The wetting model is a classical problem in surface and biomimetic science.  Wettability is determined by the balance between adhesive and cohesive forces, adhesive which is when a liquid tries to spread on a surface and cohesive when it forms into a ball.

The resultant between adhesive and cohesive forces is called the contact angle. As the tendency of a drop to spread out over a surface increases, the contact angle decreases, making the contact angle an inverse measure of wettability.

There are two classical models to examine the macroscopic contact angle on a rough or heterogeneous substrate; the Wenzel model and the Cassie–Baxter model. The Wenzel model indicates that part of the liquid completely pierces into the microstructures on the rough solid substrate and the geometric topographies amplify the hydrophobicity of hydrophobic surfaces and enhance the hydrophilicity of hydrophilic surfaces. The Cassie model deals with a substrate with several phases of different wetting properties, implying that the macroscopic contact angle is actually the weighted average function of different hydrophilicities of the materials.

Although Wenzel and Cassie models can successfully elucidate the superhydrophobic phenomenon of inhomogeneous substrates, some have doubted the accuracy of these results. They designed experiments to demonstrate that the contact line and not the contact area beneath the droplet is responsible for the macroscopic contact angle. These disputes suggest that the Wenzel and Cassie models are only valid in a certain range, but to what extent the two models apply is a key subject of investigation. 

New research says that it is the triple contact line and not the contact area of the droplet/solid interface that determines the macroscopic contact angle. Their proposed continuum model, termed the mechanism-based model, can illustrate the contact line pinning effect at some wedges or phase interfaces between different materials. 

The concept of the contact angle dates back to the pioneering contribution of Thomas Young in 1805, a milestone in the characterization of the wetting property of a perfectly smooth and homogeneous solid surface. However, many experiments have shown that most wetting and de-wetting behaviors of solids are not only related to the chemical components but also closely related the micro- and/or nano-structures of their surfaces.

The combination of wetting properties and geometric topographies of a surface lead to interesting phenomena in nature. For example, the "lotus effect", that is, the self-cleaning capability of lotus leaves, is due to the intrinsic hydrophobicity and micro/nanomorphologies of surfaces. Some aquatic creatures, such as the water strider, water spider and mosquito, resort to their superhydrophobic capacities to stand, walk and jump on still or even flowing water. This skill originates from the special micro/nano setae and nanoridge structures on their legs, which can produce extremely large driving forces to support their body weight. The Stenocara beetle of the Namib Desert is a dew-collector, possessing special surface microstructures with variable hydrophilic and hydrophobic domains on its carapace.

A new approach for achieving special wetting surfaces has been shown by varying the chemical compositions and geometrical topographies. Superhydrophobic surfaces with static contact angles larger than 150° were prepared to produce ultra-hydrophobicity with fractal or hierarchical surface structures, to mimic the biological materials. These designed rough and heterogeneous surfaces have already been put into use in various areas of industry, spanning from porous media, microfluidic devices, and self-cleaning paints to glass windows.

For these natural phenomena and superhydrophobic (or superhydrophilic) materials, a critical issue is how to predict the wetting behaviors properly, which is of great value to both fundamental science and engineering applications.

The new work does not try to invalidate the Wenzel and Cassie models, but is directed toward a further understanding of the wetting mechanism based upon the triple contact line, from a new viewpoint of continuum mechanics. This theory of wetting is constructed on a strong theoretical basis, including consideration of the force equilibrium and energy, which deals with a more general substrate, say, a rough and chemically heterogeneous solid surface. 

Using the energy formulation and considering the movable boundary condition, the authors derived the macroscopic contact angle on a rough and heterogeneous substrate, under the assumption that the Young's contact angle and interfacial energies are all field functions of the position of the contact line. The equation of the relationship shows that the macroscopic contact angle has no direct connection with the gravity of the droplet. Besides, the macroscopic contact angle strongly depends on the geometrical and chemical properties of the substrate, as it is a continuum field variable at any point. This means that different positions will have different contact angles.

Finally, the result emphasizes that the macroscopic contact angle is dependent only on the properties of the triple contact line and is independent of those of the area underneath the droplet. However, the classical Wenzel and Cassie models are both concerned with the geometrical and chemical properties of the contact zone, and not the contact point. In the following, the authors designed several substrates with special roughnesses and hydrophilicities, and compared the results predicted by the Wenzel model, Cassie model and the mechanism-based model.

The results show that the mechanism-based model deviates from the classical models greatly for these types of substrates, and this conclusion suggests that the Wenzel and Cassie models only hold in a certain range.

They say the mechanism-based model can also be adopted to explore the pinning effect due to a wedge or different hydrophilicity phases. At a sharp wedge, the derivative of the substrate curve is discontinuous, and the macroscopic contact angle does not have a determinate value. In reality, there are no pure "sharp" or "jump" points, and these sharp tips have small curvature radii. In particular, a fine AFM tip may take a radius of several tens of nanometers. The curvature at the singular point can be enlarged, and in this narrow domain, the derivative of the substrate can change from zero to the value when the triple contact line is located on the inclined side of the wedge. In the real world, this transition zone near the sharp tip is too minute to be observed.

Therefore, the triple contact line is often regarded as being "pinned" at this position, and only variation in the contact angle is macroscopically seen. Similarly, we can consider a substrate only including two phases with different contact angles. This pinning phenomenon is really a "black box" used to find of a solution employing the developed classical model. Mathematically, there is no wetting property jump at the interface of two different materials binding together. Indeed, there is also a transition zone owing to the atom diffusion, where the width is reported as several tens of micrometers. In this area, the gradient of the wetting property is very abrupt, spanning from a smaller value to a larger value, and the motion of the triple contact line cannot be observed by the naked eye.

The only phenomenon is that the contact angle at this point (in fact in this area) continuously increases with an increasing droplet volume.

This study investigated the wetting model for a general substrate from a new viewpoint of continuum mechanics, and the analyses first showed how the Wenzel and Cassie models deviate from the experimental results, and then elucidated the motion of the triple contact line. The result shows that the contact angle is independent of the gravity of the droplet and the contact area underneath the droplet, and is only dependent on the property of the triple contact line . Several substrates with typical geometric topographies and chemical components were provided as examples to demonstrate the difference between the classical models and mechanism-based models.

The derived model can also illustrate the pinning effects on a sharp wedge or at the interface between two different phases by employing continuum field theory. This continuum model also casts light on the pinning effect on a sharp wedge or at the interface between two different phases. The investigation revisits the fundamentals of wetting on rough and heterogeneous substrates, which will help in the design of super-hydrophobic materials and provide the predictions required to engineer novel microfluidic devices.

Published in SCIENCE CHINA Physics, Mech&Astro, 2012, Issue 11