Surface tension and Wettability: Tell me more!

The SPH solver of Nextflow Software now includes a surface tension model.

We are pleased to gather hereby papers written by Amaury Bannier, Product Manager, and Valentin Bonnifet, Field Application Engineer, about the various possible applications of this technology.

You may have already noticed your coffee mysteriously rising in-between the sugar grains against the basic law of gravity, or the surprising behavior of water drops on the lotus leaves that remains almost spherical instead of spreading down. Those phenomena, respectively called capillary rise and super-hydrophobicity, are manifestations of the surface tension and, more especially, of the wettability. Let’s briefly reviews explain what it means.

Surface tension

Where there is an interface between immiscible (liquid or gas) fluids, there is surface tension. Surface tension arises from the attractive forces of that exist between molecules at the interface. Liquid molecules attract each other, gas molecules too, but those on the interface are missing half of their neighbors. As a result, they experience an attraction towards their side, and a tension arises across the surface layer. This mechanical effect is called surface tension and often denoted γ.

Under the influence of the surface tension, fluid interfaces tend to contract to the smallest possible area and, conversely, increasing fluid interfaces area requires energy. Schematically, these interfaces can be thought as the elasticity of a toy balloon rubber membrane: the higher the balloon rubber elasticity γ, the higher the force that tends to reduce its area (Force = γ · Length), the higher the energy required to inflate it (Energy = γ · Surface), and the higher the pressure gap between inside and outside of the balloon (Laplace Pressure = 2 · γ · Curvature).

Surface tension explains why small drops of water are spherical and why water drips from a faucet: the spherical shape is the one that minimizes the area, and a cylindrical water stream reduces its area when deforming toward individual droplets (so-called Plateau-Rayleigh instability).


Wettability comes into play when multiple surfaces compete against each other, each one pulling in its direction. As a result of this competition, a regular contact angle is observed between those interfaces to optimize the total surface tension energy.

Consider the case of a drop of liquid (L), put on a planar solid substrate (S), and surrounded by gas (G). Three interfaces are competing: G/L, G/S and L/S. Their respective surface tensions affect the optimized contact angle Θ between the solid plane and the drop tangent:

  • Θ = 90° contact angle. This occurs when the two solid-related interfaces have the same surface tension (γ[G/S] = γ[L/S]). The 90 degrees contact angle reduces the gas/liquid interface area.
Drop on surface with 90 degrees contact angle
  • High wettability (Θ < 90°). When the gas/solid interface is more energy-greedy than the liquid/solid one (γ[G/S] > γ[L/S]), the drop spreads on the substrate so that the solid substrate is more covered by the liquid and less by the gas. When water is involved, the substrate is said hydrophilic.
Drop on hydrophilic surface
  • Low wettability (Θ > 90°). Conversely, when the liquid/solid surface tension is higher than the gas/solid one (γ[G/S] < γ[L/S]), the drop retracts to limit its interface with the solid substrate. When water is involved, the substrate is said hydrophobic.
Drop on hydrophobic surface
  • A substrate is said super-hydrophilic when the contact angle almost reaches Θ = 0°. The drop tends to spread into a thin layer to cover most of the solid. This happens when the gas/solid surface tension is high enough (γG/S > γL/S + γG/L) such that the two liquid-related interfaces are less costly than a single gas/solid interface. Similarly, a substrate is super-hydrophobic when the contact angle almost reaches Θ = 180°. The drop tends to remain almost spherical above the solid surface.

Industrial Interest

The understanding of the surface tension and wettability effects is essential for many industrial applications. This includes for instance windshield rain-repellent treatment, anti-fogging agents, waterproof concrete, self-cleaning materials, experiments under micro-gravity, microfluidic application and multiple chemical processes… Therefore, its proper modeling within numerical simulation can be of crucial interest.

Impact and runoff of water drops on inclined planes, with more or less hydrophobic materials.

Everyday life applications

One could learn a lot about fluid mechanics phenomena by putting a glance at everyday life flows. Tossing a cup of coffee, looking at the rain falling outside or playing with soap bubbles are playful and powerful possibilities among others to better understand physics. Today we focus on a flow which seems to be very simple but involves few interesting subtilities related to surface tension: the water tap closure.


The setup is quite easy to describe. Standing in front of the sink, brushing my teeth, I see a pipe letting water fall vertically under gravity. However, I notice that the resulting flow is not easy to describe. Indeed, the water jet does not fall with constant radius and I can experiment different flow regimes depending on the tap flow rate. When the flow rate is large the water jet is continuous. But when I slowly close the tap, the jet twists and splits up in droplets as shown above.

Now let’s talk about the physics behind the scene. Due to flow velocity, radius and falling length, the tap water is subject to instabilities driven by surface tension. Joseph Plateau was the first to experimentally characterize this instability in 1873, followed by Rayleigh who gave an explanation from a theorical point of view. Surface tension acts as a spring on the free surface of the stream. Disturbances travel on the free surface media and grow exponentially until the separation point is reached.

This phenomenon is not circumscribed to our bathrooms or kitchens. In order to deposit ink on paper, it is mandatory to trigger the Rayleigh-Plateau instability in an inkjet printer. Conversely, staying away from this instability during the stretching process is a challenging key point for optical fiber manufacturers due to the high radius over length ratio of such a cylinder.

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Some of the everyday flows seem to be simple but are very challenging to reproduce numerically. This water tap flow involves a complex free surface leading to expensive computations using conventional finite volume method based on VOF (Volume Of Fluid) or Level-set. The SPH (Smooth Particle Hydrodynamics) Lagrangian approach is a good candidate to deal with such flows due to its intrinsic ability to represent complex free surfaces. The above figure shows a computation result using our SPH solver including a surface tension model. The water flows at 1 m/s then the valve is progressively closed displaying different stability flow regimes.

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One can also see surface tension instability with the droplet oscillation. When droplets form, the jet breakdown excites vertical oscillation mode as depicted on the right . The droplets shape is like a donut where the middle moves up and down.

In an upcoming article we will focus on a phenomenon called hydraulic jump that arises when the jet impacts the sink.

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