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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Phi (r)=4\epsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]}

where:

• $\Phi (r)$ is the intermolecular pair potential between two particles at a distance r;

• $\sigma$ is the diameter (length), i.e. the value of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r} at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Phi (r)=0}  ;

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \epsilon }  : well depth (energy)

Reduced units:

• Density, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho ^{*}\equiv \rho \sigma ^{3}} , where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho =N/V} (number of particles Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N} divided by the volume $V$.)

• Temperature; Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T^{*}\equiv k_{B}T/\epsilon } , where $T$ is the absolute temperature and $k_{B}$ is the Boltzmann constant

Argon

The Lennard-Jones parameters for argon are $\epsilon /k_{B}\approx$ 119.8 K and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma \approx } 0.3405 nm. (Ref. 2)

This figure was produced using gnuplot with the command:

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

Features

Special points:

• $\Phi (\sigma )=0$
• Minimum value of $\Phi (r)$ at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r=r_{min}} ;

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

Critical point

The location of the critical point is (Caillol (Ref. 3))

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{c}^{*}=1.326\pm 0.002}

at a reduced density of

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho _{c}^{*}=0.316\pm 0.002} .

Triple point

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

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

Approximations in simulation: truncation and shifting

The Lennard-Jones model is often used with a cutoff radius of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2.5 \sigma} . See Mastny and de Pablo (Ref. 4) 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi (r) = c_{m,n} \epsilon \left[ \left( \frac{ \sigma }{r } \right)^m - \left( \frac{\sigma}{r} \right)^n \right]. }

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } being positive integers and $m>n$. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{m,n} } is chosen such that the minimum value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(r) } being Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{min} = - \epsilon } . 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

1. J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, 43 pp. 461-482 (1931)
2. L. A. Rowley, D. Nicholson and N. G. Parsonage "Monte Carlo grand canonical ensemble calculation in a gas-liquid transition region for 12-6 Argon", Journal of Computational Physics 17 pp. 401-414 (1975)
3. J. M. Caillol " Critical-point of the Lennard-Jones fluid: A finite-size scaling study", Journal of Chemical Physics 109 pp. 4885-4893 (1998)
4. 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)