= Solution
For a clean liquid interface at fixed <temperature> and composition, <surface tension> is the reversible free-energy cost per added <area>, $\gamma=(\partial F/\partial A)_{T,V,N}$, so $dF=\gamma\,dA$. Its units are both $\mathrm{J\,m^{-2}}$ and $\mathrm{N\,m^{-1}}$. Molecules at an interface have a different environment from those in the bulk; creating more interface changes their interactions and <entropy>. For constant $\gamma$, the interfacial <free energy> is $\gamma A$ up to a reference constant. This is not generally identical to the internal <surface energy>: the interfacial <entropy> density is $-d\gamma/dT$, giving internal-energy density $\gamma-T\,d\gamma/dT$.
A line of <length> $\ell$ in one interface transmits a tangential <force> $\gamma\ell$, perpendicular to the line. A soap film has two interfaces, giving <force> $2\gamma\ell$ on a sliding rim. Quasistatic extension by $dx$ creates total <area> $2\ell dx$, recovering that <force> from $dF=2\gamma\ell dx$. Uniform positive <surface tension> favors minimum <area> at fixed <volume>, hence an isolated spherical droplet when gravity is negligible. <curvature> gives the <Young–Laplace equation>, with outward normal chosen consistently:
$$
\boxed{\Delta p=\gamma\left(\frac1{R_1}+\frac1{R_2}\right).}
$$
A spherical liquid drop has $\Delta p=2\gamma/R$; a thin soap bubble with two similar interfaces has approximately $4\gamma/R$. The <capillary length> $\sqrt{\gamma/(\Delta\rho g)}$ compares gravity with capillarity. <surface tension> also restores <capillary waves>, rather than providing a bulk static <shear modulus>.
Interfacial contamination and added <surfactants> can change $\gamma$ substantially; a nonuniform value produces tangential <Marangoni stress>. A rim-force measurement should therefore control <temperature>, cleanliness and the number of interfaces. Capillary-rise methods additionally need the <contact angle>, because the vertical component is proportional to $\gamma\cos\theta$. An independent frame-force measurement can distinguish a contact-angle change from a true change of <surface tension>.
Back to article page
