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Radial distribution of particle separation in a liquid at small distances

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Draw schematically the radial distribution function $g(r)$ for a Lennard-Jones fluid at low and high particle densities. Discuss in both cases the behavior at small $r$. At high densities $g(r)$ has some damped oscillations. Explain their origin and what would happen at long distances.

What I know:

The radial distribution function $g(r)$ describes how density varies as a function of distance from a reference particle (Wikipedia).

Specifically in Lennard-Jones case, we have :

\[img\]https://i.imgur.com/Z41FKUL.png\[/img\]

Here we can observe that if repulsion outweighs attraction, the curve will die out on the x axis. On the contrary, if it is the other way around, the curve will grow exponentially. For the sake of clarity, I would remark y = 0 as the Ideal Gas behaviour.

I have read that $g(r)$ vanishes at short distances, because the probability of finding two particles close to each other vanishes due to the repulsive part of the potential. At high densities $g(r)$ can show some damped oscillations:

These oscillations express the preference of the particles to be found at specific distances from a reference particle at the origin. For instance in the LJ case, a first layer of particles will be localized closed to the minimum of LJ's potential, which is the origin of the first peak in $g(r)$.

This layer prevents other particles from getting close to it, which is what causes the first
minimum in $g(r)$.

What I do not know.

1)The explanation:

'These oscillations express the preference of the particles to be found at specific distances from a reference particle at the origin'.

How can particles have 'preference' to be at different distances? I think this is not a good physical explanation. I have been thinking about this and I would say that as there is a high density of particles at small distances, the repulsion force is exerted on these, which triggers such oscillations. Finally there is a point where they are so close to each other that the repulsion force provides them with an initial kinetic energy which will be equal to a final electrostatic potential energy of zero (selecting our zero of electrostatic potential energy at y=0) , and that is the moment when the curve dies out on the x axis. Do you agree with this explanation?**

2)Are we dealing with underdamped oscillations at high densities? Could you give an insight into these kind of oscillations (if what I have just said is not enough or incorrect)?

3)When we are dealing with long distances, would $g(r)$ tend to be one? What would we observe, the Ideal Gas behaviour? Why? Is it because there is no interaction between the particles?

As always, I am interested in explaining such phenomena from a physical point of view.

ADDITIONAL INFORMATION RELATED TO THE RADIAL DISTRIBUTION:

I came across the following slide:

Which compares the structure factor $S(Q)$ with the radial distribution function $g(r)$. I know that the structure factor is a mathematical description of how a material scatters incident radiation, but how is the structure factor related to the distribution function (as this slide suggests)?

asked Nov 1, 2018 in Physics Problems by Jorge Daniel (606 points)
edited Dec 12, 2018 by Jorge Daniel
There seems to have been a software change to the Pasteboard site.
1) Why $2r_0$? (I mean the factor of 2 and no other number)?

2) Yes, after seeing carefully the graphs it is highly likely that we are not dealing with harmonic oscillations. I have a problem here, as I have just seen harmonic ones. How could we confront non-sinusoidal oscillations? If we say $m \frac{d^2 x}{d t^2} = -kx$ we get the harmonic oscillations. Is there any other force law which using Newtonian mechanics gives as another kind of oscillations? If there is not, do you recommend me using quantum mechanics? I am not sure if it would be feasible, but I thought about using the quantum harmonic oscillator to make an approximation by perturbation theory.
1) $2r_0$ (twice particle radius) is the minimum distan
OK thank you Professor, I think now I understood the oscillations issue.

The following comment has nothing to do with the problem but I wanted to ask:

I am interested in learning more about classical mechanics and mechanics in general. I have been using books like Tipler, Physics for scientists and Engineers and so on, but these most of the time do not derive physics laws. So I have been looking for such a kind of book (https://physics.stackexchange.com/questions/12175/book-recommendations) and I came across Classical Mechanics by Taylor and Classical Mechanics by Goldstein.

I have been looking for a book related to mathematical methods as well. I started reading Mathematical Methods for Physicists by Arfken.

Any personal recommendations? I would appreciate it, thank you.
Sorry, I do not know many textbooks, and do not like making recommendations anyway. It is very easy to find what textbooks are available for any topic, and get recommendations, reviews and previews, from sites such as Amazon, GoogleBooks, etc. Which one suits your requirements is a very subjective decision.
I have posted some instructions for embedding images using Imgur : see http://physics.qandaexchange.com/?qa=253/how-to-upload-an-image

1 Answer

1 vote
 
Best answer

1. Preference

The spatial distribution of particles settles into a configuration which minimises potential energy. This is achieved when there is some regular structure - for example, hexagonal close packing. Gravitational potential energy is far too small on the scale of microscopic particles. The balance is between kinetic energy and electrostatic potential energy as defined by the Lennard-Jones potential.

Ideal gas particles are assumed to have no attraction at all, only "hard sphere" repulsion when they come into contact. The potential function for ideal gas interactions is therefore $U(r)=0$ for $r>2r_0$ and $U(r)=∞$ for $r≤2r_0$, where $r_0$ is the radius of particles so $2r_0$ (twice particle radius) is the minimum distance between the centres of particles.

2. The Nature of the Oscillations

These are not dynamic oscillations in the motion of particles (periodic variations with time), like a mass on a spring. So the question about over/under-damping is not relevant - although it is possible some analogy could be made with dynamic oscillations.

These are static oscillations in the distribution of inter-particle distances (oscillations in space), like the ripples of a wave function in quantum mechanics, or the ripples in sand on a beach.

$g(r)$ is a probability distribution function. The "oscillations" indicate that there are periodic values of particle separation which are more likely than average (peaks) or less likely than average (troughs). These "oscillations" are not necessarily sinusoidal (harmonic).

High density causes the "oscillations" by restricting the space which particles can move about in. When particles are forced close together they slide into relative positions where they can keep as much motion as possible - ie they form structures such as hexagonal close packing. At low densities particles have plenty of freedom to move about and can occupy all relative positions equally. All values of separation $r$ are equally likely - the probability distribution function is uniform, flat.

The oscillations are not necessarily sinusoidal. The graph you linked does look like a damped harmonic oscillation. But it is not an oscillation in time and space like a pendulum, it is a periodic variation in the probability of finding a particle at distance $r$ from an arbitrary particle. It is probably not easy to see it if you look at the particles. It is a statistical variation which shows up when you calculate average the distances, because averaging removes random variations.

The explanation you quoted suggests that the graph is statistical. It has been constructed by taking a snapshot of the jiggling particles, and measuring the distance of every particle from one aribitrary particle, which is chosen as the origin. (Perhaps this is repeated with every particle in turn being used as the origin. This is the same as measuring the distance between every pair of particles.) The distances of every particle from the origin are measured, and a histogram is plotted of the density of particles at a distance in the range $r−Δr$ to $r+Δr$ (vertical axis) vs r (horizontal axis).

For a gas you would expect a uniform density distribution.

For a 'cold' 3D cubic crystalline solid you find sharp peaks at separations r which are multiples of $1,\sqrt2, \sqrt3, 2, \sqrt5, \sqrt6, ...$ units. In fact for all $r^2=l^2+m^2+n^2$ with all possible combinations of integers $l, m, n$. In between are ranges in $r$ which have zero frequency - eg 1.25 units. These peaks and troughs extend over a large range in $r$ - ie there is long range order.

For a 'hot' crystal these peaks and the troughs in between them are broader and more rounded, like rolling hills and valleys, giving the impression of oscillations. (The unit of spacing also increases as the solid expands.)

For a liquid or amorphous solid there are peaks (and troughs) only at small values of r - ie there is short range order. At large values of r the distribution is more uniform like a gas.

answered Dec 11, 2018 by sammy gerbil (28,466 points)
edited Dec 16, 2018 by sammy gerbil
Thank you for your answer. I have updated my question, as I have a doubt regarding the radial distribution.
The slide defines the Structure Factor $S(q)$ in terms of $q$ but it does not explain what $q$ is - possibly the wave number of the radiation which is being scattered? The slide does not explain how $S(q)$ relates to $g(r)$. Possibly they are Fourier Transform pairs. If you want to understand this you will need to do some research for yourself.  Read a book or website on the subject, or enrol on a university course which teaches it.  It will be difficult but if you are trying to teach yourself this topic that's what you need to do.  Alternatively  ask John Rennie. I don't want to become your personal tutor.
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