Susanna Laurén Sep 29, ’20 < 4 min

Cassie – Baxter equation – Extreme wettability on surfaces

Young’s equation is the basis of the definition of wettability. However, it’s well known that Young’s equation is not valid on real surfaces. The theory assumes that the surface is ideal which means both chemical and topographical homogeneity. This is not achieved on real surfaces which in addition to being chemically often heterogeneous, are almost always rough at least on a nanoscale. To better describe the contact angle on real surfaces, different wetting states have been proposed. The most discussed are Wenzel state and Cassie-Baxter state.

Cassie’s law

Cassie’s law describes the contact angle of a liquid on a chemically heterogeneous surface. The law was first introduced in 1948 [1]. According to it, the apparent contact angle on a heterogeneous surface can be expressed as;

cos⁡θc1cos⁡θ1 + σ2 cosθ

, where θ1 and θ2 are the contact angles on the two chemically different areas and the σ1 and σ2 are the corresponding fractional surface areas. The equation as such has very little practical use as it is very unlikely to find the surface with very distinct chemical heterogeneity, let alone being able to identify their fractional surface areas.

Cassie-Baxter equation to describe wettability of porous surfaces

The more commonly encountered equation is that of Cassie-Baxter [2]. It describes the contact angle on a porous surface. Instead of having two different surface chemistries as in Cassie law, the surface is now composed of solid surfaces and areas where the drop is in contact with air. The equation can be written as

cos⁡θc1 cos⁡θ1 - σ2= σ1(cos⁡θ1 + 1) - 1

Since the contact angle between the liquid and air is 180° and σ2 = 1 - σ1.

A combination of Cassie – Baxter and Wenzel states leads to extreme wettabilities

Although the calculation of the theoretical contact angle with the Cassie-Baxter equation would be quite difficult, the model has been used to design surfaces with extreme wettabilities. The model shows that decreasing the contact area between the solid and the liquid (i.e. increasing the area where the liquid is in contact with air) will increase the contact angle if the contact angle on the same smooth solid is higher than 90 °.

As the highest possible contact angle on a smooth surface is around 120 °, most of the superhydrophobic surfaces utilize dual-scale roughness. On those types of surfaces, you would have for example micropillar arrays with are coated with nanoparticles to add nanoscale roughness to it. Those surfaces utilize both Cassie-Baxter and Wenzel models to reach extreme wettability.

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[1] A. B. D. Cassie, "Contact angles," Discussions of the Faraday Society, vol. 3, pp. 11-16, 1948.
[2] A. B. D. Cassie and S. Baxter, "Wettability of porous surfaces," Transactions of the Faraday Society, vol. 40, pp. 546-551, 1944.


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