Angle of Contact

Angle of Contact

When a liquid meets a solid surface, the liquid surface near the wall is generally curved. This curvature is not accidental — it arises from the competition between adhesive forces (liquid–solid) and cohesive forces (liquid–liquid). The angle of contact (or contact angle) is the angle that the tangent to the liquid surface makes with the solid surface, measured inside the liquid.

More precisely: at the point where the liquid, solid, and vapour (or air) meet — the three-phase contact line — draw the tangent to the liquid–vapour interface. The angle between this tangent and the solid surface, measured through the liquid, is the angle of contact, denoted by $\theta$.

The value of $\theta$ depends on the relative strengths of adhesion and cohesion:

• If adhesive forces dominate, the liquid spreads on the solid — $\theta$ is small (less than $90^\circ$). Such a liquid is said to wet the solid.
• If cohesive forces dominate, the liquid beads up — $\theta$ is large (greater than $90^\circ$). Such a liquid does not wet the solid.

For water and clean glass, $\theta \approx 0^\circ$ (complete wetting). For mercury and glass, $\theta \approx 140^\circ$ (non-wetting). For water and wax, $\theta \approx 107^\circ$.

---

Properties of the Angle of Contact

Two important properties of the angle of contact follow from the balance of surface tensions at the three-phase contact line.

Property 1: The angle of contact depends on the nature of the liquid and the solid in contact

This is a direct consequence of the fact that $\theta$ is determined by the intermolecular forces between the liquid molecules themselves (cohesion) and between the liquid and solid molecules (adhesion). Different liquid–solid pairs have different relative strengths of these forces, so $\theta$ varies. For example, water wets glass but not wax; mercury does not wet glass but wets clean zinc.

Property 2: The angle of contact depends on the medium above the free surface of the liquid

The liquid–vapour interface is one of the three surfaces meeting at the contact line. If the vapour above the liquid is replaced by a different gas (or by another immiscible liquid), the surface tension of the liquid–vapour interface changes. Since the balance of tensions at the contact line involves this surface tension, $\theta$ changes accordingly.

---

Derivation of the Angle of Contact from Force Balance

At the three-phase contact line, three surface tensions act along the three interfaces:

• $\sigma_{sl}$ — solid–liquid surface tension (tends to reduce the solid–liquid area)
• $\sigma_{sv}$ — solid–vapour surface tension (tends to reduce the solid–vapour area)
• $\sigma_{lv}$ — liquid–vapour surface tension (tends to reduce the liquid–vapour area)

These three forces per unit length act along the respective interfaces, tangential to them. For the contact line to be in equilibrium, the horizontal components must balance. The vertical component of $\sigma_{lv}$ is balanced by the reaction of the solid surface (the solid is rigid and does not deform).

›Proof

Consider the contact line where the three phases meet. The surface tension $\sigma_{sv}$ pulls the contact line along the solid surface, away from the liquid. The surface tension $\sigma_{sl}$ pulls it along the solid surface, toward the liquid. The liquid–vapour surface tension $\sigma_{lv}$ pulls at an angle $\theta$ to the solid surface (measured inside the liquid).

Resolve $\sigma_{lv}$ into horizontal and vertical components:

• Horizontal component: $\sigma_{lv} \cos\theta$ (pointing away from the liquid if $\theta < 90^\circ$)
• Vertical component: $\sigma_{lv} \sin\theta$ (balanced by the normal reaction of the solid)

For horizontal equilibrium:

$\sigma_{sv} = \sigma_{sl} + \sigma_{lv} \cos\theta$

Rearranging:

$\cos\theta = \frac{\sigma_{sv} - \sigma_{sl}}{\sigma_{lv}}$

This is Young's equation (or the Young–Dupré equation). It gives the equilibrium contact angle in terms of the three surface tensions.

From this equation, we see:

• If $\sigma_{sv} > \sigma_{sl}$, then $\cos\theta > 0$, so $\theta < 90^\circ$ — the liquid wets the solid.
• If $\sigma_{sv} < \sigma_{sl}$, then $\cos\theta < 0$, so $\theta > 90^\circ$ — the liquid does not wet the solid.
• If $\sigma_{sv} - \sigma_{sl} = \sigma_{lv}$, then $\cos\theta = 1$, so $\theta = 0^\circ$ — complete wetting (the liquid spreads into a thin film). …