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I am thinking about how the velocity of molecules measured from a small region of space might change as the region of inquiry moves closer to the edge of a container. Ultimately I am thinking about MR imaging with velocity phase encoding, which can (in clinical application) resolve voxels in the 1mm$^3$ range (depending on scanner, bore, patient factors and the like).

For instance, if I have a glass container filled with distilled water, sitting still on a table. It is at standard temperature and pressure. If I measure the averaged velocity of all the molecules in the container, there would be a net zero velocity; the water is not jumping out of the container.

That is not to say that the molecules are still; I understand that they will all be moving and interacting per the kinetic model; just that on a macroscopic scale the water is, on average, still. It will have an average velocity of zero and an average speed relative to the temperature and pressure.

As I reduce my region of interest from "the container" down through "1 mL" and smaller towards the minuscule, that average velocity mean of zero will be maintained until statistical variation becomes more apparent, and at the molecular scale it will break down on individual measurements, but still be maintained if averaged over time.

But as I move a region of interest towards the container wall, I wonder if there is anisotropy? That is, as I get towards the wall with the directional component of the velocity become oriented perpendicular to the container wall.

I imagine that there will be several things that happen.

  1. As you approach the edge, there will be a bias towards the edge from van der waal's forces and affected by the properties of the container (material, it's effects on surface tension I presume)
  2. Apart from the bias in point 1, molecules heading perpendicular to the wall will be unaffected; molecules heading towards the wall will be reflected (losing some of their kinetic energy); and
  3. Those on an oblique trajectory will be reflected in the plane parallel to the edge (my nomenclature might be off) again losing some energy, but also having their trajectories altered somewhat by the weak interactions as above

Please feel free to fill me in on the other interactions I am missing.

So I then question myself, if we approach the limit withe our region of interrogation, as we do so will there be a bias in the molecules' velocity perpendicular to the wall?

Will the molecules' velocities (in 2D) go from this?

     \  |  /
      \ | /
    ___\|/___
       /|\
      / | \
     /  |  \

To this?

     \ | /         |
______\|/______    |
      /|\          |
     / | \         |<-- Wall

But still averaging zero on any large scale?

Or will it be biased, having a net velocity away from the container wall, i.e.

    \              |
     \ | /         |
______\|/___       |
      /|\          |
     / | \         |<-- Wall
    /              |

Or something like this, where there could be an increase in the number

I suppose to clarify I am thinking of long range motion rather than just local motion as discussed here

To extend, how would increasing the pressure (assuming temperature was allowed to equalize) affect this? I presume that the pressure would affect both local motion and long-range motion within the liquid, but mostly in the magnitude domain, not the direction. Similarly for heat, but then also including motion within the molecule itself for non-monatomic molecules.

In summary, does the directional component of the average velocity of molecules in a liquid at rest vary at the edge of a container, does the magnitude or direction (or proportion of molecules heading in a particular direction) component vary anisotropically at the edge of the container, or is there no appreciable difference up to the edge of the container?

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  • $\begingroup$ Don't forget that the container wall is also a bunch of atoms vibrating with a thermal distribution of energies. As long as the temperature of the wall and the liquid are the same, how does this impact your answer? $\endgroup$ – Jon Custer Dec 1 '14 at 16:51
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Qualitative Reasoning

As you anticipated, the shape of the velocity distribution (what you call local motion) does not depend on the position but is dictated by the Maxwell-Boltzmann distribution. To answer in short then: the average velocity of molecules does not vary as a function of distance from the container wall (here I am using "wall" as an idealized construct: an external potential).

What does depend on the distance from walls is displacements or as you put it, long range motion. This is not the same thing as velocity. So let's focus on $\langle\mathbf{r}(t+T) - \mathbf{r}(t)\rangle$, i.e. the average displacement during the time $T$. What happens in the absence of walls? Well, the particle in a homogeneous medium is as likely to go into any direction as any other, so this quantity always vanishes. This is why one usually looks at the mean-square displacements, $\langle(\mathbf{r}(t+T) - \mathbf{r}(t))^2\rangle$, which does not vanish and can be connected to diffusion constant. We won't need to bother, though, for clearly if there is a wall no displacement can go past this and as such the distribution will become slanted resulting in nonzero average displacements.

How close to the wall must we be to see the effect? Depends. On $T$ (if the particle starting from a distance $d$ from the wall can reach the wall in $T$, the wall has no effect on its displacement distribution). And on the particulars of the medium (density and such).

Some simulations

I like to run simple simulations to figure stuff out, to give a better picture and to reinforce (or dispute) one's physically based reasoning. I'm not going to explain what the data represents in any great detail: It is from a molecular dynamics run with Lennard-Jones particles interacting with a static Lennard-Jones wall. So basically a few thousand particles in a box.

So first off, we'd probably like to see the velocity distribution into the direction perpendicular to the wall over the entire system. Note that it is Gaussian, as indeed expected. velocity distribution into the direction perpendicular to the wall

Next let's have a look at the density. I know you didn't ask for it, but here it is: density distribution near the wall

Fine, so near the wall the fluid nucleates as the wall is infinitely regular. Now let's look at the root mean square (RMS) velocity as a function of distance from the wall: (RMS) velocity as a function of distance from the wall

Right, so the velocity distribution does not scale in any way, as we reasoned earlier (don't worry about the noise near the wall, it's due to the fact that my simulations were very short and small meaning I don't have a lot of data; note that the errors coincide with the places where the density is the lowest). Moving on to mean displacement (here defined with the opposite sign as above i.e. the function can be read as: given position $z$, how much did the average particle move in $T$ to reach this point): enter image description here

The curves going lower are from longer $T$.

Boom. There you have it: Confinement has an effect on diffusion as expected.

P.S.

Finally, you might be interested in a numerical technique used in computational fluid dynamics called the lattice Boltzmann method. It, in a way, plays around with velocity distributions and near walls has to resolve the collision such that no particles can stream through.

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  • $\begingroup$ Thank-you very much for your answer? $\endgroup$ – AssidiousBlue Dec 2 '14 at 17:05

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