Evaporative thermo-fluidics and deposition patterns in surface-active droplets

This study experimentally and theoretically investigates how surfactant concentration and substrate wettability influence evaporation rates, thermo-solutal transport, and deposition patterns in sessile droplets, revealing that Marangoni solutal advection dominates the flow dynamics while viscous resistance and surfactant crowding can dampen these effects.

The Gist

Imagine you drop a single bead of water onto a surface and watch it dry. You might expect it to just shrink and vanish. But in reality, that tiny drop is a bustling city of invisible traffic, temperature changes, and chemical reactions.

This paper is like a detective story investigating what happens when we add surfactants (the same kind of molecules found in soap and shampoo) to that water drop. The researchers wanted to know: How does soap change the way a drop dries, and what does it leave behind?

Here is the story of their findings, broken down into simple concepts.

1. The Setup: The "Drying Drop" Race

The scientists put drops of water mixed with different amounts of soap (specifically SDS and CTAB) onto two different surfaces:

• The "Wet" Surface (Hydrophilic): Like a clean glass window. The water spreads out flat.
• The "Water-Repelling" Surface (Superhydrophobic): Like a lotus leaf or a waxed car. The water beads up into a perfect sphere.

They watched these drops dry using high-speed cameras, thermal cameras (to see heat), and a special laser technique called PIV (which acts like a wind tunnel for liquids, letting them see the invisible currents inside the drop).

2. The Secret Engine: The "Marangoni Conveyer Belt"

In a plain water drop, the edges dry faster than the middle. This creates a "coffee ring" effect where all the dirt gets pushed to the edge.

But when you add soap, things get crazy. Soap molecules are like little surfers that love to sit on the surface of the water. As the drop evaporates, the soap gets pushed around, creating invisible currents inside the drop.

• The Analogy: Imagine a crowded dance floor. If the music (evaporation) gets faster at the edges, people (water molecules) rush out. But the soap molecules act like a DJ who suddenly changes the beat, creating a whirlpool that pulls people back toward the center.
• The Result: This "whirlpool" (called Marangoni advection) mixes the drop up. It brings cooler water from the center to the edges, which actually makes the drop evaporate faster—but only up to a point.

3. The "Goldilocks" Zone: Not Too Little, Not Too Much

The researchers found a sweet spot for the soap concentration, related to something called CMC (Critical Micelle Concentration). Think of CMC as the point where the soap molecules are so crowded they start hugging each other and forming little balls (micelles) instead of sitting on the surface.

• Too Little Soap: The currents are weak. The drop dries slowly.
• Just Right (0.5 CMC): The soap creates the strongest whirlpools. The drop evaporates the fastest! It's like the perfect amount of wind to fill a sail.
• Too Much Soap (1 CMC): The soap molecules get too crowded. They start acting like a thick, sticky gel (increasing viscosity). The "whirlpool" gets clogged and slows down. The drop actually dries slower again because the soap is too thick to move easily.

The Verdict: On a flat glass surface, the drop dried fastest with a medium amount of soap. On the water-repelling surface, adding more soap kept speeding things up, but the physics were slightly different.

4. The Aftermath: The "Rim" and the "Fingers"

When the drop finally dries, it leaves behind a stain.

• The Coffee Ring: Without soap, you get a thick ring of dirt at the edge.
• The Soap Effect: With soap, the internal whirlpools fight against the outward flow. This creates a very thick, distinct "rim" of dried soap at the edge.
  • SDS (Anionic soap): Created a massive, wide rim (like a thick wall).
  • CTAB (Cationic soap): Created a much thinner rim.

Inside this rim, the scientists saw strange fingering patterns (like tree roots or lightning bolts). This happened because the soap flow became unstable, creating little ripples and waves as the liquid dried.

5. The "Stick-and-Slip" Dance

One of the coolest discoveries was how the edge of the drop moved.

• Plain Water: The edge stays stuck (pinned) until the very end.
• Soapy Water: The edge behaves like a person walking on a slippery floor. It gets stuck, then suddenly slips forward, gets stuck again, and slips again.
• The Result: Instead of one big ring, the drop leaves behind multiple concentric rings (like a target). This "stick-slip" behavior happens because the soap molecules are constantly changing how the water sticks to the surface.

Summary: Why Does This Matter?

This research is like learning the rules of a game that happens billions of times a day.

• For Farmers: If you spray pesticides with soap, knowing how the drop dries helps you control where the medicine lands on a leaf.
• For Printers: If you are printing with ink (which has surfactants), you need to know how the ink dries to avoid messy rings or uneven colors.
• For Engineers: Understanding these invisible currents helps in designing better cooling systems or medical coatings.

In a nutshell: Adding soap to a water drop turns a simple drying process into a complex dance of currents. Too little soap, and nothing happens. Too much, and it gets sticky. But just the right amount creates a powerful engine that speeds up drying and leaves behind unique, patterned footprints.

Technical Summary

1. Problem Statement

The evaporation of sessile droplets is a critical phenomenon in applications ranging from inkjet printing and coating to thermal management and agro-chemical delivery. While the evaporation of pure water and particle-laden droplets is well-studied, the specific role of surfactant molecules in modulating internal hydrodynamics, evaporation rates, and final deposition patterns remains complex and not fully understood.

Key gaps addressed in this research include:

• How surfactant concentration (relative to the Critical Micelle Concentration, CMC) influences evaporation kinetics on different substrates.
• The dominant transport mechanisms (thermal vs. solutal Marangoni advection, buoyancy) driving internal fluid flow.
• The relationship between internal advection patterns and the resulting deposition morphologies (e.g., ring formation, stick-slip behavior).

2. Methodology

The study employed a combined experimental and theoretical scaling-based approach to investigate aqueous droplets containing anionic (Sodium Dodecyl Sulphate, SDS) and cationic (Cetyltrimethylammonium Bromide, CTAB) surfactants.

Experimental Setup:

• Substrates: Experiments were conducted on hydrophilic (glass, $\theta \approx 40^\circ$) and superhydrophobic (SHS, $\theta \approx 155^\circ$) surfaces.
• Droplet Generation: 50 $\mu$L droplets were dispensed using a syringe pump within an acrylic chamber controlled for temperature and humidity.
• Imaging & Diagnostics:
  • Shadow Imaging: Used to track droplet geometry (contact radius, contact angle, volume) and evaporation rates.
  • Infrared Thermography (FLIR T650sc): Measured surface temperature distributions to analyze thermal gradients.
  • Particle Image Velocimetry (PIV): Utilized 10 $\mu$m fluorescent seeding particles and a 532 nm laser to quantify internal velocity fields and flow patterns.
• Variables: Surfactant concentrations were varied from 0 to 1 CMC (and beyond) to observe non-monotonic effects.

Theoretical Analysis:

• Scaling Laws: The authors derived dimensionless energy and mass balance equations to compare the magnitudes of different transport mechanisms.
• Dimensionless Numbers: Calculated Rayleigh ($Ra$), Thermal Marangoni ($Ma_T$), Solutal Marangoni ($Ma_S$), Capillary ($Ca$), and Evaporation ($Evp$) numbers to determine stability and dominant flow regimes.
• Velocity Prediction: Developed theoretical models for Marangoni velocities to compare against experimental PIV data.

3. Key Results

A. Evaporation Kinetics and Wettability

• Non-Monotonic Trends: Evaporation rates did not increase linearly with surfactant concentration.
  • Hydrophilic Surfaces: Rates increased up to 0.5 CMC and then decreased at 1 CMC. The maximum rate was observed at 0.5 CMC.
  • Superhydrophobic Surfaces (SHS): Similar non-monotonic trends were observed, but overall evaporation rates were lower due to reduced solid-liquid contact area and thermal resistance from the air cushion.
• Evaporation Modes:
  • Hydrophilic: Dominated by Constant Contact Radius (CCR) mode (pinned contact line).
  • Superhydrophobic: Dominated by Constant Contact Angle (CCA) mode (receding contact line).
• Surfactant Comparison: SDS (anionic) generally enhanced evaporation rates more effectively than CTAB (cationic), attributed to differences in interfacial activity and concentration at the CMC.

B. Internal Hydrodynamics and Transport Mechanisms

• Velocity Profiles: PIV measurements revealed that internal flow velocities increased significantly with surfactant concentration up to 0.5 CMC (reaching ~1.75 mm/s on hydrophilic and ~4.5 mm/s on SHS). At 1 CMC, velocities dropped due to viscous damping and surfactant crowding.
• Dominant Mechanism:
  • Rayleigh-Bénard Convection: Negligible ($Ra \ll Ra_{critical}$).
  • Thermal Marangoni: Present but secondary ($Ma_T$ values were low, e.g., ~90).
  • Solutal Marangoni Advection: Identified as the dominant transport mechanism. $Ma_S$ values were significantly higher (up to ~7000 on SHS at 1 CMC). The accumulation of surfactants at the droplet periphery created surface tension gradients that drove strong internal circulation.
• Viscosity Effects: As evaporation progressed, bulk surfactant concentration increased, leading to higher viscosity. This viscous resistance suppressed advection at high concentrations (1 CMC), explaining the drop in evaporation rates.

C. Deposition Patterns and Contact Line Dynamics

• Stick-Slip Behavior: Contact line velocity measurements revealed sudden fluctuations (spikes) corresponding to "stick-slip" motion. This was absent in pure water but prominent in surfactant-laden droplets.
• Multiple Rings: The stick-slip behavior led to the formation of multiple concentric rings (deposition bands) rather than a single coffee ring. This was attributed to cycles of pinning, surfactant aggregation/precipitation, and depinning.
• Rim Structure:
  • A distinct "rim" of deposited material formed at the droplet edge.
  • SDS rims were significantly wider (425 $\mu$m) than CTAB rims (100 $\mu$m) due to higher surfactant availability and stronger advection.
  • Fingering Instabilities: The rims exhibited finger-like patterns, attributed to visco-capillary instabilities and Marangoni eddies disrupting the outward capillary flow.
• Suppression of Coffee Ring: While a rim formed, the strong inward Marangoni flow partially suppressed the classic "coffee ring" effect, leading to more complex deposition morphologies including patch-type and cell-type deposits away from the rim.

4. Key Contributions

1. Mechanism Identification: The study definitively establishes that solutal Marangoni advection (driven by surfactant concentration gradients) is the primary driver of enhanced evaporation and internal flow in surfactant-laden droplets, outweighing thermal Marangoni and buoyancy effects.
2. Concentration Thresholds: It identifies an optimal surfactant concentration (~0.5 CMC) for maximizing evaporation rates, beyond which viscous damping and interfacial crowding suppress flow.
3. Wettability Coupling: It demonstrates how substrate wettability (CCR vs. CCA modes) interacts with surfactant physics to dictate evaporation lifetimes and flow stability.
4. Deposition Correlation: The work uniquely links internal advection strength to specific deposition morphologies (rim width, fingering, multiple rings), providing a predictive framework for controlling drying patterns.

5. Significance

This research provides fundamental insights into the thermo-fluidic behavior of complex fluids, which is essential for optimizing industrial processes such as:

• Inkjet Printing: Controlling droplet spreading and preventing unwanted ring stains.
• Coatings and Films: Achieving uniform deposition of active ingredients.
• Agro-chemicals: Enhancing the coverage and uptake of pesticides/fertilizers on leaf surfaces (which are often hydrophobic).
• Thermal Management: Understanding how surfactants can be used to tune evaporation rates for cooling applications.

By quantifying the interplay between surfactant concentration, substrate wettability, and internal flow dynamics, the authors offer a robust scaling-based model that can predict evaporation behavior and deposition outcomes in surface-active systems.