Translation dynamics of evaporating sessile binary-mixture droplet populations

This study combines theoretical modeling and experimental validation to demonstrate that the translational dynamics of evaporating binary-mixture droplet pairs—ranging from attraction and repulsion to "chasing"—are governed by the interplay of solutal and thermal Marangoni stresses, capillary effects, and vapour shielding, with specific outcomes determined by the droplets' initial compositions.

The Gist

Imagine a world where tiny droplets of liquid don't just sit still and evaporate; instead, they dance, chase each other, push one another away, or merge into a single giant drop. This is the story of how binary mixture droplets (drops made of two different liquids, like water mixed with morpholine or ethanol) behave when they are placed near each other on a warm surface.

The researchers behind this study built a mathematical "movie" to predict how these drops move, and they checked their movie against real-life experiments. Here is the breakdown of their findings using simple analogies.

The Stage: The Warm Table and the "Vapor Shield"

Imagine two people standing close together in a crowded room. If they both start shouting, the air between them gets filled with sound, making it harder for their voices to carry out to the rest of the room.

In the paper, the "shouting" is evaporation. When two droplets sit close together, they release vapor (gas) into the air. The space between them gets "crowded" with this vapor. This phenomenon is called "vapor shielding." Because the air between the drops is already full of vapor, the drops can't evaporate as fast on the side facing each other as they can on the outside.

The Forces at Play: The Invisible Tug-of-War

The movement of these drops is determined by a tug-of-war between three invisible forces:

1. The Capillary Force (The "Rubber Band"):
   Because the drops evaporate slower on the inside (due to vapor shielding) and faster on the outside, the shape of the drop gets lopsided. The outside edge gets thinner and curves more sharply, while the inside edge stays thicker. This creates a pressure difference, like a rubber band pulling the drops toward each other. This force usually causes attraction.

2. Thermal Marangoni (The "Heat Push"):
   Evaporation cools things down. Since the outside of the drop evaporates faster, it gets colder. The inside, shielded by vapor, stays warmer. In liquids, surface tension changes with temperature (warmer liquid has lower surface tension). This temperature difference creates a flow that pushes the liquid from the warm inside toward the cold outside. This acts like a repulsive force, pushing the drops apart.

3. Solutal Marangoni (The "Composition Push"):
   This is specific to mixtures. As the drops evaporate, the more volatile liquid (the one that turns to gas easily) disappears faster. This changes the "recipe" of the liquid inside the drop. If the recipe changes unevenly across the drop, it creates a flow driven by the difference in liquid composition. This can either pull drops together or push them apart, depending on the specific mix.

The Dance Moves: What Happens?

1. The "Attraction" Dance (Pure Drops or Low Heat)
If the drops are made of a single liquid, or if the surface isn't too hot, the "Rubber Band" (Capillary force) wins. The drops feel a gentle pull toward each other, slide across the surface, and eventually crash into one another to merge.

• Analogy: Two magnets slowly sliding together on a table.

2. The "Repulsion" Dance (High Heat)
If the surface is very hot, the "Heat Push" (Thermal Marangoni) becomes very strong. It overpowers the rubber band. The drops actively push away from each other and refuse to merge.

• Analogy: Two people on a crowded bus who suddenly decide they need more personal space and shuffle away from each other.

3. The "Chase" (Different Recipes)
This is the most interesting part. If you have two drops with different initial mixtures (e.g., one is 50% water, the other is only 10% water), something unique happens. The drop with more of the volatile ingredient (the "stronger" evaporator) starts to push the other drop.

• Analogy: Imagine a fast runner (the high-concentration drop) chasing a slower walker (the low-concentration drop). The fast runner doesn't just catch up; it seems to "herd" the slower one, pushing it forward. The paper calls this "chasing." The high-concentration drop is driven by the solutal Marangoni effect to push the other one away.

The Experiment vs. The Model

The researchers created a complex computer model to simulate these interactions. They tested it using real water-morpholine drops on a heated glass plate.

• At lower temperatures (30°C): The drops attracted each other and merged, just like the model predicted.
• At higher temperatures (60°C): The drops stayed apart, repelling each other, again matching the model.
• The "Chase": When they put a 10% water drop next to a 50% water drop, the 50% drop "chased" the 10% one.

The Bottom Line

The paper concludes that the movement of these tiny droplets isn't random. It is a precise balance of forces:

• Vapor shielding creates the uneven evaporation that starts the whole process.
• Capillary forces try to pull them together.
• Heat differences try to push them apart.
• Liquid composition differences can cause one drop to chase another.

By understanding this delicate balance, the researchers can predict whether two drops will hug, fight, or chase, simply by knowing their ingredients and how hot the surface is.

Technical Summary

Technical Summary: Translation Dynamics of Evaporating Sessile Binary-Mixture Droplet Populations

Problem Statement
The evaporation of sessile droplets is a complex phenomenon governed by factors such as volatility, interfacial tension, and thermal conductivity. While the dynamics of isolated droplets are well-studied, practical applications often involve droplet populations where mutual interactions occur. Recent observations indicate that droplets interact via the vapour phase, a phenomenon known as "vapour shielding," which suppresses evaporation rates between adjacent droplets and alters their lifetimes. Furthermore, binary mixture droplets exhibit distinct behaviors compared to pure droplets due to solutal Marangoni effects arising from concentration gradients. However, a comprehensive understanding of the translation dynamics (attraction, repulsion, and "chasing") of binary mixture droplet pairs, particularly the interplay between thermal Marangoni, solutal Marangoni, and capillary forces, remains incomplete. This study investigates the theoretical and experimental dynamics of two adjacent binary mixture droplets to elucidate these complex interactions.

Methodology
The authors propose a comprehensive, lubrication theory-based diffusion-limited model for two adjacent droplets placed on a rigid substrate. The system considers a binary mixture of a more volatile component (MVC) and a less volatile component (LVC). Key methodological features include:

• Governing Equations: The model solves coupled mass, momentum, energy, and concentration equations for the liquid phase, alongside quasi-steady diffusion equations for the vapour phase.
• Regularization: To address the moving contact line singularity, the model employs a precursor film formulation with disjoining pressure, accounting for intermolecular van der Waals interactions. This approach naturally yields constant contact radius (CCR) and constant contact angle (CCA) modes without external constraints.
• Physics Included: The model incorporates:
  • Vapour Shielding: Non-uniform evaporation caused by the accumulation of vapour between droplets.
  • Thermal Marangoni: Surface tension gradients driven by temperature differences (evaporative cooling).
  • Solutal Marangoni: Surface tension gradients driven by concentration differences of the binary components.
  • Capillary Effects: Driven by curvature and pressure gradients.
• Numerical Solution: The evolution equations are solved using the Finite Element Method (FEM) in COMSOL Multiphysics.
• Experimental Validation: Experiments were conducted using water-morpholine binary mixture droplets on heated substrates to qualitatively validate the model's predictions.

Key Contributions and Results

1. Pure Droplet Dynamics (Vapour Shielding):
   For pure droplets, the model confirms that vapour shielding creates an asymmetric evaporation flux, with lower flux on the proximal side (facing the other droplet) and higher flux on the distal side. This asymmetry induces capillary stresses that drive attraction and coalescence. However, the study identifies a competition between capillary forces (driving attraction) and thermal Marangoni stresses (driving repulsion). When thermal Marangoni numbers ($Ma$) are low, capillary forces dominate, leading to attraction. At higher $Ma$, thermal Marangoni stresses can overcome capillary forces, resulting in repulsion.

2. Binary Mixture Dynamics (Solutal vs. Thermal Marangoni):
   For binary mixtures, the dynamics are governed by the interplay of solutal Marangoni, thermal Marangoni, and capillary forces.

   • Identical Compositions: When two droplets have the same initial composition, solutal Marangoni and capillary forces generally induce attraction, while thermal Marangoni effects drive repulsion. The net motion depends on the relative strength of these forces, which is modulated by the initial composition and substrate temperature.
   • Different Compositions: A distinct "chasing" behavior is observed when droplets have different initial concentrations of the MVC. The droplet with a higher concentration of the more volatile component pushes (or chases) the droplet with a lower concentration. This motion is driven entirely by solutal Marangoni stresses arising from the concentration gradient.

3. Regime Mapping:
   The study constructs a regime map based on the initial concentration of the MVC ($X_{A,i}$) and the thermal Marangoni number ($Ma$).

   • Low $Ma$: Droplets exhibit attraction regardless of initial composition.
   • High $Ma$: Droplets with high $X_{A,i}$ may initially attract before repelling as the solutal effect diminishes with evaporation. Droplets with low $X_{A,i}$ exhibit strictly repulsive behavior.
   • Surface Tension Ratio ($\sigma_R$): The study contrasts water-morpholine ($\sigma_R < 1$, MVC has higher surface tension) with ethanol-water ($\sigma_R > 1$, MVC has lower surface tension). Ethanol-water droplets exhibit rapid spreading (Marangoni spreading), which can lead to immediate coalescence, masking translation dynamics unless thermal Marangoni stresses are strong enough to limit spreading.

4. Experimental Corroboration:
   Experiments with water-morpholine droplets qualitatively matched the model's predictions. At lower substrate temperatures (lower $Ma$), droplets converged and coalesced. At higher temperatures (higher $Ma$), droplets remained in position or repelled. Additionally, experiments with droplets of differing concentrations demonstrated the predicted "chasing" behavior.

Significance
The paper claims to provide a fundamental understanding of the mechanisms driving the translation of evaporating binary mixture droplets. By integrating vapour-mediated interactions with thermo-capillary and soluto-capillary phenomena, the model resolves the complex interplay that determines whether droplets attract, repel, or chase one another. The study highlights that while vapour shielding is a primary driver of non-uniform evaporation, the resulting motion is dictated by the specific balance of Marangoni stresses, which is highly sensitive to the binary mixture's composition and the thermal environment. The work serves to elucidate these interaction mechanisms, offering a theoretical framework that aligns with experimental observations of water-morpholine systems, while acknowledging the qualitative nature of the comparison due to the model's two-dimensional filament assumption versus the three-dimensional nature of experimental droplets.