**5. Cassie-Baxter model**

Wenzel model.

*r*, which is larger than 1.

material (*θ*<sup>0</sup> < 90°) [10–12].

**Figure 3.**

**118**

interface area is enlarged to be *s*

*21st Century Surface Science - a Handbook*

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*

roughness. There is a ratio of the sl interface area to the geometric surface area,

*s* 0

With a variation of the geometric *sl* interface area, the amount of energy

In addition, the lv interface is not affected by the surface roughness. So the

cos *<sup>θ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>r</sup> <sup>γ</sup>sv* � *<sup>γ</sup>sl* ð Þ

where *θ*<sup>w</sup> is the contact angle on the rough surface with Wenzel model and *θ*<sup>0</sup> 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***<sup>w</sup>** is larger than *θ*<sup>0</sup> for the hydrophobic material (*θ*<sup>0</sup> > 90°); and it is smaller than *θ* for the hydrophilic

*γlv*

*sl* which is equal to the "actual surface" by the

*sl* ¼ *rssl* (5)

*sl* ¼ *r γsv* � *γsl* ð Þ*dssl* (6)

cos *θ*<sup>0</sup> ¼ *r* cos *θ* (8)

cos *θ<sup>w</sup>* ¼ *r* cos *θ*<sup>0</sup> (9)

(7)

0

*γsv* � *γsl* ð Þ*ds*<sup>0</sup>

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

Taken *θ*<sup>w</sup> and *θ*<sup>0</sup> to represent *θ*' and *θ*, respectively, it is obtained:

released from it or accumulated in it is increased:

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

*Schematic of a droplet on the rough surface described by Wenzel.*

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.

$$ds'\_{sl} = f ds\_{sl} \tag{10}$$

$$ds'\_{lv} = ds\_{lv} + (\mathbf{1} - f)ds\_{sl} \tag{11}$$

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

$$f(\chi\_{sv} - \chi\_{sl})d\mathbf{s}'\_{sl} = f(\chi\_{sv} - \chi\_{sl})d\mathbf{s}\_{sl} \tag{12}$$

$$
\gamma\_{lv}d\mathbf{s}'\_{lv} = \gamma\_{lv}d\mathbf{s}\_{lv} + (\mathbf{1} - f)\gamma\_{lv}d\mathbf{s}\_{sl} \tag{13}
$$

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

$$\cos \theta' = \frac{f(\chi\_w - \chi\_d)}{\chi\_w} - (1 - f) \tag{14}$$

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

$$\cos \theta' = f(\cos \theta + \mathbf{1}) - \mathbf{1} \tag{15}$$

Taken *θ*<sup>c</sup> and *θ*<sup>0</sup> to represent *θ*' and *θ*, respectively, it is obtained:

$$\cos \theta\_{\varepsilon} = f(\cos \theta\_0 + 1) - 1 \tag{16}$$

where *θ*<sup>c</sup> is the contact angle on rough surface with Cassie-Baxter model. Eq. (16) is Cassie-Baxter equation. According to Cassie-Baxter model, only the

**Figure 4.** *Schematic of a droplet on the rough surface described by Cassie-Baxter.*

#### **Figure 5.**

*Schematic of self-similar contact line pinning. (a) A liquid droplet that rests in a Cassie-Baxter state on a hierarchical surface exhibits an apparent receding angle* θ*<sup>0</sup> r . (b) The apparent contact line of the drop is divided into many smaller first-level contact lines, each at the top of a first-level roughness feature with width* w *and spacing* s*. each of these first-level contact lines sits at the base of a first-level capillary bridge, which has a local receding contact angle* θ*<sup>1</sup> r . (c) The apparent contact line of each second-level capillary bridge is further divided into smaller second-level contact lines, each atop a second-level roughness feature. Each second-level contact line sits at the base of a second-level capillary bridge, which has a local receding contact angle* θ*<sup>2</sup> r .*

characteristics of hydrophobicity can be enhanced. *θ*<sup>c</sup> is always larger than *θ* on the rough surface [14–17].

In fact, numerous investigations have been devoted to the wettability on different surfaces, particularly for the surfaces inspired by Nature Mother [18–26]. 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*<sup>0</sup> <sup>r</sup> (the zeroth level) to *θ*<sup>1</sup> 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*<sup>2</sup> r is distinctively different from *θ*<sup>1</sup> <sup>r</sup> 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 [28–33].

**Author details**

Yeeli Kelvii Kwok

**121**

Kong, Kowloon, Hong Kong

*Wettability on Different Surfaces*

*DOI: http://dx.doi.org/10.5772/intechopen.92885*

Department of Mechanical and Biomedical Engineering, City University of Hong

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: yeelikwok@yahoo.com

provided the original work is properly cited.

#### **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.

*Wettability on Different Surfaces DOI: http://dx.doi.org/10.5772/intechopen.92885*
