# Lennard-Jones model

The Lennard-Jones potential was developed by Sir John Edward Lennard-Jones.

## Lennard-Jones potential

The Lennard-Jones potential is given by:

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

where:

• $\sigma$ : 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:

• $\Phi(\sigma) = 0$
• Minimum value of $\Phi(r)$ at $r = r_{min}$;
$\frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ...$

#### Critical point

The location of the critical point is

$T_c^* \approx 1.33$

at a reduced density of

$\rho_c^* \approx 0.32$.

Caillol (Ref. 3) reports $1.326 \pm 0.002$ and $0.316 \pm 0.002$.

#### Triple point

The location of the triple point as found by Mastny and de Pablo (Ref. 2) is

$T_{tp}^* = 0.694$

## Approximations in simulation: truncation and shifting

The Lennard-Jones model is often used with a cutoff radius of $2.5 \sigma$. See Mastny and de Pablo (Ref. 2) fa an analysis of the effect of this cutoff on the melting line.

## Related potential models

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

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

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

These forms are usually referred 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.