Multiphysics Simulation of Drop Sliding
Technical Challenge
Drop sliding on or sticking to an inclined surface is a common phenomenon — in nature, everyday life, and industrial applications from heat transfer to cleaning (Figure 1). Accurately predicting drop pinning and sliding is important to a wide range of industrial applications including windshields, water-repellent fabrics, heat exchanger surfaces, solar panels, power line insulators, salt spray drainage on ship hulls and offshore structures, and bubbles in fluidic channels.
Veryst Solution
The computational fluid dynamics (CFD) and multiphase flow experts at Veryst developed a 3D multiphase flow simulation to predict whether a drop slides or sticks on an inclined surface (Figure 1). Everyone understands the driving force: gravity. What’s commonly misunderstood is the source of the resistive force, namely contact angle hysteresis.
To explain contact angle hysteresis, we first review the basics of contact angles. The contact line is the perimeter of the wetted area under a drop where liquid, air, and substrate meet. At each point on the contact line, the angle between the substrate and the drop’s free surface (the air/liquid interface), through the liquid, is called the contact angle. On a perfectly flat, smooth, and clean (homogeneous chemistry) surface, the (theoretical) equilibrium state for a drop is that the contact angle is constant along the contact line. We call this the static equilibrium contact angle, θS (Figure 2, a).
Contact angle hysteresis arises when the contact line pins (gets stuck) on microscale or nanoscale surface imperfections (roughness or chemical impurities) resulting from natural processes or manufacturing. This pinning resists the driving force (gravity, air drag, electromagnetism, etc.). If this resistive force is strong enough, it can hold the drop in place instead of letting it slide. Specifically, for a drop to move, the advancing portion of its contact line must have a contact angle exceeding what’s called the advancing contact angle, θA, and the receding portion of its contact line must have a contact angle that is below the receding angle, θR (Figure 2, b). Both θA and θR depend on the liquid, gas (in this case, air), and substrate surface (chemistry and roughness). We developed our model in COMSOL Multiphysics®. Since COMSOL does not have built-in functionality to model contact angle hysteresis, we developed this into our model via a custom implementation.
An important consequence of contact angle hysteresis is a force of retention on the drop, which scales as σr(cosθA – cosθR), where σ is the surface tension of the air-water interface and r is the drop radius. Increasing the strength of the hysteresis, Δθ=θA−θR, increases the force of retention resisting the drop’s motion. Since the force of gravity scales with the drop volume, and hence r3, while the force of retention scales linearly with r, large drops slide while smaller drops get pinned. Our simulations recapitulate this relationship for different drop sizes and levels of hysteresis (Figure 3).
To better quantify the sliding-pinning transition, we ran our model with different levels of hysteresis to identify the critical radius separating drop sliding and pinning (Figure 4). We compare our results to Furmidge [1]’s semi-analytical prediction that includes a tunable constant with reported ranges in the literature (Figure 4, shaded band). Our simulation results fall within this band across the full range of hysteresis Δθ values tested, which validates our model.
Conclusion
Veryst successfully captured the physics of drop sliding and pinning on inclined surfaces using high-fidelity multiphase CFD simulations. Our validated results demonstrate the dependence of the critical sliding radius on contact angle hysteresis. The model serves as a predictive tool for anticipating drop sliding and pinning behavior, supporting a wide range of industrial applications.
References
[1] Furmidge, C. V. “Studies at phase interfaces. I. The sliding of liquid drops on solid surfaces and a theory for spray retention.” Journal of Colloid Science 17, no. 4 (1962): 309–324.