Laplace pressure

The Laplace pressure is the pressure difference between the inside and the outside of a curved surface that forms the boundary between two fluid regions.^[1] The pressure difference is caused by the surface tension of the interface between liquid and gas, or between two immiscible liquids.

The Laplace pressure is determined from the Young–Laplace equation given as^[2]

${\displaystyle \Delta P\equiv P_{\text{inside}}-P_{\text{outside}}=\gamma \left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}\right),}$

where ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$ are the principal radii of curvature and ${\displaystyle \gamma }$ (also denoted as ${\displaystyle \sigma }$) 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. In this case, ${\displaystyle R_{1}}$ = ${\displaystyle R_{2}}$:

${\displaystyle \Delta P=\gamma {\frac {2}{R}}}$

For a gas bubble within a liquid, there is only one surface. For a gas bubble with a liquid wall, beyond which is again gas, there are two surfaces, each contributing to the total pressure difference. If the bubble is spherical and the outer radius differs from the inner radius by a small distance, ${\displaystyle R_{o}=R_{i}+d}$, we find

${\displaystyle \Delta P=\Delta P_{i}+\Delta P_{o}=2\gamma \left({\frac {1}{R_{i}}}+{\frac {1}{R_{i}+d}}\right)={\frac {2\gamma }{R_{i}}}\left(1+{\frac {R_{i}}{R_{i}+d}}\right)={\frac {2\gamma }{R_{i}}}\left(1+{\frac {R_{i}+d}{R_{i}+d}}-{\frac {d}{R_{i}+d}}\right)={\frac {4\gamma }{R_{i}}}\left(1-{\frac {1}{2}}{\frac {d}{R_{i}+d}}\right)\approx {\frac {4\gamma }{R_{i}}}+{\mathcal {O}}(d).}$