= Surface energy
{title2=$E_{\mathrm s}=\gamma A$}
For a <liquid> interface with constant <surface tension> $\gamma$, the mechanical <surface energy> is the <energy> associated with its <surface area> $A$:
$$
E_{\mathrm s}=\gamma A+C,\qquad
E_{\mathrm s}(A_2)-E_{\mathrm s}(A_1)=\gamma(A_2-A_1).
$$
The additive constant $C$ does not affect the <work> needed to change <surface area>. For several interfaces with the same <surface tension>, $A$ means their total <surface area>; the two faces of a thin soap film must both be counted. https://arxiv.org/html/2011.11403v2[Tomas Bohr and Bernhard Scheichl, Surface Tension and Energy Conservation in a Moving Fluid, equation (13)]{ref}
With assumed $\gamma=0.072\,\mathrm{N\,m^{-1}}$, creating $1\,\mathrm{cm^2}=10^{-4}\,\mathrm{m^2}$ of one interface requires $7.2\,\mathrm{\mu J}$ of reversible <work>. In the sliding-rim example illustrated under <surface tension>, moving a $20\,\mathrm{mm}$ rim outwards by $1\,\mathrm{mm}$ creates $4.0\times10^{-5}\,\mathrm{m^2}$ across the two faces and requires $2.88\,\mathrm{\mu J}$. Independently, multiplying its $2.88\,\mathrm{mN}$ holding <force> by the displacement gives the same <work>.
This makes the preference for a round isolated droplet precise. At fixed <volume> $V$, the three-dimensional <isoperimetric inequality> gives
$$
A^3\geq36\pi V^2,
$$
with equality for a <sphere>. Thus a <sphere> minimizes <surface energy> among sufficiently regular closed shapes enclosing the same <volume>, when <surface tension> is uniform and other <energy> contributions are negligible. This is an application of the <isoperimetric inequality>, rather than a proof of it. Walls, <wetting>, and <gravity> can change the shape by adding constraints or other <energy> contributions. https://math.hmc.edu/funfacts/isoperimetric-inequality/[Francis E. Su et al., Isoperimetric Inequality, Math Fun Facts]{ref}
The differential rule $dE_{\mathrm s}=\gamma\,dA$ integrates directly to $E_{\mathrm s}=\gamma A+C$ because <surface tension> is constant. At fixed <liquid> <volume>, decreasing <surface area> can drive <viscous fluid flow>; the released <surface energy> can supply <viscous dissipation>. This mechanical bookkeeping is useful for predicting which deformations can release <energy>. If <surface tension> depends on <temperature> or composition, distinguish this mechanical area-<work> relation from the complete interfacial <thermodynamic free energy> and <internal energy>.
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