# Morphology Characteristics of the Liquid-Vapour Interface in Porous Media
## 4 Discussion
### Evolutionary Trend of l-v Interface  
As phase boundaries/contact surfaces, the liquid-vapour and solid-liquid interface both play crucial roles in the mechanical and thermal behaviour of porous media, such as interparticle stress [41], as well as enhanced vapour diffusion and heat transfer [42, 43]. The area of the l-v interface is calculated from the segmented image and classified according to the principal curvature. To mitigate the influence of sample size on the area, the liquid-vapour (l-v) interface area is divided by the solid area of the corresponding sample (with each sample having an equal porosity), resulting in a normalized interface area (a-v = S-v/S solid), which is modified in this study by considering the l-v interface [44, 45], where the upper surface of the soil sample is in direct contact with the air. The surface area of the liquid phase of the first slice (the vapour-liquid interface in direct contact with the atmosphere), i.e., a-v = (S-v + S-first)/S solid, was supplemented, and the evolution with saturation is shown in Figure 11. a-v has a small value of 0.058 at the beginning of evaporation (S-v = 85.0%) and it keeps increasing nonlinearly until it reaches the peak value of 0.176 around the saturation of 31.7%. The same trend was also found in our experiments with glass beads [10]. The evolution of the internal interface during evaporation of sandy soils is directly investigated and its effect on the evaporation rate can be discussed in relation to the curvature of the interface. Figure 11: Variation in l-v interface area with a saturation: (**a**) all internal interfaces and (**b**) interfaces categorised according to morphology. Figure 10: Connectivity analysis of liquid and vapour phase: evolution of Euler characteristic number with saturation. Forming more liquid ganglia (liquid bridges, liquid rings, etc.) in the interparticle pore space increases the interparticle forces and enhances heat transfer and evaporation fluxes. To study the trend of interparticle forces in solids, we discuss the trend of the mean curvature of l-v after classifying them according to their morphology separately and thus apply while Young-Laplace equation Laplace equation to establish the relationship between capillary pressure and mean curvature [23; 24]. Based on careful observation, the form of the l-v interface of the sample during the whole soil water evaporation process can be classified according to the principal curvature as follows: annular liquid bridge (Type III: \(\kappa_{1}\kappa_{2}<0\)), porous liquid bridge (Type I: \(\kappa_{1}>0\), \(\kappa_{2}>0\)) and attached droplet/isolated liquid ring (Type II: \(\kappa_{1}<0\), \(\kappa_{2}<0\)). Figure 12 visualizes the capillary pressure evolution of the produced ganglia within the soil as evaporation proceeds. The capillary pressure can be calculated using the Young-Laplace equation:  
\[\psi=T_{\rm s}\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)\]  
where \(\psi\) represents the capillary pressure, \(T_{\rm s}\) is the surface tension of water at the curved liquid interface, and \(R_{1}\)and\(R_{2}\) are the principal radii of curvature at the water-air interface. The Young-Laplace equation indicates that the magnitude of capillary pressure at the curved liquid interface can be determined by the liquid's surface tension and the radius of curvature. The curvature of interface can be directly calculated from accurately extracted interface, thus reflecting the evolution of capillary pressure throughout the entire evaporation process. The suction of the liquid ganglia produced in the tested soil samples did not change significantly with evaporation until the remaining stage of evaporation (\(S_{\rm w}=6.4\%\)). On the contrary, at \(S_{\rm w}=6.4\%\), there was a more pronounced increase in the samples' curvature radius due to the liquid ganglia's inability to maintain their morphology at more minor saturation levels. the l-v interface shrinks to smaller pores, corresponding to a larger capillary pressure, until it breaks down.