65.66
°
1.146052
rad
0.412088
102.8
mN/m
-42.8
mN/m
1
65.66
°
1.146052
rad
0.412088
102.8
mN/m
-42.8
mN/m
1
The Contact Angle Calculator determines the equilibrium contact angle (θ) of a liquid droplet on a solid surface using Young's equation. The contact angle is the most direct measure of surface wettability — how well a liquid spreads on a solid. Surfaces with θ < 90° are hydrophilic (water-loving), while surfaces with θ > 90° are hydrophobic (water-repelling). Extreme cases include superhydrophilic surfaces (θ < 10°, complete wetting) and superhydrophobic surfaces (θ > 150°, like lotus leaves). Contact angle measurement is crucial in coatings technology, printing, adhesion, textile treatment, biomaterial design, microfluidics, and self-cleaning surface engineering. This calculator also computes the work of adhesion, which quantifies the strength of the liquid-solid interaction.
Young's equation relates the contact angle to the three interfacial energies at the three-phase contact line:
$$\cos\theta = \frac{\gamma_{SG} - \gamma_{SL}}{\gamma_{LG}}$$
where γSG is the solid-gas (solid surface energy), γSL is the solid-liquid interfacial energy, and γLG is the liquid-gas surface tension. When γSG > γSL, the surface is hydrophilic (θ < 90°). When γSG < γSL, the surface is hydrophobic (θ > 90°).
The work of adhesion (Dupré equation combined with Young's equation):
$$W_a = \gamma_{LG}(1 + \cos\theta)$$
represents the thermodynamic work required to separate the liquid from the solid surface. High Wₐ indicates strong adhesion (good wetting), while low Wₐ indicates weak adhesion (poor wetting). For complete wetting (θ = 0°), Wₐ = 2γLG. For complete non-wetting (θ = 180°), Wₐ = 0.
The contact angle directly classifies surface wettability: superhydrophilic (θ < 10°, water spreads completely — anti-fog coatings, self-cleaning glass); hydrophilic (10° < θ < 90° — most clean metals, glass, ceramics); hydrophobic (90° < θ < 150° — waxes, many polymers, fluoropolymers); superhydrophobic (θ > 150° — lotus-effect surfaces, nanostructured coatings). The work of adhesion quantifies the practical adhesion strength. Real surfaces show contact angle hysteresis (difference between advancing and receding angles) due to roughness, heterogeneity, and contamination, which Young's equation for ideal surfaces does not capture.
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Water on clean glass has a contact angle of about 65°, indicating a hydrophilic surface. The relatively high work of adhesion (102.8 mN/m) explains why water droplets adhere well to glass surfaces.
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Water on Teflon (PTFE) has θ ≈ 112°, confirming hydrophobicity. The low surface energy of fluoropolymers makes γSL > γSG, resulting in cos(θ) < 0. The low work of adhesion means water beads up and rolls off easily.
The balance between the solid's surface energy (γSG) and the solid-liquid interfacial energy (γSL) determines wettability. If γSG > γSL + γLG, complete wetting occurs (θ = 0°). If γSG < γSL, the surface is hydrophobic (θ > 90°). High-energy surfaces (metals, glass, ceramics) tend to be hydrophilic, while low-energy surfaces (polymers, waxes, fluoropolymers) tend to be hydrophobic.
Contact angle hysteresis is the difference between the advancing angle (θₐ, measured when the droplet expands) and receding angle (θᵣ, measured when the droplet contracts): Δθ = θₐ - θᵣ. It arises from surface roughness, chemical heterogeneity, and molecular reorientation. Low hysteresis (<10°) indicates a smooth, homogeneous surface; high hysteresis indicates roughness or heterogeneity.
The Wenzel equation relates the apparent contact angle on a rough surface (θ*) to the Young's angle: cos(θ*) = r·cos(θ), where r is the roughness ratio (actual area/projected area, r ≥ 1). Roughness amplifies the inherent wettability — hydrophilic surfaces become more hydrophilic, and hydrophobic surfaces become more hydrophobic.
The Cassie-Baxter model describes a droplet sitting on top of surface features with air trapped underneath: cos(θ*) = f₁cos(θ₁) + f₂cos(θ₂), where f₁ and f₂ are the area fractions of solid and air contact. This composite interface can produce superhydrophobic behavior (θ > 150°) even with moderately hydrophobic materials, as seen on lotus leaves and engineered nanostructured surfaces.
The most common method is sessile drop goniometry: place a small droplet (1–5 μL) on the surface and measure the angle at the three-phase contact line using optical imaging and software analysis. Other methods include the captive bubble method (for underwater measurements), Wilhelmy plate method (for fibers), and capillary rise method (for porous materials).
The lotus effect is natural superhydrophobicity (θ > 150°) combined with very low contact angle hysteresis (<5°), causing water droplets to roll off easily while carrying away dirt particles (self-cleaning). Lotus leaves achieve this through hierarchical micro/nanostructures covered with hydrophobic wax crystals, creating a Cassie-Baxter state.
Contact angle helps predict biological interactions with implant surfaces. Moderate hydrophilicity (θ = 40–70°) generally promotes protein adsorption and cell adhesion for implants. Superhydrophilic surfaces promote tissue integration. Hydrophobic or superhydrophobic surfaces resist biofouling. Contact angle also predicts blood compatibility of vascular devices.
The Dupré work of adhesion (Wₐ = γLG(1 + cosθ)) is the reversible thermodynamic work per unit area to separate the liquid from the solid in the presence of vapor. It ranges from 0 (complete non-wetting, θ = 180°) to 2γLG (complete wetting, θ = 0°). Practical adhesion also depends on mechanical interlocking, surface roughness, and viscoelastic dissipation.
Contact angle provides qualitative guidance — better wetting (lower θ) generally means better adhesive spreading and stronger bonds. However, practical adhesion depends on many additional factors including mechanical interlocking, chemical bonding, interdiffusion, and failure mode. Contact angle is a necessary but not sufficient predictor of adhesive performance.
The spreading coefficient S = γSG - γSL - γLG = γLG(cosθ - 1). When S ≥ 0, the liquid spontaneously spreads to form a thin film (complete wetting). When S < 0, the liquid forms a droplet with a finite contact angle. The more negative S is, the higher the contact angle and worse the wetting.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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