Internal flow and concentration in neighbouring evaporating binary droplets and rivulets

This paper numerically and theoretically investigates how proximity-induced shielding and Marangoni forces affect the internal flow symmetry and concentration profiles of evaporating binary droplets and rivulets, revealing that while asymmetry generally diminishes over time, the influence of surface tension gradients on flow stagnation differs between two-dimensional rivulets and three-dimensional droplets due to the latter's additional azimuthal flow.

The Gist

The Big Picture: The "Crowded Room" Effect

Imagine you are at a party, and you are holding a glass of water with a drop of food coloring in it. If you stand alone in a large, empty room, the water evaporates evenly from all sides, and the food coloring mixes smoothly.

Now, imagine you are standing right next to a friend who is holding an identical glass. Suddenly, things get complicated. The air right between you two becomes humid and "stuffy" because both of you are releasing water vapor. This creates a shielding effect: the air between you is so full of moisture that neither of you can evaporate as fast as you would if you were alone.

This paper studies exactly what happens when two droplets (or long strips of liquid called "rivulets") of a special mixture (water and a non-evaporating chemical) sit close together and dry out. The researchers wanted to know: How does this "crowded" environment change the flow of liquid inside the droplets?

The Cast of Characters

1. The Droplets/Rivulets: Think of them as tiny puddles. Some are round like raindrops (droplets), and some are long, thin strips like a river on a sidewalk (rivulets).
2. The Mixture: They are made of water (which evaporates) and a thick, sticky chemical called 1,2-hexanediol (which stays behind). As the water leaves, the sticky chemical gets more concentrated.
3. The "Skin" Tension (Marangoni Effect): Imagine the surface of the droplet is like a tight drum skin. Where the water evaporates faster, the skin gets tighter (higher surface tension). Where it evaporates slower, the skin is looser. The liquid inside naturally flows from the loose skin to the tight skin, like a crowd moving toward an exit. This is called Marangoni flow.
4. The "Coffee Stain" Effect: Usually, when a coffee drop dries, the liquid rushes to the edges, leaving a ring of dirt. This is the "coffee stain." In this study, the Marangoni flow is so strong it usually overpowers the coffee stain, keeping the mixture well-mixed.

The Main Discovery: The "Stagnation Point"

Inside these drying droplets, the liquid swirls around like a tiny whirlpool. Usually, if a droplet is alone, the swirl is perfectly symmetrical (like a spinning top).

But when two droplets are neighbors, the "stuffy" air between them breaks the symmetry. The liquid flow gets pushed away from the neighbor. The researchers tracked a specific spot called the Stagnation Point.

• The Metaphor: Imagine a river flowing toward a waterfall. If you put a big rock in the middle, the water splits and flows around it. The exact spot where the water stops moving forward and starts swirling around the rock is the "stagnation point."
• The Finding: In a crowded pair of droplets, this "stopping point" doesn't stay in the middle. It gets pushed toward the center of the pair. The researchers wanted to know exactly how far it moves and why.

The Two Different Worlds: Strips vs. Drops

The researchers found that the shape of the liquid matters immensely. They compared Rivulets (long strips) and Droplets (round drops).

1. The Rivulets (The 2D Strips)

Think of a rivulet as a long, straight river. The flow can only go left, right, up, or down. It's a flat, two-dimensional world.

• The Surprise: For these strips, the position of the stagnation point does not care how strong the "skin tension" (Marangoni force) is. Whether the flow is weak or super strong, the stagnation point stays in the same spot relative to the distance between the strips and the angle they make with the ground.
• The Analogy: It's like a train on a single track. No matter how fast the engine (Marangoni force) pushes, the train can only stop at the same spot because the track (geometry) dictates where it can go.

2. The Droplets (The 3D Spheres)

Now, imagine a round drop. It has an extra dimension: it can swirl around (azimuthally), like a spinning top.

• The Surprise: For round drops, the stagnation point does move depending on how strong the "skin tension" is. If the flow gets stronger, the stagnation point shifts significantly.
• The Analogy: A round drop is like a busy roundabout. If the traffic (flow) gets heavier, the cars (liquid) don't just go straight; they swirl around the center in complex patterns. The extra freedom to spin in circles makes the system much more sensitive to the strength of the force driving it.

Why Did They Do This? (The "Why It Matters")

This isn't just about puddles drying on a table. This research is crucial for high-tech industries:

• Inkjet Printing: If you print two dots of ink next to each other, they might dry unevenly, ruining the picture.
• Spray Cooling: When cooling a hot engine with a spray of liquid, knowing how the droplets interact helps cool it more efficiently.
• Pesticide Delivery: Ensuring that droplets spread evenly on a leaf so the poison works correctly.

The Takeaway

The researchers built a "simplified map" (a mathematical model) to predict how these droplets behave without needing to run super-expensive, slow computer simulations every time.

• For long strips (rivulets): The flow is predictable and depends mostly on how close they are and how steep the drop is. The strength of the internal flow doesn't change the outcome much.
• For round drops: The flow is chaotic and sensitive. The strength of the internal swirling forces changes everything.

In short: When droplets hang out together, they change each other's behavior. If they are long strips, they are stubborn and predictable. If they are round drops, they are sensitive and reactive. Understanding this helps engineers design better printers, sprays, and coatings.

Technical Summary

1. Problem Statement

The paper investigates the evaporation dynamics of neighboring binary liquid bodies (specifically droplets and rivulets), a scenario common in applications like inkjet printing, spray cooling, and pesticide delivery. Unlike isolated droplets, neighboring liquid bodies interact through the gas phase, creating a shielding effect where the vapor concentration near the liquid surface increases, reducing the local evaporation rate.

This shielding effect breaks the symmetry of the evaporation-driven flows and concentration fields within the liquids. For binary mixtures (e.g., water and 1,2-hexanediol), selective evaporation of the volatile component induces surface tension gradients, driving Marangoni flows. The authors aim to understand how key parameters—contact angle ($\theta$), inter-droplet/rivulet distance ($b$), and the Marangoni number ($Ma$)—govern the symmetry breaking and the position of the internal flow stagnation point.

2. Methodology

The study employs a multi-scale approach combining full numerical simulations with simplified analytical and quasi-stationary models.

• Governing Equations: The authors solve the full transient system of equations for the liquid phase (Navier-Stokes, advection-diffusion) and the gas phase (Laplace equation for vapor diffusion). They assume low contact angles, negligible gravity, and that only one component evaporates.
• Numerical Simulations: Full 2D transient simulations were performed for neighboring rivulets (cylindrical drops) using a sharp-interface Lagrangian–Eulerian finite element method (pyoomph). This served as the benchmark.
• Quasi-Stationary Model: Given the slow evaporation of the specific binary mixture, a quasi-stationary assumption was introduced. This assumes the internal flow and concentration fields reach a steady state at each instant, provided Marangoni mixing overwhelms the "coffee-stain" effect (solute accumulation at the rim).
• Lubrication Approximation: To reduce computational cost and enable analytical derivation, the authors applied a lubrication approximation (thin-film limit) to the quasi-stationary equations. This vertically averages the flow and concentration, incorporating Taylor-Aris dispersion to account for mixing.
• Evaporation Flux Models: Analytical expressions for local evaporative flux were derived for isolated and neighboring rivulets/droplets, accounting for the shielding effect.

3. Key Contributions

1. Symmetry Breaking Characterization: The paper quantifies the asymmetry in neighboring evaporating binary systems by tracking the stagnation point ($x_0$) of the internal flow (the point where the two vortices merge).
2. Model Hierarchy Validation: The authors successfully validated a computationally efficient quasi-stationary model against full transient simulations and further reduced it to a lubrication model. They demonstrated that the quasi-stationary model is highly accurate for high Marangoni numbers.
3. Analytical Derivation: For rivulets, the authors derived an explicit analytical expression for the stagnation point shift (Eq. 6.20) using the lubrication model, showing its dependence on contact angle and distance but independence from the Marangoni number.
4. Distinction between 2D and 3D Geometries: A critical contribution is the identification of a fundamental difference between 2D rivulets and 3D droplets regarding Marangoni dependence.

4. Key Results

A. Rivulets (2D Geometry)

• Stagnation Point Behavior: As evaporation proceeds and the contact angle decreases, the stagnation point ($x_0$) migrates toward the center of the rivulet.
• Parameter Dependence:
  • Marangoni Number ($Ma$): The stagnation point position is independent of the Marangoni number. The asymmetry is driven purely by the geometric shielding effect and the resulting concentration gradients.
  • Contact Angle ($\theta$) and Distance ($b$): The asymmetry increases with larger contact angles and decreases with larger inter-rivulet distances. Higher contact angles confine the vapor more, enhancing the shielding effect and asymmetry.
• Model Accuracy: The lubrication model captures the general trends but underestimates the stagnation point shift at larger contact angles because standard analytical flux models often assume zero contact angle.

B. Droplets (3D Geometry)

• Marangoni Dependence: Unlike rivulets, the stagnation point in droplets depends strongly on the Marangoni number.
• Mechanism: In 3D droplets, Marangoni flow has an azimuthal component (in addition to radial and vertical). This extra degree of freedom leads to a non-linear evolution of the concentration field. As $Ma$ increases, enhanced mixing alters the concentration gradients, shifting the stagnation point.
• Vertical Gradients: There is a significant vertical gradient in the stagnation point position; the point at the substrate shifts differently than the point at the liquid-gas interface as $Ma$ increases.

C. Validation

• The quasi-stationary model showed "perfect agreement" with transient simulations for rivulets when $Ma$ was sufficiently high to suppress the coffee-stain effect.
• The lubrication model agreed well with transient results for evaporation rates at later stages (low contact angles) but struggled with the stagnation point location at early stages due to approximations in the evaporative flux and vertical gradients.

5. Significance and Implications

• Theoretical Insight: The work clarifies why 2D models (rivulets) cannot fully capture the dynamics of 3D droplets in binary mixtures. The presence of azimuthal flow in droplets introduces a non-trivial dependence on Marangoni forces that is absent in 2D.
• Practical Application: Understanding the stagnation point shift is crucial for controlling particle deposition (e.g., in inkjet printing). If the stagnation point shifts asymmetrically, particles may deposit off-center, leading to defects in printed patterns.
• Modeling Efficiency: The study provides a validated, computationally cheaper framework (quasi-stationary lubrication model) for predicting internal flows in neighboring evaporating systems, provided the contact angle is small and Marangoni mixing is strong.
• Future Work: The authors note that analytical expressions for evaporative flux with non-zero contact angles are currently missing in literature and are necessary to fully capture the contact angle dependence in lubrication models.

In summary, this paper establishes that while geometric shielding drives asymmetry in both 2D and 3D evaporating binary systems, the Marangoni number plays a negligible role in 2D rivulets but a dominant role in 3D droplets due to the additional azimuthal flow freedom.