# Surface tensions

Surface tensions arise from the imbalance of molecular forces at an interface. A net force at an interface implies that work must be done to expand the surface; that is, the surface tensions can be though of as forces integrated over distances or the changes in energy between a molecule completely surrounded by molecules and a molecule only partially surrounded by others.

 If the cohesion energy per molecule is inside a liquid: $U$ Then the cohesion energy per molecule at the surface is $\frac{U}{2}$ If the size of a molecule is a then it occupies an area of $a^{2}$ Therefore the surface tension is of the order $\sigma =\frac{U}{2a^{2}}$ If the liquid is near its boiling point then $U\approx kT$ Or the surface tension is about $\sigma =\frac{kT}{2a^{2}}$

$\text{at }25^{0}\text{ }\sigma =\frac{2\times 10^{-21}J}{a^{2}}\text{ For }a=3Ang\text{ }\sigma =20{}^{mJ}\!\!\diagup\!\!{}_{m^{2}}\;$

Witten (p. 155) proposes a surface-tension scaling relation, $\sim \frac{\alpha }{{kT}/{\delta A}\;}$, where $\delta A$ estimates the area of each flexible unit of the liquid:

Witten, Fig. 16.7

Hydrogen bonding in water and metallic bonding in metals raise the energies of interaction considerably above kT and hence all have higher surface tensions.