# Lennard-Jones model

The **Lennard-Jones** intermolecular pair potential is a special case of the Mie potential and takes its name from Sir John Edward Lennard-Jones (Ref. 1). The Lennard-Jones model consists of two 'parts'; a steep repulsive term, and
smoother attractive term, representing the London dispersion forces. Apart from being an important model in its-self,
the Lennard-Jones potential frequently forms one of 'building blocks' of may force fields,

## Contents

## Functional form

The Lennard-Jones potential is given by

where

- is the intermolecular pair potential between two particles or
*sites* - is the diameter (length),
*i.e.*the value of at which - is the well depth (energy)

In reduced units:

- Density: , where (number of particles divided by the volume )
- Temperature: , where is the absolute temperature and is the Boltzmann constant

The following is a plot of the Lennard-Jones model for the parameters 120 K and 0.34 nm. See argon for different parameter sets.

This figure was produced using gnuplot with the command:

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

## Special points

- Minimum value of at ;

## Critical point

The location of the critical point is (Caillol (Ref. 2))

at a reduced density of

- .

Vliegenthart and Lekkerkerker (Ref. 4) have suggested that the critical point is related to the second virial coefficient via the expression

## Triple point

The location of the triple point as found by Mastny and de Pablo (Ref. 3) is

- (liquid); (solid)

## Approximations in simulation: truncation and shifting

The Lennard-Jones model is often used with a cutoff radius of . See Mastny and de Pablo (Ref. 3) for an analysis of the effect of this cutoff on the melting line.

## m-n Lennard-Jones potential

It is relatively common to encounter potential functions given by:

with and being positive integers and .
is chosen such that the minimum value of being .
Such forms are usually referred to as **m-n Lennard-Jones Potential**.
For example, the 9-3 Lennard-Jones interaction potential is often used to model the interaction between
the atoms/molecules of a fluid and a continuous solid wall.
On the '9-3 Lennard-Jones potential' page a justification of this use is presented.

## Radial distribution function

The following plot is of a typical radial distribution function for the monatomic Lennard-Jones liquid (here with and kcal/mol at a temperature of 111.06K:

## Equation of state

*Main article: Lennard-Jones equation of state*

## Virial coefficients

*Main article: Lennard-Jones model: virial coefficients*

## Phase diagram

*Main article: Phase diagram of the Lennard-Jones model*

## Mixtures

## Related models

- Lennard-Jones model in 1-dimension (rods)
- Lennard-Jones model in 2-dimensions (disks)
- Lennard-Jones model in 4-dimensions
- Lennard-Jones sticks
- 9-3 Lennard-Jones potential
- 10-4-3 Lennard-Jones potential
- Stockmayer potential
- Mie potential

## References

- J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society,
**43**pp. 461-482 (1931) - J. M. Caillol " Critical-point of the Lennard-Jones fluid: A finite-size scaling study", Journal of Chemical Physics
**109**pp. 4885-4893 (1998) - 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) - G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics
**112**pp. 5364-5369 (2000)