# Molecular model of liquids

The macroscopic behavior of gasses and solids follow very intuitively from the description of matter as quasi-spherical molecules interacting with each other with attractive and repulsive forces.

Things like pressure or crystals can be nicely explained with the model of little balls bouncing around or being tightly packed.

I struggle to develop an intuition for liquids though.

The main grief I'm given by the fact that liquids are barely compressible, suggesting a near constant mean distance of the molecules.

So we're looking for a microscopic picture of particles with a near constant mean distance where the particles are still free enough to allow fluid macroscopic behavior.

That seems a bit odd: Clearly if all particles had the same distance, that would be a crystal, so that's not right. So there must be a way in which they can have different distances, but not on average - which is what's so odd about it.

So what's going on?

EDIT: I'd like to make it a little bit more clear what I find so puzzling in this picture.

Imagine a hexagonal crystal (2D for the sake of simplicity). Can you imagine any kind of movement of a subset of vertices that wouldn't substantially change the distance of at least some neighboring vertices from the constant distance we had before? (Maybe such movement exists, maybe it exists only in three dimensions, but if so it's somewhat conterintuitive.)

But if we allow the distances to substantially change, what then guarantees the incompressibility?

• In a liquid, the average distance between two neighboring atoms is not so different from that of a crystal, but the arrangement of the atoms in space is disordered. – valerio Jan 17 at 11:54
• But what does such an arrangement look like? And in particular, how does the change of the arrangement look like when the liquid moves? That's much harder to get a grasp on than for gasses and solids. – John Jan 17 at 14:24
• Following your edit, I'm not completely sure what you are asking. One measure of the resistance to compression is the bulk modulus which is similar for liquids and solids. Resistance to shear is measured by the shear modulus which is zero for liquids and gases, and nonzero for solids. These can be defined in terms of fairly simple deformations of the material. Is that what you are after? – user197851 Jan 17 at 14:43
• @LonelyProf Ideally I'd love to have an animation about the trajectories of about a few dozen or so neighboring molecules in a moving liquid. Artistic problems aside, how would you move the molecules without changing average distances? Don't you think that's totally counter-intuitive? – John Jan 17 at 14:51
• I have edited my answer in an attempt to address your comments and the edited version of your question. – user197851 Jan 17 at 15:09

To try and explain this, since you are mostly interested in the distances between atoms, let me show a picture of the pair distribution function for the Lennard-Jones liquid, at a fairly low temperature (the black curve). This was calculated from a molecular dynamics (computer) simulation. With the origin of coordinates on an (arbitrary) atom, $$g(r)$$ shows the probability of finding another atom at a distance $$r$$, divided by the same quantity for an ideal gas having the same density $$\rho$$. So the combination $$\rho g(r)$$ is a "local density" of other atoms, around a typical atom. You'll notice a sharp peak at $$r\approx\sigma$$ (where $$\sigma$$ is the atomic diameter) which indicates a fairly well defined shell of nearest neighbours. Then there are subsidiary peaks at $$r\approx 2\sigma$$, $$3\sigma$$ etc. This is somewhat similar to what you see in a solid: more-or-less well defined next-nearest-neighbour, and next-next-nearest-neighbour distances. At large distance, though, the function settles down to a constant value: there are no long-range positional correlations in a liquid. In a solid, the peaks would be sharper, the structure more complicated, and the long distance correlations would not disappear; also the function would depend on direction in a solid, whereas it is isotropic in a liquid. So, there are neighbours with (roughly) the same nearest-neighbour distance as in a solid, there is also some local structure in the liquid as you look further away from a given atom, and this (together with the high overall density $$\rho$$, which for liquids is quite similar to solids) all contributes to making liquids fairly incompressible, similar to solids.

For comparison, I've shown in red the same quantity for a low-density Lennard-Jones gas, at the same temperature. There is a peak around $$r\approx\sigma$$ which is due to the attractive part of the Lennard-Jones potential. It's a bit higher, and a bit broader, than in the liquid. However, this is (again) just relative to the ideal gas at the same density. The actual density of neighbouring atoms, in a gas, is low: most atoms are not near other atoms. And also, there are no subsidiary peaks.

Don't forget that a liquid can be continuously converted into a gas, by raising the temperature above the critical point, then reducing the pressure, then (if you like) reducing the temperature again. We identify them as two separate phases because it is also possible to see a sharp phase transition between them, if we try to do this interconversion below the critical point. So there will always be a region of the phase diagram in which the properties (including the compressibility) are intermediate between the values we associate with liquids/solids and gases.

EDIT following OP change to question, and OP comments.

Other atomic liquids have fairly similar-looking plots of $$g(r)$$. For solids, I roughly described the differences in the appearance of $$g(r)$$ in my answer. I don't have a plot for a solid immediately to hand.

There are two main points here. Firstly, compressing a solid, or a liquid, will push the atoms, which are already close, even closer. This is resisted by the repulsive part of the interatomic potential: the energy goes up rapidly as the atoms begin to overlap. Hence the bulk modulus of a solid or a liquid is quite high, and we loosely speak of these phases as being "incompressible".

The second point is that the atoms in a liquid are continuously in motion, at thermal equilibrium. It is still perfectly possible to define a distribution of distances, as an average, and this is what is represented by $$g(r)$$. Quantities such as the pressure, and the bulk modulus, can be calculated as integrals over $$r$$, involving $$g(r)$$ and the interaction potentials (such as Lennard-Jones). The atomic motion is averaged over, in order to get $$g(r)$$. A computer simulation (such as molecular dynamics, in which the atoms actually move around following Newton's equations of motion) can be used to calculate $$g(r)$$ for a liquid; there are also some fairly complicated theories for this function.

You seem to be starting from an assumption that there is a particular configuration of atoms with well defined distances, and then you seem to be worrying about the fact that they are not fixed. When considering a liquid, we cannot assume that the atoms are fixed; but also we cannot ignore their interactions. This is why liquids are interesting, and why simple models (such as a harmonic solid, or an ideal gas) do not apply.

• Thank you for taking so much time for this, but I could have been clearer - it's not just about the distance. I've added a bit more explanation to my question. – John Jan 17 at 14:37
• Thinking about this graph, would a solid and other liquids look similar? – John Jan 17 at 14:46
• Hmm, kind of unsatisfying, I'll have to brood on this. In any case I appreciate your help. – John Jan 17 at 15:17