Simulations of inertial liquid-lens coalescence with the pseudopotential lattice Boltzmann method

This study employs the pseudopotential multi-component lattice Boltzmann method to numerically investigate the coalescence dynamics of low-viscosity liquid lenses across a wide range of contact angles, revealing quantitative agreement with experiments for small angles, validating thin-sheet theoretical models up to approximately 40 degrees, and demonstrating that initial 3D bridge growth is independent of the equilibrium contact angle.

The Gist

Imagine you are watching two tiny, floating droplets of oil on a pond. Suddenly, they touch. Instead of just sitting there, they instantly merge, forming a single, larger drop. This process is called coalescence. It happens everywhere: in rain clouds, in your morning coffee, and in high-tech inkjet printers.

This paper is a deep dive into what happens when these droplets aren't just floating freely in the air, but are "lenses" sitting right on the boundary between two different liquids (like oil floating on water). The researchers wanted to understand the physics of this merge, especially when the droplets are shaped like thick, round lenses rather than flat puddles.

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

1. The Tool: A Digital "Pixel" Universe

To study this, the scientists didn't just use a microscope; they built a digital universe inside a supercomputer. They used a method called the Lattice Boltzmann Method.

Think of this like a video game grid. Instead of smooth, continuous water, the computer sees the liquid as millions of tiny, invisible pixels. The rules of the game (physics) tell these pixels how to bounce off each other, stick together, and flow. By running this "game" millions of times per second, they could watch the droplets merge in slow motion, seeing details that are too fast for human eyes or even real-world cameras to catch.

2. The Experiment: The "Bridge" Between Two Worlds

When two droplets touch, they don't instantly become one big blob. First, a tiny bridge of liquid forms between them, like a tiny rope connecting two islands. This rope then rapidly thickens and widens until the two islands become one.

The researchers asked: How does the shape of the droplets change how fast this bridge grows?

• The "Flat" Droplets (Small Angles): Imagine two droplets that are very spread out, almost like flat pancakes sitting on the water.
• The "Round" Droplets (Large Angles): Imagine two droplets that are very round and tall, like little domes or beads.

3. The Discovery: The "Thin Sheet" Rule Breaks

For the flat, pancake-like droplets, the scientists found that existing math formulas (called "thin-sheet equations") worked perfectly. It's like predicting how a thin sheet of paper folds; the math is simple and accurate.

However, when they tested the round, dome-shaped droplets (large contact angles), the old math failed.

• The Analogy: Imagine trying to predict how a thick, round beach ball merges with another beach ball using the same rules you'd use for a flat piece of paper. The rules don't work because the shape is too thick and curved.
• The Result: The old formulas overestimated how fast the bridge would grow for these round droplets. The real physics was more complex.

4. The 3D Surprise: The "Independent Growth"

When they simulated this in 3D (making it look like a real sphere rather than a 2D slice), they found something fascinating about the bridge.

• The Height vs. Width Race: As the bridge grows, it gets taller (height) and wider (radius).
• The Finding: For the round droplets, the width of the bridge grew at the exact same speed, regardless of how round the droplets were. It was as if the bridge's width had its own "autopilot" that didn't care about the droplet's shape.
• The Twist: The height, however, did depend on the shape. Rounder droplets made the bridge shoot up faster.

This means that at the very beginning of the merge, the bridge doesn't grow in a perfect, predictable ratio (like a circle expanding). Instead, it grows in a weird, non-linear way where the width and height are doing their own thing before they eventually sync up.

5. Why Does This Matter? (The Real World)

You might wonder, "Who cares about merging oil droplets?"

• Inkjet Printing: When printers spray ink, tiny droplets hit a wet surface. If they merge too fast or too slow, the picture looks blurry. Understanding these rules helps engineers print sharper images.
• Fog Harvesting: In dry places, we try to catch water from fog. The water droplets on the mesh need to merge and fall off efficiently. If they get stuck or merge poorly, we lose water.
• Making New Materials: Scientists use these liquid merges to mix chemicals for making medicines or electronic parts. Knowing exactly how they mix helps them control the final product.

The Bottom Line

This paper is like a map for the "first second" of a liquid merge. The researchers showed us that while we have good maps for flat droplets, the round ones are a different terrain entirely. They proved that our old math tools need an upgrade for rounder shapes, and they provided a new, more accurate digital model to help engineers design better printers, water collectors, and materials in the future.

In short: They used a supercomputer to watch tiny liquid droplets hug, and they learned that round hugs are much more complicated than flat ones!

Technical Summary

1. Problem Statement

The coalescence of liquid lenses (droplets floating at a liquid-liquid interface) is a critical process in applications such as inkjet printing, fog harvesting, and wet-on-wet coating. While the coalescence of suspended droplets and sessile droplets on substrates has been extensively studied, the dynamics of liquid-lens coalescence, particularly for lenses with large contact angles and at later stages of the process, remain underexplored.

Existing theoretical models, such as those based on thin-sheet equations, are generally limited to small contact angles ($\theta < 40^\circ$) and assume the lubrication approximation (film thickness $\ll$ characteristic length). There is a lack of comprehensive numerical data validating these models across a wide range of contact angles and investigating the transition from non-linear to linear growth regimes in three dimensions.

2. Methodology

The authors employed the Pseudopotential Multi-Component Lattice Boltzmann Method (PMLB) to simulate the fluid dynamics of liquid lens coalescence in both 2D and 3D.

• Numerical Framework:
  • Model: Shan-Chen pseudopotential model with a D3Q19 lattice.
  • Collision Operator: Single Relaxation Time (SRT) BGK scheme.
  • Interfacial Physics: A mean-field interaction force is introduced between fluid components to generate surface tension. The "effective mass" formulation ensures numerical stability at high densities.
  • Advantages: Compared to Color-Gradient LBM (CGLB) or Cahn-Hilliard-Navier-Stokes (CHNS) approaches, PMLB offers lower computational cost and simpler implementation, though with slightly less flexibility in surface tension tuning.
• Simulation Setup:
  • System: Three fluid components (two lens fluids and one surrounding fluid) with equal viscosity and density.
  • Regime: Simulations were conducted in the inertial regime (low Ohnesorge number, $Oh \ll 1$), where inertia dominates over viscosity.
  • Dimensions:
    • 2D: Used to validate against experimental data and thin-sheet theory. Initial lens height $H=1000$ lattice units.
    • 3D: Used to investigate cross-sectional shapes and radius growth. Initial lens height $H=100$ lattice units (due to computational constraints).
  • Parameters: A wide range of equilibrium contact angles ($\theta$) was tested, from $22^\circ$ to $90^\circ$.

3. Key Contributions

1. Extension to Large Contact Angles: The study extends numerical analysis beyond the small-angle limit, investigating coalescence dynamics for lenses with large contact angles ($\theta > 40^\circ$) where traditional thin-film theories fail.
2. Validation of Thin-Sheet Theory: The authors quantitatively validate the range of applicability for thin-sheet equations, confirming their accuracy for $\theta < 40^\circ$ but demonstrating their breakdown at larger angles.
3. 3D Dynamics Discovery: In 3D simulations, the authors identified a distinct decoupling between bridge height and radius growth at the initial stage, a phenomenon not observed in sessile droplet coalescence.
4. Cross-Sectional Evolution: The paper characterizes the evolution of the bridge's cross-sectional shape, showing a transition from a non-linear to a linear dependency between bridge height and radius.

4. Key Results

A. Single Lens Equilibrium (Validation)

• The PMLB model accurately reproduced the equilibrium shape of a single liquid lens (spherical cap) across various surface tension ratios.
• Simulation results for lens height profiles matched analytical solutions derived from Neumann's law and geometric constraints.

B. 2D Coalescence Dynamics

• Bridge Growth Scaling: For small contact angles ($\theta < 40^\circ$), the bridge height ($h_0$) follows the theoretical scaling law $h_0 \sim t^{2/3}$, consistent with the inviscid thin-sheet theory.
• Breakdown at Large Angles: For $\theta = 59^\circ$, the thin-sheet theory overestimates the bridge growth because the lubrication assumption (thin film) is violated. Conversely, for $\theta = 90^\circ$ (suspended droplets), the theory underestimates growth.
• Self-Similarity: The bridge profiles and horizontal velocity profiles exhibit self-similar dynamics, collapsing onto universal curves when rescaled by time and contact angle. This confirms the validity of similarity solutions in the inertial regime.
• Oscillations: Unlike the inviscid theoretical prediction, the simulations showed damped velocity oscillations due to viscous effects in the numerical model.

C. 3D Coalescence Dynamics

• Decoupled Growth: In the initial stage, the growth of the bridge radius ($r_0$) is independent of the equilibrium contact angle. This contrasts with sessile droplets, where radius growth is coupled to contact angle.
  • Explanation: The rapid coalescence speed at a fluid-fluid interface prevents the cross-section from relaxing immediately to the equilibrium contact angle shape.
• Cross-Sectional Shape: The bridge cross-section maintains a spherical cap shape, but the instantaneous contact angle ($\theta_c$) of this cross-section is initially lower than the equilibrium angle ($\theta_{eq}$).
• Transition Regime:
  1. Initial Stage: $\theta_c$ increases rapidly as the bridge radius grows faster than the height (non-linear dependency).
  2. Intermediate Stage: $\theta_c$ reaches a plateau, indicating a transition to a linear dependency between bridge height and width ($r_0/h_0 = \text{const}$).
  3. Final Stage: Oscillations occur before the system relaxes to the equilibrium contact angle.

5. Significance and Implications

• Theoretical Limits: The work clearly defines the boundaries of thin-sheet theory, establishing that it is only quantitatively accurate for liquid lenses with contact angles up to approximately $40^\circ$.
• Methodological Validation: It demonstrates the robustness of the Pseudopotential LBM for ternary fluid systems, providing a computationally efficient alternative to more complex phase-field or color-gradient methods for studying multiphase flows.
• Industrial Applications: The findings provide critical timescale estimates for liquid lens coalescence. This is directly applicable to optimizing wet-on-wet inkjet printing and coating processes, where the competition between coalescence and evaporation determines the final deposition pattern and film quality.
• Future Research: The paper highlights the need for a new theoretical model that can predict the critical conditions for the transition from non-linear to linear growth regimes in 3D, a gap that remains to be filled.