Young's equation (${\gamma }_{SV}={\gamma }_{SL}+{\gamma }_{LV}\cos \theta$)

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.