# Wetting and Roughness: Part 1

Original entry: Alexander Epstein, APPHY 226, Spring 2009

**Wetting and Roughness: Part 1**

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 1, we look at the introductionWetting Without Roughnessand consider the sectionRougness

## Soft matters

### Brief Intro

One of the authors of our course text, David Quere, authored this lovely review paper, which focuses on how the roughness of a solid affects its wettability. Controlling wettabilty of a solid surface is of great interest in many engineering contexts. The extremes, complete wetting and complete drying, are particularly desirable. Windshields should have a uniform film to maintain the glass transparency, and to remove dust particles under gravity or air flow. Surfaces that must have no liquid contamination should force drops to be spherical and virtually without contact, so that they roll off.

In the classical case of flat and chemically homogeneous solids explored by Young and Laplace, the relationship between the three interfacial surface tensions (equivalent to surface energies) determines the wetting condition. As Marangoni explained, a film extends from a liquid reservoir on a wetting solid, replacing <math>\gamma_{SA}</math> by <math>(\gamma_{SL} + \gamma_{LA})</math> if the spreading coefficient

<math>S = \gamma_{SA}\ - \gamma_{SL}\ - \gamma_{LA}\ >\ 0</math> .

For *S*> 0 a drop spreads (Fig. 1a), while oppositely it coalesces into a spherical cap (Fig. 1b). Now, the chemical (or Young-Dupre) contact angle on a homogeneous flat surface is defined by our favorite equation:

<math>\gamma_{SA} = \gamma_{SL} + \gamma_{LA}\ cos\theta</math>

The contact angle is fixed by the chemical nature of the three phases if the solid surface is flat. If the solid is rough, the rules of the game change. This Quere considers shortly.

To digress briefly, we can also define a drying parameter

<math>D\ = \gamma_{SL} - \gamma_{SA} - \gamma_{LA}</math>

If *D* > 0 the contact line will be withdrawn by surface forces until a film of air comes between the solid and the liquid. In other words, complete drying of the surface occurs. Interestingly this happens if water on freshly cleaned glass (high <math>\gamma_{SA}</math>) has an air bubble at the water-glass interface. And in the Leidenfront effect, *D* goes to zero. This is when a liquid is placed on a solid surface whose temperature greatly exceeds the liquid's boiling point, and the liquid rests on a film of its own vapor, therefore not contacting the solid. However D is never > 0 for any liquid on any **flat** solid, i.e., complete drying of a liquid resting on a **flat** solid is not possible. The emphasis is on "flat" because superhydrophobic rough surfaces enable complete drying (dewetting).

Before we consider roughness of surfaces, let's briefly consider *wicking*, or capillary draw into a porous material. The simple wicking parameter tells us whether a liquid will be drawn into a capillary tube:

<math>W\ = \gamma_{SA} - \gamma_{SL}</math>

Wicking or penetration occurs if *W* > 0, as shown in Fig. 2a, which corresponds to the contact angle being less than 90°, or the Laplace pressure behind the advancing meniscus being lower than the atmospheric pressure. When *S* > 0, *W* > 0 as well, and what happens is that a molecular thickness liquid film precursor advances along the tube walls (Fig. 1a), reducing interfacial energy; then the meniscus advances on the prewet tube surface, driven by the reduction of liquid-air interface. In this complete wetting scenario (not a bad approximation for a clean glass tube), <math>W = \gamma_{LA}</math>

The opposite scenario happens if wicking is energetically unfavorable, or *W* < 0. Then liquid is expelled from the tube or pore (Fig. 2b). This idea is the basis for a porous solid that can remain filled with air even when immersed in water or other liquid. Without leaving room temperature, we can then approach Leidenfrost effect behavior! Only with surface roughness are water repellency and other specific properties possible.

### Rough Solids: Contact Angle Hysteresis and Wenzel

Most solids are rough at the micron scale either naturally or from processing steps. Lamination, for example, can produce striations or grooves; compaction of grains results in grain-scale roughness; and coating processes can produce microdrops when the coating film dewets. The much rarer solids that are in fact molecularly flat include floated glass; here the surface roughness is from thermal fluctuation of a liquid-liquid interface and therefore only on the scale of angstroms.

Why are chemical or topographic defects on surfaces important in wetting? As we learned in class, defects can pin a contact line and generate contact angle hystersis. A droplet that is sessile on an inclined surface does not move because its advancing and receding contact angles, <math>\theta_a > \theta_r</math>, are different. Equivalently, the asymmetry in contact angles <math>\Delta\theta = \theta_a - \theta_r</math> (which can vary from nearly zero to nearly <math>\theta_a</math>) creates a Laplace pressure difference between the front of the drop (high curvature) and the rear (smaller curvature) that opposes gravity for a small droplet. Quere points out that there is a certain range of possible contact angles for a given liquid-rough solid system, as shown in Figure 3. A topographic defect such as the edge of a groove or tip will have a characteristic angle <math>\phi</math>, and the maximum contact angle when the defect is present is <math>\pi - \phi + \theta</math>. Hence any angle between <math>\theta</math> and <math>\pi - \phi + \theta</math> is possible.

**Two important conclusions:**

1. A groove acts as a nonwetting defect and a tip as a wetting defect.

2. A drop deposited on a rough surface stops spreading when it is surrounded by nonwetting defects, and it stops dewetting upon evaporation when it is pinned on wetting defects. These motions happen in jump-pin-jump fashion.

We can now exploit these principles to guide a flowing liquid along a predefined route on a rough surface. In nature, this is observed in a species of desert beetle that condenses and collects moisture from the air to survive. Of course, this is also a problem, for example on dirty windows and shower mirrors, where sessile droplets collect and compromise optical function.

#### Wenzel Model

Clearly roughness affects the contact angle hysteresis. But it also impacts the apparent (or macroscopic) contact angle such that it can be very different from that predicted by Young-Dupre. Wenzel first noted this phenomenon and introduced a simple geometrical argument based on the roughness factor, *r*:

<math>r = \frac{\text{actual surface area}}{\text{apparent or projected area}}</math>

The equilibrium apparent contact angle <math>\theta^{*}</math> is reached as the drop spreads and the contact line follows all the rough features of the surface, displacing all air with liquid (Figure 4). An energy balance can be written for displacement of the contact line:

<math>dE = r(\gamma_{SL} - \gamma_{SA})\ dx + \gamma_{LA}\ dx\ cos\theta^{*}</math>

We see that roughness increases both the solid energies by a factor *r*. The equilibrium of <math>dE = 0</math> gives the apparent contact angle:

<math>cos\ \theta^{*} = r\ cos\theta</math>

and if the solid is flat (*r* = 1) this recovers the Young-Dupre equation. The Wenzel relation shows that roughness amplifies both hydrophilicity and hydrophobicity, leading to the great interest in roughness as a mechanism to study extreme wetting. It is far from a quantitative model, however. The roughness factor can be made arbitrarily large, but experiments show that complete wetting and complete drying cannot be induced merely by large roughness. Such behavior is not observed because Wenzel assumptions are not always satisfied. Pinning of the contact line on defects and residual liquid in troughs can impede droplet motion. Thus the Wenzel state has very low receding angles and giant hysteresis <math>(\Delta\theta \sim \theta_a)</math>. An important point to remember therefore is that the scale of the drop should be much larger than the scale of the defects to directly use the Wenzel model.

## References

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