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 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,

Functional form

The Lennard-Jones potential is given by

${\mathrm{\Phi }}_{12}(r)=4ϵ[{(\frac{\sigma }{r})}^{12}-{(\frac{\sigma }{r})}^6]$

where

• $r:=|{\mathbf{r}}_1-{\mathbf{r}}_2|$
• ${\mathrm{\Phi }}_{12}(r)$ is the intermolecular pair potential between two particles or sites
• $\sigma$ is the diameter (length), i.e. the value of $r$ at which ${\mathrm{\Phi }}_{12}(r)=0$
• $ϵ$ is the well depth (energy)

In reduced units:

• Density: ${\rho }^{*}:=\rho {\sigma }^3$, where $\rho :=N/V$ (number of particles $N$ divided by the volume $V$)
• Temperature: $T^{*}:=k_{B}T/ϵ$, where $T$ is the absolute temperature and $k_B$ is the Boltzmann constant

The following is a plot of the Lennard-Jones model for the parameters $ϵ/k_B\approx$ 120 K and $\sigma \approx$ 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

• ${\mathrm{\Phi }}_{12}(\sigma )=0$
• Minimum value of ${\mathrm{\Phi }}_{12}(r)$ at $r=r_{min}$;

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

Critical point

The location of the critical point is

$T_c^{*}=1.326\pm 0.002$

at a reduced density of

$\underset{c}{\overset{*}{\rho }}=0.316\pm 0.002$.

Vliegenthart and Lekkerkerker have suggested that the critical point is related to the second virial coefficient via the expression

$B_2{|}_{T=T_c}=-\pi {\sigma }^3$

Triple point

The location of the triple point as found by Mastny and de Pablo (Ref. 4) is

$T_{tp}^{*}=0.694$

$\underset{tp}{\overset{*}{\rho }}=0.84$ (liquid); $\underset{tp}{\overset{*}{\rho }}=0.96$ (solid)

Approximations in simulation: truncation and shifting

The Lennard-Jones model is often used with a cutoff radius of $2.5\sigma$, beyond which ${\mathrm{\Phi }}_{12}(r)$ is set to zero. See Mastny and de Pablo for an analysis of the effect of this cutoff on the melting line.

n-m Lennard-Jones potential

It is relatively common to encounter potential functions given by:

${\mathrm{\Phi }}_{12}(r)=c_{n,m}ϵ[{(\frac{\sigma }{r})}^n-{(\frac{\sigma }{r})}^m].$

with $n$ and $m$ being positive integers and $n>m$. $c_{n,m}$ is chosen such that the minimum value of ${\mathrm{\Phi }}_{12}(r)$ being ${\mathrm{\Phi }}_{min}=-ϵ$. Such forms are usually referred to as n-m 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. Another example is the n-6 Lennard-Jones potential, where $m$ is fixed at 6, and $n$ is free to adopt a range of integer values. The potentials form part of the larger class of potentials known as the Mie potential.

Radial distribution function

The following plot is of a typical radial distribution function for the monatomic Lennard-Jones liquid (here with $\sigma =3.73\AA$ and $ϵ=0.294$ kcal/mol at a temperature of 111.06K):

Typical radial distribution function for the monatomic Lennard-Jones liquid.

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

Perturbation theory

The Lennard-Jones model is also used in perturbation theories, for example see: Weeks-Chandler-Anderson perturbation theory.

Mixtures

• Binary Lennard-Jones mixtures
• Multicomponent Lennard-Jones mixtures

Related models

• 8-6 Lennard-Jones potential
• 9-3 Lennard-Jones potential
• 9-6 Lennard-Jones potential
• 10-4-3 Lennard-Jones potential
• n-6 Lennard-Jones potential
• Kihara potential
• Lennard-Jones model in 1-dimension (rods)
• Lennard-Jones model in 2-dimensions (disks)
• Lennard-Jones model in 4-dimensions
• Lennard-Jones sticks
• Mie potential
• Soft sphere potential
• Stockmayer potential