Colloidal Suspensions can have Non-Zero Angles of Repose below the Minimal Value for Athermal Frictionless Particles

Experiments on dense colloidal silica suspensions in rotating microfluidic drums reveal that while thermal agitation causes the smallest particles to exhibit zero angle of repose, larger particles arrest at finite angles that remain below the minimal value for athermal frictionless granular materials, a phenomenon explained by a rheological model linking the arrest to a crossover between glass and jamming transitions driven by the competition between gravitational and thermal pressures.

The Gist

Imagine you have a bucket of sand. If you pour it onto a table, it forms a little hill. If you tilt the table, the sand slides down until it finds a stable slope. That stable slope is called the angle of repose.

For normal, dry sand (which is "athermal," meaning it doesn't feel the tiny jiggles of heat), this angle is usually quite steep—about 30 degrees. Even if you use perfectly smooth, frictionless glass beads, physics says they can't form a pile flatter than about 5.8 degrees. Below that, gravity wins, and the pile collapses into a flat puddle.

The Big Question:
What happens if the particles are tiny? So tiny that they are constantly being bumped around by water molecules (like a crowd of people jostling a single person in a mosh pit)? This is called thermal agitation or Brownian motion.

Scientists have long known that if the particles are super small (like dust), this constant jiggling acts like a giant mixer. It keeps the pile moving forever, flattening it completely. The angle of repose becomes zero.

But what about the "Goldilocks zone" in between? The particles that are big enough to settle, but small enough to still feel the heat-jiggles? Do they flatten out completely, or do they stop at a tiny, non-zero angle?

The Experiment: The Micro-Tumbler
The researchers built a tiny, high-tech version of a cement mixer. Imagine a drum made of clear plastic, only 100 micrometers wide (about the width of a human hair). They filled these drums with water and silica particles (tiny glass beads) of different sizes, ranging from 2 to 7 micrometers.

They spun these drums to mix the particles, then stopped them. Gravity pulled the particles to the bottom, forming a pile. Then, they watched what happened over days, weeks, and even a month.

The Discovery: The "In-Between" Angle
Here is the magic they found:

1. The Super-Tiny Beads (The "Dust"): For the smallest particles, the water molecules were jiggling them so hard that the pile never stopped moving. It slowly crept and flattened until it was perfectly flat. Angle = 0°.
2. The Medium Beads (The "Sweet Spot"): As they used slightly larger beads, something surprising happened. The piles did stop moving, but they didn't stop at the "normal" 5.8-degree limit. They stopped at a tiny, non-zero angle (between 0.6° and 3.7°).
   • Think of it like this: Imagine a group of people trying to stand on a slippery slope. If they are tiny and being pushed by a crowd (thermal energy), they slide down. If they are huge and heavy, they lock together and stand firm at a steep angle. But if they are medium-sized, they are heavy enough to stay put, but the crowd's jostling is just enough to let them slide a little bit, settling at a very gentle slope that is flatter than any normal sand pile could ever be.
3. The Heavy Beads: As the particles got even bigger, the angle of the pile grew steeper, eventually approaching that 5.8-degree limit of normal sand.

Why Does This Matter?
This is a bridge between two worlds of physics:

• Colloids: The world of tiny particles where heat (jiggling) rules.
• Granular Materials: The world of sand and rocks where gravity and friction rule.

Usually, these two worlds are treated as completely separate. This experiment showed that there is a smooth transition zone where the rules of both worlds mix. The particles found a "happy medium" where they form a pile, but a flatter one than anyone thought possible for solid materials.

The "Creep" Factor
One of the coolest parts of the study was how slow it was. For the medium-sized beads, the pile didn't stop instantly. It "crept" down the slope at a snail's pace.

• For the smallest beads, it took hours to flatten.
• For the medium beads, it took weeks to settle into that tiny angle.
• The researchers had to be clever: they couldn't wait years to see if a pile would flatten, so they tilted the drums to different starting angles to "trick" the system into revealing its final resting spot faster.

The Conclusion
The scientists proved that you can have a pile of solid particles that is stable but incredibly flat—flatter than the "minimum" angle physics textbooks said was possible for frictionless sand. It turns out that when you mix the jiggling of heat with the weight of gravity, you get a new kind of material behavior that sits right in the middle of the liquid and solid worlds.

In short: Even solid piles can be "soft" if they are small enough and jiggly enough, settling at angles we never thought they could reach.

Technical Summary

1. Problem Statement

The study addresses a fundamental gap in the rheology of complex fluids: the transition between colloidal (thermal) and granular (athermal) regimes.

• Granular Limit: In athermal, frictionless granular materials, the angle of repose ($\theta_r$) is determined by geometric interlocking, with a theoretical minimum value of $\theta_{ath} \approx 5.8^\circ$. Friction typically increases this to $\sim 30^\circ$.
• Colloidal Limit: For very small particles ($d \le 2 \, \mu\text{m}$), thermal agitation (Brownian motion) dominates over gravity. This promotes slow creep flows that eventually flatten any pile, resulting in a zero angle of repose ($\theta_r = 0^\circ$).
• The Open Question: The behavior of dense suspensions in the intermediate regime—where particles are large enough to sediment but small enough to be influenced by thermal fluctuations—remains experimentally unexplored. Specifically, it is unknown if a stable, non-zero angle of repose exists that is lower than the minimal athermal limit ($\theta_{ath}$), or if the system transitions directly from $\theta_r = 0$ to $\theta_r \ge \theta_{ath}$.

2. Methodology

The researchers employed a microfluidic approach to systematically probe the crossover between thermal and athermal behaviors.

• Experimental Setup:
  • Device: A microfluidic rotating drum platform consisting of an array of 20 PDMS drums ($D=100 \, \mu\text{m}$, $W=50 \, \mu\text{m}$).
  • Materials: Monodisperse silica particles ($d \approx 1.93$ to $7.00 \, \mu\text{m}$) suspended in ultrapure water. The particles possess a negative surface charge, creating electrostatic repulsion that prevents direct solid-solid contact (effectively frictionless).
  • Observation: The drums were mounted vertically on a motorized stage and imaged via a horizontal microscope (up to 75 Hz) to track pile evolution over periods up to one month.
• Control Parameter:
  • The ratio of particle weight to thermal agitation was quantified by the Gravitational Péclet number ($Peg$):
    $$Peg = \frac{mgd}{k_B T}$$
    where $m$ is the buoyant mass, $g$ is gravity, $d$ is diameter, and $k_B T$ is thermal energy.
  • The study covered $Peg$ values from $\approx 15$ to $\approx 2500$.
• Measurement Protocol:
  • Flow-to-Arrest: Piles were tilted to an initial angle ($\theta_{start}$) and monitored until flow ceased.
  • Angle-Sweeping: For high $Peg$ values where relaxation times are prohibitively long, the team used a "sweeping" protocol. They initiated flow at progressively smaller angles ($\theta_{start} < 5^\circ$) to find the threshold where the pile stops flowing within the experimental timeframe.
  • Validation: The independence of the final angle from the initial tilt and drum geometry was verified.

3. Key Contributions

1. Experimental Discovery: The first experimental demonstration that dense colloidal suspensions can arrest at a finite, non-zero angle of repose that is strictly below the minimal athermal limit ($\theta_{ath} \approx 5.8^\circ$).
2. Bridging Regimes: The study successfully maps the continuous evolution of $\theta_r$ from $0^\circ$ (colloidal limit) to values approaching but remaining below $5.8^\circ$ (granular limit) as $Peg$ increases.
3. Validation of Rheological Model: The results provide the first experimental validation for a recent phenomenological model (Billon et al., 2023) that describes the arrest dynamics as a crossover between glass and jamming transitions.

4. Key Results

• Low $Peg$ Regime ($Peg \lesssim 35$):
  • Thermal agitation is sufficient to drive creep flows that completely flatten the pile.
  • Result: $\theta_r = 0^\circ$. No yield stress is observed.
• Critical Transition ($Peg \approx 68$):
  • A critical threshold exists where the system transitions from fluid-like to solid-like behavior.
  • Above this threshold, piles stop flowing at a finite angle.
• Intermediate/High $Peg$ Regime ($68 < Peg \lesssim 2500$):
  • The angle of repose $\theta_r$ increases with $Peg$.
  • Key Finding: $\theta_r$ remains strictly less than $\theta_{ath}$.
    • At $Peg \approx 68$, $\theta_r \approx 0.6^\circ$.
    • At $Peg \approx 2548$, $\theta_r \approx 3.7^\circ$.
  • This confirms the existence of a "thermal-to-athermal crossover" where the yield stress is non-zero but the geometric interlocking is not yet fully dominant.
• Model Agreement:
  • The experimental data fits the theoretical model of Billon et al. (Eq. 2 in the paper) with a single fitting parameter ($\alpha = 2.4$) relating the dimensionless pressure $\tilde{\Pi}$ to $Peg$.
  • The model predicts that the emergence of a non-zero $\theta_r$ corresponds to the system crossing the glass transition volume fraction ($\phi_G \approx 0.58$) as pressure increases.
  • Confocal imaging of a low-$Peg$ pile confirmed $\phi \approx 0.558$, just below $\phi_G$, supporting the hypothesis that increasing pressure (via larger particles) pushes the system into the glassy/jammed state.

5. Significance

• Theoretical Implications: The findings challenge the classical view that the minimal angle of repose for frictionless particles is fixed at $\approx 5.8^\circ$. They demonstrate that thermal fluctuations can suppress the yield stress to create stable, low-angle piles, effectively bridging the rheology of colloidal glasses and granular materials.
• Mechanism of Arrest: The study confirms that the arrest in these systems is driven by a competition between thermal pressure and granular pressure. As granular pressure increases relative to thermal energy, the system transitions from a fluid (glass transition) to a jammed solid, but the "softness" of the thermal component allows for angles lower than the purely geometric limit.
• Broader Context: This work provides a crucial experimental benchmark for models of yield stress in complex fluids and offers insights into how mechanical vibrations (analogous to thermal agitation) might influence the stability of granular piles, a topic previously lacking macroscopic predictive models.

In conclusion, the paper establishes that thermal agitation does not merely eliminate the angle of repose; it creates a distinct intermediate regime where stable, non-zero angles exist below the traditional granular limit.