52. Pressure under curved liquid surface

In its effort to reduce, the surface film of the liquid creates additional pressure inside the liquid. For this reason, the pressure inside the soap bubble is greater than the atmospheric pressure. This overpressure depends on the curvature of the liquid surface or the soapy film. The easiest way to find the overpressure \(p\) for a spherical surface.

We will increase the size of the spherical drop by injecting liquid into it with a very thin syringe. Let the volume of the droplet increase from \(V_1\) to \(V_2\), and the increase in the volume of \(\Delta{V} ~= ~V_2-V_1 \) is very small (Fig. 1). As is known (see \(\S{12}\)), the work of the pressure forces when the volume changes by \(\Delta{V}\) equals \(p\Delta{V}\). In this case, the work of the overpressure forces \(\Delta{W} ~= -p_s\Delta{V}\).

The work \(\Delta{W}\) is negative because the overpressure force acting on the surface element is pointing in the opposite direction to the displacement of the surface element. It is equal to the change in potential (surface) energy taken with the opposite sign

\( \Delta{W} ~= ~-p_s\Delta{V} ~= ~-\Delta{U_s} ~= ~-\sigma{(A_2-A_1)} \)

where \(A_1\) and \(A_2\) are the starting and ending surface area of the drop.
Consequently,

\( p_s\Delta{V} ~= ~\sigma\Delta{A} \) (5-2)

Let's calculate the ball volume change and ball surface area change with small radius \(\Delta{r}\) increase.

\( \Delta{V} ~= ~\frac{4\pi}{3}(r+\Delta{r})^3 - \frac{4\pi}{3}r^3 ~\approx ~4\pi{r^2}\Delta{r} \) (5-3)

\( \Delta{A} ~= ~4\pi(r+\Delta{r})^2 - 4\pi{r^2} ~\approx ~8\pi{r}\Delta{r} \) (5-4)

Here we have neglected members proportional to \(\Delta{r^2}\) and \(\Delta{r^3}\) because they are small. By placing \((5-3)\) and \((5-4)\) in equation \((5-2)\), we get the desired result.

The overpressure inside a spherical droplet is

\( p_s ~= \frac{2\sigma}{r} \) (5-5)

Therefore, the presence of a spherical shape of the interface surface of two bodies environment leads to the appearance of additional pressure \((5-5)\) in one of the body environment.