Lennard-Jones model

Lennard-Jones potential

The Lennard-Jones potential is given by

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

where:

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

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))

Special points:

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

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 potential is often use to model the interaction between the atoms/molecules of a fluid and a continuous solid wall.