Ideal wet two-dimensional foams and emulsions with finite contact angle

Simulations using the Surface Evolver demonstrate that ideal two-dimensional foams and emulsions with finite contact angles develop spontaneous inhomogeneities (flocculation) at high liquid fractions, with disordered foams exhibiting steady growth of this instability without requiring external perturbation.

The Gist

Imagine a crowded dance floor where everyone is holding hands, forming a perfect honeycomb pattern. This is a dry foam. Now, imagine someone starts pouring a little bit of water onto the floor. The dancers (bubbles) get a bit squishy, and the spaces between them fill with liquid. This is a wet foam.

For a long time, scientists thought that if you kept adding water, the dancers would just get squishier and squishier, but they would stay in their neat, organized lines. They assumed the "glue" holding the dancers together was perfectly uniform.

But this paper says: "Not so fast!"

The researchers discovered that if the "glue" between the bubbles has a specific property (called a finite contact angle), the neat dance floor eventually falls apart. Instead of staying in a perfect grid, the bubbles start clumping together, leaving huge empty puddles of water in between. In the world of emulsions (like salad dressing or cream), this clumping is called flocculation.

Here is the breakdown of their discovery using simple analogies:

1. The "Sticky" Glue

In a normal foam, the surface tension (the "skin" of the bubble) is the same everywhere. But in this study, the researchers looked at foams where the "skin" between two bubbles is slightly weaker than the skin between a bubble and the surrounding water.

Think of it like this:

• Normal Foam: The bubbles are like magnets that repel each other slightly. They want to stay evenly spaced.
• This Foam: The bubbles have a "sticky" side. When they get close, they want to stick together more than they want to stay apart.

2. The Ordered Dance (The Hexagonal Foam)

First, the scientists looked at a perfectly organized foam (like a honeycomb).

• The Setup: They slowly added water.
• The Result: At first, everything looked fine. But once they added just a little bit too much water, the perfect grid became unstable. It was like a house of cards that was holding its breath; it needed a tiny nudge (a perturbation) to fall.
• The Collapse: Once nudged, the bubbles suddenly rearranged themselves. They stopped being a grid and started forming clumps separated by cracks of pure liquid. The energy of the system dropped, meaning the clumped state was actually more comfortable for the bubbles than the neat grid.

3. The Chaotic Dance (The Disordered Foam)

Next, they looked at a messy, random foam (like bubbles in a real glass of beer).

• The Setup: They added water to this messy crowd.
• The Result: This time, they didn't even need a nudge. The chaos happened spontaneously.
• The Process: As water was added, small groups of bubbles started sticking together. These groups grew bigger and bigger, swallowing up their neighbors. Meanwhile, the liquid between the groups grew into massive, empty lakes.
• The Analogy: Imagine a crowded party where people start hugging in small groups. Suddenly, these groups merge into one giant, massive hug in the center of the room, leaving the rest of the room empty and filled with punch.

4. Why Does This Matter?

The paper explains that for a long time, scientists ignored this "sticky" effect because they thought the contact angle was zero (perfectly smooth). But in the real world, it's rarely zero.

• In Food and Cosmetics: This explains why your mayonnaise might separate, or why your face cream might clump up. The droplets are "sticky" and want to form clusters (flocculation) rather than staying evenly distributed.
• The "Wet Limit": Scientists used to think there was a maximum amount of water a foam could hold before it broke. This paper says that with sticky bubbles, that limit doesn't really exist in the way we thought. Instead of just getting wetter, the foam changes its entire structure, turning into a "clump surrounded by a lake."

The Takeaway

The main lesson is that small changes in how bubbles touch each other can lead to massive changes in how the whole system behaves.

If you have a system of bubbles (or droplets) that are slightly "sticky," adding more liquid won't just make them wetter; it will cause them to collapse into clusters, leaving large gaps of liquid behind. It's a reminder that in the world of soft materials, order is fragile, and a little bit of stickiness can turn a perfect pattern into a chaotic clump.

Technical Summary

Here is a detailed technical summary of the paper "Ideal wet two-dimensional foams and emulsions with finite contact angle" by Cox et al.

1. Problem Statement

The paper addresses a gap in the theoretical understanding of two-dimensional (2D) foams and emulsions, specifically regarding the role of finite contact angles ($\theta$) between liquid films and Plateau borders.

• Context: Traditional models of 2D foams often assume a zero contact angle ($\theta = 0$), where liquid films meet Plateau borders tangentially. However, in real systems (such as surfactant solutions with electrolytes), finite contact angles can exist (up to $\sim 17^\circ$).
• Phenomenon: In liquid-liquid emulsions, finite contact angles lead to "flocculation" (spontaneous clustering of droplets). The authors investigate whether this phenomenon occurs in ideal 2D foams and how it affects the system's stability and energy landscape as the liquid fraction ($\phi$) increases.
• Objective: To determine how finite contact angles alter the "wet limit" (the maximum liquid fraction before the foam structure breaks down) and to analyze the onset of structural inhomogeneities (clustering/cracking) in both ordered and disordered foam systems.

2. Methodology

The study relies heavily on numerical simulations using Ken Brakke's Surface Evolver software, which minimizes surface energy subject to geometric constraints.

• System Setup:
  • Ordered Foams: Simulated as monodisperse hexagonal lattices (256 cells) with periodic boundary conditions.
  • Disordered Foams: Simulated as polydisperse systems ($N=1500$ bubbles) generated via Voronoi construction with periodic boundary conditions.
  • Physical Model: The foam is treated as incompressible. The model distinguishes between two interface types:
    1. Plateau borders: Liquid-filled regions with bulk interfacial tension $\gamma$.
    2. Thin films: Gas-liquid-gas interfaces with tension $2\gamma_f$.
  • Contact Angle Implementation: The contact angle $\theta$ is defined by the ratio of tensions: $\theta = \cos^{-1}(\gamma_f / \gamma)$. The simulations vary $\theta$ from $\approx 2.6^\circ$ to $15.9^\circ$.
• Simulation Protocol:
  • Systems are initialized as "dry" foams ($\phi \approx 0.03$).
  • The liquid fraction $\phi$ is incrementally increased (steps of 0.001) up to $\phi \approx 0.25$ by expanding the system area while keeping bubble areas constant.
  • Topological Transitions: The software detects when film lengths approach zero ($10^{-4}$) or when Plateau border sides overlap, triggering discrete topological changes (merging or splitting of Plateau borders) to maintain energy minimization.
  • Perturbation: For ordered foams, a small random displacement of vertices is applied near critical liquid fractions to trigger instability.

3. Key Contributions

• Re-evaluation of the Wet Limit: The authors demonstrate that the concept of a single "wet limit" (maximum stable liquid fraction) is ill-defined for foams with finite contact angles.
• Distinction Between Order and Disorder:
  • Ordered Foams: Inhomogeneity (clustering) is a metastable phenomenon. The ordered hexagonal structure remains locally stable beyond the minimum energy point but requires an external perturbation to transition to a lower-energy inhomogeneous state.
  • Disordered Foams: Inhomogeneity develops spontaneously and steadily as $\phi$ increases, without the need for external perturbation.
• Flocculation Mechanism: The paper establishes that finite contact angles create effective attractive forces between bubbles (analogous to droplet flocculation in emulsions), driving the system toward phase separation (large liquid pools and bubble clusters).

4. Key Results

A. Ordered Hexagonal Foams

• Critical Liquid Fractions: Two critical values of $\phi$ are identified for a given $\theta$:
  1. $\phi_0^\theta$: The liquid fraction where energy is minimized (zero compression).
  2. $\phi_m^\theta$: The maximum liquid fraction where bubbles remain in contact (point contacts).
• Metastability: For $\phi > \phi_0^\theta$, the ordered hexagonal structure is metastable. The system can lower its energy by forming inhomogeneities (cracks or liquid pools).
• Triggering Instability: In simulations, the transition from a homogeneous ordered state to an inhomogeneous state requires a perturbation. Once triggered, the system forms "cracks" where bubbles separate, and the energy plateaus at a lower level than the metastable homogeneous branch.

B. Disordered Foams

• Spontaneous Inhomogeneity: Unlike ordered foams, disordered foams spontaneously undergo topological transitions (film rupture and Plateau border merging) as $\phi$ increases.
• Growth of Inhomogeneity:
  • The average Plateau border area grows sublinearly.
  • The area of the largest Plateau border ($A_{max}^{pb}$) grows exponentially with $\phi$.
  • This indicates a "flocculation" process where a few large liquid pools form while most bubbles remain in small clusters.
• Energy Behavior: The energy per bubble decreases as $\phi$ increases (due to the formation of lower-energy inhomogeneous structures) and eventually saturates to a roughly constant value.
• Contact Angle Dependence:
  • Smaller contact angles lead to more frequent topological transitions and faster growth of inhomogeneity.
  • Larger contact angles result in higher overall energy for a given $\phi$.

5. Significance and Implications

• Theoretical Impact: The findings challenge the standard assumption that 2D foams with finite liquid fractions remain homogeneous up to a specific wet limit. It suggests that finite contact angles fundamentally alter the thermodynamic stability of foams.
• Industrial Relevance: The results provide a theoretical basis for understanding flocculation in emulsions and "sticky" foams used in food and cosmetics. The spontaneous clustering observed in disordered simulations mirrors real-world instability in emulsions.
• Future Directions: The authors suggest that these results should prompt a re-examination of ideal disordered foam models (e.g., statistics of bubble rearrangements and avalanches) to include finite contact angles. They also call for experimental validation, noting a current lack of 2D data to compare with these simulations.

In summary, the paper establishes that finite contact angles induce a spontaneous tendency toward inhomogeneity (flocculation) in 2D foams, particularly in disordered systems, rendering the traditional concept of a sharp wet limit invalid.