**2. Theoretical aspects on powder wettability**

#### **2.1 Basics on contact angle**

When a water drop enters in contact with a surface, or a liquid comes in contact with a vertical and infinitely wide plate or porous solid and powders, and rises by capillary effect, the contact angle gives an insight of these wetting phenomena. With a macroscopic view, the contact angle represents the angle formed at the intersection of the interfaces liquid-solid and liquid-vapor by applying a tangent line at a point from the so-called "three-phase contact line", where the solid, liquid, and vapor phases coexist (see **Figure 1a**). Contact angles are not only limited to the liquid-vapor interface on a solid but also applicable to the liquid-liquid interface on a solid. Depending on the media (surface or porous structure), the contact angle measurement and the tool to be used are different.

#### **2.2 Flat surface (Young's equation)**

On an ideal flat surface (partial wetting), Young's equation allows the determination of the liquid contact angle in equilibrium θe. It represents the static contact angle value based on the interfacial energy balance between the solid, liquid, and gas surrounding the liquid at the triple line contact (**Figure 2**). Young described first the entire phenomenon in 1805, which was mathematically formalized by Dupré while also discovered elsewhere by Laplace. Thus, this Young-Dupré law is commonly called "Young-Laplace equation" [46–49].

#### **2.3 Rough surfaces (Wenzel model & Cassie-Baxter model)**

In the case of non-ideal flat surfaces (roughness added), two contact angle measurements are involved. On the one hand, the Wenzel model adds roughness to

**Figure 2.**

*Flat solid surface cases: (a) Young's equation; (b) liquid drop with an equilibrium contact angle (θe).*

**Figure 3***.*

*Rough surfaces described by the Wenzel model: (a) apparent contact angle θ<sup>w</sup> determination, θ<sup>e</sup> being the equilibrium value used in Eq. (1), and RW* ≥ *1 represents the ratio between the real surface area over the projected area; (b) drop in Wenzel state.*

*Wettability of Probiotic Powders: Fundamentals, Methodologies, and Applications DOI: http://dx.doi.org/10.5772/intechopen.106403*

#### **Figure 4.**

*Rough surfaces described by the Cassie-Baxter model: (a) Cassie-Baxter contact angle θCB determination, ƒ<sup>A</sup> and ƒ<sup>B</sup> being the fractions of surfaces A and B; (b) drop in Cassie-Baxter state.*

complete Young equation (Eq. (1)), and on the other hand, the Cassie-Baxter model depicts surfaces with chemical heterogeneities.

#### *2.3.1 Wenzel model*

The Wenzel model illustrates how the drop is in contact through the roughness of surfaces that Young's equation does not take into account. Thus, the drop is pinned on the surface, forming an apparent value of contact angle, which is different to the equilibrium one (**Figure 3**). This angle is modified by a factor called the Wenzel roughness *R*<sup>w</sup> [50–54].

Such phenomenon appears when the solid surface is completely in contact with liquid under the droplet. However, air can be trapped below the drop that tends to lower its energy (e.g. fakir and nails board). In this situation, the drop contact angle is described by the Cassie-Baxter model.

### *2.3.2 Cassie-Baxter model*

The Cassie-Baxter model describes the rough surface as a succession of two different surfaces SA and SB with a resulting wettability that depends on two contact angles θ<sup>A</sup> and θ<sup>B</sup> [53, 55, 56]. In this case, the contact angle θCB is a function of the fractions of surfaces A, a first-level roughness feature with solid width and air spacing structure, and surfaces B, a second level roughness at the base of a second-level capillary bridge (**Figure 4**), as defined in Eq. (2).

Theoretical studies show the importance of roughness and its impact on the Wenzel to Cassie-Baxter transition. In fact, the double scale roughness is not always the key parameter to enhance the transition. This also depends on RW factor and the decrease of *ƒ<sup>A</sup>* [57–59].

#### **2.4 Capillary rise phenomenon (Washburn & Darcy models)**

Trees, oil extracting, ink absorption, liquids and sponge, sugar and coffee are among common applications where one can witness a capillary rise phenomenon. It has been described many years, even centuries, ago. When a both-side opened tube is approached to a liquid reservoir surface, the liquid tends to fill the tube. The thinner the tube, the higher the liquid rises inside. The height or height square (or the

equivalent mass) versus time is always monitored. This curve has two parts, a dynamic component (rising) and a steady state component. At certain time, the liquid reaches a limit height (h) represented by Jurin's law (Steady state) represented by Eq. (3) [48, 60].

$$h = \frac{2\chi\cos\theta\_{\epsilon}}{\rho \text{g}R} \tag{1}$$

where *θ<sup>e</sup>* is the contact angle, *R* the radius of the tube, *ρ* the density of the liquid, *γ* the surface tension, and *g* the gravity.

#### *2.4.1 Lucas-Washburn model*

The dynamic part of the capillary rise is driven by the Lucas-Washburn equation (LWE). As it is a time-dependent phenomenon [5, 48, 61–66], the equation analytical expression including the gravity and inertia parts is:

$$\frac{2}{R}\text{y.}\cos\left(\theta\_{\epsilon}\right) = \rho \text{gh} + \frac{8}{R^2}\eta h. \frac{dh}{dt} + \rho. \left(h\frac{d^2h}{dt^2} + \left(\frac{dh}{dt}\right)^2\right) \tag{2}$$

where *R* represents the radius of the tube, *ρ* the density of the liquid, *γ* the surface tension, *g* the gravity, and *η*: viscosity.

By neglecting the gravity and inertia and assuming h to 0 when time equals 0, and using the Taylor's expansion [67], the LWE is obtained from the above equation:

$$h = \sqrt{\frac{2t}{b}}\tag{3}$$

where

$$b = (8/R2)/\_{\rm L}Rcos(e)\_{\rm L}$$

By replacing the height *h* with the mass *m* of the liquid through the relation *<sup>m</sup>* <sup>¼</sup> *<sup>ρ</sup>:πR*<sup>2</sup> h [68, 69], the following equation can be obtained:

$$
\gamma m(t)2 = \frac{\rho 2\pi 2R^5 \gamma \cos \theta}{2\eta} \text{\*} t \tag{4}
$$

This Lucas-Washburn's equation possesses limitation because the inertial forces are neglected.

#### *2.4.2 Darcy model*

The capillary rise previously presented does not take into account the permeability, even though there is a capillary constant in the LWE (geometry of the porous material). However, the approximation done with the critical radius is not enough to determine the permeability of materials (porosity). Darcy's law describes thus the wicking in porous media.

*Wettability of Probiotic Powders: Fundamentals, Methodologies, and Applications DOI: http://dx.doi.org/10.5772/intechopen.106403*

Darcy's discovery in 1856 was based on the study of the Dijon fountains, the ground water, and permeability by defining through theoretical and experimental works a universal formula related to the flux of water in sand [48, 70–72]. By changing the pressure (height of the reservoir), a linear relation between the flux and the pressure was observed, which was at the origin of Darcy's law, as follows:

$$h(t).\frac{dh(t)}{dt} = \frac{k}{\eta}.\frac{\chi}{R} \tag{5}$$

where *k* being the porous material permeability (can be related to viscosity), *R*: radius of the section, *η*: viscosity, and *γ*: surface tension.

By integrating the Darcy equation, it is easy by using the previous arguments to write

$$h = \sqrt{\frac{2.t}{b\_{\text{eff}}}} \tag{6}$$

where

$$b\_{\rm eff} = \frac{\eta R}{k.\gamma} = \frac{\frac{8}{R^2}\eta}{\frac{2}{R}\gamma.\cos\left(\theta\_\epsilon\right)} = \frac{4\eta}{\gamma.R.\cos\left(\theta\_\epsilon\right)}\tag{7}$$

From Eqs. (8) and (9), we can deduce

$$h = \sqrt{\frac{2.k.\chi.t}{R\_{\text{eff}}}}\tag{8}$$

This equation indicates that the permeability *k* is proportional to h<sup>2</sup> and *Reff* represents the effective radius. From the kinematics of the imbibition in a porous media, the effective radius can thus be extracted, characterizing its permeability or equivalently its porosity.
