**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:

$$(\chi\_{sv} - \chi\_{sl})d\mathfrak{s}\_{sl} = \chi\_{lv}d\mathfrak{s}\_{lv} \tag{2}$$

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

For the system applied on a hydrophobic surface as shown in **Figure 2b**, with the

cos *<sup>θ</sup>* <sup>¼</sup> *dslv dssl*

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

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

� *γsv* � *γsl* ð Þ�*ds* ð Þ¼ *sl γlvdslv* (4)

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

is independent of materials and surface tension.

*Drop of liquid on solid surfaces when the equilibrium has not been reached.*

**4. Effect of surface roughness on contact angle**

this kind of surface.

**Figure 2.**

*Wettability on Different Surfaces*

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

wettability [2–8].

**117**

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:

$$\cos \theta = \frac{ds\_{lv}}{ds\_{sl}} \tag{3}$$

#### **Figure 2.**

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

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

where *ds*sl and *ds*lv mean a slight variation in the area of sl interface and lv interface,

cos *<sup>θ</sup>* <sup>¼</sup> *dslv dssl*

respectively. By combining with Eq. (1), the contact angle can be expressed by:

*γsv* � *γsl* ð Þ*dssl* ¼ *γlvdslv* (2)

(3)

shown in **Figure 1b**.

*Contact angle on various surfaces.*

*21st Century Surface Science - a Handbook*

**Figure 1.**

**3. Dynamic equilibrium**

be described by the following equation:

**116**

*Drop of liquid on solid surfaces when the equilibrium has not been reached.*

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.

$$-(\chi\_w - \chi\_{sl})(-d\mathfrak{s}\_{sl}) = \chi\_{lv}d\mathfrak{s}\_{lv} \tag{4}$$

$$\cos\theta = \frac{d\mathfrak{s}\_{lv}}{d\mathfrak{s}\_{sl}}$$

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 [2–8].

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 *s* 0 *sl* which is equal to the "actual surface" by the roughness. There is a ratio of the sl interface area to the geometric surface area, *r*, which is larger than 1.

$$
\mathfrak{s}'\_{sl} = \mathfrak{rs}\_{sl} \tag{5}
$$

**5. Cassie-Baxter model**

*Wettability on Different Surfaces*

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

than 1 in Cassie-Baxter model.

among the interfaces is changed:

**Figure 4.**

**119**

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

*sl* ¼ *fdssl* (10)

*sl* ¼ *f γsv* � *γsl* ð Þ*dssl* (12)

� ð Þ 1 � *f* (14)

*lv* ¼ *dslv* þ ð Þ 1 � *f dssl* (11)

*lv* ¼ *γlvdslv* þ ð Þ 1 � *f γlvdssl* (13)

cos *θ*<sup>0</sup> ¼ *f*ð Þ� cos *θ* þ 1 1 (15)

cos *θ<sup>c</sup>* ¼ *f*ð Þ� cos *θ*<sup>0</sup> þ 1 1 (16)

*ds*0

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

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

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

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

*γlv*

With a variation of the profile of the droplet, the amount of energy transited

*ds*0

*γsv* � *γsl* ð Þ*ds*<sup>0</sup>

*γlvds*<sup>0</sup>

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

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

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

$$r(\chi\_{sv} - \chi\_{sl})d\mathfrak{s}'\_{sl} = r(\chi\_{sv} - \chi\_{sl})d\mathfrak{s}\_{sl} \tag{6}$$

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:

$$\cos \ \theta' = \frac{r(\chi\_{sv} - \chi\_{sl})}{\chi\_{lv}} \tag{7}$$

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

$$r\cos\ \theta' = r\cos\theta\tag{8}$$

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

$$r\cos\theta\_w = r\cos\theta\_0\tag{9}$$

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

**Figure 3.** *Schematic of a droplet on the rough surface described by Wenzel.*
