# Wetting and Roughness: Part 3

Authors: David Quere

Annu. Rev. Mater. Res. 2008. 38:71–99

#### Soft matter keywords

microtextures, superhydrophobicity, wicking, slip

By Alex Epstein

### Abstract from the original paper

We discuss in this review how the roughness of a solid impacts its wettability. We see in particular that both the apparent contact angle and the contact angle hysteresis can be dramatically affected by the presence of roughness. Owing to the development of refined methods for setting very well-controlled micro- or nanotextures on a solid, these effects are being exploited to induce novel wetting properties, such as spontaneous filmification, superhydrophobicity, superoleophobicity, and interfacial slip, that could not be achieved without roughness.

In Part 3, we examine the sectionsSuperhydrophobicityandSpecial Properties

## Soft matters

If a hydrophobic solid is rough enough, the liquid will not conform to the solid surface as assumed by the Wenzel model, and instead air pockets will form under the liquid and support it. This is the Cassie state. It is observed if the energy of the liquid-vapor interfaces is lower than the energy of wetting the solid. In the case of our beloved micro/nanoposts, we can assume that the liquid-air interfaces are flat (since the Laplace pressure can be assumed zero at the bottom of the drop) and that the wet surface area <math> \sim\ (r - \phi_s)</math> and liquid-air area <math> \sim\ (1 - \phi_s)</math>. The Cassie state is favored when

<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math>

and the corresponding critical Young angle relation is

<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math>

For very rough solids (<math>\scriptstyle{r \gg 1}</math>), <math>\scriptstyle{cos\ \theta_c\ \to\ 90^{\circ}}</math>, and the criterion for air trapping is satisfied since we already assume chemical hydrophobicity (<math>\scriptstyle{\theta > 90^{\circ}}</math>). Materials with long hairs can have a roughness of 5 to 10, and a beautiful example of this is the *Microvelia* water strider pictured in Figure 12. Its legs have high aspect ratio hydrophobic hairs that trap air in a Cassie state, allowing the insect to skate on water.

The Cassie state in the above case can be considered stable. However, lower roughness factors lead to the critical angle criterion not being met, and the Cassie state can then be metastable. As long as the drop does not nucleate a contact point with the bottom of the rough surface, the air will remain trapped.

**An obvious but important fact:** the more air under the drop in a Cassie state, the closer the apparent angle <math>\scriptstyle{\theta^{\circ}}</math> is to 180°, or no contact. Any deviation from 180° is diagnostic of the fraction of solid surface area in contact with the liquid. From an energy balance, the equilibrium apparent contact angle <math>\scriptstyle{\theta^{*}}</math> is:

<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math>

For example, if <math>\theta = 110-120°</math>, and <math>\phi_s = 5-10%</math>, the apparent angle is 160-170°. This condition must be accompanied by the presence of edges on the posts (or more generally of large slopes on the rough surface). Re entrant designs make more robust Cassie states and allow even hydrophilic surfaces can trap air under liquid!

### References

1. Quere, David. Wetting and Roughness. Annu. Rev. Mater. Res. 2008. 38:71-99

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