Collapse of a hemicatenoid bounded by a solid wall: instability and dynamics driven by surface Plateau border friction

This study presents numerical simulations and experimental measurements demonstrating that the collapse of a hemicatenoid soap film bounded by a solid wall is driven by viscous dissipation within surface Plateau borders, a mechanism distinct from the inertia-dominated collapse of free catenoids and relevant to bubble fragmentation in confined geometries.

The Gist

Imagine a soap film stretched between two rings. If you pull the rings too far apart, the film in the middle gets thin, wobbles, and suddenly snaps, collapsing into a tiny bubble. This is a classic physics problem that scientists have studied for over a century. Usually, this snap happens so fast that the air rushing in and the film's own weight are the main forces at play, while the "stickiness" (viscosity) of the soapy water barely matters.

But this paper explores a twist on that classic experiment.

The Setup: A Soap Film with a Wall

Instead of just two rings, the researchers placed a flat glass plate right through the middle of the soap film, splitting it into two halves. Think of it like slicing a bagel perfectly in half with a knife, but the "knife" is a solid wall the soap film is stuck to.

Now, instead of a full ring of soap, you have two "half-rings" (hemicatenoids) attached to the glass. When they pull the rings apart, these half-films still want to collapse, but they can't just snap freely. Their edges are sliding along the glass wall.

The Problem: The "Sticky" Edge

Here is the key discovery: In this new setup, the collapse isn't driven by the air rushing in. Instead, it's driven by friction.

Imagine the edge of the soap film where it touches the glass is like a runner on a track.

• The Old Way (3D Catenoid): The runner is on a frictionless ice rink. They sprint forward, and the speed depends on how hard they push (surface tension) and how heavy they are (air inertia). The stickiness of their shoes doesn't matter much.
• The New Way (Hemicatenoid): The runner is now dragging their feet through thick mud (the glass wall). The speed of the collapse depends entirely on how "slippery" or "sticky" that mud is.

The researchers call this moving edge a Surface Plateau Border (SPB). As the film collapses, this border has to slide along the glass. The paper argues that the resistance the border feels (friction) is what controls how fast the film shrinks.

The Experiment: Testing the "Mud"

To test this, the team made soap films with different levels of "mud" (viscosity). They added glycerol to the soap water to make it thicker and stickier.

• Thick Soap: When the soap was very thick, the film collapsed slowly.
• Thin Soap: When the soap was thinner, it collapsed faster.

This proved that, unlike the classic 3D version, the thickness of the liquid matters a lot here. The friction of the edge sliding on the wall is the boss of the show.

The "Martini Glass" Shape

As the film collapses, it doesn't just shrink evenly. It gets weirdly shaped. The researchers found that just before the film snaps, the neck of the soap film flattens out and looks like an upside-down Martini glass.

They measured the angle of this glass shape and found it was almost exactly the same (about 67–68 degrees) whether the soap was thick or thin, and whether it was a full ring or a half-ring. This suggests that the shape of the collapse is dictated by geometry (the rules of the wall and the rings), while the speed is dictated by the friction.

The Computer Model

The team built a computer simulation to match their real-world experiments. They tried different mathematical rules for how much friction the edge feels. They found that the rule that worked best was one where the friction increases in a specific, non-linear way as the edge moves faster. This rule fits with the idea that the soap film has "mobile" ingredients (surfactants) that make the surface act like a stress-free, slippery skin, but the interaction with the wall still creates drag.

Why This Matters (According to the Paper)

The paper concludes that understanding how these "half-films" collapse helps explain how bubbles break apart in very tight spaces. Specifically, they mention:

1. Porous Materials: Like foam inside rocks or soil.
2. Microfluidic Devices: Tiny machines that manipulate fluids in channels.

In these tight spaces, bubbles often get squeezed against walls, and their behavior is governed by the same friction rules the researchers discovered.

In short: The paper shows that when a soap film is stuck to a wall, it doesn't collapse like a free-floating balloon. It collapses like a runner dragging their feet in mud, where the stickiness of the liquid and the friction against the wall determine the speed, even though the final shape of the collapse remains a predictable "Martini glass."

Technical Summary

Technical Summary: Collapse of a Hemicatenoid Bounded by a Solid Wall

Problem Statement
This study investigates the dynamics of a soap film hemicatenoid collapsing under the influence of surface tension, specifically when the film is bisected by a solid glass wall. While the collapse of a full catenoid supported by two rings is a classic problem where surface tension balances air inertia (with film viscosity playing a minor role), the introduction of a solid wall creates a distinct class of boundary-driven dynamics. In this configuration, the film is bounded by two stationary semicircular frames and two movable Surface Plateau Borders (SPBs) that slide along the glass plate. The central problem is to understand how the frictional forces arising from viscous dissipation within these moving SPBs govern the collapse, contrasting this with the inertial dynamics of the bulk 3D catenoid.

Methodology
The research employs a combined approach of high-speed experimental observation and numerical simulation:

• Experimental Setup: Full catenoids were formed between two coaxial circular rings ($R=4$ cm). A glass plate was inserted through the diameters of the rings to create two independent hemicatenoids. The rings were separated beyond the critical instability threshold ($d/R > 0.663$). Experiments utilized TTAB soap solutions with varying glycerol concentrations to adjust viscosity ($\eta$) from 1.0 to 77 mPa·s while maintaining constant surface tension. High-speed imaging (up to 4,000 fps) captured the temporal evolution of the collapse.
• Numerical Model: A model was developed based on the separation of time scales: the bulk film motion is fast (inertial), while the SPB motion is slow (viscous). The film is approximated as a minimal surface at every instant, constrained by the moving contact lines. The dynamics of the SPBs are governed by a balance between the capillary driving force ($f_\gamma \cos \Psi$) and a viscous frictional force ($f_v$). The friction law is modeled as $f_v \sim Ca^n$, where $Ca$ is the capillary number and $n$ is an exponent representing the dissipation mechanism (ranging from $1/2$ to $2/3$). Simulations were performed using the Surface Evolver software, updating the SPB position based on the local contact angle $\Psi$ and the derived velocity law.

Key Results

1. Viscosity Dependence: Unlike the 3D catenoid collapse, which is largely independent of film viscosity, the hemicatenoid collapse dynamics show a strong dependence on viscosity. The contraction speed decreases as viscosity increases, confirming that the motion is dominated by SPB friction rather than air inertia.
2. Friction Law Exponent: By comparing experimental data (neck radius evolution and contraction speed) with simulations using different friction exponents ($n=1/2, 2/3, 1$), the study finds the best consistency with $n=2/3$. This exponent corresponds to systems with mobile surfactants and stress-free interfaces, implying that dissipation occurs primarily within the SPBs.
3. Geometric Evolution: The collapsing neck evolves from a circular cross-section to a flattened shape, eventually forming a "Martini glass" configuration (two conical films connected by a quasi-cylindrical region) just before pinch-off. The characteristic angle $\theta^*$ of this configuration is approximately $67.5^\circ$, consistent with the 3D catenoid case, suggesting that the large-scale shape is governed by intrinsic geometric features rather than the specific friction law.
4. Pinch-off Dynamics: A discrepancy was observed near the moment of pinch-off: simulations predict a constant maximum contraction speed until the transition, whereas experiments show a speed maximum followed by a decrease to zero. This is attributed to the complex dynamics of trapped air in the neck, a phenomenon also noted in 3D catenoid collapses.
5. Critical Viscosity: The study identifies a critical viscosity ($\eta_c \approx 40$ mPa·s) where the transition between inertial and frictional regimes occurs, consistent with an Ohnesorge number ($Oh$) of unity.

Significance and Claims
The paper claims to provide a quantitative explanation for the collapse of soap films in confined geometries where the boundary moves as a consequence of surface evolution. The primary contribution is the demonstration that for hemicatenoids, the dynamics are controlled by the frictional properties of the Surface Plateau Borders rather than the bulk film viscosity or air inertia.

The authors state that this understanding helps explain the fragmentation of bubbles in highly confined environments, such as porous materials and microfluidic devices, where similar SPB friction mechanisms control foam generation and bubble size distribution. Furthermore, the study connects these dynamics to other topological transitions in soap films, such as the reconnection of SPBs during the collapse of a Möbius strip, suggesting a broader relevance to boundary singularities in capillary surfaces. The work validates the use of nonlinear friction laws ($f \sim Ca^{2/3}$) in modeling the motion of surfactant-laden interfaces in contact with solid walls.