Unjamming in a 3D Granular System: The Micromechanical Role of Friction in Force Distributions and Rheological Properties

Using Discrete Element Method simulations, this study investigates how interparticle friction influences the evolution of force distributions, coordination numbers, and packing density during the unjamming transition of 3D granular systems driven by random particle extraction.

The Gist

Imagine a giant, 3D jar filled with thousands of tiny marbles. This jar is sitting on a table, and gravity is pulling everything down. In this paper, scientists are asking a simple but tricky question: What happens when you start pulling marbles out from the very bottom of the jar?

Usually, when we think of sand or grains, we think of them as solid (like a sandcastle) or liquid (like pouring sand from a bucket). But there's a weird middle ground called "jamming." Think of it like a traffic jam. When cars are packed tightly, they can't move (solid). If you clear a few cars, they might still be stuck. But eventually, if you remove enough, the whole line starts to flow (liquid).

The researchers wanted to see exactly how a pile of sand "un-jams" (turns from solid to flowing) when you pull pieces out from the bottom, rather than shaking it or pouring it.

Here is the story of their experiment, explained simply:

1. The Setup: The "Digital Sandcastle"

Instead of using real sand (which is messy and hard to track), they built a digital world using a computer.

• They created a box with 30,000 virtual spheres (like marbles).
• They let them fall and settle under gravity, just like real sand.
• Then, they started a "controlled demolition." Every few seconds, they randomly picked 30 marbles from the bottom half of the box and deleted them.
• The Goal: Watch how the remaining marbles react. Do they just sit there? Do they slide down? Do they collapse?

2. The Two Acts of the Play

The computer simulation showed that the sand behaves in two distinct "acts":

• Act 1: The Quiet Phase. At first, when they start removing marbles, nothing dramatic happens. The sand just settles a tiny bit. The remaining marbles are still holding hands tightly. It's like removing a few people from a crowded elevator; everyone just shifts slightly, but the elevator stays still.
• Act 2: The Collapse. Suddenly, the system hits a tipping point. The sand stops just "settling" and starts flowing. The whole column of sand starts to sink at a steady speed. It's as if the "floor" of the elevator suddenly vanished, and everyone slides down together.

3. The Secret Ingredient: Friction (The "Stickiness")

The scientists played with a variable called friction.

• Low Friction (Smooth Marbles): Imagine Teflon-coated marbles. They slide past each other easily.
• High Friction (Rough Marbles): Imagine marbles covered in sandpaper. They grip each other tightly.

The Discovery:

• Rough marbles can hold together even when the pile is less dense. They are like a group of people holding hands tightly; they can stand even if the crowd is spread out a bit.
• Smooth marbles need to be packed very tightly to stay stable. If you pull a few out, they slide apart immediately.

The researchers found a mathematical rule: The stickier the marbles, the lower the density can get before the whole thing collapses.

4. The "Force Chains": The Invisible Skeleton

This is the coolest part. In a pile of sand, not all marbles carry the same weight.

• Imagine a group of people standing in a circle. If one person leans on another, that person leans on the next, creating a chain of support.
• In the sand, these are called Force Chains. They are invisible lines of marbles that carry the heavy load, while other marbles just hang around doing nothing.

What happened when they pulled marbles out?

• At first, the "skeleton" of force chains was strong and complex.
• As they removed marbles, the skeleton got weaker. The chains broke.
• Eventually, the remaining chains became very "unfair." A tiny few chains had to carry almost all the weight, while most chains carried nothing.
• The scientists used a fancy math tool (called the Gini Coefficient, usually used to measure wealth inequality) to measure this. They found that at the moment of collapse, the "inequality" of the forces hit a specific, universal number, regardless of how sticky the marbles were. It's like saying, "No matter how you build the tower, it always breaks when the weight distribution gets this specific kind of unfair."

5. The Big Picture: Why Does This Matter?

You might wonder, "Who cares about pulling marbles out of a computer box?"

This helps us understand real-world disasters:

• Landslides: When soil loses its grip and slides down a hill.
• Sinkholes: When the ground suddenly gives way.
• Asteroids: How loose piles of rock in space behave when they get hit.

The paper shows that even though the "trigger" (pulling marbles from the bottom) is different from other ways sand moves (like shaking it), the rules of the collapse are surprisingly similar. Whether you shake a jar of sand or pull the bottom out, the sand follows the same "un-jamming" rules once it hits that critical point.

The Takeaway

The scientists proved that friction is the hero that keeps granular materials (like sand, grains, or rocks) stable. But once you remove enough material, the "force chains" that hold the structure together break down in a very predictable way. The system goes from a solid block to a flowing liquid, and it does so by concentrating all the stress onto a few remaining "hero" particles before finally giving up.

It's a bit like a game of Jenga: you can pull out blocks for a while, and the tower stands. But eventually, you reach a point where the remaining blocks are holding everything, and the next move makes the whole thing crash. This paper tells us exactly how "sticky" the blocks need to be to delay that crash.

Technical Summary

1. Problem Statement

The paper investigates the unjamming transition in three-dimensional (3D) granular systems composed of frictional spheres. While jamming transitions (solid-to-liquid) are well-studied in shear-driven or gravity-free decompression scenarios, the specific mechanics of unjamming under gravity with a controlled reduction in packing fraction remain less understood.

The authors address the following challenges:

• How does the system lose mechanical stability when particles are removed rather than sheared?
• How does the interparticle friction coefficient ($\mu_p$) influence the critical packing fraction ($\phi_c$) and coordination number ($Z_c$) at the point of unjamming?
• What are the micromechanical changes in force networks (force chains) and stress heterogeneity as the system approaches marginal stability?

2. Methodology

The study employs Discrete Element Method (DEM) simulations using the open-source software MercuryDPM.

• System Setup:

  • Particles: $N_0 = 30,000$ spherical particles ($d=1$ mm, density $\rho=2500$ kg/m$^3$).
  • Geometry: Confined in a rectangular box ($L_x = L_y = 24d$) with solid walls.
  • Physics: Particles interact via a linear spring-dashpot model (normal and tangential) with Coulomb friction. Rolling friction is included to account for surface roughness.
  • Gravity: Acts vertically, creating an anisotropic stress state.

• Driving Protocol (Particle Extraction):

  • Instead of shear, the system is driven out of equilibrium by randomly removing particles from the bottom half of the container at regular intervals ($\Delta t_e = 0.02$ s).
  • This "extraction-driven" protocol simulates a controlled loss of support, allowing the system to evolve from a jammed (solid-like) state toward an unjammed (liquid-like) state.
  • The static interparticle friction coefficient $\mu_p$ is varied as the primary control parameter ($\mu_p \in [0.2, 0.9]$).

• Key Metrics:

  • Macroscopic: Number of active particles ($N_K$), center of mass height ($z_{cm}$), packing fraction ($\phi$), and coordination number ($Z$).
  • Microscopic: Normal force distributions ($P(f_n)$), plasticity index ($\zeta = f_t / \mu_p f_n$), and force chain statistics.
  • Statistical Tool: The Gini coefficient is applied to the plasticity index and force chain intensities to quantify the heterogeneity of stress distribution.

3. Key Contributions

1. Novel Unjamming Protocol: The authors introduce a systematic, extraction-driven approach to study unjamming in 3D gravity-confined systems, distinct from traditional shear-flow or isotropic compression methods.
2. Gini Coefficient Application: A novel quantification of contact force heterogeneity is achieved by applying the Gini coefficient to the plasticity index, providing a robust metric for stress distribution inequality.
3. Friction-Dependent Critical States: The study establishes a quantitative relationship between interparticle friction and the critical structural parameters ($\phi_c$ and $Z_c$) at the unjamming point, showing consistency with rheological frameworks despite the different driving mechanism.
4. Force Network Evolution: Detailed characterization of how force chains fragment and how stress heterogeneity evolves (increasing) as the system approaches the critical point.

4. Key Results

A. Dynamical Regimes

The system exhibits two distinct regimes during particle extraction:

1. Initial Regime (Jammed): Particle motion is limited. The center of mass ($z_{cm}$) rises linearly (due to bottom removal), and the packing fraction ($\phi$) decreases linearly. Rearrangements are localized.
2. Late Regime (Unjamming/Unstable): The system enters a state of sustained, collective rearrangement. $z_{cm}$ begins to decrease linearly (collapse), and the number of active particles ($N_K$) saturates. The system maintains a constant, critical packing fraction ($\phi_c$) despite continued particle loss.

B. Critical Structural Parameters

• Critical Packing Fraction ($\phi_c$): As $\mu_p$ increases, $\phi_c$ decreases. The relationship follows a rational function:
  $$\phi_c = \frac{p_1 \mu_p + p_2}{\mu_p + q_1}$$
  This indicates that higher friction allows the system to remain stable at lower densities (closer to Random Loose Packing).
• Critical Coordination Number ($Z_c$): $Z_c$ decreases as $\mu_p$ increases. The relationship between $Z_c$ and $\phi_c$ aligns with the Random Loose Packing (RLP) prediction by Song et al., confirming that friction reduces the number of contacts required for stability.

C. Force Distributions and Heterogeneity

• Normal Forces: The distribution of normal forces ($P(f_n)$) transitions from an exponential decay (characteristic of dense states) to a "stretched exponential" with a wider tail ($c \approx 0.8$) as unjamming approaches. This indicates a reorganization where remaining forces become more heterogeneous.
• Plasticity Index ($\zeta$): The mean plasticity index $\langle \zeta \rangle$ decreases with higher $\mu_p$. The Gini coefficient of $\zeta$ ($G_\zeta$) decreases over time but stabilizes at a critical value dependent on $\mu_p$, suggesting that higher friction leads to more heterogeneous load sharing (more "strong" chains).
• Force Chains:
  • The number of force chains and their average intensity decrease as the system approaches unjamming.
  • Universal Saturation: The Gini coefficient of force chain intensities ($G_F$) converges to a universal value of $\approx 0.284$ for all friction coefficients at the unjamming point. This suggests that the final state of maximal stress heterogeneity is governed by geometric constraints and the loss of stability rather than frictional dissipation.

5. Significance

• Theoretical Consistency: The study demonstrates that despite the fundamentally different driving mechanism (extraction vs. shear), the friction-dependent trends in critical packing fraction and coordination number are structurally consistent with established rheological models (e.g., $\mu(I)$ rheology) and shear-flow experiments.
• Mechanism of Failure: It clarifies that unjamming in gravity-confined systems occurs through the fragmentation and weakening of the force network rather than the growth of long-range correlations. The system loses stability by becoming unable to sustain its own weight as the contact network thins.
• Practical Applications: The findings are relevant for understanding natural and industrial processes involving granular erosion, silo discharge, and localized failure in granular assemblies (e.g., landslides, asteroid regolith dynamics), where material is removed rather than sheared.

In conclusion, the paper provides a comprehensive micromechanical picture of how friction dictates the path to marginal stability in 3D granular media, bridging the gap between extraction-driven unjamming and classical rheological descriptions.