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This question is inspired by this question/answer pair: Is this formula for the energy of a configuration of 3 fluids physically reasonable?

Consider three immiscible fluids forming contact surfaces, where none of the three can make a lubrication layer for the other two (the surface energy between fluid 1 and fluid 2 is not decreased by putting a thin layer of fluid 3 inbetween them, and likewise for the other two permutations). In this case, if all the contact surface tensions are positive, you have a minimum energy when all the surfaces are flat.

But there is a separate line tension for the 3-fluid interface itself. Can this line tension be negative? If it is negative, the line would like to wriggle, but the surface tension will require that the wriggles straighten themselves out as quickly as possible, to make the surface energy least. In this case, the minimizing energy configuration seems to be a very rapidly wriggling curve which is only infinitesimally different at the atomic scale from straight line. This suggests that a negative line-tension always renormalizes to exactly zero line-tension at long wavelengths. Is this correct?

Are there experimental or computational 3-phase line interfaces with a negative line tension? Do they renormalize to a zero line tension limit? Does this mean that zero line tension is a common observation?

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I should delete this question--- but I read a paper of Widom's which contradicts some of the conclusions here (he said that negative line-tension is possible and occurs in some situations, and I thought this would lead to crumpling and renormalization to zero tension). Perhaps he is right and I am missing something, I need to think more, but I still think the crumpling and renormalization to 0 tension is right. –  Ron Maimon May 25 '12 at 10:27
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1 Answer

This is really a comment, but it got a bit long to put in the comment box.

Consider two immiscible fluids (of different densities) A and B forming the usual two phase system with a flat interface, and introduce a drop of a third immiscible fluid, C, at the interface so it forms a lens. The outer edge of the lens gives you the 3 fluid line, and at this line the three surface tensions must balance, otherwise the drop contact angle will change, and the line move, until the surface tensions balance.

Doesn't this mean the line tension is always zero? The only way the line tension could be non-zero is if the system cannot reach equilibrium i.e. the drop of C either rolls up or spreads infinitely, and in these cases there is no line of contact between the three fluids.

After some thought:

Look at the system above in another way. The way you'd calculate the line tension is to calculate the energy required to increase/decrease the line length. But you can't do this without changing the interfacial surface areas. So you need to make some perturbation and calculate the energy change, then calculate the change in the surface energies and subtract this. What you're left with is the contribution from the line tension.

But the way you calculate the conformation of the system is to minimise the surface energies, so once you've subtracted off the change in interfacial energy you're left with zero. So once again you reach the conclusion that the line tension must be zero.

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This is just not true. The line tension can be positive. You can see this in a computer simulation, and there are no doubts. It is not zero in general. I am asking whether it can be negative. I will probably delete the question and incorporate it into the answer, because I am determining the answer, and it everything I said is just correct, no question needed. –  Ron Maimon Mar 26 '12 at 16:40
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