🔬 condensed matter

Interface crossing behavior of prolate microswimmers: thermo and hydrodynamics

Using large-scale lattice Boltzmann simulations, this study reveals that the ability of prolate microswimmers to cross liquid-liquid interfaces depends on a competition between thermodynamic interfacial forces and active hydrodynamic forces, where a critical capillary number determines whether swimmers are trapped or successfully traverse the boundary.

Original authors: Rishish Mishra, Harish Pothukuchi, Harinadha Gidituri, Juho Lintuvuori

Published 2026-06-08

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CC BY 4.0

Original authors: Rishish Mishra, Harish Pothukuchi, Harinadha Gidituri, Juho Lintuvuori

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a microscopic world where tiny, self-propelled bacteria swim through liquids. Now, picture these swimmers trying to cross a boundary between two different liquids, like oil and water. This paper explores what happens when these microscopic "swimmers" hit that invisible wall.

Here is the story of their journey, explained simply:

The Characters and the Setting

The Swimmers: Think of them as tiny, cigar-shaped (prolate) robots or bacteria. They have a motor that pushes them forward at a constant speed.
The Boundary: This is the interface between two liquids that don't mix. In the real world, this is like the surface between oil and water.
The Goal: The swimmers want to cross from one side to the other.

The Great Tug-of-War

When a swimmer approaches this liquid boundary, it faces a battle between two opposing forces:

The "Active" Force (The Engine): This is the swimmer's own motor. It wants to push straight through the wall, just like a car trying to drive through a foggy patch.
The "Thermodynamic" Force (The Sticky Trap): This is a bit more subtle. Imagine the liquid boundary is like a stretched rubber sheet. When the swimmer touches it, it tries to "wet" the surface. Because of the shape of the swimmer and the nature of the liquids, the surface tension acts like a sticky trap. It wants to grab the swimmer and hold it right at the surface, much like a piece of dust gets stuck on a spiderweb. This is called the Pickering effect.

The Outcome: Trapped or Crossing?

The paper uses powerful computer simulations to see who wins this tug-of-war. The result depends on two main things:

How fast the swimmer is going.
The angle at which it hits the wall.

Scenario A: The Trap (The "Sticky Web")
If the swimmer is moving slowly or hits the wall at a shallow angle (like a stone skipping across water), the "sticky" force wins. The swimmer gets stuck at the boundary. It can't break through. Instead, it gets rotated by the forces until it lies flat, parallel to the surface, like a leaf floating on a pond. It is effectively trapped.

Scenario B: The Breakthrough (The "Bullet")
If the swimmer is moving very fast or hits the wall at a steep angle (almost straight on), its engine is strong enough to overpower the sticky trap. It punches through the boundary and keeps swimming on the other side.

The "Critical Capillary Number"

The researchers created a simple rule (a formula) to predict the outcome. Think of it as a "tipping point."

If the ratio of the swimmer's speed to the "stickiness" of the liquid is low, it gets trapped.
If the ratio is high, it crosses.

They found that their computer predictions matched real-world experiments with Bacillus subtilis bacteria almost perfectly. Even though the bacteria in the lab were in a slightly different type of liquid mixture, the physics of "trapped vs. crossing" was the same.

The Spin: Why Do They Turn?

One of the most interesting findings is how the swimmer turns when it gets stuck.

Hydrodynamic Spin: As the swimmer moves, it creates currents in the fluid (like a boat creating a wake). These currents push the swimmer to turn parallel to the wall.
Thermodynamic Spin: The "stickiness" of the surface also pulls on the swimmer, encouraging it to lie flat to minimize the energy of the surface.

It's a double whammy: both the water currents and the surface stickiness work together to rotate the swimmer until it is lying flat against the interface.

The Bottom Line

This paper explains that whether a microscopic swimmer gets stuck at a liquid boundary or crosses it is a simple game of physics: Engine Power vs. Surface Stickiness.

Slow/Shallow Angle? You get stuck and lie flat.
Fast/Steep Angle? You break through.

The researchers successfully modeled this behavior, proving that the "sticky" nature of the liquid surface (thermodynamics) is just as important as the swimmer's speed (hydrodynamics) in deciding its fate.

Technical Summary: Interface Crossing Behavior of Prolate Microswimmers

Problem Statement
The motility of microorganisms in aqueous environments is heavily influenced by feedback mechanisms from their surroundings, including fluid-fluid interfaces. While hydrodynamic interactions between microswimmers and interfaces are well-documented, the role of thermodynamic trapping forces (specifically the Pickering effect) in the dynamics of active particles crossing clean liquid-liquid interfaces remains less explored. Recent experiments by Cheon et al. (2024) observed that rod-shaped Bacillus subtilis bacteria could either cross or become trapped at an isotropic-nematic liquid-liquid interface depending on their velocity and initial projection angle. This study aims to investigate the trapping dynamics of prolate (rod-like) microswimmers at clean liquid-liquid interfaces, rationalizing the competition between interfacial thermodynamic forces and active hydrodynamic forces.

Methodology
The authors employed large-scale 3D lattice Boltzmann simulations to model the interaction of a prolate microswimmer with a clean liquid-liquid interface.

Swimmer Model: The microswimmer was modeled as a solid prolate particle (aspect ratio $AR = 3$) using the spheroidal squirmer model under Stokes flow. The swimmer was defined as a "pusher" ( $\beta = -1$ ) to mimic Bacillus subtilis, with a prescribed slip velocity on its boundary.
Fluid Interface: The binary fluid mixture was modeled using a Ginzburg-Landau free energy functional coupled with the Cahn-Hilliard equation for phase evolution and the Navier-Stokes equation for fluid velocity. The interface was treated as neutrally wetting (contact angle $\approx 90^\circ$ ).
Simulation Parameters: Simulations were conducted at low Reynolds numbers ( $Re \sim O(10^{-2})$ ). The capillary number ( $Ca = \eta u_0 / \sigma_0$ ) was varied by either changing the interfacial tension ( $\sigma_0$ ) or the swimming speed ( $u_0$ ). The initial projection angle ( $\phi_0$ ) of the swimmer relative to the interface was systematically varied.
Analysis: The study analyzed particle trajectories, orientation dynamics, and velocity fluctuations. A theoretical framework was developed based on an energy balance between the thermodynamic trapping energy (derived from the reduction of interfacial area) and the active energy required to cross the interface.

Key Contributions and Results

Phase Diagram of Crossing vs. Trapping: The simulations established a phase diagram in the space of capillary number ($Ca$) and initial alignment angle ( $\phi_0$ ). The results demonstrate that neutrally wetting swimmers can either cross the interface or become trapped, depending on the competition between active swimming forces and interfacial tension.
- Trapping: Occurs when interfacial (thermodynamic) forces dominate, typically at low $Ca$ or shallow initial angles. The swimmer aligns parallel to the interface.
- Crossing: Occurs when active forces dominate (high $Ca$), allowing the swimmer to traverse the interface.
- Validation: The numerical phase boundary aligns remarkably well with experimental observations of Bacillus subtilis crossing isotropic-nematic interfaces (Cheon et al., 2024), despite the different physical mechanisms proposed for the trapping force in the two studies.
Thermodynamic Trapping Mechanism: The study proposes that the trapping is driven by the Pickering effect. For a prolate swimmer, the energy cost to remove the particle from the interface depends on the contact area, which varies with the alignment angle $\phi$ . The authors derived a critical capillary number ( $Ca^*$ ) by balancing the thermodynamic trapping energy ( $\Delta E_{th} \propto \sigma_0 \cdot \text{Area}(\phi)$ ) against the active work done by the swimmer. This analytical prediction ( $Ca^*$ ) successfully defines the boundary between trapped and crossing states observed in simulations.
Reorientation Dynamics: The study reveals that the reorientation of swimmers parallel to the interface is driven by a combined torque with both hydrodynamic and thermodynamic components.
- Hydrodynamic Torque: Arises from the force dipole ( $B_2$ ) of the pusher swimmer and scales with the capillary number.
- Thermodynamic Torque: Arises from the gradient of the interfacial free energy ( $\tau_{thermo} \sim -\partial \Delta E_p / \partial \phi$ ) and depends on the alignment angle.
- Observations: Swimmers initially oriented at shallow angles may exhibit an initial reorientation toward the interface normal (due to far-field hydrodynamics) before being trapped and realigned parallel to the interface. The rate of reorientation increases linearly with $Ca$.
Potential Energy Landscape: By integrating velocity fluctuations and calculating free energy differences, the authors mapped the trapping potential. The potential is parabolic near the interface, with a depth that remains relatively constant for a fixed interfacial tension, confirming the dominance of thermodynamic trapping at low $Ca$.

Significance and Claims
The paper claims to provide a unified theoretical and numerical explanation for the interface crossing behavior of rod-like microswimmers. Its primary significance lies in:

Mechanism Identification: Demonstrating that the trapping of neutrally wetting prolate swimmers is dominated by the Pickering effect (thermodynamic reduction of interfacial energy), contrasting with previous experimental interpretations that attributed trapping to spring-like forces from interfacial deformation.
Predictive Capability: The derived critical capillary number based on an energy balance equation accurately predicts the transition between trapped and crossing states, showing excellent agreement with both the authors' simulations and independent bacterial experiments.
Dual Torque Origin: Establishing that the reorientation of microswimmers at interfaces is not solely a hydrodynamic phenomenon (as often assumed for spherical squirmers) but involves a significant thermodynamic component for non-spherical particles.

The authors conclude that their model, while differing in the specific physical origin of the trapping force compared to the Cheon et al. experiments, successfully captures the macroscopic dynamics and state transitions of microswimmers at liquid-liquid interfaces.