Lennard-Jones model
The Lennard-Jones intermolecular pair potential was developed by Sir John Edward Lennard-Jones in 1931 (Ref. 1).
Functional form
The Lennard-Jones potential is given by:
where:
- is the intermolecular pair potential between two particles at a distance r;
- is the diameter (length), i.e. the value of at ;
- : 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
The following is a plot of the Lennard-Jones model for the parameters 120 K and 0.34 nm. See also argon for appropriate parameter sets.
![](/SklogWiki/images/thumb/6/6f/Lennard-Jones.png/400px-Lennard-Jones.png)
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 (Caillol (Ref. 2))
at a reduced density of
- .
Triple point
The location of the triple point as found by Mastny and de Pablo (Ref. 3) 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. 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.
Related pages
- Phase diagram of the Lennard-Jones model
- Lennard-Jones model: virial coefficients
- Lennard-Jones equation of state
- Lennard-Jones sticks
- Lennard-Jones disks
- 9-3 Lennard-Jones 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)