Revisited Cassie's law to incorporate microstructural capillary effects

The equilibrium contact angle and the receding contact angle of water droplets suspended on surfaces comprising arrays of equidistant, uniformly sized square micropillars has been measured with goniometry. Surfaces with distinct pillar size and spacing were fabricated via photolithography and deep reactive-ion etching prior to hydrophobization via the molecular vapor deposition of perflourodecyltrichlorosilane (FDTS). The surfaces exhibited superhydrophobic properties and the measured equilibrium contact angle was compared with the prediction of the Cassie equation based on the measured contact angle on a flat FDTS surface and the surface morphology. A poor agreement between the experimental data and the data predicted by the Cassie equation was found when the spacing between structures was less than the width of the pillars. For more closely spaced structures, the deviation between the measured and predicted values increased. In this roughness region, the measured angle is unchanged by the spacing but the receding angle continues to be dependent on the surface structure. A microscopic examination of the interface between the surface and the droplet revealed that the liquid-gas portion of the contact line was distorted at the pillar edges. The extent of the distortion could not be accurately quantified but it was shown that if the capillary region was assumed to be semicircular and extending half of the width of a pillar in to the liquid-gas region of the contact line, that the contact angle could be predicted well. Moreover, a good prediction of the experimental data of a prior study of droplets on closely spaced circular polydimethylsiloxane micropillar arrays is presented.

Figure 1

(a) Side-on goniometer view of droplet on microstructured surface, (b) SEM image of square pillar array, and (c) schematic of goniometer setup.

Figure 2

One-dimensional schematic of (a) contact line of droplet with capillary bridge near droplet contact line. (b) Schematic of assumed contact lines for (left) Cassie model, (middle) spacing greater than pillar width, and (right) interacting capillary bridges. Below: roughness used for each case. Optical microscopy image of contact line of water droplet on $10\times 10\mu {\mathrm{m}}^2$ square pillar arrays with (c) 5 μm and (d) 10 μm spacing showing a curvature of the liquid-vapor interface at the contact line.

Figure 3

Equilibrium contact angle as a function of the solid area fraction; experimentally measured from arrays of FDTS coated square pillars (green circles), Cassie's law prediction for experimental FDTS surfaces (broken green line), CBE-modified Cassie's law prefiction for experimental FDTS surfaces (solid green line), data from Bhushan et al. [25] (red and blue circles), Cassie's law prediction for Bhushan et al. (broken red line), and CBE-modified Cassie's law prediction for Bhushan et al. (solid red line).

Figure 4

Initial contact angle (circles, solid black line) and receding angle (triangles, dashed line) plotted against solid area fraction. The blue region illustrates the contact angle hysteresis.