Oil lens capillary balance ($h^2=2\Gamma /(({\rho }_w-{\rho }_o)g)$)

For a broad, thin oil lens below a flat ice-water interface, define the effective interfacial energy $\Gamma ={\gamma }_{io}+{\gamma }_{ow}-{\gamma }_{iw}$ from the relevant surface tensions. At fixed oil volume $\mathcal{V}=Ah$, the energy is $E=\Gamma \mathcal{V}/h+\frac{1}{2}({\rho }_w-{\rho }_o)g\mathcal{V}h$. Setting its derivative to zero gives $h^2=2\Gamma /[({\rho }_w-{\rho }_o)g]$ for $\Gamma >0$, and the positive second derivative proves a minimum. A contact angle measured through the oil gives $\Gamma ={\gamma }_{ow}(1-\cos \vartheta )$. Oil-ice tension alone does not determine this effective quantity. Complete wetting or rough underside pockets invalidate this finite, flat-lens approximation. With densities $1025$ and $850\mathrm{kg}{\mathrm{m}}^{-3}$, a lens below $1\mathrm{cm}$ requires $0<\Gamma <0.08584\mathrm{N}{\mathrm{m}}^{-1}$.