Lennard-Jones model

Lennard-Jones potential

The Lennard-Jones potential, developed by Sir John Edward Lennard-Jones, is given by

$\Phi (r)=4\epsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]$

where:

• $\Phi (r)$ is the intermolecular pair potential between two particles at a distance r;

• $\sigma$ : diameter (length);

• $\epsilon$ : well depth (energy)

Reduced units:

• Density, $\rho ^{*}\equiv \rho \sigma ^{3}$, where $\rho =N/V$ (number of particles $N$ divided by the volume $V$.)

• Temperature; $T^{*}\equiv k_{B}T/\epsilon$, where $T$ is the absolute temperature and $k_{B}$ is the Boltzmann constant

Argon

The Lennard-Jones parameters for argon are $\epsilon /k_{B}\approx$ 119.8 K and $\sigma \approx$ 0.3405 nm. (Ref. ?)

This figure was produced using gnuplot with the command:

plot (4*120*((0.34/x)**12-(0.34/x)**6))

Features

Special points:

• $V(\sigma )=0$

• Minimum value of $V(r)$ at $r=r_{min}$;

${\frac {r_{min}}{\sigma }}=2^{1/6}\simeq 1.12246...$

Related potential models

It is relatively common the use of potential functions given by:

$V(r)=c_{m,n}\epsilon \left[\left({\frac {\sigma }{r}}\right)^{m}-\left({\frac {\sigma }{r}}\right)^{n}\right].$

with $m$ and $n$ being positive integer numbers and $m>n$, and $c_{m,n}$ is chosen to get the minumum value of $V(r)$ being $V_{min}=-\epsilon$

These forms are usually refered to as m-n Lennard-Jones Potential.

The 9-3 Lennard-Jones interaction potential is often use to model the interaction between the atoms/molecules of a fluid and a continuous solid wall. In (9-3 Lennard-Jones potential) a justification of this use is presented.

Other dimensions

• 1-dimensional case: Lennard-Jones rods.
• 2-dimensional case: Lennard-Jones disks.

References

1. J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, 43 pp. 461-482 (1931)