Antiplane shear

The Laplace pressure is the pressure difference between the inside and the outside of a curved surface. The pressure difference is caused by the surface tension of the interface between liquid and gas.

The Laplace pressure is determined from the Young–Laplace equation given as

$\mathrm{\Delta }P\equiv P_{\text{inside}}-P_{\text{outside}}=\gamma (\frac{1}{R_1}+\frac{1}{R_2}),$

where $R_1$ and $R_2$ are the radii of curvature and $\gamma$ is the surface tension. Although signs for these values vary, sign convention usually dictates positive curvature when convex and negative when concave. The Laplace pressure is commonly used to determine the pressure difference in spherical shapes such as bubbles or droplets. When this is the case the radii of curvature are equal ($R_1$ = $R_2$) and the equation simplifies to,

$\mathrm{\Delta }P=\frac{2\gamma }{R},$

where

• P_inside is the pressure inside the bubble or droplet
• P_outside is the pressure outside the bubble or droplet
• $\gamma$ (also denoted as $\sigma$) is the surface tension
• R is the radius of the bubble or droplet

A common example of use is finding the pressure inside an air bubble in pure water, where $\gamma$ = 72 mN/m when at 25°C (298 K). The extra pressure inside the bubble is given here for three bubble sizes:

Bubble diameter (2r) (µm)	$\mathrm{\Delta }P$ (Pa)	$\mathrm{\Delta }P$ (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 µm, 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.

See also

• Ostwald ripening
• Kelvin equation
• Laplace number

References

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