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My question is fairly straightforward, is a curved interface the only way a surface tension-induced pressure jump can form over that interface, or is it also possible to occur over a flat surface? Obviously, the Young-Laplace equation requires a curved interface to balance a pressure jump, which is given by:

$$\Delta p = \sigma \kappa,$$

where $\sigma$ is surface tension and $\kappa$ is interface curvature. This makes total sense for problems like a balloon where the rubber must curve for the tension to generate a net force opposite to the pressure force. However, I recently confused myself a bit after concluding that the balloon problem is not a great analog to a liquid-gas interface. For a liquid interface, the actual molecules on the surface are the same as in the bulk, they are just missing some neighbors on one side, unlike for a ballon where there are rubber molecules on the "interface" (I recognize the thickness is orders of magnitude greater than that of a liquid-air interface) and gas molecules inside. This means that even for a flat interface, there is a net inward force on a fluid-fluid interface, which should generate a balancing pressure in the bulk --- this is different for a balloon where only a tension force tangent to the interface will exist rather than also having a net inward force. Has this pressure been observed in experiments, and if so, how is it expressed mathematically, as the Young-Laplace equation will clearly not capture it?

And on this note, due we ever observe a "pressure" in a solid object due to the net interface forces into the bulk? It seems like even a solid block should be slightly (VERY slightly) compressed by this effect.

What am I missing here?

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  • $\begingroup$ > "This makes total sense for problems like a balloon where the rubber must curve for the tension to generate a net force opposite to the pressure force." Beware, balloon does not obey the surface tension equation; pressure difference is approximately independent of the balloon size. $\endgroup$ Commented Aug 4, 2022 at 16:35
  • $\begingroup$ I might not be understanding your point, but this seems demonstrably false. The two-balloon experiment clearly indicates that pressure depends on the radius of the balloon. $\endgroup$
    – mrmudd
    Commented Aug 4, 2022 at 16:55
  • $\begingroup$ It depends somewhat on radius, but not in the way the surface tension of a liquid-gas interface does. The dependence is weaker and not as simple. $\endgroup$ Commented Aug 4, 2022 at 16:57
  • $\begingroup$ I don't disagree that there are additional complexities with the balloon problem pertaining to the elasticity of the rubber material, but I do not find your commentary particularly useful or even mostly correct. The pressure jump from the inside to outside of the balloon, absolutely depends on the curvature of the surface - not just somewhat. Again, it's a bit more complex because of the solid mechanics, but the analogy does not require us to really delve into those for my simple question. $\endgroup$
    – mrmudd
    Commented Aug 4, 2022 at 17:28
  • $\begingroup$ See experimental results in Fig.1 here researchgate.net/publication/… . $\endgroup$ Commented Aug 4, 2022 at 17:33

2 Answers 2

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This means that even for a flat interface, there is a net inward force on a fluid-fluid interface, which should generate a balancing pressure in the bulk

If the surface is perfectly flat, there cannot be a pressure differential. Because if there was one, any macroscopic patch of the liquid close to the surface would experience different forces from the remaining liquid and gas, which would accelerate the patch and disturb the mechanical equilibrium.

If the surface is curved, non-zero pressure differential can exist because it is counterbalanced by the surface forces acting on the patch boundary; their resultant points towards the center of the curvature.

even for a flat interface, there is a net inward force on a fluid-fluid interface, which should generate a balancing pressure in the bulk

Not in the macroscopic sense of the word "force", like when we are talking about surface tension force. Surface tension force acts in the plane of the interface; it does not act in direction normal to the interface.

You are probably thinking about a single molecule that is attracted by inter-molecular forces towards the liquid when it is far enough from it or just in the interface layer but far enough from other molecules so repulsion is not important. This can happen at any time for part of the molecules in the interface, but not all of them.

But then you are writing as if all molecules there experience only this unidirectional attraction towards the liquid and that this manifests as some macroscopic force on the interface layer pushing it towards the liquid.

This is simply not correct, or at least, is an incomplete account of the microscopic inter-molecular forces there.

There are also repulsive inter-molecular forces. In equilibrium, one macroscopic result of all these forces, attractive and repulsive, is that net external force on the molecules in the interface layer is zero. If the interface is flat, on any element of the interface layer there is just as much force due to rest of the liquid pointing towards the liquid as there is force due to rest of the liquid in the opposite direction. This is necessary for the equilibrium to exist. So there is no "surface tension" force normal to the interface, and there is nothing to balance.

Maybe it will help to imagine that molecule in the interface layer being repelled from the liquid body is compressing the liquid body (increase of liquid pressure), and molecule being attracted is expanding the liquid body (decrease of pressure). Both kinds of molecules are present in the interface layer, so it is possible these two effects cancel the effect on the liquid pressure, as we know is necessary for the equilibrium.

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  • $\begingroup$ You are just repeating my concession on why Young-Laplace makes sense, but you are not addressing my concerns that there is net inward attractive force on the molecules at the surface. How is this force balanced? $\endgroup$
    – mrmudd
    Commented Aug 4, 2022 at 16:54
  • $\begingroup$ OK, I have expanded my answer to address your concern. $\endgroup$ Commented Aug 4, 2022 at 19:53
  • $\begingroup$ This is closer to the answer I'm looking for; however, your description of repulsive forces balancing the attractive force is exactly what I'm referring to. You even go as far to call it a "pressure" in your last paragraph. So then your answer should be yes, a flat interface does require a pressure to balance the inward attractive force. This pressure is the result of the repulsive intermolecular forces. $\endgroup$
    – mrmudd
    Commented Aug 4, 2022 at 22:06
  • $\begingroup$ Of course there is non-zero pressure in the liquid, but this has the same value as in the gas, as explained. $\endgroup$ Commented Aug 4, 2022 at 22:21
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I think what the discussion and question may be about, especially the last part concerning if solids have a "pressure" is termed the surface free energy. There is nice casual discussion at this website pointing out the surface free energy is the equivalent of surface tension for solids. So you can certainly have a flat surface like a solid with surface free energy.

This is one reason you have different contact angles for water on a surface. In some types of crystal growth people talk about balancing the surface free energy and the surface tension. In very old papers you will also find people talking about the surface tension of ionic crystals.

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