# Wetting and Roughness: Part 2

Original entry: Alexander Epstein, APPHY 226, Spring 2009

**Wetting and Roughness: Part 2**

Authors: David Quere

Annu. Rev. Mater. Res. 2008. 38:71–99

## Contents

#### Soft matter keywords

microtextures, superhydrophobicity, wicking, slip

### 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 2, we look at the sectionsMicrotextured SolidsandHemiwicking

## Soft matters

During the past decade, the use of microtextured solids and more recently nanotexturing has been popular to induce surface wetting properties that cannot otherwise be obtained. Roughness of the surface changes the unique Young angle to a range of possible angles and generates an apparent angle in the surface plane that is different from the local angle at the contact line.

Quere asserts that three factors are responsible for the sudden resurgence of interest in this area.

1. Late 1990s research from the Kao Corporation showing large contact angles of liquids on fluorinated rough surfaces. Note that this was similar to results reported in the 1940s. 2. Papers by Neinhuis and Barthlott in Germany reporting the variety of surface features found on hydrophobic plants, such as the lotus. Animal studies followed. 3. Developments in micro/nanofabrication techniques that allowed more sophisticated designs to be studied, as inspired by (1) and (2).

Briefly, the Kao experiment plotted relation of the apparent contact angle <math>\theta^{*}</math> of various liquids on a rough fluorinated surface against the expected Young angle of each liquid on a flat fluorinated surface. The S-shaped curve, seen in Figure 5, describes the amplifying effect of roughness on hydrophilicity and hydrophobicity. The first, steeper slope on the right side follows Wenzel's roughness closely. The second, smaller slope is the superhydrophilic regime, in which Wenzel breaks down because hemiwicking of surrounding surface cavities (as considered below) leads to the droplet sitting on both solid and liquid.

### Micro/Nanotexture Inspirations from Nature

As early as AD 77, Pliny the Elder reported the beading of water drops on woolly plant leaves, the first reported observation of superhydrophobicity. Truly systematic studies of natural microtextures, however, have only happened in the last decade.

Plant leaves often features bumps on the scale of 10-50 <math>\mu</math>m, and some, including the now famous lotus, also have a finer 100 nm scale of features (Fig. 6b). The fractal geometry appears to contribute to the superhydrophobicity and self-cleaning ability of the leaf. The mechanism of the fractal surface is debated still, but phenomenologically it provides both a high contact angle and a low hysteresis. Thus drops on the leaves have very high mobility and roll off with ease. Remarkably, the rice leaf has an anisotropic arrangement of papillae that direct the flow of water along preferred directions. Other noteworthy example of superhydrophobic surfaces in nature are the feathers of pigeons and ducks; cicada, butterflies; and the leg setae of water striders in Fig. 6c that rest on trapped air (c.f., Leidenfrost effect) as they travel on water. Mosquitoes' eyes are completely drying due to a pattern of 100 nm bumps (Fig. 6d).

#### Trying to Synthesize

Quere notes that we can make a superhydrophobic surface by a very crude technique in the garage: take a piece of glass to a sooty flame, and the dark soot coating will provide microroughness and plenty of carbon to repel water. Obviously most of today's research uses sophisticated techniques to create surface patterns.

### Hemiwicking

Quere introduces the term "hemiwicking" to describe an imbibition phenomena in rough surfaces that is similar but different from classical wicking. As a liquid film progresses through surface micro/nanostructures such as those shown in Figure 7, at least one additional side is exposed to air. Thus, we have another liquid-air interface. This is different from the Wenzel model, in which the cavities of the surface are filled as though they are capillary tubes while areas not under the drop remain dry.

The simplest example of a hemiwicking microtexture is a groove of width *w* and depth <math>\delta</math>, shown in Figure 8. Such grooves can be exploited, as in the rice leaf, to achieve directional wetting.

Hemiwicking will occur if the solid is wetting (<math>\gamma_{SL} < \gamma_{SA}</math>) and if this energy change overcomes the additional liquid-vapor interface formed on top. This liquid-vapor interface will be flat to minimize area, so the surface energy change is:

<math>dE\ = (\gamma_{SL} - \gamma_{SA})(2\delta + w )\ dx + \gamma_{LA}\ w\ dx</math>

The Young equation provides that the liquid progression is favorable (d*E* < 0) if <math>\theta , \theta_c</math>, where

<math>cos\ \theta_c = \frac{w}{2\delta + w}</math>

The right-hand side of this equation varies from 0 to 1 depending on the groove cross-section aspect ratio <math>\delta / w</math>. Important conclusions:

High aspect ratio (narrow and deep) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> high <math>\longrightarrow</math> spontaneous hemiwicking Low aspect ratio (shallow and wide) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> low <math>\longrightarrow</math> difficult to hemiwick

#### Pillars or Posts

Micro/nanoposts are the bread and butter surfaces that we study and apply in the Aizenberg Group. So it is helpful to realize that similar arguments hold for a solid decorated with posts (Fig. 7a) as with grooves. The two characteristics of this surface are pillar density <math>\phi_s</math> and roughness *r*. Hemiwicking through a forest of posts is seen in Figure 9: a film of ethanol permeates the posts in a circle beyond the footprint of a drop. In some cases, the film conforms to the symmetry of the post pattern and will be square or hexagonal, as explored by the Stone Group [2].

The impregnating front propagates as shown in Figure 10. Liquid coats the solid on an area proportional to <math>r - \phi_s</math>, while the liquid-vapor interfacial area is proportional to <math>1 - \phi_s</math>. With *dx* larger than the scale of the post pitch *p*, the energy of hemiwicking in a forest of posts is:

<math>dE\ = (\gamma_{SL} - \gamma_{SA})(r - \phi_s)\ dx + \gamma_{LV}(1 - \phi_s)\ dx</math>

As with grooves, there is a maximum Young angle <math>\theta_c</math> below which progression of the liquid is spontaneous.

<math>cos\ \theta_c = (1 - \phi_s)(r - \phi_s)</math>

The take-away message here is that we can tune the invasion of liquid by controlling the geometry of the forest. For rare defects (small <math>\phi_s</math>), <math>cos\ \theta_c \approx 1/r</math>: the rougher the substrate, the more favorable is hemiwicking. In comparison to grooves, the liquid front in a forest of posts must somehow be activated to achieve the jumps in Fig. 10. For wetting liquids, this is facilitated by the menisci, which form around each post, allowing the liquid to reach the next row. In other cases, the contact line can remain pinned in a metastable Wenzel state, and additional external energy such as vibration is need to nucleate contact with the next row of posts. This means that an equilibrium "fried egg" configuration of a drop coexisting with a finite ring of film is possible--the edge of this fried egg is represented in Figure 11.

Looking back at Fig. 5, the second low-slope regime (<math>\theta < \theta_c</math> results from the hemiwicking effect. The low slope of the apparent angle is due to the fact that the drop sits on a composite surface consisting mainly of liquid; therefore the solid roughness has little effect on <math>\theta^{*}</math>. In this situation, moving the contact line by d*x* eliminates solid-air interfaces on a surface fraction <math>\phi_s</math>, and liquid-air interfaces are eliminated on a fraction <math>1 - \phi_s</math>. This tweaks the earlier energy statement, and the apparent angle becomes

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

### Hemiwicking dynamics

If wetting of the liquid is complete (<math>S > 0, \theta = 0</math>), a molecular precursor film propagates first, and then the hemiwicking film follows to lower the total surface energy by reducing liquid-vapor interface. Then the driving force for hemiwicking is <math>\gamma(r -1)</math>, depends only on *r*, and vanishes on a flat surface (*r* = 1). Since the surface defects are small, the main competing force is viscous, and it scales as <math>\eta V x</math>, denoting *x* as the impregnated distance and <math>\eta</math> as the viscosity. The force balance recovers the same time scaling as the Washburn law inside a porous medium:

Hemiwicking distance <math>x\ \propto\ \sqrt{t}</math>

Equivalently, we can apply the classic diffusion law <math>x^2 = Dt</math> to quantify the diffusion constant *D*. For wetting liquids and posts of height *h*, pitch *p*, and radius *b*, the wicking force scales as <math>\gamma b h/p^2</math>. The situation is slightly different for low and high aspect ratio posts. For short posts, the viscous resistance is fixed by the depth *h* of the flow and

<math>\text{Low aspect ratio posts:}\ D \sim\ (\gamma /\eta)(b h^2 /p^2)</math>

which is easily tuned by the post height. [3] For tall posts, most of the viscous resistance is around the posts and the coefficient is

<math>\text{High aspect ratio posts:}\ D \sim \gamma b/ \eta</math>

To conclude: Fixing the height of the pillars does not only influence the film dynamics, it also selects its thickness, since surface energy favors a film thickness which matches the pillar height.

### References

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

2. Courbin L, et al. Imbibition by polygonal spreading on microdecorated surfaces. *Nat. Mater.* 2007, **6**, 661-664

3. C. Ishino, et al. Wicking within forests of micropillars, 2007 *EPL* **79**, 56005