain of jamming physics scales with disorder.
- Decoupling from Glass Transition: The jamming density can approach the close-packing limit ( $\phi_{cp}$ ) as $\eta \to 0$ . This extends the "J-line" beyond the maximum jamming packing fractions predicted by mean-field (MF) theories for ideal glasses (e.g., $\phi_{GCP} \approx 0.87$ in 2D). This suggests a fundamental decoupling between jamming criticality (driven by contact-level disorder) and glass transition physics (driven by particle-level structure).
Mechanism of Random Nonaffine Elasticity:
The transition to jamming criticality is driven by random nonaffine elasticity arising from contact-level disorder.
- Nonaffinity Analysis: In near-crystalline packings, nonaffine deformation distributions exhibit six-fold symmetry at high pressures due to lattice constraints. However, below $p_j$ , the distribution becomes isotropic, indicating the system responds flexibly, free from particle-level geometric order.
- Anisotropy Crossover: The six-fold anisotropy parameter $C_6$ drops sharply near $p_j$ , marking a mechanism change where elastic responses become dominated by random nonaffinity regardless of the underlying particle-scale order.
Universal Coordination-Number Distribution:
A universal probability distribution of the coordination number ( $f_z$ ) is uncovered in marginally jammed states, independent of particle-level structure.
- Specific Values: In 2D harmonic systems, the distribution converges to $f_3 \approx 0.308$ , $f_4 \approx 0.416$ , $f_5 \approx 0.237$ , and $f_6 \approx 0.039$ .
- Implications: This distribution represents an additional constraint on the mechanical network beyond the global Maxwell criterion ( $Z_c = 2d$ ). It arises from strong steric constraints in low-dimensional jammed configurations, which are absent in MF models and likely explain deviations in scaling exponents (e.g., $\delta=2.5$ in 2D vs. MF prediction of 2).
Contact Hyperuniformity vs. Density Hyperuniformity:
- Contact Hyperuniformity: The spatial fluctuation of the coordination number is strongly suppressed near the jamming transition for both near-crystalline and fully disordered packings, exhibiting $\sigma^2_Z(L) \sim L^{-1}$ and $S_Z(q) \sim q^{1.5}$ . This confirms contact hyperuniformity is a universal feature of marginally jammed solids.
- Density Hyperuniformity: In contrast, density hyperuniformity is not a generic feature. While quenched binary systems show effective hyperuniformity at intermediate scales, spectral density saturates or upturns at large scales. Near-crystalline packings show even less density hyperuniformity, suggesting it is not intrinsically linked to jamming criticality across the order-disorder spectrum.

Significance and Claims
The paper claims to provide a unified understanding of the organizing principles in marginally jammed solids by demonstrating that jamming criticality is not restricted to disordered systems but extends to the limit of perfect crystalline order.

Redefining Amorphous Solids: The findings suggest a shift in defining amorphous solids from particle-level structure to contact-level mechanical self-organization. The universal coordination-number distribution and contact hyperuniformity serve as robust signatures of emergent rigidity.
Theoretical Implications: The results highlight the limitations of mean-field and lattice models, which fail to capture the nonperturbative many-body correlations and steric constraints present in physical dimensions. The study posits that contact-level disorder, rather than particle-level glassiness, is the primary driver of jamming criticality.
Scope: The work complements principles of mechanical self-organization and underscores the broad relevance of jamming physics, suggesting that the "J-line" is not bounded by the glass transition but is instead determined by the interplay of polydispersity and pressure.

Technical Summary: Universal Jamming Criticality and Self-Organizing Principles from Disorder to the Limit of Perfect Crystalline Order

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