Instabilities of ring-rivulets: Impact of substrate wettability

This paper uses numerical simulations to demonstrate that tuning substrate wettability patterns, such as annular bands and radial contact angle gradients, allows for precise control over the stability, breakup dynamics, and resulting droplet morphology of ring-rivulets.

The Gist

Imagine you have a tiny, perfect ring of water sitting on a table. If the table is just a plain, ordinary surface, this water ring is unstable. It's like a balloon that's about to pop, but instead of popping, it tries to fix itself. Depending on how wide the ring is compared to its size, it will do one of two things:

1. Snap into beads: If the ring is thin and narrow, it will break apart into a string of separate water droplets, like a necklace of pearls.
2. Collapse into a puddle: If the ring is thick and wide, it will shrink inward, pulling all the water into a single, round puddle in the center.

This paper is like a recipe book for controlling that water ring. The researchers used computer simulations to see what happens if you change the "texture" of the table (the substrate) underneath the water. They found that by painting specific patterns on the table, they could force the water ring to do exactly what they wanted, even if it wanted to do something else naturally.

Here is how they did it, using simple analogies:

1. The "Velcro Track" (The Annular Band)

Imagine the water ring is a runner on a track. On a normal track, the runner might get tired and stop in the middle (collapse) or trip and scatter (breakup).

The researchers put a special strip of "Velcro" (a ring of extra-sticky surface) right under the water.

• The Result: This sticky ring acts like a fence. It stops the water from collapsing into the center puddle. No matter how wide the ring is, it must break apart into droplets because the "Velcro" holds it in place.
• The Control: By making the "Velcro" stickier or less sticky compared to the rest of the table, they could control exactly how many droplets formed. It's like adjusting the tension on a guitar string to get a specific number of notes.

2. The "Hill and Valley" (The Radial Gradient)

Next, they changed the table so that the "stickiness" changed gradually as you moved from the center outward, like a gentle hill or a valley.

• The "Valley" (Sloping Downward): Imagine the table gets stickier as you move toward the center. This acts like a slide. The water ring feels a strong pull toward the middle. Even if the ring was wide and usually would have broken into beads, this "slide" forces it to rush inward and collapse into a single puddle.
• The "Hill" (Sloping Upward): Imagine the table gets less sticky as you move toward the center (or stickier as you move outward). This acts like a hill the water has to climb. If the ring tries to collapse inward, it hits a "wall" of resistance. This stops the collapse completely, keeping the ring stable and intact, even if it's wide and unstable on a normal table.

The Big Picture

The main takeaway is that the shape and behavior of these liquid rings aren't just about the water itself; they are heavily influenced by the surface they sit on.

• Uniform Surface: The water follows its natural instincts (breakup or collapse).
• Patterned Surface: The researchers can "program" the surface to act as a traffic cop. They can tell the water, "You must break into 10 droplets," or "You must stay as one big ring," or "You must rush to the center."

By simply changing the chemical "texture" of the table in specific patterns, they gained total control over whether the water ring breaks apart, collapses, or stays stable, and exactly how many droplets it creates.

Technical Summary

Technical Summary: Instabilities of Ring-Rivulets: Impact of Substrate Wettability

Problem Statement
The paper addresses the dewetting dynamics of ring-shaped liquid rivulets deposited on partially wettable substrates. While the breakup of straight rivulets is a well-studied problem governed by Rayleigh-Plateau instability, ring rivulets present additional complexity due to their non-uniform curvature and the distinct curvatures of their inner and outer contact lines. Previous theoretical and numerical work has established that the fate of a ring rivulet—whether it collapses into a single central droplet or breaks up into multiple droplets—is primarily dictated by the aspect ratio ($\psi_0$, defined as the ratio of the rivulet width to its radius) and the substrate contact angle. However, the potential to actively control these instability pathways and the resulting dewetting morphologies through engineered substrate wettability patterns remains largely unexplored for ring geometries. The study investigates whether spatially varying contact angles can be used to manipulate the stability, breakup timescales, and final droplet configurations of ring rivulets.

Methodology
The authors employ numerical simulations based on the thin film equation (TFE), solved using a lattice Boltzmann method (LBM) via the Swalbe solver. The governing equation describes the evolution of the film thickness $h(x,t)$:
$$\partial_t h(x,t) = \nabla \cdot (M_\delta(h) \nabla p)$$
where the pressure $p$ includes capillary forces ($-\gamma \nabla^2 h$) and a disjoining pressure term $\Pi(h, \theta)$ that accounts for fluid-solid interactions and allows for spatially varying equilibrium contact angles $\theta(x)$. The mobility function $M_\delta(h)$ incorporates an effective slip length $\delta$ to regularize the contact line singularity.

Initial conditions consist of a toroidal cap profile centered at the origin, which is relaxed to an equilibrium shape and perturbed with Gaussian noise. The study systematically varies:

1. Uniform Substrates: Constant contact angles $\theta_0$ and aspect ratios $\psi_0$.
2. Annular Band Patterns: A region of lower contact angle $\theta_a$ confined within the initial rivulet width, surrounded by a background of higher contact angle $\theta_b$.
3. Radial Gradients: Linear contact angle profiles pointing either inward ($\theta$ decreases toward the center) or outward ($\theta$ increases toward the center).

Key Results

• Uniform Substrates:

  • Breakup vs. Collapse: A critical aspect ratio $\psi_0 \approx 0.2$ discriminates between breakup and collapse. For $\psi_0 < 0.2$, the ring breaks into multiple droplets ($n_d > 1$); for $\psi_0 > 0.2$, it collapses into a single central droplet.
  • Droplet Count: In the breakup regime, the number of droplets $n_d$ generally follows the linear stability analysis (LSA) prediction $n_d \approx \pi / (2\psi_0)$, particularly for small contact angles. However, for larger contact angles ($\theta_0 > 10^\circ$), $n_d$ becomes largely independent of $\psi_0$, suggesting the process is driven by reduced substrate wettability.
  • Timescales: Breakup times ($\tau_b$) increase with $\psi_0$ for very small contact angles but become independent of $\psi_0$ for larger angles. Collapse times ($\tau_c$) scale as $\tau_c \sim \psi_0^{-2}$, a relationship derived phenomenologically and confirmed by simulations.

• Annular Band Patterns (Lower Contact Angle Band):

  • Suppression of Collapse: Depositing the rivulet on an annular band of lower contact angle (surrounded by a higher contact angle background) effectively removes the collapse mode. The ring undergoes breakup for all tested aspect ratios.
  • Control of Morphology: The contrast between the band and background contact angles serves as a control parameter. Increasing the mismatch causes the number of formed droplets to deviate from the homogeneous prediction and become independent of the initial aspect ratio.
  • Stability: The pattern increases the stability of the rivulet against rupture, resulting in longer breakup times compared to uniform substrates.

• Radial Gradients:

  • Inward Gradient (Wettability increases toward center): This configuration accelerates the retraction of the ring. Steep gradients can induce a rapid collapse that prevents breakup entirely. Shallower gradients allow the ring to break up while shrinking, leading to droplets that eventually merge at the center. This enables transient control over droplet formation near the center.
  • Outward Gradient (Wettability decreases toward center): This gradient acts as a stabilizing force against the natural tendency of wide rings ($\psi_0 > 0.2$) to collapse. It can stabilize the principal radius of the ring, preventing hole closure and allowing the deposition of larger liquid volumes in a ring shape.

Significance and Claims
The paper claims that by tuning substrate wettability patterns, it is possible to exert precise control over the stability and dewetting morphology of ring rivulets. Specifically:

1. Mode Selection: It is possible to selectively remove the collapse mode (via annular bands) or the breakup mode (via outward gradients), thereby dictating whether the system evolves into a single droplet or a droplet array.
2. Morphological Control: The number and position of final droplets can be manipulated not just by the initial geometry, but by the contact angle contrast and gradient direction.
3. Timescale Engineering: Wettability patterns can alter the characteristic time scales of dewetting, either accelerating retraction or stabilizing the ring against collapse.

The authors conclude that these findings offer a mechanism to control the fate of ring rivulets, which are precursors to circular droplet patterns. They suggest that future work could explore time-dependent wettability patterns to further manipulate instabilities and discover new dewetting pathways, though they do not propose specific experimental implementations beyond the numerical framework presented.