Lennard-Jones model

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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]


  •  \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 .)


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))


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 

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.

Other dimensions

See also


  1. J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, 43 pp. 461-482 (1931)
  2. 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)
  3. J. M. Caillol " Critical-point of the Lennard-Jones fluid: A finite-size scaling study", Journal of Chemical Physics 109 4885-4893(1008)