Contact Angle, Work of Adhesion & Spreading Calculator

Understanding the contact angle and work of adhesion calculator — concept and principle

The angle a liquid drop makes with a solid shows how strongly the liquid is attracted to that solid. This calculator turns a measured contact angle and the liquid's surface tension into the work of adhesion, the spreading coefficient and a wetting class. It is used in coating, adhesive bonding, printing and in checking whether a plasma or chemical surface treatment worked.

How it is calculated

• Work of adhesion: W_a = γ(1 + cos θ), the Young-Dupré equation.
• Spreading coefficient: S = γ(cos θ − 1).
• Wetting class: θ < 10° spreads; 10–90° hydrophilic / wets; 90–150° hydrophobic; 150° and above superhydrophobic.

γ is the liquid surface tension in mN/m (equal to mJ/m²) and θ the equilibrium contact angle in degrees. Results are in mN/m.

The principle

Three interfacial tensions pull on the contact line: solid-vapour γ_sv, solid-liquid γ_sl and liquid-vapour γ. Young's equation balances them: γ_sv = γ_sl + γ cos θ. The work of adhesion, the energy to separate a unit area of liquid from the solid, is W_a = γ_sv + γ − γ_sl. Substituting Young's equation removes the solid terms, which cannot be measured directly, leaving γ(1 + cos θ). The spreading coefficient S = γ_sv − γ_sl − γ compares adhesion with the liquid's own cohesion (2γ) and becomes γ(cos θ − 1), which is zero at complete wetting and negative for any finite angle.

Worked example

1. Inputs: water, γ = 72.8 mN/m, θ = 70°.
2. cos 70° = 0.34202.
3. W_a = 72.8 × (1 + 0.34202) = 97.699 mN/m.
4. S = 72.8 × (0.34202 − 1) = −47.901 mN/m.
5. 70° lies between 10° and 90°: hydrophilic / wets.

How to read the result

W_a runs from 0 (θ = 180°) to 2γ (θ = 0°; 145.6 mN/m for water). The closer to 2γ, the better the wetting; a more negative S means the drop beads up. Clean glass gives water angles of a few degrees, many plastics roughly 70–100°, fluoropolymers above 100°. After a surface treatment, a falling angle and rising W_a mean better wettability.

Common mistakes

• Treating an advancing or receding angle as the equilibrium angle; hysteresis can be tens of degrees.
• Using pure-water surface tension for a liquid containing surfactant.
• Reading S as the strength of a cured adhesive joint.

Frequently asked questions

Are surface tension and surface energy the same?

For liquids, yes; mN/m equals mJ/m². A solid's surface energy cannot be measured directly and is estimated from angles of several test liquids.

How do I find the solid's own surface energy?

Measure angles with at least two liquids of known polar and dispersive parts, such as water and diiodomethane, and apply a model like Owens-Wendt.

Why is S never positive here?

A positive S means complete spreading, so no angle exists to measure; computed from an angle, S is at most zero.

Sources: T. Young, "An Essay on the Cohesion of Fluids", Philosophical Transactions of the Royal Society, 1805 · A. W. Adamson and A. P. Gast, Physical Chemistry of Surfaces, Wiley