Stability of Cassie-Baxter wetting states on microstructured surfaces

A stable Cassie-Baxter (CB) wetting state is indispensable for the superhydrophobicity of solid surfaces. In this paper, we analyze the equilibrium and stability of CB wetting states. Using an energy approach, the stability criteria of CB wetting states are established for solid surfaces with either two- or three-dimensional symmetric microstructures. A generic method is presented to calculate the critical pressure at which the CB state on a microstructured solid surface collapses. The method holds for microstructures with arbitrary generatrix, and explicit solutions are derived for a few representative microstructures with a straight or circular generatrix. In addition, some possible strategies are proposed to design surface structures with stable CB wetting states from the viewpoints of geometry and chemistry.

Figure 1

Schematic of the model. (a) In the CB wetting state, the microstructure on the solid surface is partly wetted by the liquid. (b) When the pressure difference $\mathrm{\Delta }p$ across the liquid-vapor interface varies, the three-phase contact line (red point) moves accordingly. The equilibrium path is then defined as the moving trajectory of the contact position during the variation in $\mathrm{\Delta }p$.

Figure 2

CB wetting of a 2D microstructure. The unit cell is symmetric with respect to the $z$ axis, and its generatrix is expressed by a continuous function $x=f\left(z\right)$.

Figure 3

CB wetting state on a 3D protruding microstructure. (a) The side view and (b) the top view. For a close-packed microstructure, the hexagonal outer boundary of the unit cell model is simplified as a circular one in (c).

Figure 4

Example of the wetting behavior under varying pressure. (a) The cross-section shape of the solid microstructure, and the distributions of different equilibrium paths. The variations in (b) the applied pressure $\mathrm{\Delta }p/p_0$, (c) the SDFE ${\overline{G}}_{2\mathrm{D}}^{\prime \prime }$, and (d) the curvature $\kappa$ of surface generatrix with dimensionless contact position ${\overline{z}}_{\mathrm{c}}$. The black dashed line representing the zero value stands for the boundary between stable and unstable properties. Stable paths (e.g., paths 2 and 4) are marked in solid curves, while dashed curves stand for unstable paths (e.g., paths 1 and 3).

Figure 5

Variations in (a) the equilibrium positions ${\overline{z}}_{\mathrm{c}}$ and (b) the SDFE ${\overline{G}}_{\mathrm{pr}}^{\prime \prime }$ of 3D protruding microstructures with a sinusoidal generatrix, where the distance $r_0$ varies from 20 to 60 μm.

Figure 6

Equilibrium position and critical pressure difference for 2D microstructure with a straight generatrix. (a) Dimensionless equilibrium positions ${\overline{z}}_{\mathrm{c}}$ with respect to $\mathrm{\Delta }p$ and $r_0$. (b) Maximal and minimal pressure differences (curves) predicted by Eq. (C1) are consistent with the numerical results (symbols).

Figure 7

Equilibrium positions and critical pressure differences for 2D microstructures with a circular generatrix. (a) The dimensionless equilibrium positions ${\overline{z}}_{\mathrm{c}}$ with respect to $\mathrm{\Delta }p$ and $r_0$. (b) The maximal and minimal pressure differences predicted by Eq. (D2) are identical to the numerical calculations.

Figure 8

Equilibrium positions and critical pressure differences for 3D protruding structures with a circular generatrix. (a) The dimensionless equilibrium positions ${\overline{z}}_{\mathrm{c}}$ with respect to $\mathrm{\Delta }p$ and $r_0$. (b) The theoretical (curves) and numerical (symbols) results of the maximal and minimal pressure differences. The simulation results are obtained by using the treatment where the outer boundary of a unit cell is circular (open squares) or hexagonal (crosses).

Figure 9

Curvature effect on the wetting stability. (a) If the wetting state under $\mathrm{\Delta }p=0$ is located in the red region with a negative curvature, the whole equilibrium path in the range between $\mathrm{\Delta }{P}_{min}$ and $\mathrm{\Delta }{P}_{max}$ will be stable. Otherwise, (b) the equilibrium path containing the state $\mathrm{\Delta }p=0$ will be unstable on the generatrix with $\kappa >0$.

Figure 10

(a) Schematic of a nonsmooth microstructure. (b) A 3D $\mathsf{T}$-shaped microstructure. Its wetting property can be analyzed by dividing the generatrix into three piecewise-smooth regions, 1 (green), 2 (red), and 3 (orange). The stable three-phase contact line can be located either in path II of region 2 or at position ${\mathrm{D}}_1$.