Wetting and Contact-Angle Hysteresis: Density Asymmetry and van der Waals Force

A droplet depositing on a solid substrate leads to the wetting phenomenon, such as dew on plant leaves. On an ideally smooth substrate, the classic Young’s law has been employed to describe the wetting effect. However, no real substrate is ideally smooth at the microscale. Given this fact, we introduce a surface composition concept to scrutinize the wetting mechanism via considering the liquid-gas density asymmetry and the fluid-solid van der Waals interaction. The current concept enables one to comprehend counterintuitive phenomenon of contact-angle hysteresis on a smooth substrate and increase of contact angle with temperature as well as gas bubble wetting.

Figure 1

(a) Sketch for the surface composition concept when a liquid droplet ($L$) is deposited on a solid substrate ($S$) in a gas phase ($G$). The average volume fraction of the liquid species penetrating into solid at the microscale is denoted by $\varphi$. Between the dashed lines: $L\text{−}G$ diffuse interface. Gray, solid; cyan, water; orange, gas. (b) Schematic sectional view in the cylinder coordinate $(r,\phi ,z)$.

Figure 2

Energy landscape $E({\varphi }_L,{\varphi }_G)$ (top) and the map of the Young’s contact angle $\theta ({\varphi }_L,{\varphi }_G)$ (bottom) for three types of wetting, classified as (a), (b), and (c). In type (a), there is only a local minimum in the energy landscape (i) and the contact angle at the minimum state is zero (ii). In types (b) and (c), two local minima appear in the energy landscape, resulting in two Young’s contact angles ${\theta }_1$ and ${\theta }_2$. The parameters in (a), (b), and (c) are $(\mathrm{\Delta }p-\mathrm{\Delta }\chi )/\chi =2$, 0.5, and 0, respectively. The contact angle is calculated according to the Young’s equation with $\varrho =1$.

Figure 3

Contact angle. (a) The contact angle ${\theta }_1$ (cyan) and ${\theta }_2$ (orange) as a function of the ratio $(\mathrm{\Delta }p-\mathrm{\Delta }\chi )/\chi$ for different scaling factors, $\varrho =1$, 0.5, and 0.25. Stars: numerical simulation of the phase-field model. The labels (i), (ii), and (iii) highlight the setups for the simulation in (c). (b) Comparison with the Cassie-Wenzel theory for $\varrho =0.25$. The roughness factor $\mathrm{\Upsilon }$ in the Wenzel equation and the parameter $\mathrm{\Phi }$ in the Cassie equation are set to be 1.2 and 0.8, respectively. (c) Simulation results for ${\theta }_1$ (cyan) and ${\theta }_2$ (orange) with $(\mathrm{\Delta }p-\mathrm{\Delta }\chi )/\chi =-1$ (i), $-0.5$ (ii), and 0 (iii) when $\varrho =0.25$.