## Abstract

Wettability has been explored for 100 years since it is described by Young’s equation in 1805. It is all known that hydrophilicity means contact angle (θ), θ < 90°; hydrophobicity means contact angle (θ), θ > 90°. The utilization of both hydrophilic surfaces and hydrophobic surfaces has also been achieved in both academic and practical perspectives. In order to understand the wettability of a droplet distributed on the textured surfaces, the relevant models are reviewed along with understanding the formation of contact angle and how it is affected by the roughness of the textured surface aiming to obtain the required surface without considering whether the original material is hydrophilic or hydrophobic.

### Keywords

- wettability
- droplet
- hydrophilic
- hydrophobic
- surface tension
- contact angle
- textured surface
- Wenzel model
- Cassie-Baxter model

## 1. Introduction

It is well known that when a small droplet of liquid is deposited on the solid surface, it forms a shape with a contact angle to the solid. This phenomenon is firstly described by Young in 1805, and he proposed that surface energy is the interaction between the forces of adhesion and the forces of cohesion which determine whether the wetting occurs or not (i.e., the spreading of a liquid over a surface) [1]. If it does not occur the complete wetting, the liquid in a bead shape will be formed. In the same time, as a function of the surface energies, a contact angle is defined in the system.

When the liquid wets the solid, three different interfacial boundary surfaces, viz., solid-air (sv), solid-liquid (sl), and liquid-air (lv), are involved. The contact angle, which is included between the interfaces of sl and lv, has to reach a certain value to satisfy the equilibrium state of the three interfacial tensions. It is all known that there are two requirements for the equilibrium.

## 2. Static equilibrium

The first requirement for keeping a balance of the three interfacial tensions in horizontal direction is described by Young’s Eq. (1):

where **γ** denotes the interfacial tension in the denomination of the force per unit length, or of the energy per unit area, which are equivalent in measuring the surface energy density, and *θ* is the contact angle at a location where the tangent along an lv interface intersects the solid surface as shown in Figure 1. For the surface of solid with high surface energy, *γ*_{sv} > *γ*_{sl}, *γ*_{lv} directs to the side of *γ*_{sl} and forms a contact angle smaller than 90°. This kind of surface is known to be hydrophilic as shown in Figure 1a. For a solid with low surface energy, *γ*_{sv} < *γ*_{sl}, *γ*_{lv} directs to the side of *γ*_{sv} and forms a contact angle larger than 90° which is known to be hydrophobic as shown in Figure 1b.

## 3. Dynamic equilibrium

Another requirement is the dynamic equilibrium determined by the interface energy which can be calculated from *γ* × *s*, where *s* is the area of interface. It should be noted that for a droplet of liquid with certain volume resting on a solid surface, it has the smallest lv interface when the contact angle is 90° (i.e., the droplet is a hemisphere as shown by the blue quarter circle in Figure 2); and whether its sl interface spreads (i.e., *θ* decreases) when *θ* < 90° or contract to be more like a sphere (i.e., *θ* increases) when *θ* > 90°, the lv interface area increases. Firstly, considering a droplet on a hydrophilic solid surface as shown in Figure 2a, the shape of the droplet has not reached equilibrium. With the spreading of the liquid, the area of both the sl interface and the lv interface will increase simultaneously. Because *γ*_{sv} > *γ*_{sl} on hydrophilic surface, the increment of the sl interface area means the conversion from the sv interface to the sl interface. The process involves a release in energy from the sv interface to the sl interface; as a result, the increment of the lv interface area implies a consumption of energy. When the energy changes caused by these two contrary factors are equal, the shape of the droplet will settle and the contact angle will achieve the final value of *θ*. This energy equilibrium can be described by the following equation:

where *ds*_{sl} and *ds*_{lv} mean a slight variation in the area of sl interface and lv interface, respectively. By combining with Eq. (1), the contact angle can be expressed by:

It should be noticed that *ds*_{lv}/*ds*_{sl} is the area changing rate of the lv interface with the sl interface increasing; it is only determined by the shape of the droplet. Eq. 3 shows the relationship between the contact angle and the profile of the droplet and is independent of materials and surface tension.

For the system applied on a hydrophobic surface as shown in Figure 2b, with the effect of the contracting of liquid, the area of the sl interface decreases with increasing lv interface. Because *γ*_{sv} < *γ*_{sl} on hydrophobic surface, the decrement of the sl area involves a release of energy to the increasing lv interface area. When the dynamic equilibrium of energy is reached, Eqs. (2) and (3) can also be applied on this kind of surface.

It should be noted that (*γ*_{sv}-*γ*_{sl}) and *ds*_{sl} are negative on hydrophobic surface.

## 4. Effect of surface roughness on contact angle

It should be noticed that there distinctively exists a difference between the geometric surface and the actual surface and their interface is not ideal as a proposed model in the textbooks. Actually, the surface of any real solid is not a perfect plane. Due to the surface roughness, the real area of the actual surface is larger than the so-called ideal (geometric) surface. Consequently, the surface roughness affects the contact angle and the contact angle distinctively varies with the surface roughness. As a result, in order to keep the equilibrium, the profile of a droplet will vary with the effect of the surface roughness. For studying *θ*’ (new contact angle) distributed on the real rough surface and the effect of its roughness on the relevant wettability, Wenzel and Cassie-Baxter proposed two different models to explain as a key effective factor how solid surfaces with the real geometry features affect the wettability [2, 3, 4, 5, 6, 7, 8].

Wenzel model.

According to the model described by Wenzel in 1936 [9], the solid surface completely contacts with liquid under the droplet as shown in Figure 3. The *sl* interface area is enlarged to be *r*, which is larger than 1.

With a variation of the geometric *sl* interface area, the amount of energy released from it or accumulated in it is increased:

In addition, the lv interface is not affected by the surface roughness. So the equilibrium with the new contact angle of *θ*’ can be expressed by:

Compared with Eq. (1), *θ*’ can be depicted as:

Taken *θ*_{w} and *θ*_{0} to represent *θ*’ and *θ*, respectively, it is obtained:

where *θ*_{w} is the contact angle on the rough surface with Wenzel model and *θ*_{0} is the original contact angle according to the ideal smooth surface. Eq. 9 is the Wenzel equation. It shows that when Wenzel model is applied, *r* > 1, the morphology of the surface always magnifies the underlying wetting properties. *θ*_{w} is larger than *θ*_{0} for the hydrophobic material (*θ*_{0} > 90°); and it is smaller than *θ* for the hydrophilic material (*θ*_{0} < 90°) [10, 11, 12].

## 5. Cassie-Baxter model

In 1944, Cassie applied and explored Wenzel equation on porous materials [13]. According to Cassie-Baxter model, air can be trapped below the drop as shown in Figure 4. The area of the *sl* interface is reduced by the surface roughness while a part of that transits to the lv interface in indentations. The ratio of the actual *sl* interface area to the geometric surface area is represented by *f*, which is smaller than 1 in Cassie-Baxter model.

With a variation of the profile of the droplet, the amount of energy transited among the interfaces is changed:

The equilibrium with the new contact angle of *θ*’ can be expressed by:

Compared with Eq. (1), *θ*’ can be calculated as:

Taken *θ*_{c} and *θ*_{0} to represent *θ*’ and *θ*, respectively, it is obtained:

where *θ*_{c} is the contact angle on rough surface with Cassie-Baxter model. Eq. (16) is Cassie-Baxter equation. According to Cassie-Baxter model, only the characteristics of hydrophobicity can be enhanced. *θ*_{c} is always larger than *θ* on the rough surface [14, 15, 16, 17].

In fact, numerous investigations have been devoted to the wettability on different surfaces, particularly for the surfaces inspired by Nature Mother [18, 19, 20, 21, 22, 23, 24, 25, 26]. Paxson et al. [27] fabricated a surface with the hierarchical textures initiated by lotus leaves and revealed the relevant mechanism of the variation or evolution of the adhesion force per unit length of the projected contact line distributed on natural textured surfaces. Results show that the adhesion force varies with the pinned fraction of each level of hierarchy.

Figure 5 shows a droplet sitting on a textured surface in a Cassie-Baxter state. It depicts the real contact line of the droplet, which is changed into many smaller lines. Meanwhile, the contact angle also changes from *θ*_{0}^{r} (the zeroth level) to *θ*_{1}^{r} (the first level of hierarchy) as shown in Figure 5b. If the contact line is divided into much smaller lines, viz., the second level of hierarchy, the related contact angle *θ*_{2}^{r} is distinctively different from *θ*_{1}^{r} of the first level of hierarchy as shown in Figure 5c. These phenomena will be kept on until a homogeneous wetting interface achieved when reaching a level *n*. Consequently, the contact angle either increases or decreases by adding multiple length scales of roughness at all smaller levels depending on the pinned fraction of each level of hierarchy, which is critical for designing surfaces with various adhesion [28, 29, 30, 31, 32, 33].

## 6. Conclusion

The droplet on a solid surface will exhibit a certain value of contact angle to achieve the equilibrium of the interfacial tensions. In addition, surface roughness will influence the contact angle, based on Wenzel’s and Cassie-Baxter’s theories, with the assumption of overhangs. It reveals that the contact angle can be controlled by the intentionally fabricated textured surfaces, and the surface with the fabricated textures can be changed from hydrophilic to hydrophobic, and vice versa, without considering whether the original material is hydrophilic or hydrophobic.