**3. Fundamentals of Liquid Penetration**

η , η ∆ In order to be able to understand mechanisms that influence the ultrasound signal during penetration, a fundamental understanding of liquid penetration dynamics is necessary. In general, penetration of a liquid into a porous substrate takes place by capillary flow into the pores. Assuming cylindrical pores, the flow can be described with the Lucas–Washburn equation. It can be derived from the Poiseuille equation for laminar flow: [30]

∆

$$\frac{\text{dh}}{\text{dt}} = \frac{\text{r}^2 \Delta \text{p}}{8 \eta \text{h}} \text{ ,} \tag{3}$$

γ θ γ θ where h (m) is the distance travelled by the liquid, r (m) the capillary radius, η (Ns/m<sup>2</sup> ) the liquid viscosity, and ∆p (N/m<sup>2</sup> ) the driving pressure difference. If gravity effects are negligible—as is the case for small pores in the micrometer range like in paper [31]—the driving pressure difference is the capillary pressure. It can be described with the Laplace equation: [32]

$$\Delta \mathbf{p} = \frac{\mathbf{\dot{\gamma}}\_{\rm L} \cos \theta}{\mathbf{r}} \,, \tag{4}$$

θ θ θ

With γ<sup>L</sup> (N/m) being the surface tension of the liquid and θ ( ◦ ) the contact angle of the liquid with the solid. Substituting Equation (4) into Equation (3) and integrating over time leads to the Lucas–Washburn Equation:

$$\mathbf{h} = \sqrt{\mathbf{f}} \cdot \sqrt{\frac{\mathbf{r} \mathbf{\gamma}\_{\rm L} \cos \theta}{2 \eta}} \tag{5}$$

This equation indicates two important relations. First, it shows that the penetration depth (and thus the penetration volume) is proportional to the square root of time. Second, the penetration is driven by the contact angle between the liquid and the solid. Equation (4) shows that the capillary pressure is only positive if cos θ > 0, i.e., θ < 90◦ . If θ > 90◦ the liquid is not sucked into the capillaries, but instead an external pressure needs to be applied so that the liquid enters the pores. Therefore, the contact angle between the liquid and the solid is the key criterion to determine if capillary penetration is possible.

Penetration into a real porous substrate is much more complex than penetration into cylindrical pores. Nevertheless, the Lucas–Washburn equation well illustrates the fundamental requirements for penetration in porous materials.

If capillary penetration is not possible, the liquid in contact with a porous substrate can still enter the pores as vapor, via surface wetting, by penetration within the porous paper fibers, or by diffusion. That is especially relevant if the substrate interacts with the liquid, as is the case for aqueous liquids and paper. The transport of water vapor through paper is most likely primarily driven by surface diffusion [30]. That means that water can enter the paper also if it is heavily sized and the contact angle is more than 90◦ . However, in that case there will be no liquid front filling the pores as is the case for capillary penetration. Instead water molecules will diffuse along or within the fiber surface, causing them to swell.
