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I just realized there's something extremely basic about surface tension that I don't understand. Surface tension is a property of the interface between two different materials, such as water and air, or water and glass, or water and oil. According to standard introductory textbooks, surface tension creates a force per unit length of $\gamma$ "along the surface" between two materials.

However, "the surface" is not a real physical object. Presumably, this force should be acting on one of the two materials, or possibly both. In the cases of water and air, and water and glass, the surface tension forces clearly act on the water, as the air is chemically inert and the glass is not free to move. But what about two fluids, like water and oil?

Specifically, if I consider the surface between water and oil, then which of the following is true?

  • There is a tension force per unit length of $\gamma$ along the surface, acting on the water.
  • There is a tension force per unit length of $\gamma$ along the surface, acting on the oil.
  • Both of the previous two are true simultaneously. (But then the energy needed to increase the surface area by $dA$ would be $2 \gamma \, dA$, which is incorrect.)
  • The tension forces per unit length acting on the water and oil are both nonzero, and add up to $\gamma$. (But then what determines how much force acts on each individually? Why do people never talk about this split, and how would one look up the values?)
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  • $\begingroup$ It might help to conceptualize if you recognize that in real systems, the surface is a real physical object. The transition between the two media can never be truly discontinuous, there is always a "selvedge" or "verge" where the medium is neither of the media. $\endgroup$
    – garyp
    Aug 8, 2020 at 16:01

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The best way to think about surface tension is as an energy per unit area stored in the boundary of a fluid. For example, a soap bubble becomes a sphere because the total surface-energy is minimized when the surface area is minimized, and the minimum area surface encapsulating a fixed volume is a sphere.

In the case of water and oil, we must recognize that water and oil do not mix. Their intermolecular forces are such that the oil binds to oil, water binds to water, but oil does not bind to water. In other words, the minimum energy configuration of water and oil is for the water to be bound up together and the oil to be bound up together.

Finally, to answer your question, we recognize that the surface tension is sourced by the binding energies of the fluids. Since the dominant binding energies are the water-water and oil-oil binding energies, the self-surface tension dominates, and we can safely neglect any water-oil surface tension.

In a more general circumstance, where the water-oil binding energy exceeds the self-binding energy of either fluid, the minimum energy configuration is no-longer clumps of fluid, but a perfect mixture of the two. In other words, if the inter-fluid binding energy dominates, the surface will simply dissolve away, leaving you with a solution.

Edit: The binding energy at the surface interface is the oil-water binding energy added up over the surface interface. These inter-fluid forces in the weak-coupling limit are typically the result of van der waals forces.

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  • $\begingroup$ I agree with what you're saying, but my point is, if I wanted to specifically find out how much force acted on the water, and how much on the oil, what numbers would I have to look up? $\endgroup$
    – knzhou
    Aug 8, 2020 at 4:34
  • $\begingroup$ Then your specific question is best thought of as the friction between the two surfaces, not a surface tension. The answer to this question is difficult in terms of microscopic quantities (these would be van der waals forces). You'd best perform an experiment. $\endgroup$
    – Guy
    Aug 8, 2020 at 4:36
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To your "bullet" #3: I think the answer depends on how $\gamma$ is defined. If the surface work is written as $\delta W_a = \gamma dA$ then that refers only to one side of the interface. For example, a soap bubble of radius $r$ will have pressure difference between the inside near the center of the bubble and within the soap layer $\Delta p_a = \frac {2\gamma}{r}$ but the pressure difference between the inside of the bubble and outside air will be $\Delta p_b = \frac {4\gamma}{r}$. See for detail in Guggenheim: THERMODYNAMICS, An Advanced Treatment for Chemists and Physicists, page 52-53, eqs 1.61.1-4.

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