Surface tensions

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

<math>\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}}\;</math>

Witten (p. 155) proposes a surface-tension scaling relation, <math>\sim \frac{\alpha }{{kT}/{\delta A}\;}</math>, where <math>\delta A</math> 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.

An example: Spore ejection

Fungi present an elegant application of surface tension forces when they eject spores. For instance, mushrooms need to launch spores away from the gills on their underside to be carried away by the wind, to produce new mushrooms. The process has four main steps, which are illustrated below (as shown on the Australian National Botanic Gardens website):


The sequence is essentially a conversion from energy stored as surface tension to the kinetic energy of a moving spore: 1) the spore secretes a small amount of sugar molecules, which lead to the condensation of water near the attachment point (2). The resulting droplet of water increases in size (3), until it comes into contact with a thin coating of water on the spore's surface. At that point, surface tension pulls the water in the droplet around the surface of the spore, shifting the center of mass forward and launching the spore away from the gill's surface (4). This process is so effective that accelerations as high as 20,000g are possible in some fungi, as studied with high speed photography [1].

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