# Lennard-Jones model

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

## Contents

## Lennard-Jones potential

The Lennard-Jones potential is given by:

where:

- is the intermolecular pair potential between two particles at a distance r;

- : diameter (length);

- : well depth (energy)

Reduced units:

- Density, , where (number of particles divided by the volume .)

- Temperature; , where is the absolute temperature and is the Boltzmann constant

## Argon

The Lennard-Jones parameters for argon are 119.8 K and 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:

- Minimum value of at ;

#### Critical point

The location of the critical point is

at a reduced density of

- .

Caillol (Ref. 3) reports and .

#### Triple point

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

## Approximations in simulation: truncation and shifting

The Lennard-Jones model is often used with a cutoff radius of . 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:

with and being positive integer numbers and , and is chosen to get the minimum value of being

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.

#### Other dimensions

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

## See also

- Phase diagram of the Lennard-Jones model
- Lennard-Jones model: virial coefficients
- Lennard-Jones equation of state

## References

- J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society,
**43**pp. 461-482 (1931) - Ethan A. Mastny and Juan J. de Pablo "Melting line of the Lennard-Jones system, infinite size, and full potential", Journal of Chemical Physics
**127**104504 (2007) - J. M. Caillol " Critical-point of the Lennard-Jones fluid: A finite-size scaling study", Journal of Chemical Physics
**109**4885-4893(1008)