Pinch-off of non-Brownian rod suspensions: onset of heterogeneity and effective extensional viscosity

This study demonstrates that the pinch-off of density-matched non-Brownian rod suspensions reveals a continuum breakdown governed by rod length rather than diameter, leading to an effective extensional viscosity that increases with both volume fraction and aspect ratio while following a modified scaling law for spherical particles.

The Gist

Imagine you are squeezing a tube of toothpaste. If it's just plain toothpaste, it flows smoothly and snaps off cleanly. But what if you mixed in thousands of tiny, rigid sticks (like tiny pieces of spaghetti or fiberglass)? How would that mixture behave when you try to pull it apart?

This paper is a scientific investigation into exactly that question. The researchers wanted to know: At what point does a mixture of liquid and tiny rods stop acting like a smooth, uniform fluid and start acting like a messy collection of individual particles?

Here is the story of their discovery, broken down into simple concepts.

1. The Experiment: The "Pendant Drop" Test

The scientists didn't use a toothpaste tube; they used a syringe to let a drop of liquid hang from the tip, like a water droplet on a faucet. Then, they let gravity pull the drop down. As the drop stretches, the "neck" connecting it to the syringe gets thinner and thinner until it snaps.

They filled this liquid with nylon fibers (tiny rods) of different lengths and thicknesses. Some were short and fat (like a toothpick), and others were long and skinny (like a strand of spaghetti).

2. The Three Acts of the Breakup

When they watched the drop break apart using high-speed cameras, they saw a dramatic three-act play, similar to what happens with sand or balls, but with a twist because the particles are rods:

• Act 1: The Smooth Sailing (The "Equivalent Fluid")
  At first, the drop stretches smoothly. Even though it's full of rods, it acts like a single, thick liquid. The rods are all mixed in evenly. If you were a tiny ant inside the drop, you wouldn't see individual sticks; you'd just feel a thick, gooey fluid.
• Act 2: The Traffic Jam Breakup (The "Dislocation")
  As the neck gets thinner, things get chaotic. The rods can't all fit in the narrowing space anymore. They start to push each other aside, creating empty pockets where there are no rods at all. The liquid starts to flow only through these empty pockets. It's like a crowded hallway where people suddenly realize they can only squeeze through if they stop holding hands and let the gaps form. The drop starts to thin out much faster here because the "traffic" (the rods) has cleared out of the way.
• Act 3: The Final Snap (The "Pure Liquid")
  Eventually, the neck becomes so thin that no rods are left in it. It's just pure liquid again. The drop snaps off exactly like a drop of water would.

3. The Big Discovery: Length Matters More Than Width

The most surprising finding was about which size of the rod matters most.

You might think that the thickness of the rod (its diameter) determines when the mixture gets messy. But the researchers found that it's actually the length of the rod that controls the chaos.

• The Analogy: Imagine trying to fit a long, thin rope through a small hole. Even if the rope is very thin, its length makes it hard to fit through without tangling or bunching up.
• The Result: The point where the mixture stops acting like a smooth fluid depends on the length of the rods, not their thickness. If the rods are long, the mixture breaks down into chaos much earlier (when the neck is still relatively thick). If the rods are short, the mixture stays smooth for longer.

4. Viscosity: How "Thick" is the Soup?

The scientists also measured how "thick" (viscous) the mixture was.

• They found that adding more rods makes the liquid thicker, just like adding more sugar makes tea sweeter.
• However, the "thickness" measured by stretching the drop (extensional viscosity) was different from the "thickness" measured by spinning it in a machine (shear viscosity).
• The Metaphor: Think of stirring honey with a spoon (shear) versus pulling a strand of honey apart (extension). They feel different! The rods align differently depending on how you move them. In this study, the "stretching" test showed that long rods make the liquid feel much thicker than short rods do.

5. Why Does This Matter?

You might wonder, "Who cares about breaking a drop of liquid with tiny sticks?"

Actually, this happens everywhere in the real world:

• 3D Printing: When printing with materials containing fibers, the nozzle needs to handle these "breakups" without clogging or making a mess.
• Painting and Coating: When you spray paint or coat a surface, the liquid stretches and breaks into droplets. If the paint has fibers in it, understanding how they break apart helps make smoother, more even coats.
• Biological Fluids: Our blood and mucus contain fiber-like structures. Understanding how they flow and break helps us understand diseases and drug delivery.

The Bottom Line

This paper tells us that when you mix long, rigid sticks into a liquid, the liquid behaves smoothly at first. But as you stretch it, the sticks eventually get in each other's way, creating empty spaces that cause the liquid to snap apart faster.

The key takeaway? It's the length of the sticks, not their width, that decides when the smooth flow turns into a chaotic mess. This helps engineers design better materials for everything from industrial sprays to medical treatments.

Technical Summary

1. Problem Statement

The study addresses the fundamental question of when a particulate suspension ceases to behave as a homogeneous continuum fluid and begins to exhibit discrete, heterogeneous behavior. While this phenomenon is well-characterized for suspensions of non-Brownian spheres, it remains poorly understood for anisotropic particles, specifically rigid fibers.

Key challenges with fibers include:

• Multiple Scales: Fibers possess two characteristic lengths (diameter $D$ and length $L$), raising the question of which scale governs the onset of heterogeneity.
• Microstructure Complexity: Fibers introduce rotational degrees of freedom and complex microstructures (from disordered to flow-aligned states) that depend heavily on flow history and boundary conditions.
• Extensional Flow Gap: Most rheological data for fibers comes from shear flows. There is a lack of understanding regarding fiber behavior in extensional flows, such as capillary pinch-off, which is critical for processes like spraying, coating, and printing.

2. Methodology

The authors investigated the pinch-off dynamics of pendant drops formed from suspensions of rigid nylon fibers in a viscous, density-matched liquid.

• Materials:
  • Particles: Nylon fibers with two diameters (23 µm and 50 µm) and lengths ranging from 100 µm to 1900 µm, resulting in aspect ratios ($\lambda = L/D$) from 2.1 to 84.
  • Fluid: A Newtonian mixture of water, poly(ethylene glycol) ether, and Zinc Chloride. It was density-matched ($\rho \approx 1120$ kg/m³) to the nylon to eliminate sedimentation and buoyancy effects. The viscosity was $\eta_0 \approx 270$ mPa·s.
• Experimental Setup:
  • Pinch-off: Drops were extruded from a 5.5 mm capillary. High-speed imaging (Phantom VEO 710) captured the neck thinning and breakup.
  • Rheometry: Parallel-plate rheometry (Anton Paar MCR 302) was used to measure shear viscosity at shear rates of 1 s⁻¹ and 100 s⁻¹ for comparison.
• Analysis Approach:
  • The thinning neck thickness $h(t)$ was tracked over time.
  • The dynamics were compared to a pure liquid baseline. A linear time mapping ($\tau \sim \eta_r$) was applied to collapse the suspension data onto the pure liquid curve to identify the effective extensional viscosity ($\eta_r$) and the critical thickness ($h^*$) where the continuum description breaks down.

3. Key Results

A. Three Regimes of Pinch-off

High-speed imaging revealed three distinct regimes, similar to those observed in spherical suspensions:

1. Equivalent-Fluid Regime: At early times (large neck thickness), the suspension behaves as a homogeneous Newtonian liquid with an effective viscosity.
2. Dislocation Regime: As the neck thins, fluctuations in particle volume fraction appear. Deformation localizes in regions with fewer particles (lower viscosity), causing the suspension to "dislocate" (rods separate).
3. Interstitial Liquid Regime: Once the neck is devoid of rods, the dynamics revert to the thinning of the pure interstitial liquid.

B. The Critical Threshold ($h^*$) and Scaling Law

The transition from the equivalent-fluid regime to the dislocation regime occurs at a critical neck thickness $h^*$.

• Scaling: The authors found that $h^*$ scales with the rod length ($L$), not the diameter. The data collapses onto a master curve described by the scaling law derived previously for spheres:
  $$\frac{h^*}{L} \sim \eta_r^{1/3} \left( \frac{\ell_c}{L} \right)^{2/3}$$
  where $\ell_c$ is the capillary length.
• Implication: The relevant discrete scale for the breakdown of continuum rheology in rod suspensions is the particle length. The diameter has negligible direct effect on $h^*$.

C. Effective Extensional Viscosity

• The effective extensional viscosity ($\eta_r$) extracted from the pinch-off experiments increases with both volume fraction ($\phi$) and aspect ratio ($\lambda$).
• Comparison to Shear Viscosity: While the extensional viscosity follows similar trends to shear viscosity, the absolute values differ due to the different flow kinematics (extensional vs. shear) and evolving rod orientation.
• Modeling: Both extensional and shear viscosities are well-described by Mills' equation:
  $$\eta_r = \frac{1-\phi}{(1-\phi/\phi^*)^2}$$
  where $\phi^*$ is a critical volume fraction parameter.

D. Critical Volume Fraction ($\phi^*$) Dependence

The parameter $\phi^*$, representing the maximum packing fraction, was found to be not a universal constant but dependent on the flow type and particle geometry.

• Aspect Ratio Dependence: $\phi^*$ decreases monotonically as the aspect ratio $\lambda$ increases.
• Empirical Law: This dependence is captured by a sigmoid function:
  $$\phi^*(\lambda) = \phi^*_\infty + \frac{2(\phi^*_0 - \phi^*_\infty)}{1 + e^{\lambda/\lambda_c}}$$
• Flow Dependence: The fitted parameters for $\phi^*$ differ between the parallel-plate rheometer (shear flow) and the pinch-off experiment (extensional flow), indicating that the microstructure (alignment) selected by the flow significantly impacts the effective packing limit.

4. Significance and Conclusions

• Continuum Breakdown: The study confirms that pinch-off is a sensitive probe for detecting the breakdown of continuum rheology in anisotropic suspensions.
• Length Scale Control: It establishes that for rigid rods, the rod length (not diameter) is the governing scale for the onset of heterogeneous thinning.
• Unified Framework: Despite the complexity of fiber microstructures, the thinning dynamics can be described using methods developed for spheres, provided the correct length scale ($L$) is used.
• Practical Application: The findings provide a predictive framework for industrial processes involving fiber suspensions (e.g., coating, 3D printing, fiber spinning), where understanding the transition from homogeneous flow to particle-induced instability is crucial for controlling product quality.

In summary, the paper bridges the gap between spherical and fibrous suspension rheology in extensional flows, demonstrating that while microstructure matters, the geometric scaling laws for continuum breakdown remain robust when adapted for particle length.