Engineering & Science
Wenzel vs Cassie-Baxter Rough-Surface & Superhydrophobic Contact Angle
Calculate apparent droplet contact angle on textured surfaces in both the pinned Wenzel state cos(θ_W) = r * cos(θ_Y) and the lotus-leaf air-pocket Cassie-Baxter state cos(θ_CB) = f_s * (cos(θ_Y) + 1) - 1.
How lotus-leaf micropillars turn a 108° fluoropolymer into a 153.7° superhydrophobic surface!
Flat smooth fluoropolymers never exceed an intrinsic Young contact angle θ_Y of ~115°–120°—yet water beads roll effortlessly off lotus leaves at >150°! Surface texture unlocks two distinct wetting regimes: in Robert Wenzel’s (1936) homogeneous state, liquid penetrates every groove so surface roughness ratio r >= 1 amplifies intrinsic wettability via cos(θ_W) = r * cos(θ_Y) (making hydrophilic θ_Y < 90° MORE wettable and hydrophobic θ_Y > 90° MORE repellent)!
In A. B. D. Cassie and S. Baxter’s (1944) fakir state, droplets rest atop micropillar tips of solid area fraction f_s (0 < f_s <= 1) while air pockets occupy (1 - f_s) underneath: cos(θ_CB) = f_s * (cos(θ_Y) + 1) - 1! Take a smooth θ_Y = 108° coating textured to roughness r = 1.80 and pillar top fraction f_s = 0.15: pinned Wenzel wetting reaches θ_W = 123.80°, whereas suspended Cassie-Baxter air-pocket wetting leaps past the 150° superhydrophobic threshold to θ_CB = 153.68°!
| Surface Texture & Intrinsic θY Regime | Governing Wetting Law | Apparent Contact Angle | Droplet Adhesion Behavior |
|---|---|---|---|
| Hydrophobic Wenzel State (θY = 108°, r = 1.80) | cos(θW) = r · cos(θY) | 123.80° | Pinned in Grooves (High Hysteresis) |
| Lotus Cassie-Baxter State (θY = 108°, fs = 0.15) | cos(θCB) = fs(cos θY + 1) - 1 | 153.68° (>150°!) | Self-Cleaning Roll-Off State |
| Hydrophilic Wenzel State (θY = 65°, r = 1.80) | cos(θW) = r · cos(θY) | 40.47° (< 65°!) | Roughness Enhances Spreading! |
Wenzel Homogeneous Roughness Contact Angle θ_W (degrees)
Clamps r * cos(θ_Y) inside [-1, +1] (since extreme roughness triggers complete 0° hemi-wicking or 180° saturation) and converts back to degrees.
=IF(OR(B2 < 0, B2 > 180, B3 < 1), "Invalid Wetting Inputs", ROUND(DEGREES(ACOS(MAX(-1, MIN(1, B3 * COS(RADIANS(B2)))))), 2))Cassie-Baxter Air-Pocket Composite Contact Angle θ_CB (degrees)
Evaluates DEGREES(ACOS(f_s * (COS(RADIANS(θ_Y)) + 1) - 1)) for micropillar solid fraction 0 < f_s <= 1.
=IF(OR(B2 < 0, B2 > 180, B4 <= 0, B4 > 1), "Invalid Wetting Inputs", ROUND(DEGREES(ACOS(B4 * (COS(RADIANS(B2)) + 1) - 1)), 2))Try the example
Use the original values or change them to test your understanding.