Two-Color LIF investigation of mixing during droplet impact onto a thin liquid film

This paper presents a two-color laser-induced fluorescence (2C-LIF) technique to simultaneously measure film thickness and scalar concentration during droplet impact on thin liquid films, enabling the quantification of mixing homogeneity and the identification of transport regime transitions across various flow conditions and fluid compositions.

The Gist

Imagine you are watching a single raindrop fall into a shallow puddle. To the naked eye, it looks like a simple splash: a drop hits, a little crown of water flies up, and then everything settles. But underneath that surface, a chaotic, high-speed dance is happening. The water from the drop is swirling, mixing, and colliding with the water already in the puddle.

This paper is about a team of scientists who built a special "magic camera" to watch that invisible dance in extreme detail.

The Problem: The "Black Box" of Mixing

For a long time, scientists could see where the water went (the splash shape), but they couldn't easily see how well the new drop mixed with the old water. It was like watching a magician mix two colors of paint in a jar, but the jar was opaque. You could see the jar shake, but you couldn't tell if the colors were blending perfectly or just sitting in separate blobs.

Previous methods were like looking at a black-and-white photo: they could tell you "this area has some drop-water" and "this area has puddle-water," but they couldn't tell you the exact ratio. Was it 50/50? 90/10? They also couldn't measure the depth of the water and the mixing at the same time without doing two separate, messy experiments.

The Solution: The "Two-Color" Flashlight

The researchers developed a technique called Two-Color Laser-Induced Fluorescence (2C-LIF). Think of it like this:

1. The Setup: They put a thin layer of water (the "puddle") on a glass plate.
2. The Tracers: They added two different "glow-in-the-dark" dyes to the water.
   • Dye A (The Drop): The falling drop contains a special dye that glows Red.
   • Dye B (The Puddle): The puddle contains a different dye that glows Blue.
3. The Magic Camera: They shine a bright green laser light on the scene. Both dyes glow, but they glow in different colors.
   • A super-fast camera splits the light into two channels: one looking only for Red, the other only for Blue.
   • By comparing how bright the Red is versus how bright the Blue is at every single pixel, the computer can instantly calculate two things at once:
     • How deep the water is (based on how much total light is coming through).
     • How much drop-water is mixed in (based on the ratio of Red to Blue).

It's like having a camera that can simultaneously tell you the depth of a swimming pool and exactly how much blue food coloring has spread through the water, all in a split second.

What They Discovered: The Dance of the Drop

Using this high-tech camera, they watched drops hit thin films of water at different speeds and thicknesses. Here is what they saw:

• The Vortex Ring (The Donut): When the drop hits, it doesn't just splash up; it creates a swirling ring of water underneath, like a donut made of liquid spinning on its side. This "donut" is the main engine that mixes the drop with the puddle.
• The Speed Matters:
  • Slow drops: The mixing is gentle. The "donut" stays together, creating neat, concentric rings of mixed water, like ripples in a pond.
  • Fast drops: The "donut" gets squashed and breaks apart. The mixing becomes chaotic and turbulent, scrambling the colors together much faster.
• The Thickness Matters: If the puddle is very thin, the bottom of the glass stops the swirling "donut" early, limiting how far the mixing spreads. If the puddle is deeper, the swirl can travel further out.

The "Homogenization" Score

To measure how well the mixing worked, the scientists invented a score called the Coefficient of Variation (CV).

• Imagine you have a jar of marbles with red and blue ones. If they are perfectly mixed, the jar looks purple everywhere. If they are separated, you see big red patches and big blue patches.
• The CV score measures how "patchy" the mixture is. A high score means the colors are still separated (bad mixing). A low score means they are blended (good mixing).

They found that the mixing happens in two stages:

1. The Chaotic Stage: The drop hits, and the swirling "donut" violently smashes the two liquids together. The CV score drops quickly.
2. The Calm Stage: Once the swirling stops, the liquids finish mixing slowly through a process called diffusion (like how a drop of ink slowly spreads in still water). The CV score levels off.

The Twist: The "Sour" Puddle

Finally, they tested what happens if the puddle isn't just water, but a mix of water and alcohol (like a very weak cocktail).

• Alcohol and water behave differently. When they mix, they create invisible "tension" forces (like a tight skin) that pull the liquid around.
• This created a new kind of mixing called Marangoni flow. Instead of just the drop's impact causing the swirl, the chemical difference between the alcohol and water started pulling the liquid around on its own. This made the mixing last longer and behave differently than in pure water.

Why Does This Matter?

You might wonder, "Who cares about a drop hitting a puddle?"
Actually, this happens everywhere in industry:

• Spray Painting: When a car is painted, thousands of droplets hit the surface. If they don't mix perfectly with the layer underneath, the paint will look streaky or peel off.
• Pharmaceuticals: When making medicine, droplets of active ingredients need to mix perfectly with a liquid base to ensure every pill has the right dose.
• 3D Printing: Some printers spray liquid layers that must merge perfectly.

By understanding exactly how drops mix with thin films, engineers can design better sprays, smoother paints, and more reliable medicines. This paper gave them the "magic camera" to finally see the invisible dance and learn the rules of the game.

Technical Summary

1. Problem Statement

Droplet impact on thin liquid films is a critical process in industrial applications such as spray coating, pharmaceuticals, and optical manufacturing. While extensive research exists on the macroscopic outcomes of these impacts (e.g., splashing, crown formation), there is a significant gap in the quantitative understanding of the internal mixing dynamics, particularly during the early, inertia-dominated stages.

Previous studies relied on:

• Dye-based visualization: Often providing only binary (dyed vs. undyed) information or qualitative spatial extent, lacking depth-resolved concentration data.
• Single-color Laser-Induced Fluorescence (LIF): While capable of measuring concentration or thickness, it cannot resolve both simultaneously from a single measurement. This necessitates sequential experiments, increasing uncertainty and experimental effort.
• Limitations: Existing methods fail to capture the transient coupling between film thickness deformation and scalar concentration transport in real-time.

2. Methodology

The authors developed and applied a Two-Color Laser-Induced Fluorescence (2C-LIF) technique to simultaneously reconstruct local film thickness and scalar concentration fields.

Experimental Setup:

• Fluids: Water droplets impacting thin films of pure water and binary ethanol-water mixtures (10% and 30% ethanol).
• Parameters: Reynolds numbers ($Re$) of 1800, 3000, and 4200; Weber numbers ($We$) of 24, 54, and 110; and dimensionless film thicknesses ($\delta = h/D$) of 0.09, 0.22, and 0.36.
• Optical System:
  • Excitation: Continuous-wave green LED ($\lambda \approx 532$ nm).
  • Tracers: Two fluorescent dyes were used: Rhodamine B (RhB) and Rhodamine 6G (Rh6G).
  • Labeling Strategy:
    • Pure Water: RhB in both droplet and film; Rh6G only in the film.
    • Ethanol-Water: Both dyes in droplet and film (fixed concentrations) to isolate thickness and ethanol fraction effects, as Rh6G fluorescence is sensitive to ethanol concentration.
  • Detection: Two high-speed CMOS cameras (6000 fps) capturing two spectral bands (561 nm and 624 nm) simultaneously via dichroic mirrors and band-pass filters.

Calibration and Reconstruction:

• The fluorescence signal depends on light intensity, quantum yield, path length (film thickness), and concentration.
• The system measures total intensity ($S = I_1 + I_2$) and intensity ratio ($R = I_2/I_1$).
• Pixel-wise Calibration: A 3D polynomial regression model was fitted for each pixel to map $(S, R)$ to local film thickness ($h$) and concentration ($C$). This accounts for optical non-uniformities (vignetting) and non-linear fluorescence responses.
• Validation: Reconstruction errors were validated against known references, showing mean absolute errors of ~24 $\mu$m for thickness and $1.67 \times 10^{-6}$ M for concentration in water films.

3. Key Contributions

1. Simultaneous Measurement: First demonstration of simultaneous, spatiotemporally resolved reconstruction of film thickness and scalar concentration in a single experiment, eliminating the need for sequential measurements.
2. Quantitative Mixing Metric: Introduction of the Coefficient of Variation (CV) of the concentration field as a global metric to quantify mixing homogeneity and identify the transition from convective to diffusion-dominated regimes.
3. Empirical Correlation: Formulation of an empirical scaling law relating the plateau value of the CV (final mixing state) to the Reynolds number and film thickness.
4. Extension to Multi-component Systems: Successful application of the 2C-LIF method to binary ethanol-water films, capturing complex transport mechanisms like Marangoni stresses.

4. Key Results

Film Thickness Dynamics:

• The 2C-LIF technique successfully resolved transient film deformations, including the formation of annular rims, central cavities, and upward-directed jets.
• Higher impact velocities (higher $We$) led to more pronounced rim elevations and earlier onset of crown structures due to dominant inertial forces.

Mixing Dynamics (Pure Water):

• Vortex-Driven Transport: Mixing is initially driven by the formation and evolution of a droplet-induced vortex ring.
  • Low $Re$: Axisymmetric ring-shaped mixing patterns.
  • Intermediate $Re$: Azimuthal perturbations lead to vortex ring destabilization and finger-like structures.
  • High $Re$: Strong coupling between the rising jet and the vortex ring, altering the mixing pattern.
• CV Evolution: The CV decreased over time, indicating homogenization.
  • The decay rate was faster for higher impact velocities.
  • A temporary increase in CV was observed during jet formation due to transient heterogeneity.
• Regime Transition: The study identified the transition time ($t_d$) from inertia-driven convection to diffusion-controlled mixing. Higher velocities prolonged the convective phase.
• Scaling Law: The final mixing state (plateau CV) was correlated as:
  $$CV_{pl} = A_0 Re^{-\alpha} \delta^{-\beta} + A_{floor}$$
  Where increasing $Re$ improves mixing (lower CV), and thinner films ($\delta$) show larger normalized concentration variations.

Binary Ethanol-Water Films:

• Marangoni Effects: The presence of ethanol introduced surface tension gradients, generating Marangoni stresses.
• Altered Dynamics: Unlike pure water, where coherent vortex rings dominated, ethanol films exhibited additional small-scale convective motions.
• Prolonged Mixing: Solutal Marangoni stresses extended the convective mixing phase, delaying the transition to the diffusion-dominated regime compared to pure water films.

5. Significance

This work provides a robust experimental framework for directly quantifying species transport in thin liquid films during droplet impact. By decoupling thickness and concentration measurements, the 2C-LIF method offers unprecedented insight into the early-time mixing processes that dictate coating uniformity and material properties in industrial applications. The ability to characterize these dynamics in both single-component and complex multi-component (binary) systems lays the groundwork for optimizing spray coating processes and understanding fluid behavior in micro-fluidic and biological contexts.