A contaminant-concentration-dependent surface tension does not explain the absence of solutal Marangoni flow in evaporating droplets

This study demonstrates through combined experiments and modeling that the absence of predicted Marangoni flows in evaporating droplets is not caused by standard contaminant-concentration-dependent surface tension effects, but rather indicates that Marangoni stresses are effectively suppressed altogether, with observed flows being driven entirely by natural convection.

The Gist

Imagine a tiny drop of water sitting on a table, slowly evaporating into the air. For decades, scientists have been trying to figure out how the liquid moves inside that drop as it dries up.

Think of the drop like a miniature, invisible whirlpool. The big mystery is: What is the engine driving this whirlpool?

The Old Theory: The "Soap" Problem

For a long time, the textbook answer was that the drop should be churning violently. As the water evaporates, it leaves behind salt, sugar, or alcohol, creating uneven concentrations. This should create a "surface tension gradient"—imagine the surface of the drop acting like a stretched rubber sheet that is tighter in some spots than others. This tension should pull the liquid around, creating a fast, powerful flow (called Marangoni flow).

However, when scientists actually looked at these drops, the liquid was moving incredibly slowly—thousands of times slower than the math predicted.

The usual explanation was: "Ah, the water isn't pure. There must be invisible soap or dirt (contaminants) on the surface. This 'soap' acts like a brake, smoothing out the tension and stopping the flow."

The New Discovery: It's Not Just a Brake; It's a Different Engine

This new paper says that explanation is wrong. The authors didn't just say, "We need better soap models." They said, "The whole idea that contaminants are just slowing down the flow is incorrect."

Here is the simple breakdown of their findings:

1. The "Gravity" Engine is the Real Driver
The authors tested drops with salt, glycerol (a thick syrup), and ethanol. They found that the liquid inside wasn't moving because of surface tension at all. Instead, it was moving because of gravity (natural convection).

• The Analogy: Imagine a pot of soup on a stove. The hot soup at the bottom rises, and the cooler soup at the top sinks, creating a loop. In these tiny drops, the evaporation makes the liquid heavier or lighter in different spots, and gravity pulls it down or pushes it up, creating a slow, steady loop. This matches what they saw in the experiments perfectly.

2. The "Backwards" Flow
The most shocking part of the study is that in some cases, the liquid was moving in the opposite direction of what the "surface tension" theory predicted.

• The Analogy: If you have a river flowing downhill, and you suddenly see the water flowing uphill, you know something is wrong with your map. The "surface tension" map said the water should flow one way, but the "gravity" map said the other. The water followed gravity, ignoring the tension.

3. Why "Soap" Can't Explain It
The authors tried to use the old "soap" models to explain why the flow was so slow or why it reversed direction. They ran computer simulations adding "soap" (surfactants) to the mix.

• The Result: The soap models failed completely.
  • If you add soap to stop the flow, the math says the liquid should just stop moving. It doesn't explain why the liquid would start moving in the opposite direction.
  • It's like trying to explain why a car is driving backward by saying, "The brakes are too strong." Strong brakes stop a car; they don't make it drive in reverse. The "soap" models simply cannot account for the flow reversing direction.

The Conclusion

The paper concludes that we have been looking at the wrong mechanism. The reason we don't see the super-fast "Marangoni" flow in these drops isn't because contaminants are acting as a weak brake. It's because the surface tension gradient is effectively suppressed entirely or doesn't manifest in the way we think it does.

Instead of a fast, tension-driven race, the drop is dominated by a slow, gravity-driven dance. The "soap" theory is a dead end; the real story is that the surface tension forces are being completely overridden by something else (likely the way the contaminants interact with the surface in a way we don't yet understand), leaving gravity as the only thing moving the liquid.

In short: The drop isn't a high-speed race car with its brakes on; it's a slow-moving elevator driven by gravity, and the "soap" explanation for why it's slow doesn't make sense.

Technical Summary

Technical Summary: Contaminant-Dependent Surface Tension and the Absence of Solutal Marangoni Flow

Problem Statement
Theoretical models of evaporating droplets consistently predict Marangoni flows (driven by surface-tension gradients) with velocities on the order of millimeters per second. However, experimental observations in millimetric aqueous droplets under ambient conditions typically reveal flow velocities orders of magnitude smaller (micrometers per second). While this discrepancy is frequently attributed to the presence of surface contaminants that counteract surface-tension gradients, the precise mechanism by which these contaminants suppress Marangoni stresses remains unclear. Furthermore, existing studies often attribute internal recirculation in thin droplets exclusively to Marangoni convection, potentially overlooking the role of natural convection driven by density gradients.

Methodology
To investigate this discrepancy, the authors combined Particle Image Velocimetry (PIV) experiments with a coupled hydrodynamic and solute transport model.

• Experimental Setup: The study utilized sessile and pendant droplets containing aqueous solutions of sodium chloride, glycerol, or ethanol. For ethanol-water mixtures, the gas phase was saturated with ethanol vapor to prevent evaporation of the volatile component, isolating solutal effects. Internal velocity fields were measured using a thin laser sheet and polystyrene tracers, with light refraction corrected via ray tracing.
• Numerical Modeling: A mathematical model was developed where water vapor transport in the air is treated as purely diffusive. Inside the droplet, inertia is neglected, and the solutal Boussinesq approximation is applied to account for density gradients. The model incorporates Marangoni stresses via surface-tension gradients in the interfacial stress condition and is fully coupled with advection-diffusion transport of the solute.
• Contamination Modeling: The authors tested standard contamination models, including:
  1. Insoluble surfactants that linearly reduce surface tension.
  2. Soluble surfactants.
  3. Surface rheology modeled via the Boussinesq-Scriven law (treating the interface as a 2D Newtonian fluid with surface shear and dilatational viscosity).

Key Results

1. Dominance of Natural Convection: Experimental velocity fields for all three solutions (salt, glycerol, ethanol) in both sessile and pendant configurations were quantitatively reproduced only when Marangoni effects were neglected in the simulations ($Ma = 0$). Simulations including surface-tension variations based on literature values failed to match experimental data, often by an order of magnitude, and in some cases predicted the wrong flow direction.
2. Flow Reversal Evidence: In specific configurations (e.g., sessile glycerol-water droplets and pendant salt-water/ethanol-water droplets), the density gradient and the surface-tension gradient point in opposite directions. In these cases, the experimental surface velocity aligned with the density gradient (buoyancy) and opposed the surface-tension gradient.
3. Gravity as the Driver: By comparing sessile and pendant drops, the authors confirmed that reversing the direction of gravity reversed the flow direction, proving that the recirculation is driven by buoyancy (natural convection) rather than Marangoni stresses. This also rules out thermal Marangoni effects associated with evaporative cooling.
4. Failure of Contamination Models:
   • Insoluble Surfactants: While fitting a surfactant-induced Marangoni number ($Ma_\Gamma$) could reproduce flow magnitudes when the velocity and tension gradients were parallel, it failed when they were anti-parallel. Theoretical analysis showed that to reverse the flow direction, the surfactant gradient would need to be negative, which contradicts the physical requirement that surfactant transport must follow the flow (positive gradient) under quasi-steady conditions.
   • Soluble Surfactants: Similar inconsistencies arise; there is no clear mechanism to produce the necessary surfactant distribution to reverse the flow without invoking interfacial advection, which leads to the same contradiction as the insoluble case.
   • Surface Rheology: Models incorporating surface viscosity (Boussinesq-Scriven law) also failed. To suppress Marangoni flow and achieve a shear-free condition, the model predicted an inward radial velocity for sessile glycerol droplets, directly contradicting the outward velocity observed experimentally.

Key Contributions

• The study demonstrates that in evaporating aqueous droplets containing salt, glycerol, or ethanol, the internal flow is driven entirely by natural convection, not Marangoni convection.
• It provides experimental evidence where surface velocity opposes the theoretical surface-tension gradient, a phenomenon that standard contamination models cannot explain.
• It mathematically proves that neither the reduction of surface tension by contaminants nor surface rheological effects can account for the observed flow reversal or the complete absence of Marangoni flow.

Significance and Claims
The authors conclude that the macroscopic manifestation of Marangoni stresses is effectively suppressed altogether, rather than merely reduced by contaminants. They argue that the standard assumption that contaminants simply counteract surface-tension gradients is insufficient to explain the experimental reality.

The paper posits that simply assuming negligible surface-tension gradients in models (while retaining buoyancy effects) is sufficient to reproduce all experimental observations. This suggests that contamination or another undetermined interfacial phenomenon prevents the macroscopic manifestation of the gradient in the first place. The authors emphasize that while they do not question the existence of solutal Marangoni convection in general (citing other studies where it is observed), the conventional models used to characterize surface contamination fail to explain its absence in these specific evaporating droplet scenarios. Consequently, the fundamental question of why Marangoni convection is rarely observed in evaporating droplets requires theories that go beyond current contamination models.