Modelling wetting-bouncing transitions of droplet impact on random rough surfaces

This study utilizes volume of fluid simulations to investigate droplet impact on random hydrophobic surfaces, revealing that while maximum spreading decreases linearly with increasing roughness and contact time remains constant, the interplay between Weber number and surface roughness governs wetting-bouncing transitions and delays bouncing with larger roughness.

The Gist

Imagine you are at a summer picnic, and you accidentally drop a single water droplet from your soda can onto the ground. What happens next? Does it splatter and stick? Does it bounce off like a tiny rubber ball? Or does it shatter into smaller drops?

This paper is essentially a high-speed, computer-generated movie of that exact moment, but with a twist: instead of a flat sidewalk, the scientists are dropping droplets onto surfaces that look like tiny, random mountain ranges.

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

1. The Setup: The "Bumpy" Trampoline

The researchers used a supercomputer to simulate water droplets hitting surfaces with different levels of "bumpiness" (roughness).

• The Smooth Surface: Think of a perfectly polished glass table.
• The Rough Surface: Think of a surface covered in microscopic sandpaper or a fractal landscape (like a tiny, jagged mountain range). They created surfaces ranging from barely bumpy to very jagged (from 2 to 50 micrometers—imagine a human hair is about 70 micrometers, so these are incredibly small hills).

They also varied how hard the droplet was thrown (the speed), which is like changing the force of your drop.

2. The Three Possible Outcomes

When the droplet hits the ground, it can do one of three things, depending on how fast it's going and how bumpy the ground is:

• The "Stuck" Landing (No Bouncing): If the drop is slow or the ground is very bumpy, the droplet hits, spreads out like a pancake, and then just stays there. It gets stuck.
• The "Perfect" Bounce: If the drop is moving at just the right speed and the ground isn't too bumpy, the droplet hits, flattens, and then springs back up into the air like a super-bouncy ball, leaving the surface completely dry.
• The "Shattered" Bounce: If the drop is moving very fast and the ground is smooth, it hits, flattens, and then snaps back up—but it's so energetic that it breaks apart, leaving behind tiny satellite droplets (like a water balloon popping).

The Big Surprise: The researchers found that bumpiness acts like a brake. The bumpier the surface, the harder it is for the droplet to bounce. A very bumpy surface can stop a droplet from bouncing even if it's moving fast.

3. The "Re-Spreading" Mystery

Here is a weird thing they found. On very bumpy surfaces, the droplet sometimes behaves like a confused gymnast.

1. It hits and spreads out.
2. It starts to pull back (retract) to bounce.
3. But then, it stops! It gets stuck on a tiny "mountain" peak, and then suddenly spreads out again before finally deciding to bounce or stay.
   It's like a runner tripping over a rock, stumbling forward, and then taking another step before recovering.

4. The "Magic Time" Rule

This is the most fascinating part of the paper.
Usually, you'd think that if you throw a ball harder, it stays in the air longer. Or if you throw it on a rougher surface, it might stick longer.
But not for water droplets.

The scientists discovered that the amount of time a droplet stays in contact with the surface is almost exactly the same, no matter what.

• Whether the surface is smooth or bumpy.
• Whether the drop is moving slowly or fast.
• The contact time is constant.

Think of it like a metronome. No matter how hard you hit the drum, the rhythm stays the same. The droplet hits, flattens, and leaves in a fixed amount of time (about 3.9 times a specific "natural time" of the drop). This is a huge discovery because it means engineers can predict exactly how long a drop will stick to a surface just by knowing the size of the drop, without needing to know the speed or the texture of the ground.

5. Why Does This Happen? (The "Air Bubble" Secret)

Why do bumpy surfaces stop the bounce?
When a droplet hits a bumpy surface, it traps tiny pockets of air underneath it, like a trampoline made of air bubbles.

• On a smooth surface, the water spreads evenly.
• On a bumpy surface, the water gets caught in the "valleys" of the roughness. It gets messy, the edges of the water (the contact line) become jagged and irregular, and the trapped air acts like a cushion that absorbs the energy needed to make the drop bounce.

Why Should We Care?

This isn't just about water droplets; it's about designing better surfaces.

• Anti-Icing: If you want to stop ice from forming on an airplane wing, you want water to bounce off instantly. This paper tells us how to make surfaces that encourage bouncing.
• Spray Painting & 3D Printing: If you are spraying paint or printing circuits, you want the droplet to stick and spread evenly, not bounce off. This research helps us design surfaces that catch the droplets.
• Self-Cleaning: Raindrops bouncing off a car windshield take dirt with them. Understanding the "bouncing" threshold helps make cars that stay cleaner.

The Takeaway

The paper tells us that surface texture is a powerful switch. By changing how "bumpy" a surface is, we can control whether a water drop sticks, bounces, or shatters. And the coolest part? No matter how you change the speed or the bumps, the droplet always seems to have a "timer" that tells it exactly when to leave the surface.

Technical Summary

Here is a detailed technical summary of the paper "Modelling wetting-bouncing transitions of droplet impact on random rough surfaces."

1. Problem Statement

Droplet impact on rough surfaces is critical for applications such as inkjet printing, spray cooling, coating, and 3D printing. While droplet dynamics on smooth and patterned surfaces are well-studied, the mechanisms governing impact on random rough surfaces remain incompletely understood. Specifically, there is a lack of comprehensive understanding regarding how random surface morphology influences:

• The transition between wetting (deposition) and bouncing (rebound).
• The temporal evolution of the triple contact line.
• Internal flow dynamics and air entrapment.
• The scaling laws for maximum spreading and contact time across varying roughness and impact velocities.

2. Methodology

The authors employed Volume of Fluid (VoF) numerical simulations to investigate droplet impact dynamics on random hydrophobic surfaces.

• Numerical Framework:
  • An improved geometric VoF approach was used to capture the sharp gas-liquid interface.
  • Governing equations included the continuity equation, momentum equation (including surface tension via the Continuum Surface Force model), and a dynamic contact angle model (Seebergh et al.) to account for stick-slip effects.
  • Interface curvature was calculated using a geometric fitting approach (Jibben et al.).
• Surface Generation:
  • Random fractal rough surfaces were generated using the Weierstrass–Mandelbrot (W-M) function.
  • Six Root-Mean-Square (RMS) roughness values ($R_q$) were tested: 0, 2, 5, 10, 20, 30, and 50 $\mu$m.
• Simulation Parameters:
  • Droplet: Diameter $D_0 = 1.6$ mm, Equilibrium contact angle $\theta_e = 100^\circ$ (hydrophobic).
  • Impact Conditions: Five impact velocities ($U_0$) corresponding to Weber numbers ($We$) ranging from 5.7 to 12.9.
  • Total Cases: 35 numerical simulations covering the matrix of 5 $We$ values and 7 $R_q$ values.
• Validation: The model was validated against experimental data (Rioboo et al.) and other numerical studies, showing superior accuracy in predicting spreading factors compared to existing models.

3. Key Results

A. Impact Outcomes

Three distinct impact regimes were identified based on $We$ and $R_q$:

1. No Bouncing (Deposition): Occurs at low $We$ or high $R_q$. The droplet deposits on the surface.
2. Complete Bouncing: Occurs at moderate $We$ and low-to-moderate $R_q$. The droplet rebounds entirely without breaking up.
3. Bouncing with Breakup (Receding Breakup): Occurs at high $We$ and low $R_q$. The droplet rebounds but fragments into satellite droplets during the retraction phase.

B. Spreading Dynamics

• Maximum Spreading Factor ($\beta_m$): $\beta_m$ decreases linearly as surface roughness ($R_q$) increases for a given $We$.
• Scaling Laws: Two modified scaling laws were proposed to predict $\beta_m$:
  • Low Roughness ($R_q \le 20$ $\mu$m): $\beta_m = 1.14(1.38 + We)^{0.46}$
  • High Roughness ($R_q \ge 30$ $\mu$m): $\beta_m = 1.11(1.55 + We)^{0.40}$
  • Significance: The reduction in the exponent (from 0.46 to 0.40) indicates that high roughness hinders inertial spreading, shifting the regime from inertia-dominated to roughness-inhibited.
• Re-spreading: For surfaces with $R_q \ge 30$ $\mu$m, a "re-spreading" phenomenon was observed during the recoiling phase, where the droplet spreads again after initial retraction.

C. Contact Time

• Invariance: A key finding is that the droplet contact time ($\tau_c$) is independent of both the Weber number and the surface roughness (for $R_q \le 20$ $\mu$m).
• New Scaling: For hydrophobic surfaces ($\theta_e = 100^\circ$), the contact time is proposed as:
  $$\tau_c = 3.9 \sqrt{\frac{\rho R^3}{\sigma}}$$
  This contrasts with superhydrophobic surfaces where $\tau_c/\tau \approx 2.5$.

D. Microscopic Insights

• Contact Line Dynamics:
  • On smooth surfaces, the triple contact line remains a near-perfect circle.
  • On rough surfaces ($R_q > 10$ $\mu$m), the contact line becomes a 3D irregular curve. The dimensionless contact line length increases with roughness, while circularity decreases significantly (from ~1.0 to ~0.88).
• Air Entrapment: Higher roughness promotes air bubble entrapment ($A_b$), reducing the effective wetting area ($A_w/A_t$). This shifts the system from the Wenzel state immediately upon contact.
• Internal Flow:
  • Smooth surfaces exhibit symmetric internal flow and velocity vectors.
  • Rough surfaces induce disordered asymmetry in the velocity field near the contact line. Increased roughness enhances internal flow disturbances and instabilities.

4. Wetting-Bouncing Transition Mechanism

The transition is governed by the joint effect of $We$ and $R_q$:

• Increasing $We$ promotes bouncing (provides sufficient inertia to overcome adhesion).
• Increasing $R_q$ delays the transition to bouncing (increases viscous dissipation and energy loss).
• Breakup: Requires high inertia (high $We$) but low roughness to minimize energy dissipation, allowing rapid retraction that leads to instability and fragmentation. High roughness stabilizes the droplet, preventing breakup even at high $We$.

5. Significance and Contributions

• Mechanistic Understanding: The study elucidates how random surface morphology alters internal flow dynamics and contact line behavior, providing a physical basis for the wetting-bouncing transition.
• Predictive Tools: The proposed linear scaling laws for maximum spreading and the invariant contact time formula offer robust tools for designing surfaces.
• Applications: These findings are directly relevant to optimizing:
  • Water-repellent and anti-icing surfaces: By controlling contact time and rebound behavior.
  • Spray coating and inkjet printing: By managing spreading factors and avoiding breakup.
  • Agricultural spraying: By regulating droplet retention on rough plant leaves.
• Future Directions: The authors suggest extending this framework to superhydrophobic surfaces and moving substrates (translational or rotational motion).