Laplace pressure
The Laplace pressure is the pressure difference between the inside and the outside of a bubble or droplet. The effect is caused by the surface tension of the interface between liquid and gas.
The Laplace pressure is given as[1]
\[\Delta P \equiv P_\text{inside} - P_\text{outside} = \gamma\left(\frac{1}{R_1}+\frac{1}{R_2}\right),\]
where in the case of \(R_1\) being equal to \(R_2\), \( \Delta P = \frac{2 \gamma}{R} , \)
where
- Pinside is the pressure inside the bubble or droplet
- Poutside is the pressure outside the bubble or droplet
- \(\gamma\) (also denoted as \(\sigma\)) is the surface tension
- R is the radius of the bubble
This relation can be readily derived from the Young–Laplace equation.
A common example of use is finding the pressure inside an air bubble in pure water, where \(\gamma\) = 72 mN/m. The extra pressure inside the bubble is given here for three bubble sizes:
| Bubble diameter (2r) (microns) | \(\Delta P\) (Pa) | \(\Delta P\) (atm) |
|---|---|---|
| 1000 | 288 | 0.00284 |
| 3.0 | 96000 | 0.947 |
| 0.3 | 960000 | 9.474 |
A 1 mm bubble has negligible extra pressure. Yet when the diameter is ~ 3 microns, the bubble has an extra atmosphere inside than outside. When the bubble is only several hundred nanometers, the pressure inside can be several atmospheres. One should bear in mind that the surface tension in the numerator can be much smaller in the presence of surfactants or contaminants. The same calculation can be done for small oil droplets in water, where even in the presence of surfactants and a fairly low interfacial tension \(\gamma\) = 5–10 mN/m, the pressure inside 100 nm diameter droplets can reach several atmospheres. Such nanoemulsions can be antibacterial because the large pressure inside the oil droplets can cause them to attach to bacteria, and simply merge with them, swell them, and "pop" them.[2]
See also
References
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