Asymmetric freezing of a sliding droplet on an inclined surface

This study employs numerical simulations based on the lubrication approximation to demonstrate that the asymmetric morphology of a droplet freezing on an inclined surface is primarily governed by the interplay between sliding-induced deformation, substrate wettability, and inclination, where early-time dynamics and the competition between gravity and capillarity determine the formation of tilted ice cusps and contact-angle hysteresis.

The Gist

Imagine a tiny drop of water sliding down a cold, tilted windowpane. Usually, when water freezes, it forms a neat, symmetrical spike at the top, like a little ice mountain. But what happens if that drop is moving while it freezes? Does it stay symmetrical, or does it get squished and twisted?

This paper uses advanced computer simulations to answer that question. The researchers created a virtual world where they could watch a water droplet slide down a cold, inclined surface and freeze in real-time. Here is the story of what they found, explained simply:

The Setup: A Sliding Drop on a Tilted Stage

Think of the droplet as a tiny, wet marble rolling down a ramp. The ramp is the "inclined surface," and the cold air is the freezer. In the real world, this happens on airplane wings, wind turbines, or even just a cold car windshield.

The researchers wanted to see how three main forces played a game of tug-of-war inside the drop:

1. Gravity: Pulling the drop down the ramp.
2. Surface Tension (Capillarity): Trying to keep the drop together in a tight, round ball (like a soap bubble).
3. Freezing: The ice forming from the bottom up, locking the shape in place.

The Big Discovery: The "Frozen Memory"

The most surprising thing they found is that movement matters.

If a drop sits still and freezes, it makes a symmetrical spike. But if the drop is sliding when it starts to freeze, the final ice shape is asymmetrical. It's like taking a photo of a runner mid-stride and freezing them in place; the shape is stretched and tilted, not perfectly balanced.

The researchers call this a "frozen memory." Even if the drop stops sliding just a split second before it fully freezes, the shape it had while it was moving gets locked into the ice. The final ice spike doesn't point straight up; it leans toward the direction the drop was sliding.

The Tug-of-War: Gravity vs. The "Ice Floor"

As the drop slides, gravity tries to stretch it out, making the front (the part leading the way) bulge and the back (the tail) thin out.

• On a steep slope: Gravity wins easily. The drop stretches out like taffy, and the final ice spike leans heavily forward.
• On a wet surface: If the surface is very "sticky" (highly wetting), the water spreads out more. Interestingly, the researchers found that sometimes, as the ice starts to form, the remaining liquid water actually gets pulled backward (up the slope) for a brief moment, fighting against gravity. It's like a rubber band snapping back before the ice locks it in.

The "Ice Cusp" (The Pointy Tip)

When a drop freezes, it often forms a sharp point at the top, called a "cusp."

• The Angle: The researchers found that the angle of this pointy tip changes based on how steep the slope is and how "wettable" the surface is.
• The Rule: The steeper the slope and the more the water likes to spread on the surface, the more the tip leans over.
• The "Freezing Speed" Factor: They also tested how fast the water freezes. If the water freezes very quickly (high "Stefan number"), the ice locks the shape in before gravity has time to stretch it out. This results in a smaller, less tilted spike. If it freezes slowly, gravity has more time to stretch the drop, creating a more dramatic lean.

Why This Matters (According to the Paper)

The paper explains that for a long time, scientists studied freezing drops that were sitting still. This new study shows that moving drops are a completely different beast. You can't just take the rules for stationary drops and apply them to sliding ones.

The researchers built a mathematical "recipe" (a model) that successfully predicts exactly how these sliding drops will look once they turn to ice. They found that the early moments of freezing are the most critical; that's when the drop is still liquid and mobile, and that's when gravity does its most work to distort the shape.

Summary in a Nutshell

• Stationary drops freeze into symmetrical shapes.
• Sliding drops freeze into lopsided, tilted shapes because they get "stretched" by gravity while they are still liquid.
• The faster they freeze, the less time gravity has to stretch them, so the shape stays closer to the original.
• The steeper the slope, the more the final ice spike leans.

The paper concludes that to understand how ice forms on moving surfaces (like aircraft or power lines), we must account for the drop's motion, not just the temperature. The shape of the ice is a permanent record of how the water was moving when it turned to solid.

Technical Summary

1. Problem Statement

The study addresses a critical gap in the understanding of droplet freezing dynamics. While extensive research exists on sessile (stationary) droplets freezing on horizontal surfaces, real-world applications (e.g., aircraft wings, wind turbines, power lines) often involve droplets that slide on inclined surfaces before or during the freezing process.

Key questions addressed include:

• How does the transient motion of a droplet under gravity influence the final frozen morphology?
• How do substrate inclination, wettability, and freezing kinetics interact to create asymmetry in the frozen structure?
• What are the mechanisms governing the formation of the characteristic "ice cusp" (pointed tip) on inclined surfaces compared to horizontal ones?

2. Methodology

The authors developed a unified theoretical and numerical framework based on the lubrication approximation (valid for thin droplets where height $H \ll$ length $L$).

• Governing Equations: The model couples hydrodynamics (mass and momentum conservation) with heat transfer (energy conservation) for three phases: liquid water, solid ice, and the substrate.
  • Hydrodynamics: Uses the Navier-Stokes equations simplified for thin films, incorporating gravity ($g \sin \alpha$), capillarity (surface tension), and disjoining pressure (to model contact line motion and avoid singularities via a precursor film).
  • Thermodynamics: Solves the heat conduction equation for the liquid, ice, and substrate. The phase change is modeled using the Stefan condition at the liquid-ice interface, accounting for latent heat release.
• Numerical Approach:
  • The dimensionless equations are discretized using the Galerkin finite element method.
  • The nonlinear system is solved iteratively via the Newton-Raphson method with an implicit Euler time-stepping scheme featuring adaptive time steps.
  • Validation: The model was validated against:
    1. Non-freezing sliding droplets (comparing with Park & Kumar, 2017).
    2. Horizontal freezing droplets (comparing with Zadravzil et al., 2006).
    3. Experimental tip angles reported by Marin et al. (2014) and Starostin et al. (2022, 2023).
• Key Parameters: The study systematically varies:
  • Inclination Angle ($\alpha$): 0° to 75°.
  • Wettability: Equilibrium contact angle ($\theta_{eq}$) ranging from 12° to 30.6°.
  • Effective Bond Number ($Bo_m = Bo \sin \alpha$): Ratio of gravitational to surface tension forces.
  • Stefan Number ($Ste$): Ratio of sensible heat to latent heat (governing freezing rate).

3. Key Contributions

1. First Mechanistic Insight into Moving Droplet Freezing: Unlike previous studies assuming fixed droplets, this work captures the self-consistent evolution of a droplet that slides, deforms, and freezes simultaneously.
2. Decomposition of Forces: The study uniquely decomposes liquid motion into capillary-driven and gravity-driven components, revealing how they compete during different stages of freezing.
3. Discovery of Transient Reverse Motion: The model predicts a counter-intuitive phenomenon on highly wetting substrates where the liquid exhibits transient motion opposite to gravity during early freezing stages due to capillary retraction.
4. Quantification of Asymmetry: The work establishes a physical framework linking substrate tilt and wettability to the asymmetry of the frozen cusp and contact angle hysteresis.

4. Key Results

A. Asymmetry and Cusp Formation

• Inclination Effect: On inclined surfaces, the frozen droplet is highly asymmetric. The "ice cusp" (the pointed tip formed by volume expansion) tilts toward the advancing side (downslope).
• Cusp Angle ($\theta_{Cusp}$): The angle of the cusp relative to the vertical increases with the effective Bond number ($Bo_m$). Higher inclination leads to greater deviation from the vertical.
• Wettability Influence:
  • High Wettability (Low $\theta_{eq}$): Droplets flatten and elongate significantly. The advancing contact line often pins, while the receding line retracts.
  • Low Wettability (High $\theta_{eq}$): Droplets retain a more compact shape, resisting gravitational deformation.
  • Non-monotonic Hysteresis: The difference between advancing and receding contact angles ($\theta_a - \theta_r$) exhibits complex behavior. For highly wetting substrates, the hysteresis decreases after freezing, whereas for less wetting substrates, it increases.

B. Dynamics of Contact Lines

• Early-Stage Dominance: The asymmetry is established primarily in the early stages of freezing ($t/t_f < 0.1$) when the liquid volume is large and gravity dominates.
• Late-Stage Symmetry: As freezing progresses and the liquid volume diminishes, capillary forces dominate, restoring symmetry. Consequently, the final tip angle converges to a universal value (~143°) regardless of inclination, consistent with experimental observations.
• Pinning and Retraction: On the advancing side, the contact line often pins due to the formation of an ice "mesa." On the receding side, capillary forces drive retraction, thinning the liquid layer.

C. Effect of Stefan Number ($Ste$)

• Freezing Rate: Increasing $Ste$ (faster freezing) accelerates the solidification process.
• Suppression of Motion: High $Ste$ limits the time available for the droplet to slide and deform under gravity. This results in:
  • Reduced cusp angle (less deviation from vertical).
  • Reduced contact angle hysteresis.
  • Less overall morphological asymmetry.

D. Counter-Intuitive Phenomenon

• On highly wetting substrates ($\theta_{eq} \approx 15^\circ$), the model predicts a transient negative velocity (motion upslope) during early freezing. This occurs when the freezing front passes a region of negative curvature, reducing capillary pressure at the receding edge and pulling liquid against gravity.

5. Significance and Implications

• Anti-Icing and Thermal Management: Understanding how motion alters frozen morphology is crucial for designing surfaces for aircraft, wind turbines, and power lines where ice accretion can be catastrophic. The findings suggest that surface wettability and inclination must be tuned together to control ice shape.
• Material Fabrication: The ability to predict and control the asymmetric freezing of droplets is relevant for the controlled fabrication of porous materials and microstructures.
• Theoretical Advancement: The study challenges the assumption that freezing dynamics are solely governed by thermodynamics and static geometry, highlighting the critical role of hydrodynamic memory (the history of droplet motion) in determining the final solid state.

In conclusion, this paper provides a comprehensive physical framework demonstrating that the freezing of sliding droplets is a distinct regime from sessile droplet freezing, governed by a complex, time-dependent interplay between gravity, capillarity, and solidification kinetics.