Surface Fraction Dependence of Contact Angles Induced by Kinks in the $$Triple Line

On periodic superhydrophobic surfaces the receding contact angle often scales with the surface fraction, as expected from a simple rule of mixture, the Cassie relation. However, it has been argued that energy averaging breaks down owing to line pinning, and that line fraction scaling should apply instead. From experiments and simulations we show that proper inclusion of triple line defects introduce surface fraction scaling in the line depinning threshold. In contrast to the Cassie relation and in agreement with the data, this dependence is strongly nonlinear due to triple line elasticity.

Figure 1

Top views of representative surfaces: the periodic post patterns are (a) square lattice, (b) rectangular lattices, (c) hexagonal lattices. Also shown: (d) a nonperiodic, axisymmetric surface with concentric circles with constant radial and tangential spacings. The scale bar is $50\mu \mathrm{m}$.

Figure 2

Receding contact angle (plotted as effective adhesion) as a function of the line fraction, for different lattice types. The dashed line is the proportionality relation predicted by the line model [12]. For the rectangular lattices, angles measured along the $x$ direction (see schematics) scale very well with the line fraction model ($L=L_x$). The more symmetric lattices also obey a proportionality relation reasonably well. However, the angles measured along the $y$ direction for the rectangular lattices are at variance with the line model since $L=L_y$ is constant. Also shown are the values for the nonperiodic, axisymmetric surfaces, which scale with the line fraction, superimposing with the rectangular lattices in the $x$ direction for low line fractions.

Figure 3

Effective adhesion as a function of the surface fraction for various lattices. All the angles collapse on one master curve, which is a roughly linear function of the surface fraction. The proportionality relation postulated by the Cassie equation does not hold. The dashed line is a guide for the eye. Effective adhesion for axisymmetric surfaces is significantly larger, and does not fall on the same curve (inset).

Figure 4

Effective adhesion numerically calculated for block depinning in the $x$ direction on a rectangular lattice. Line fraction $\lambda =d/L_x$ varies while $L_y=20\mu \mathrm{m}$. The dashed line is the prediction of the line model. Also shown are the simulations for a triple line with kink, calculated for both $x$ ($L=L_x$) and $y$ ($L=L_y$) directions. A schematics of the geometry of the kink is shown on the right.

Figure 5

Effective adhesion numerically calculated for a triple line with a kink depinning on various lattices, plotted as a function of surface fraction $\varphi$. The linear function (long dashed line) is a guide to the eye. Also shown is the prediction from an analytical model for depinning of the triple line with kink [Eq (2)—short dashed line]. This first order expansion breaks down at larger surface fractions (dotted line).