9.6.4. Drops and Bubbles

One consequence of surface tension is that free liquid drops and bubbles are spherical if effects of gravity can be neglected. You must have seen this especially clearly in small drops just formed in a high-speed spray or jet, and in soap bubbles blown by most of us in childhood. Why are drops and bubbles spherical? What keeps soap bubbles stable?

As we have been saying repeatedly, a liquid-air interface has energy, so for a given volume the surface with minimum energy is the one with the least area. The sphere has this property.

Drops and bubbles take a spherical shape due to surface tension, which is the cohesive force that pulls liquid molecules together at the surface. When unaffected by gravity, a sphere has the smallest possible surface area for a given volume. This minimizes the surface energy of the liquid, making spherical shapes energetically favorable.

Think about blowing up a balloon. As you inflate it, the air inside tries to equalize pressure on all sides, pushing out against the rubber. The balloon naturally tries to form a perfect sphere because that shape minimizes surface area relative to its volume—just like how drops of water behave when they fall.

Another interesting consequence of surface tension is that the pressure inside a spherical drop is more than the pressure outside. Suppose a spherical drop of radius r is in equilibrium. If its radius increases by ∆r. The extra surface energy is

$[4\pi(r + \Delta r)^{2} - 4\pi r^{2}] S_{ha} = 8\pi r \Delta r S_{ha}$ If the drop is in equilibrium this energy cost is balanced by the energy gain due to expansion under the pressure difference (Pi – Po) between the inside of the bubble and the outside. The work done is

$W = (P_1 - P_0) 4 \pi r^2 \Delta r$ so that

$(P_i - P_o) = \frac{2 S_{ln}}{r}$.

For a spherical drop, the internal pressure (Pi) is greater than the external pressure (Po) due to the curvature created by surface tension. The equation shows how the force exerted by surface tension (Sla) balances the pressure difference across the surface of the drop. This relationship is key to understanding why small drops can hold together and maintain their shape despite the forces acting on them.

Imagine blowing air into a balloon. As the balloon expands, the pressure inside increases because of the extra air volume that must be contained. Similarly, in tiny water droplets or soap bubbles, the pressure inside needs to be higher due to surface tension's effects, helping to keep them intact against external pressures.

A bubble differs from a drop and a cavity; in this it has two interfaces. Applying the above argument we have for a bubble $(P_1 - P_0) = \left( \frac{4 S_{lh}}{r} \right)$. This is probably why you have to blow hard, but not too hard, to form a soap bubble. A little extra air pressure is needed inside!

Unlike a drop, which has one liquid-air interface, a bubble has two: one inside (where the air is) and one outside (where the liquid is). This means for a bubble, the internal pressure is even more elevated (by a factor related to the surface tension) than for a droplet. That’s why you need to apply a bit more force when blowing a bubble because you have to create enough air pressure to overcome this surface tension and form the double-surfaced structure.

Remember when you were a child trying to blow big bubbles? You often had to find the right balance when blowing; too hard and the bubble would pop, but just the right amount and it would expand beautifully. This required understanding the forces at play—much like understanding how the pressure influences the stability of a bubble.

One consequence of the pressure difference across a curved liquid-air interface is the well-known effect that water rises up in a narrow tube in spite of gravity.

This phenomenon, known as capillary rise, occurs in narrow tubes due to the cohesive and adhesive forces acting on the liquid. Water is drawn into the tube against gravity because the adhesive force between the water molecules and the tube’s surface (which is greater than the cohesive forces within the liquid) causes the liquid to climb. The greater the surface tension, the higher the liquid will rise in the tube.

Think of a straw in a glass of water. When you sip, the liquid rises up the straw, showing how liquid can counteract gravity due to these interactions. Similarly, if you dip a thin straw into water, you'll notice it draws water higher than what you'd see with a wider object; this highlights the importance of surface tension and adhesion in capillary action.

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