Contact angle (${\theta }_c$)

The contact angle is measured through a liquid between its interface and a solid substrate at their contact line. In a gently sloping two-dimensional film on a horizontal substrate, its small-angle value is the limiting magnitude of the height slope. A zero contact angle means a tangential approach to the substrate; it does not imply bounded curvature, as the limiting zero-flux thermocapillary film profiles illustrate.

The planar interfacial balance ${\gamma }_{SV}-{\gamma }_{SL}={\gamma }_{LV}\cos \theta$ for the equilibrium contact angle through the liquid, with interfacial free energies defined per actual area. Roughness and retained air require additional area accounting.

A contact line can remain pinned while the observed contact angle changes between receding and advancing limits. Surface roughness and chemical heterogeneity can produce this hysteresis. Slowly adding or removing liquid from a sessile drop measures advancing and receding angles separately. An observed pinned angle therefore need not be the single equilibrium angle predicted by Young's equation for a perfectly homogeneous surface.

For a smooth homogeneous solid in thermodynamic equilibrium with a liquid and vapor, Young's equation relates their three interfacial tensions to the contact angle measured through the liquid. A virtual increase $dA$ of wetted area changes the interfacial free energy by $({\gamma }_{SL}-{\gamma }_{SV}+{\gamma }_{LV}\cos \theta )dA$. Setting this to zero proves the equation. Roughness, pinning and chemical heterogeneity can make an observed apparent angle differ from this ideal equilibrium value. The equation does not determine each solid–liquid surface energy from a contact angle alone, but two liquids on the same unchanged dry substrate allow the solid–vapor term to be eliminated by subtraction.

Subtracting Young's equations for two liquids on the same solid eliminates the unknown solid–vapour interfacial energy. The result assumes equilibrium angles and an unchanged solid surface. Swelling, contamination or contact-angle hysteresis can invalidate this cancellation as a comparison of equilibrium material properties.

Combining reversible work of adhesion per area, $W={\gamma }_{SV}+{\gamma }_{LV}-{\gamma }_{SL}$, with Young's equation gives the displayed relation. It assumes the same smooth equilibrium interfaces; contact-angle hysteresis and dissipation are not included.