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Paper explaining why surface tension is parallel to the interface:

Away from the surface there is perfect force balance due to the symmetry around a molecule. Near the interface, however, the up-down symmetry is broken. To restore the force balance in the vertical direction, the upward repulsive arrow (dashed) has to balance the downward attractive arrow (solid). In the direction parallel to the interface, the symmetry is still intact, thus automatically ensuring a force balance parallel to the interface. This balance means that along the direction parallel to the interface, there is no reason why the attractive forces should have the same magnitude as the repulsive forces. In practice, the attractive forces are stronger, giving rise to a positive surface tension force.

This was the diagram provided in the paper:

enter image description here

This is my intuition for the question title: ( Is the density greater on the interface or less dense? )

Consider the molecule at the interface. The force balance on this molecule is broken creating a net force downwards that minimises surface area. However, because this molecule doesn't keep accelerating downwards it must eventually be balanced by the repulsive forces of molecules below. At this point where the attractive and repulsive forces are balanced the molecules must be more compact. Hence I think the surface of the fluid must be more dense than the bulk. Because the surface is denser than there exists more attractive forces giving rise to surface tension.

Is the above intuition correct and that a fluid's surface is indeed denser than the bulk? Or is there a flaw in the explanation above? If so, an alternative explanation would be extremely helpful.


Edit: There are varying explanations that seem to contradict themselves. Some answers state that the density is lower on the interface and that molecules attract each causing the surface to be under tension. Other answers state that the density is greater at the interface. Can someone tell me the true conclusion? To me, a higher density seems more intuitive because the attractive forces between molecules should decrease with distance. The central part of my question is to understand the true origin of surface tension at the molecular level.

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  • $\begingroup$ If you look at this kind of problem from a highly simplified static perspective of molecules repelling each other, then increased repulsion doesn't necessarily mean increased density. For example, imagine molecules in a staggered, triangular lattice. If you slide a row over (lining those molecules up with the ones above and below) you've increased the repulsion that row will feel, but the density is the same. $\endgroup$
    – Alex K
    Jan 9, 2023 at 2:41
  • $\begingroup$ I think my explanation was slightly vague. What I meant to say was the surface molecules experience a net force down. Now, this net force must be balanced otherwise the molecule would continue to accelerate downwards. This net force could be balanced by the surface molecules coming closer to molecules below as the repulsive forces increases as molecules come closer together. I'm saying to balance the surface molecules net downwards force the molecules must be closer together to exhibit stronger repulsive forces. Hence the surface would be more dense. Could this intuition be true? $\endgroup$ Jan 9, 2023 at 7:33
  • $\begingroup$ the same scheme is also discussed here, it may help: physics.stackexchange.com/q/517915/226902 $\endgroup$
    – Quillo
    Jan 9, 2023 at 11:44
  • $\begingroup$ "This net force could be balanced by the surface molecules coming closer to molecules below as the repulsive forces increases as molecules come closer together." I agree that it's possible, but I was using the lattice example to show how you can get reduced distance without increased density. In a fluid where everything is moving, you could get increased mean repulsion just by virtue of molecules taking statistically different paths. $\endgroup$
    – Alex K
    Jan 9, 2023 at 19:31

4 Answers 4

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To the Density Question:

Whether the fluid density increases or decreases depends on the other phase involved. In your example the figure shows a liquid/vapour interphase. Since the density changes continuously from liquid to vapour, the liquid density has to decrease from it's liquid bulk value to it's vapour value. See the fig. below for a density profile of liquid n-Hexane in equilibrium with it's vapour. It shows the density plotted vs the vertical distance z. z=0 is here set to the vapour bulk density, z=infinity is the liuid bulkd density. An "s-curve" starting at low densities (vapour) and changing continuously to high density (liquid) also shows the size of the interface. The higher the temperature, the larger the interface. This goes up to the critical point, at which the interface gets infinitely large, or, equivalently, no difference between vapour and liquid can be detected anymore. Plot of the Density Profile of n-Hexane at 3 different Temperatures (own work, based on square gradient theory)

To the Surface Tension Question: As already explained by others in this thread, in the bulk of a liquid phase one molecule has more neighbours to interact with than in the interface region. The bulk phase therefore is energetically more favourable as the potential energy of the system decreases. A ball drops down to earth as it's potential energy decreases (until repulsion from the ground kicks in), similarly, molecules approach each other until repulsion and attraction equal out, see L-J-Potential for example. Now, in the interphase region, due to the density decrease from bulk liquid to vapour, there are less neighbours for the molecules to interact with, therefore it requires energy to bring molecules from the (energeitcally favoured) bulk to the (less favoured) interphase. This means the system energy increases which makes it unvafourable. Thus, the system tries to decease it's interfacial area with the other phase in order to lower it's energy (e.g. the reason for spherical drop form, since spheres have the smallest surface/volume ratio).

Edit:
Since Sources were asked for: You'll find many if you search for "square gradient theory" or "density profile liquid/vapour" or sth similar.
However, the theory to describe density profiles thermodynamically was originally developed by Van der Waals in 1893.
It was extended by Cahn, Hilliard (1958)
A nice book, which summarizes the results as well as the historical development of interfaceial thermodynamics is that by Rowlinson and Widom (1982)

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This completely depends on the inertia and entropy of the fluid. An air boundary in a barrell of a double overhead ocean wave has a very different surface density compared to say the skin of liquid on a shallow puddle. The closer to chaotic motion should lean towards less dense... ie distance divers will have air jets in pools to break the surface tension so if they land wrong its no issue.. where as a surfer may struggle to jump and kick his way through the skin effect on a huge wave :)

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    $\begingroup$ I guess to add.. the faster someone falls and hits water, the denser the water seems. It boils down to the fact that water is incompresable and can only move so fast.. A puddle at rest is still in motion, random walk and all, thus chaotic in entropy. A resting puddles surface is less or near as dense. Add randomness and its less dense. Add order and the density rises $\endgroup$ Jan 11, 2023 at 9:39
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    $\begingroup$ Also to add I feel that my answer mostly associates with water and polar molcules. Cant make assumptions about how surfing a non polar solvent might go lol $\endgroup$ Jan 11, 2023 at 9:43
  • $\begingroup$ Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Jan 11, 2023 at 10:36
  • $\begingroup$ if you want to add to your answer, it is preferable to do so in the body rather than through comments as comments are subject to deletion. $\endgroup$ Jan 11, 2023 at 13:22
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    $\begingroup$ I will edit properly in the future! $\endgroup$ Jan 11, 2023 at 20:44
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The origin of surface tension is the intermolecular attraction of fluids. The actual answer to the question should be 'lower'. Just take a look at it, why is the density lower at the surface of a fluid.

Consider a round drop of a fluid (I am neglecting the presence of gravity and a container). Every molecule attracts every molecule in that droplet that results in a net attractive force towards the center of the drop. It is just like how the gravitational force of earth attracts everything to the core. I also want to replace the term 'downward force' with 'inward force' for a droplet. Now, the inward force on the molecules inside the bulk is greater because they are closer to the center while the inward force on the surface molecules is lower as they are far from the center.

As the intermolecular attraction force is weak on surface molecules and they feel repulsion from only one side, the molecules have greater kinetic energy than of the bulk molecules. In the bulk, molecules of the fluid experience strong intermolecular attraction and repulsion from every side that causes their kinetic energy to be lower. Having greater kinetic energy implies that surface molecules have greater average distance between them ('average' is used because its impossible to calculate distance between every pairs of molecule). And greater average distance means greater volume which implies less density.

Moreover, surface tension is mainly noticed in liquid and the liquid surface is not a sharp cutoff. There is liquid-vapour interface at the fluid surface. Surely, there is so much density difference between fluid and its vapour. The former has high density and the latter has lower density.

That is why the fluid surface has slightly less density than the bulk.

enter image description hereImage source

Origin of surface tension:

There is always a cohesive force between the molecules of the fluid and so the free surface of the fluid always try to stay with the least surface area. The molecules at fluid surface has the highest potential energy (Just like an object at the surface of earth has higher potential energy than any other object inside earth). If the density is higher at the surface, that means there exist a very high potential energy at the surface and the droplet will blow up or collapse to get lower potential in this case. But we see that the droplet remains the same, i.e it doesn't blow up or collapse. That means it always try to reduce its surface area and it has lower density at the surface. With lower surface area and lower density the potential energy stays at a balance. There is a nice wikipedia article on surface tension.

As of it, the fluid surface stays stretched and behaves like a streched curtain. If a line is imagined on the fluid surface, the fluid on each side of the line try to move far from each other. So, a tension generates on each side of the line. That is surface tension.

So, the cohesive force or the intermolecular attraction creates surface tension in a fluid.

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  • $\begingroup$ Does this lower density on the surface create surface tension? Or how is it that the intermolecular attractions give rise to surface tension? $\endgroup$ Jan 11, 2023 at 7:28
  • $\begingroup$ @QuinGardinerBax That isn't the density that causes surface tension. See the edit. $\endgroup$ Jan 11, 2023 at 9:00
  • $\begingroup$ @QuinGardinerBax Yes, increased intermolecular attraction causes increased surface tension. $\endgroup$ Jan 11, 2023 at 12:21
  • $\begingroup$ Great edit. Could you provide some rescources like papers to reference your answer? $\endgroup$ Jan 12, 2023 at 9:06
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    $\begingroup$ @QuinGardinerBax I provided an Wikipedia article on surface tension. The origin of surface tension mentioned here is almost the same but I found no papers on the density of fluid surface. That maybe because no one thought about it before or never made a theory on it. $\endgroup$ Jan 12, 2023 at 9:42
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Sorry for the late reply, but I felt that the existing answers are really unsatisfactory, or outright wrong, and that a better answer is needed.

In particular, the papers and books that assert unequivocally that the density must be lower right at the surface is in direct variance with theoretical expectations and computer simulations (links below). Instead, the short and downvoted answer by Proffessor Chaos[sic] is closer to the truth, as I will be explaining in this answer. Sadly, I was not able to find direct experimental observation of this, which is not too surprising, given how difficult it would be to measure this.


The first problem starts with the picture that is inside the OP. I am not disagreeing with the fact that the potential between two atoms is closer to the picture inside Debanjan Biswas's answer, that is made of Coulomb-fermion repulsion when too close, and some attraction farther away. This means that, yes, the forces on each atom would be sometimes attractive, and sometimes repulsive. However, such a Zen-like picture, neither here nor there, is not of any use in the theoretical understanding of the situation. We should, instead, take the time average (and space average) of these rapid fluctuations, and define the averaged forces that is observed by each atom, the thing that is constant, and thus much more amenable to theoretical analysis.

In this way, each atomic bond can only either pull or push, you should not be having both. Then we apply Newtonian mechanics analysis on the system. If, at the surface, there is a net force inwards or outwards, then there is no hope for mechanical stability. We must thus never draw a force with any inwards or outwards component at the surface, only along the surface. This is already a huge problem for the picture in the OP.

Now, we want to know if the atoms deep in the bulk are pulled or pushed. After all, symmetrically pulling apart or pushing together, the atom can be balanced. So, are the materials holding together usually pulling at each other, or pushing?

Here, we need to note that air pressure is very small. We can essentially consider the solid-air or liquid-air boundary to be close to a solid-vacuum or liquid-vacuum boundary. If the atoms deep in the bulk are pushing each other apart, then the lack of pressure due to the vacuum boundary should be met with continued expansion of the solid / liquid. As such, we deduce that in a usual solid / liquid, the interatomic forces are usually pulling, as expected of the concept of condensation. Hence, condensed matter. i.e. we are usually operating in the attractive region of the picture in Debanjan Biswas's answer.

This is also the case at the surface. The atoms at the surface, as I have reasoned, should not be having inward or outwards forces, only along the surface. Here, they should be pulling at each other, hence tension, rather than pushing at each other. This is also expected, since if you tried to push stuff into each other parallel to the surface, then the expected behaviour should be that of buckling/bending.

What does this all mean? It means that we should be expecting that the outermost surface atoms to be somewhat displaced inwards into the bulk than is the usual bulk distances, i.e. higher density at the surface. This is because the surface atoms do not have atoms outside the solid to pull them outwards, and so they settle a little closer than usual, achieving a new equilibrium position, where, on the average, there is no more pulling inwards.

Why, then, are people saying the opposite? Well, of course they would, if they conflate the multiple-unit-cell-averaging needed to talk about the density, with the only-one-unit-cell-averaging. The averaging cell near the surface will be half outside the solid / liquid, and thus only half the density in the bulk, and that effect is going to be so dominant, that you will only see a dip in the density plots, not a tiny peak that the tiny displacements would have shown.


In practice, this is particularly horrible to study, both experimentally and theoretically. This is because most surfaces would have to undergo surface reconstruction, and then you cannot directly compare the surface from the bulk.

This is why we have to cherish the cases whereby things are simple, and then we can see this theoretical prediction in action. Here is a simulation of three surfaces of Iron, and you can see that for all 3 types of surfaces, the first layer moves into the bulk. The 2nd and 3rd layers might move into or out of the bulk, making the determination of the density even more a headache to compute and measure, but the theoretical prediction is verified.

In Fig.2 of this arXiv paper, you can also see that there are some vague spikes in density right before the densities plummet, showing that there is some slight increase in density at the surfaces. It is clearly not easy to compute things tight enough to even observe these effects at all. Fig.2 of this paper is also barely able to see the slight increase in density right before the dip (slightly more pronounced at the right edge than the left), but is more relevant to you if you are interested in surface tension.

So, again, the expectation should be that the surface should be slightly denser, but this is strongly tempered by the phenomenon of surface reconstructions. Sadly, life is just not that easy.

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