Cohesion-induced hysteresis and breakdown of marginal stability in jammed granular materials

This study reveals that cohesive interactions in jammed granular materials violate marginal stability, generating excess rigidity and causing pronounced hysteresis in the shear modulus during compression and decompression cycles, even when the underlying interparticle forces are history-independent.

The Gist

The Big Picture: When Sticky Sand Acts Weird

Imagine you have a bucket of sand. If you just dump it on the floor, it's loose and flows like a liquid. But if you squeeze it tight, the grains lock together, and the pile becomes hard like a rock. This transition from "flowing sand" to "solid rock" is called jamming.

Scientists have spent decades studying how this works. They found that for normal, dry sand (which only pushes away when it touches), the rules are surprisingly simple and predictable. If you know how much you are squeezing it (the pressure), you can predict exactly how stiff it will be.

But what happens if the sand is wet?
Wet sand has a special property: the water creates a tiny "glue" (cohesion) that makes the grains stick together. This paper investigates what happens when you jam this "sticky" sand.

The researchers discovered something surprising: Sticky sand breaks the rules. Even if you squeeze it to the exact same pressure as dry sand, it behaves differently depending on how you got there. It has a "memory."

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The Experiment: The Push and Pull

To understand this, imagine a game of "Push and Pull" with a pile of marbles.

1. The Dry Marbles (Repulsive):
   Imagine marbles that repel each other (like magnets with the same pole facing out).

   • Pushing: You squeeze them together. They get harder and harder.
   • Pulling: You let them expand. As soon as the squeezing stops, they lose their stiffness immediately.
   • The Result: If you plot their stiffness against the pressure, the path you take to get there doesn't matter. The graph is a single, clean line.

2. The Sticky Marbles (Cohesive):
   Now, imagine the marbles have a tiny bit of Velcro on them.

   • Pushing: You squeeze them. They get stiff.
   • Pulling: You start to let them expand. Here is the magic trick: Even when you stop squeezing (zero pressure), the pile stays stiff!
   • The Result: If you plot the stiffness, you get a loop. The path you took to get to a specific pressure matters. If you just started squeezing, the pile is soft. If you were squeezing hard and then let go, the pile is still stiff.

The Analogy:
Think of Dry Sand like a stack of smooth stones. If you stop pushing them, they slide apart and become loose immediately.
Think of Sticky Sand like a pile of wet clay. If you squeeze it into a ball and then let go, it holds its shape. It stays "solid" even without you holding it tight.

The Mystery: Why Does This Happen?

The researchers asked: Why does the sticky sand remember the history of how it was squeezed?

They used a mathematical tool called Effective Medium Theory (EMT). Think of this as a "crystal ball" that predicts how a messy pile of particles should behave based on the average number of neighbors each particle touches.

For dry sand, the crystal ball says: "The system is Marginally Stable."

• Analogy: Imagine a house of cards. It's balanced on a knife's edge. It's just barely standing. If you remove one card, it falls. If you add one, it gets stronger. In this state, the "pressure" and the "number of connections" are locked together in a perfect dance. You can't change one without changing the other.

For sticky sand, the crystal ball says: "The system has Broken the Rules."

• Analogy: Now imagine that same house of cards, but the cards are glued together. You can pull the cards apart a little bit, and the glue holds them up. The house is no longer on a knife's edge; it's over-stabilized.
• Because of this "glue," the particles can hold onto each other even when the pressure drops to zero (or even becomes negative, like a suction). The system has excess rigidity. It is stronger than the math predicted it should be.

The "Hysteresis" (The Memory Effect)

The paper uses the word Hysteresis. In everyday terms, this means lag or memory.

• Dry Sand: No memory. If I tell you the pressure is 5 units, you know exactly how stiff it is.
• Sticky Sand: Has memory. If I tell you the pressure is 5 units, you have to ask: "Did you just start squeezing it, or were you squeezing it hard and then let go?"
  • If you just started squeezing, it's softer.
  • If you let go from a hard squeeze, it's stiffer.

The researchers proved that this memory exists because the "glue" allows the particles to form a stable network that doesn't rely solely on being squeezed. The particles "remember" the connections they made when they were squeezed tight, and they refuse to let go even when the pressure drops.

The Conclusion: A New Rulebook

The main takeaway is that for a long time, scientists thought "Pressure" was the only thing that mattered to describe how jammed materials behave.

This paper says: That's only true for dry sand.

For sticky materials (like wet sand, mud, or even some biological tissues), pressure isn't enough. You also need to know the history of the material. The "glue" breaks the perfect balance (marginal stability) that exists in dry systems, creating a material that is tougher and more stubborn than expected.

In a nutshell:

• Dry Jamming: Like a perfect balance scale. Predictable, no memory.
• Sticky Jamming: Like a tangled ball of yarn with Velcro. It holds its shape even when you stop pulling, and it remembers how you tangled it up. This "memory" makes it much harder to predict how it will behave.

Technical Summary

1. Problem Statement

The physics of disordered solids near the jamming transition (the point where a granular assembly acquires rigidity) is well-understood for purely repulsive particles. In these systems, mechanical properties like the shear modulus ($G$) exhibit "marginal stability," meaning they are uniquely determined by the packing fraction ($\phi$), pressure ($p$), or coordination number ($Z$). Specifically, for repulsive particles, $G \sim p^{1/2}$, and the history dependence (hysteresis) observed during compression and decompression vanishes when properties are plotted against pressure.

However, cohesive interactions (attractive forces) are ubiquitous in real-world granular materials (e.g., wet sand, colloids). It remains unclear whether the standard theoretical frameworks (like Effective Medium Theory, EMT) and the concept of marginal stability apply to cohesive systems. The central problem is whether cohesive packings retain intrinsic history dependence even when characterized by pressure, and if so, what is the underlying microscopic mechanism.

2. Methodology

The authors employed a combination of Discrete Element Method (DEM) simulations and Effective Medium Theory (EMT) to investigate this problem.

• Simulation Setup:

  • System: Frictionless, monodisperse particles in a 3D cubic box with periodic boundary conditions.
  • Interaction Potential: A piecewise linear force law featuring a repulsive core, a short-range attractive tail, and a cutoff. The force $f(r)$ is strictly history-independent (no microscopic hysteresis).
  • Protocol: The system underwent $\phi$-controlled compression (increasing packing fraction) and decompression (decreasing packing fraction).
  • Measurements: At mechanical equilibrium, the authors measured pressure ($p$), shear modulus ($G$), and coordination number ($Z$). They also analyzed the vibrational density of states (vDOS).
  • Parameters: They compared purely repulsive systems ($\alpha = 0$) with cohesive systems ($\alpha > 0$, where $\alpha$ controls attraction strength).

• Theoretical Framework:

  • Effective Medium Theory (EMT): The authors extended EMT to cohesive packings. They analyzed the Hessian matrix (stability matrix) of the system, decomposing it into contributions from repulsive/stabilizing contacts and locally destabilizing contacts (where attractive forces create negative stiffness).
  • Stability Analysis: They examined the condition for marginal stability ($p = p_c(Z)$), where $p_c$ is the critical pressure, and investigated how attractive interactions alter the relationship between $G$, $p$, and $Z$.

3. Key Contributions

1. Discovery of Persistent Hysteresis: The study demonstrates that unlike repulsive systems, cohesive granular packings exhibit pronounced hysteresis in the shear modulus ($G$) even when plotted against pressure ($p$).
2. Breakdown of Marginal Stability: The authors identify that cohesive interactions violate the condition of marginal stability. In repulsive systems, $p$ and $Z$ are uniquely linked ($p = p_c(Z)$); in cohesive systems, this link breaks down, allowing multiple stable states for a given $Z$ and $p$.
3. Theoretical Extension of EMT: They successfully extended EMT to cohesive systems, showing that while the functional form of the shear modulus remains similar, the presence of attractive forces generates excess rigidity due to the deviation from marginal stability.
4. Scaling Relation Verification: They derived and numerically verified a generalized scaling law connecting the excess rigidity to the degree of marginal stability breakdown.

4. Key Results

A. Mechanical Hysteresis and Negative Pressure

• Repulsive Particles: During decompression, rigidity ($G$) vanishes exactly when pressure ($p$) reaches zero. The $G(p)$ curve collapses onto a single master curve ($G \propto p^{1/2}$), eliminating history dependence.
• Cohesive Particles:
  • Negative Pressure: During decompression, the system maintains a finite packing fraction and rigidity even when $p < 0$ (tension).
  • Persistent Hysteresis: When plotting $G$ vs. $p$, the compression and decompression paths do not collapse. $G$ remains finite as $p \to 0$ during decompression, whereas it starts from zero during compression. This indicates that pressure is not a unique state variable for cohesive particles.

B. Breakdown of Marginal Stability

• Pressure vs. Coordination Number: For repulsive particles, data points lie exactly on the marginal stability line $p = p_c(Z)$. For cohesive particles, data points systematically lie below this line ($p < p_c(Z)$).
• Implication: The unique relation between $p$ and $Z$ is broken. The coordination number $Z$ retains a "memory" of the compaction history, preventing $G$ from being determined solely by $p$.
• Vibrational Density of States (vDOS):
  • In marginally stable (repulsive) systems, the Hessian spectrum shows an excess of low-frequency soft modes compared to the unstressed Hessian.
  • In cohesive systems, this excess is strongly suppressed. The spectra of the stressed and unstressed Hessians nearly coincide, providing direct evidence that marginal stability is broken.

C. Quantitative Scaling Law

The authors derived a generalized scaling relation for the excess rigidity:
$$\frac{G - G_0(Z)}{G_0(Z)} = \sqrt{\frac{p_c(Z) - p}{p_c(Z)}}$$
Where:

• $G_0(Z)$ is the "bare" shear modulus (valid at marginal stability).
• $p_c(Z)$ is the critical pressure for marginal stability.
• The term $(p_c - p)/p_c$ represents the deviation from marginal stability.

Result: Numerical simulations for various attraction strengths ($\alpha$) and protocols (both $\phi$-controlled and pressure-controlled) collapse perfectly onto this theoretical prediction without fitting parameters. This confirms that the hysteresis is a direct consequence of the system operating in a regime of excess rigidity caused by the breakdown of marginal stability.

5. Significance

• Fundamental Physics: This work fundamentally reshapes the understanding of the jamming transition in cohesive materials. It proves that attractive interactions fundamentally alter the stability landscape, moving the system away from the marginal stability that characterizes repulsive jamming.
• State Variables: It establishes that for cohesive granular materials, pressure alone is insufficient to characterize the mechanical state; the coordination number (or contact network history) is an essential additional state variable.
• Theoretical Robustness: The successful application of EMT to cohesive systems provides a unified theoretical framework that explains why cohesive packings are "stiffer" than predicted by repulsive-only models at the same pressure.
• Practical Implications: These findings are crucial for modeling wet granular materials, powders, and biological tissues where cohesion plays a role, explaining why such materials often exhibit complex, history-dependent mechanical behaviors (like retaining shape under tension) that standard repulsive models cannot predict.