When a liquid meets a solid, the interface can spread across the surface or remain as a bead. This behavior is called wetting and reflects the balance among the solid-vapor, solid-liquid, and liquid-vapor interfaces.

At the edge of a droplet, the contact angle $\theta$ is measured through the liquid between the solid surface and the tangent to the liquid-vapor interface.

• small $\theta$ corresponds to strong wetting;
• large $\theta$ corresponds to weak wetting;
• $\theta<90^\circ$ gives $\cos\theta>0$;
• $\theta>90^\circ$ gives $\cos\theta<0$.

Young's relation for equilibrium wetting

For an ideal, chemically uniform, smooth solid with an equilibrium contact line, horizontal interfacial-force balance gives

$$\boxed{\gamma_{SV}=\gamma_{SL}+\gamma_{LV}\cos\theta},$$

where

• $\gamma_{SV}$ is the solid-vapor interfacial energy per area;
• $\gamma_{SL}$ is the solid-liquid interfacial energy per area;
• $\gamma_{LV}$ is the liquid-vapor surface tension.

Equivalently,

$$\boxed{\cos\theta=\frac{\gamma_{SV}-\gamma_{SL}}{\gamma_{LV}}}.$$

The contact angle is therefore not a property of the liquid alone. It depends on the entire solid-liquid-surrounding-fluid combination and on the condition of the surfaces.

Curved meniscus in a circular capillary

Consider a narrow vertical circular tube of radius $r$ dipped into a liquid. The liquid-vapor meniscus meets the wall at contact angle $\theta$.

In the ideal spherical-meniscus approximation, geometry gives the meniscus radius of curvature

$$R=\frac{r}{\cos\theta}.$$

The Young-Laplace pressure jump across one spherical liquid-vapor interface is

$$\Delta p=\frac{2\gamma_{LV}}{R}.$$

Substituting the geometric relation gives

$$\boxed{\Delta p=\frac{2\gamma_{LV}\cos\theta}{r}}.$$

For a wetting liquid, $\cos\theta>0$, so the pressure immediately beneath the curved meniscus is lower than the surrounding gas pressure by this magnitude. For a non-wetting liquid with $\theta>90^\circ$, the curvature reverses and the pressure relation drives a depression rather than a rise.

Capillary rise from hydrostatic balance

If the liquid rises a height $h$ above the external free surface, hydrostatics requires a pressure difference

$$\Delta p=\rho gh,$$

where $\rho$ is the liquid density.

Combining this with the curvature pressure gives

$$\rho gh=\frac{2\gamma_{LV}\cos\theta}{r},$$

so

$$\boxed{h=\frac{2\gamma_{LV}\cos\theta}{\rho g r}}.$$

This is the ideal capillary-rise equation for a circular tube.

It shows that capillary effects become stronger as the tube becomes narrower.

Worked example

Take water with

$$\gamma_{LV}=0.072,\mathrm{N/m},$$

$$\rho=1000,\mathrm{kg/m^3},$$

inside a clean tube of radius

$$r=0.50,\mathrm{mm}=5.0\times10^{-4},\mathrm m,$$

and approximate the contact angle by

$$\theta=0^\circ.$$

Then

$$h=\frac{2(0.072)\cos0^\circ} {(1000)(9.81)(5.0\times10^{-4})} \approx2.94\times10^{-2},\mathrm m.$$

Thus

$$\boxed{h\approx2.9,\mathrm{cm}}.$$

Halving the tube radius would double the ideal rise height.

Real surfaces

Young's relation describes an ideal equilibrium contact line. Real surfaces can be rough, chemically heterogeneous, contaminated, oxidized, or dynamically moving. The observed contact angle can therefore depend on whether the contact line is advancing or receding, producing contact-angle hysteresis.

Temperature, dissolved substances, and surfactants can also change interfacial tensions. These effects matter in soldering, brazing, coatings, adhesives, porous-media infiltration, microfluidics, and many biological systems.

Wetting determines the contact geometry; surface tension and curvature determine the pressure jump; hydrostatics then determines how far a liquid column rises or falls.