Stokes flows in a sessile hemispherical drop due to evaporation and surface tension gradient

This paper presents analytical solutions for viscous flows in an evaporating hemispherical droplet under both no-slip and full-slip boundary conditions, revealing a rigid relationship between evaporation and surface tension gradients that redefines the critical Marangoni number and highlights the sensitivity of flow regimes to liquid-substrate interactions.

The Gist

The Big Picture: The "Coffee Ring" Mystery

Imagine you spill a drop of coffee on a table and let it dry. As the water evaporates, the coffee particles get swept to the edge, leaving a dark ring. This is the famous "Coffee Ring Effect."

For a long time, scientists thought this happened because the liquid simply flowed outward to replace the water that was evaporating. They called this the Deegan flow (or "compensatory flow"). It's like a crowd of people rushing to the exit of a theater because the seats are disappearing; they just want to keep the shape of the crowd intact.

However, this paper asks a tricky question: Is that outward flow the only thing happening? Or is there a second, invisible force pulling the liquid around in circles?

The Two Forces at Play

The author, Peter Lebedev-Stepanov, looks at a droplet (specifically a perfect half-sphere) and identifies two competing forces:

1. The "Evaporation Rush" (Deegan Flow): As the top of the drop dries out, liquid from the center rushes to the edge to fill the gap. This is the "coffee ring" driver.
2. The "Slippery Slide" (Marangoni Flow): When a liquid evaporates, it cools down (like sweat cooling your skin). If one part of the drop is cooler than another, the surface tension changes. Think of surface tension as the "skin" of the water. If the skin is tighter in one spot, it pulls the liquid like a rubber band. This creates swirling currents, often called Marangoni convection.

The Main Discovery: The "No-Slip" Trap

The paper's biggest finding is about how the drop interacts with the table (the substrate).

Scenario A: The Sticky Table (No-Slip Condition)
Imagine the liquid is glued to the table. It cannot slide.

• The Problem: If the liquid is glued down, the "Evaporation Rush" (Deegan flow) forces the liquid to move in a way that accidentally creates a temperature difference.
• The Result: You cannot have the coffee ring effect without also having the swirling Marangoni currents. They are "married" together. The math shows that if the liquid is stuck to the table, the temperature gradient (the "slippery slide" force) is automatically generated by the evaporation itself. You can't separate them.
• The Metaphor: It's like trying to pedal a bicycle where the wheels are frozen to the ground. To move forward, you have to twist the handlebars so hard that the bike starts spinning in circles. You can't just go straight; the two motions are locked together.

Scenario B: The Slippery Table (Slip Condition)
Now, imagine the table is covered in ice or oil, and the liquid can slide freely.

• The Solution: When the liquid can slide, the "Evaporation Rush" and the "Slippery Slide" become independent.
• The Result: You can have a pure outward flow (just the coffee ring) without the swirling currents, or you can have strong swirling currents without the outward flow. They are no longer forced to be together.
• The Metaphor: Now the bicycle wheels are free. You can pedal straight forward without the bike spinning in circles. The two motions are separated.

Why Does This Matter?

The author argues that many previous experiments might have been confused because they assumed the liquid was "stuck" (no-slip) to the surface.

• The "Critical Moment": The paper suggests there is a tipping point. At low evaporation rates, the liquid is "stuck," and the flows are mixed up. But as evaporation speeds up, the friction (shear stress) near the bottom of the drop gets so high that the liquid might suddenly start to slide.
• The Transition: When this sliding starts, the "coffee ring" flow and the "swirling" flow separate. This changes the entire structure of how the drop dries.

The "Temperature" Twist

The paper also does some heavy math to show that for the "stuck" scenario to work, the temperature inside the drop has to be very specific.

• If the drop is small and evaporating slowly, the temperature difference needed to drive the swirling currents is tiny (almost non-existent).
• However, if the drop is larger or evaporating faster, the temperature difference becomes significant, and the "swirling" (Marangoni) flow takes over, potentially breaking the "stuck" condition and causing the liquid to slide.

The Takeaway

This paper is a wake-up call for scientists studying drying drops (used in inkjet printing, medical diagnostics, and making thin films).

1. Don't assume the drop is stuck: The way the liquid touches the table changes everything.
2. The "Coffee Ring" isn't simple: It might be a mix of two different flows that are secretly linked.
3. Watch for the slide: If you increase the heat or evaporation, the liquid might suddenly stop sticking and start sliding, completely changing the pattern of how the drop dries.

In short: The paper reveals that the "glue" between the drop and the table dictates whether the liquid flows in a simple outward rush or a complex dance of swirls. If you want to control how a drop dries (like in printing a circuit board), you need to know if your liquid is "stuck" or "sliding."

Technical Summary

1. Problem Statement

The paper investigates the hydrodynamic flow within a small, slowly evaporating, sessile hemispherical droplet with a pinned contact line. The study focuses on the interplay between two distinct flow mechanisms:

• Deegan (Capillary) Flow: An outward flow driven by evaporation-induced volume loss, responsible for the "coffee ring effect."
• Marangoni Flow: A flow driven by surface tension gradients ($\nabla \sigma$), typically caused by temperature or concentration variations (e.g., anisotropic cooling).

The central problem is to determine the analytical solutions for these flows under Stokes flow conditions (low Reynolds number) and to analyze how the nature of the liquid-substrate boundary condition (specifically no-slip vs. full-slip) dictates the relationship between evaporation and Marangoni convection.

2. Methodology

The author employs an analytical approach based on "first principles" using the following framework:

• Geometry: A hemispherical droplet on a flat horizontal substrate. The shape is assumed to be a spherical segment (valid for Bond number $Bo \ll 1$).
• Governing Equations: The axisymmetric, stationary Stokes equations for an incompressible fluid are solved in spherical coordinates $(r, \theta)$.
  • Momentum equations (radial and polar components).
  • Continuity equation.
• Mathematical Technique: The velocity components and stream function are expanded in series of Legendre polynomials ($P_l(\cos \theta)$). The coefficients of these series are determined by applying specific boundary conditions.
• Boundary Conditions:
  1. Substrate: Two scenarios are analyzed:
     • No-slip: Velocity is zero at the substrate ($\theta = \pi/2$).
     • Full-slip: Shear stress is zero, allowing the liquid to slide (tangential velocity is non-zero).
  2. Droplet Surface:
     • Kinematic condition relating radial velocity to the evaporation flux.
     • Dynamic condition relating tangential velocity to the surface tension gradient (Marangoni stress).
• Heat Transfer: The paper analyzes the temperature field under two regimes:
  • Low Peclet number ($Pe \ll 1$): Molecular conduction dominates.
  • High Peclet number ($Pe \gg 1$): Convective heat transfer dominates.

3. Key Contributions and Results

A. The No-Slip Regime (Rigid Coupling)

When the no-slip condition is applied to the substrate:

• Inseparability of Flows: The analytical solution reveals a rigid, mathematical relationship between the evaporation rate ($J$) and the surface tension gradient ($\beta$).
• Non-Zero Marangoni Number: Even in the absence of an external imposed temperature gradient, the Deegan flow (capillary flow) inherently generates a specific surface tension gradient to satisfy the no-slip boundary condition.
• Critical Finding: Under no-slip conditions, the Deegan flow and Marangoni flow are inseparable. The flow cannot be classified purely as a compensatory capillary flow because it inherently possesses a non-zero Marangoni number ($Ma \approx 0.0017$ for water droplets).
• Flow Structure: The streamlines exhibit a complex, "fractalized" structure where flow bundles are tightly combined (visualized in Fig. 2 of the paper).
• Thermal Constraint: This result implies that for a no-slip system, the temperature regime inside the droplet is strictly constrained by the evaporation rate. The system cannot support an independent Marangoni convection without violating the no-slip condition or the steady-state assumption.

B. The Slip Regime (Flow Separation)

When the slip condition is allowed (liquid slides over the substrate):

• Decoupling: The evaporation rate ($J$) and the surface tension gradient ($\beta$) become independent parameters.
• Separation of Flows: The total flow can be analytically separated into two distinct components:
  1. Pure Deegan (Compensatory) Flow: Driven solely by evaporation, with no surface tension gradient ($\beta = 0$).
  2. Pure Marangoni Flow: Driven solely by the surface tension gradient, independent of the evaporation rate.
• Flow Structure: The streamlines are simpler and lack the fractal complexity seen in the no-slip case (visualized in Fig. 3).

C. Estimation of the Marangoni Number

The paper derives an effective Marangoni number ($Ma$) for the Deegan flow under no-slip conditions:
$$Ma \approx \frac{\rho c_p D (1-\chi)}{\lambda}$$
For a typical water droplet ($R \approx 100 \mu m$), this yields a very small value ($Ma \approx 0.0017$). This suggests that while the flows are mathematically coupled under no-slip, the thermal driving force for Marangoni convection generated purely by evaporation is negligible compared to typical experimental conditions where external heating or solute gradients exist.

4. Significance and Implications

• Reinterpretation of Critical Marangoni Number: The study challenges the traditional view of the critical Marangoni number as a simple threshold for the onset of convection. Instead, it suggests that the transition from "capillary flow" to "Marangoni convection" is fundamentally linked to the breakdown of the no-slip condition.
• Mechanism of Transition: The authors propose a physical scenario where:
  1. At low evaporation rates, the no-slip condition holds, and flows are coupled (Deegan + weak Marangoni).
  2. As evaporation increases, viscous shear stress near the substrate increases.
  3. Eventually, the shear stress may overcome the adhesion forces, causing the contact line to transition from no-slip to partial/full slip.
  4. This transition decouples the flows, allowing for the emergence of a self-sustaining, developed Marangoni convection regime.
• Experimental Guidance: The work urges experimentalists to pay close attention to the liquid-substrate boundary conditions. The sensitivity of the flow structure to the slip/no-slip transition is a critical factor in understanding the dynamics of evaporating droplets in applications like inkjet printing, self-assembly, and evaporative lithography.

Conclusion

The paper provides a rigorous analytical proof that the distinction between Deegan flow and Marangoni flow is not merely a matter of magnitude but is fundamentally determined by the boundary condition at the substrate. Under no-slip conditions, evaporation and surface tension gradients are inextricably linked, whereas slip conditions allow them to act as independent drivers. This offers a new theoretical framework for understanding the onset of Marangoni convection in evaporating droplets.