🔬 condensed matter

Self-organization and memory formation in two-dimensional jammed deformable matter under cyclic compression

This study demonstrates that cyclic compression of two-dimensional jammed deformable ring assemblies drives self-organization into either ordered reversible states or hysteretic limit cycles that encode robust memory of training history, with macroscopic hysteresis arising from directionally asymmetric non-affine microscale deformations.

Original authors: Rahul Nayak, Satyavani Vemparala, Pinaki Chaudhuri

Published 2026-05-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rahul Nayak, Satyavani Vemparala, Pinaki Chaudhuri

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 crowded dance floor filled with thousands of flexible, rubbery rings. In this study, scientists watched what happens when they squeeze this crowd together and then let them go, over and over again. They wanted to see if the crowd could "learn" a specific way to move and remember it, even when pushed harder later.

Here is the story of what they found, broken down into simple concepts:

1. The Setup: Squeezing the Squishy Crowd

The researchers built a computer simulation of 1,000 rings. These aren't hard steel hoops; they are soft and bendy, like rubber bands.

The Squeeze: They slowly squished the rings together. At first, the rings just bump into each other. But once they get packed tight enough (a point called "jamming"), the pressure spikes.
The Buckle: If they keep squeezing past that point, the rings don't just get tighter; they start to buckle and fold in on themselves, like a soda can being crushed. This makes the whole crowd surprisingly soft and squishy again, even though it's packed tight.

2. The Two Types of Crowds

The scientists tested two different kinds of crowds to see how they reacted to being squeezed and released repeatedly (cyclic compression).

Crowd A: The Identical Twins (Monodisperse)

Who they are: Every ring is exactly the same size and stiffness.
What happened: At first, the crowd was messy and disordered. But as they kept squeezing and releasing, the rings started to "anneal" (like metal cooling down). They organized themselves into a perfect, neat honeycomb pattern.
The Result: Eventually, the crowd found a "perfect path." When squeezed and released, they moved back and forth along the exact same track with almost no wasted energy. They became highly ordered and reversible.

Crowd B: The Mixed Bag (Polydisperse)

Who they are: The rings vary in size or stiffness. Some are stiff, some are floppy; some are big, some are small.
What happened: Because they are all different, they couldn't organize into a perfect honeycomb. They remained messy and disordered.
The Result: Instead of finding a perfect path, they settled into a "stable loop." Every time they were squeezed and released, they traced the exact same messy shape. They didn't become reversible, but they became predictable.

3. The Superpower: Robust Memory

This is the most exciting part. The "Mixed Bag" crowd developed a memory.

The Training: The scientists "trained" the crowd by squeezing them to a certain limit. The crowd learned a specific squishy shape.
The Test: Then, they tried to break the memory. They squeezed the crowd much harder than before (overdriving) or squeezed them in smaller, nested loops.
The Outcome: Even after being pushed to extremes, when the scientists returned to the original training limits, the crowd immediately snapped back to the exact same path they had learned. They remembered their training perfectly.

The Analogy: Imagine a group of people trying to walk through a crowded hallway.

If everyone is the same height, they eventually line up perfectly and walk in a straight, efficient line.
If everyone is different heights, they can't line up perfectly. But, if you push them through the crowd a few times, they learn a specific "dance" of dodging and weaving. Even if you push them harder later, they will still know exactly how to dodge to get back to that same dance.

4. How Did They Remember? (The Secret Mechanism)

You might think the rings remembered by locking their neighbors in place, like a rigid puzzle. But the paper says something more subtle is happening.

The Skeleton Stays Put: The "skeleton" of the crowd—the list of which ring is touching which—stays almost exactly the same. The connections don't break and re-form constantly.
The Shape Shifts: The memory is stored in the shape of the individual rings. When squeezed, the rings bend and twist in specific, asymmetric ways depending on whether they are being pushed in or pulled out.
The Metaphor: Think of a crowd of people holding hands in a circle. If you push them, they don't let go of each other's hands (the network stays intact). Instead, they lean forward or backward depending on the direction of the push. The "memory" isn't in who is holding hands, but in the specific way their bodies are leaning.

5. Why This Matters (According to the Paper)

The paper concludes that this ability to be squishy (deformable) is actually a superpower.

In rigid crowds, if you push too hard, things break and the memory is lost.
In these soft, deformable rings, the rings absorb the stress by bending themselves. This protects the "hand-holding" network from breaking.
This allows the system to be both soft (easy to squeeze) and smart (able to remember complex patterns), creating a new type of material that can be "trained" to remember mechanical states.

In short: By using soft, bendy rings that can change shape, the scientists created a system that can learn a mechanical routine and remember it perfectly, even when pushed to its limits, because the rings absorb the stress by bending rather than breaking their connections.

Technical Summary: Self-organization and memory formation in two-dimensional jammed deformable matter under cyclic compression

Problem Statement
The jamming transition, which describes the emergence of rigidity in disordered particle collections, is well-characterized for rigid point particles interacting via repulsive potentials. However, many physical and biological constituents (e.g., cells, microgels, polymeric rings) possess internal degrees of freedom allowing for significant shape deformation. In these deformable assemblies, the bulk mechanical response is determined by a coupling between inter-particle contacts and the elastic energy stored in shape deformations. While previous studies have explored the dynamics of such systems, it remains an open question how particle deformability influences the capacity of jammed assemblies to self-organize and encode mechanical memory under cyclic driving, particularly in regimes where standard granular assumptions (small strains, fixed shape) break down. This work specifically investigates how the interplay between shape deformability and the persistence of interstitial voids (non-confluent systems) affects self-organization and memory retention.

Methodology
The authors employ extensive numerical simulations of two-dimensional athermal assemblies of deformable soft matter modeled as ring polymers. The system consists of $N=1000$ rings interacting via a combination of Weeks-Chandler-Andersen (WCA) potentials for non-bonded monomers and finite extensible nonlinear elastic (FENE) potentials for bonded monomers. Ring backbone flexibility is controlled by an angular potential.

Three distinct model systems are studied:

Monodisperse (M0): All rings have identical monomer counts ( $n_m=100$ ) and angular stiffness ( $K_\theta=100$ ).
Stiffness-polydisperse (K30): Rings have $n_m=100$ but varying angular stiffness ( $K_\theta$ sampled from a uniform distribution with 30% polydispersity).
Size-polydisperse (P30): Rings have $K_\theta=100$ but varying monomer counts ( $n_m$ sampled from a uniform distribution with 30% polydispersity).

The mechanical response is probed using a quasistatic compression protocol. Initial unjammed configurations are generated by quenching equilibrium supercooled states to inherent structures. The system is then compressed and decompressed by rescaling the simulation box, followed by energy minimization. Cyclic driving is applied by repeating compression-decompression cycles to observe convergence to limit cycles. Key observables include pressure ( $P$ ), bulk modulus ( $B$ ), mean contact number ( $Z$ ), mean asphericity ( $\langle a \rangle$ ), and topological defects via Voronoi analysis.

Key Results

Jamming and Buckling: Upon compression, all systems exhibit a jamming transition characterized by a rapid increase in pressure and bulk modulus. Beyond a critical density ( $\rho_b$ ), the systems undergo "buckling," where rings deform from circular to rod-like shapes. This leads to a pronounced mechanical softening (a drop in the bulk modulus) as compressive stress shifts from inter-particle contacts to intra-particle shape deformations. This softening is unique to deformable particles with unconstrained annular geometry.
Monodisperse Systems (Self-Organization): In monodisperse systems, cyclic compression acts as an athermal annealing process. The system progressively eliminates structural defects, converging to a nearly reversible limit cycle passing through an ordered, nearly crystalline state (triangular lattice). The jamming density ( $\rho_J$ ) and buckling density ( $\rho_b$ ) converge, narrowing the gap between them. The asymptotic state features rigid ordered domains separated by geometrically frustrated boundaries where coupled translations and rotations occur.
Polydisperse Systems (Persistent Hysteresis): In contrast, stiffness- and size-polydisperse systems converge to stable, hysteretic limit cycles that remain disordered.
- The stiffness-polydisperse system forms a coarse-grained granular structure with polycrystalline grains separated by defect lines.
- The size-polydisperse system remains inherently amorphous due to heterogeneous shape responses.
- Despite the disorder, these systems exhibit robust memory retention. They can reproduce trained cycles even after being subjected to "overdriving" (cycles with amplitudes exceeding the training amplitude) and nested sub-cycles.
Origin of Hysteresis and Memory:
- Topological Stability: The contact network (defined by ring centers) remains largely intact during cycling, with only a negligible fraction of Voronoi bonds breaking. This topological stability protects the encoded memory against overdriving.
- Sub-Topological Irreversibility: The macroscopic hysteresis observed in polydisperse systems does not arise from contact rewiring but from directionally asymmetric non-affine deformations at the monomer scale. During compression and decompression, the system traverses different paths on a rough local energy landscape within the "frozen" contact network.
- Memory Mechanism: The ability of rings to absorb compressive stress through shape change prevents cascading plasticity, preserving the self-organized topology. The memory is encoded in the specific configuration of monomer-level reconfigurations within the topological basin.

Significance and Claims
The paper demonstrates that deformable particle assemblies constitute a distinct class of trainable jammed materials. The authors claim that the deformability of the particles acts as a mechanical buffer, absorbing stress through shape change rather than contact rewiring. This mechanism allows the system to maintain a robust, self-organized topological network even under extreme driving conditions.

The study establishes that:

Memory Robustness: Structural memory in these systems is highly robust, surviving overdriving and nested cycles, unlike in rigid-particle systems where memory is often erased by large-amplitude driving.
Mechanism of Hysteresis: Macroscopic hysteresis in disordered, deformable packings originates from sub-topological irreversibility (directionally asymmetric non-affine deformations) rather than topological rearrangements.
Design Principle: The findings suggest a design principle for soft materials that combine high mechanical compliance (squishiness) with robust memory formation, where stable topology and internal configurational fluctuations operate on separate levels of organization.

The work does not propose new experimental applications or future implications beyond characterizing these mechanical behaviors and their underlying physical mechanisms in the context of jammed soft matter and glassy dynamics.

1. The Setup: Squeezing the Squishy Crowd

2. The Two Types of Crowds

3. The Superpower: Robust Memory

4. How Did They Remember? (The Secret Mechanism)

5. Why This Matters (According to the Paper)

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