Viscoelastic Droplet Impact on Surfaces with Sharp Wettability Contrast: Coupled Influence of Relaxation Time and Surface Tension

This numerical study employs a high-fidelity 3D OpenFOAM solver to demonstrate that increasing viscoelastic relaxation time significantly enhances droplet spreading and reduces height, whereas higher surface tension suppresses expansion, while sharp wettability contrasts on hybrid surfaces induce asymmetric spreading and distinctive equilibrium morphologies driven by the coupled effects of elastic energy storage and capillary forces.

The Gist

Imagine you are watching a drop of honey fall onto a table. Now, imagine that table is half covered in a sticky, super-absorbent sponge (the "hydrophilic" side) and the other half is covered in a slick, non-stick Teflon pan (the "hydrophobic" side).

This paper is a high-tech, computer-simulated experiment to see what happens when a drop of stretchy, gooey liquid (like a polymer solution or thick paint) hits this split table. The researchers wanted to understand two main things:

1. How "stretchy" the liquid is (how long it remembers being pulled).
2. How "tight" the liquid holds itself together (surface tension).

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

1. The "Memory" of the Liquid (Relaxation Time)

Think of the liquid not just as a fluid, but as a bunch of tiny, tangled rubber bands floating in water.

• Low "Memory" (Short Relaxation Time): If the rubber bands snap back instantly, the drop acts more like water. It hits, spreads out, and then quickly settles down.
• High "Memory" (Long Relaxation Time): If the rubber bands are slow to snap back, the liquid acts like a bouncy ball made of slime. When it hits the table, it stretches out further and wider because it stores up "elastic energy" (like pulling back a slingshot) before it lets go.

The Result: The researchers found that making the liquid "stretchier" (increasing the relaxation time) made the drop spread out about 13% wider. It was like the drop had extra spring in its step, allowing it to cover more ground before it stopped.

2. The "Tightness" of the Drop (Surface Tension)

Now, imagine the drop is made of a material that really wants to stay in a tight ball (high surface tension) versus one that is happy to spread out flat (low surface tension).

• High Surface Tension: The drop is like a tight rubber band trying to shrink. When it hits the table, it fights against spreading. It stays taller and doesn't flatten out as much.
• Low Surface Tension: The drop is like a loose sheet of plastic. It spreads out easily and flattens down.

The Result: When they made the liquid "tighter" (higher surface tension), the drop spread out slightly less (about 1% less) but stayed taller (about 3% taller). It was like the drop said, "I'm not going to flatten out; I'm going to bounce back up!"

3. The "Split Personality" Table (Hybrid Wettability)

This is where things get really cool. The table wasn't uniform; it was half sticky (sponge) and half slippery (Teflon).

• The Effect: When the drop hit the middle, it couldn't decide what to do. The sticky side pulled the liquid toward it, while the slippery side pushed it away.
• The Shape: Instead of a perfect circle, the drop turned into a lopsided shape.
  • Top View: It looked like a dustpan (flat on one side, scooped out on the other).
  • Side View: It looked like a shoe (flat on the bottom, but with a raised "toe" on the slippery side).

The liquid migrated heavily toward the sticky side, leaving the slippery side behind. The "stretchy" nature of the liquid made this lopsided shape even more dramatic because the rubber bands inside the drop kept pulling it toward the sticky side even after it stopped moving.

4. Why Does This Matter?

You might wonder, "Who cares about a drop of slime on a split table?"

Actually, this is huge for technology!

• Inkjet Printing: If you are printing with special inks (which are often stretchy), you need to know exactly how they will land. If the paper has different textures, you don't want the ink to run off the page or form weird blobs.
• Spray Coating: When spraying paint or pesticides, you want the liquid to stick exactly where you aim it, not bounce off or spread too thin.
• Microfluidics: In tiny medical devices, controlling how a drop moves is essential for testing blood or drugs.

The Big Takeaway

The researchers built a super-accurate computer model to show that how stretchy a liquid is and how tight its surface is work together with how sticky the surface is to decide the final shape of a drop.

• More stretchy liquid = Bigger, wider splash.
• Tighter liquid = Smaller splash, taller drop.
• A split surface = A weird, "dustpan" shape that leans toward the sticky side.

By understanding these rules, engineers can design better surfaces and fluids for everything from printing high-resolution photos to creating better medical sprays.

Technical Summary

1. Problem Statement

The impact dynamics of droplets on solid surfaces are critical for applications such as inkjet printing, spray coating, and microfluidics. While Newtonian fluid behavior is well-understood, real-world fluids (polymer solutions, bioaerosols) often exhibit viscoelasticity, which significantly alters spreading, retraction, and rebound behaviors.

Current literature faces three key limitations:

1. Most studies assume homogeneous wettability, ignoring the complex dynamics of surfaces with sharp wettability gradients (hybrid surfaces).
2. The coupled influence of gravitational forces (via the Eötvös number) and viscoelasticity is rarely examined systematically.
3. The nonlinear coupling between elastic stress relaxation, contact-line pinning, and sharp wettability discontinuities remains uncharacterized.

This study addresses these gaps by investigating how fluid viscoelasticity (modeled by relaxation time) and surface tension interact with sharp wettability contrasts (hydrophilic vs. superhydrophobic regions) to dictate droplet morphology and dynamics.

2. Methodology

The authors developed a high-fidelity, three-dimensional numerical framework using OpenFOAM (specifically the rheoInterFoam solver from the RheoTool package).

• Physical Model:
  • Fluid: A viscoelastic droplet (Oldroyd-B constitutive model) impacting a surrounding Newtonian medium (air).
  • Surface: A hybrid surface divided into two zones: Zone 1 (Hydrophilic, $WCA = 0^\circ$) and Zone 2 (Superhydrophobic, $WCA = 160^\circ$), creating a sharp discontinuity at the impact center.
  • Parameters: Droplet diameter $D = 2$ cm, impact velocity $U = 4$ m/s.
• Governing Equations:
  • Incompressible Navier-Stokes equations coupled with the Volume-of-Fluid (VOF) method for interface tracking.
  • Stress Tensor: Decomposed into Newtonian solvent and polymeric viscoelastic stress ($T$), governed by the Oldroyd-B equation with relaxation time $\lambda$.
  • Contact Angle: A velocity-dependent dynamic contact angle model (Kistler's Hoffman function) was implemented to capture hydrodynamic effects on the contact line.
• Numerical Techniques:
  • Log-conformation formulation: Used to ensure numerical stability at high relaxation times.
  • Discretization: First-order implicit Euler for time; high-resolution schemes (CUBISTA, van Leer) for convective terms.
  • Validation: The solver was validated against experimental data and benchmark numerical studies for both Newtonian and viscoelastic impacts.

3. Key Contributions

• First Fully Coupled Framework: This study provides the first 3D simulation framework that simultaneously resolves Oldroyd-B viscoelasticity, dynamic contact angle hysteresis, and sharp wettability discontinuities under gravitational loading.
• Systematic Parametric Study: It isolates and analyzes the coupled effects of relaxation time ($\lambda$) and surface tension ($\sigma$) on droplet dynamics across uniform and hybrid surfaces.
• Discovery of Unique Morphologies: The identification of distinct "dustpan-like" (top view) and "shoe-like" (side view) equilibrium shapes resulting from the interplay of viscoelasticity and wettability gradients.

4. Key Results

A. Influence of Relaxation Time ($\lambda$)

Increasing the relaxation time from 0.02 s to 0.12 s significantly enhances elastic energy storage:

• Spreading: Maximum spreading diameter increased by 12.6–12.9% (from ~24.97 mm to ~28.17 mm).
• Height: Minimum droplet height decreased by 16.6%, indicating a flatter, more extended profile.
• Mechanism: Longer $\lambda$ delays viscous dissipation, allowing the droplet to spread further before recoiling. On hybrid surfaces, this elastic memory sustains deformation, delaying the return to equilibrium.
• Hybrid Surface Dynamics: The sharp wettability contrast ($0^\circ$ vs. $160^\circ$) induces asymmetric spreading. Fluid migrates toward the hydrophilic region, creating a lateral capillary pressure gradient. Viscoelasticity amplifies this asymmetry, leading to the unique "dustpan" and "shoe" morphologies.

B. Influence of Surface Tension ($\sigma$)

At a fixed relaxation time ($\lambda = 0.05$ s), increasing surface tension from 0.05 N/m to 0.15 N/m:

• Spreading: Suppressed maximum spreading by ~1.1% (from 27.21 mm to 26.90 mm).
• Height: Increased minimum height by 3.3% (from 2.12 mm to 2.20 mm).
• Mechanism: Higher $\sigma$ increases capillary restoring forces, which suppress radial expansion and promote faster vertical recovery (recoil).
• Stress Distribution: Higher surface tension leads to more localized viscoelastic stresses and a deeper "dustpan" curvature, whereas lower tension allows broader, more diffuse deformation.

C. Energy and Stress Evolution

• Kinetic Energy: Higher $\lambda$ prolongs kinetic energy fluctuations and delays equilibrium. On hybrid surfaces, energy decay is asymmetric: the hydrophilic side dissipates energy quickly, while the hydrophobic side sustains recoil longer.
• Normal Stress Difference ($N_1$): Higher $\lambda$ results in higher and more persistent $N_1$, indicating stronger elastic memory. The stress distribution is highly asymmetric on hybrid surfaces, with peaks localized near the wettability discontinuity.

5. Significance and Implications

This research elucidates the intricate interplay between elastic stresses, capillary action, and surface wettability gradients. The findings offer critical insights for:

• Advanced Surface Design: Engineers can tailor surface wettability patterns to control droplet directionality and final shape in inkjet printing and 3D printing.
• Process Optimization: Understanding how relaxation time and surface tension affect spreading and rebound allows for better control in spray coating and microfluidic mixing, reducing waste and improving throughput.
• Fundamental Physics: The study bridges the gap between Newtonian impact theory and complex fluid dynamics, specifically highlighting how viscoelasticity modifies the response to heterogeneous surface conditions.

In summary, the paper demonstrates that viscoelasticity does not merely alter transient dynamics but fundamentally reshapes the final equilibrium morphology of droplets on complex surfaces, providing a quantitative basis for optimizing droplet-based technologies.