Lennard-Jones model

2009-04-09T02:56:11Z

66.65.105.254: /* Triple point */

The '''Lennard-Jones''' [[intermolecular pair potential]] is a special case of the [[Mie potential]] and takes its name from [[ Sir John Edward Lennard-Jones KBE, FRS | Sir John Edward Lennard-Jones]]
<ref>[http://dx.doi.org/10.1088/0959-5309/43/5/301 J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, '''43''' pp. 461-482 (1931)] </ref>.
The Lennard-Jones [[models |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

:<math> \Phi_{12}(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] </math>

where
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>
* <math> \Phi_{12}(r) </math> is the [[intermolecular pair potential]] between two particles or ''sites''
* <math> \sigma </math> is the diameter (length), ''i.e.'' the value of <math>r</math> at which <math> \Phi_{12}(r)=0</math>
* <math> \epsilon </math> is the well depth (energy)
In reduced units:
* Density: <math> \rho^* := \rho \sigma^3 </math>, where <math> \rho := N/V </math> (number of particles <math> N </math> divided by the volume <math> V </math>)
* Temperature: <math> T^* := k_B T/\epsilon </math>, where <math> T </math> is the absolute [[temperature]] and <math> k_B </math> is the [[Boltzmann constant]]
The following is a plot of the Lennard-Jones model for the parameters <math>\epsilon/k_B \approx</math> 120 K and <math>\sigma \approx</math> 0.34 nm. See [[argon]] for different parameter sets.
[[Image:Lennard-Jones.png|400px|center]]
This figure was produced using [http://www.gnuplot.info/ gnuplot] with the command:

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

==Special points==
* <math> \Phi_{12}(\sigma) = 0 </math>
* Minimum value of <math> \Phi_{12}(r) </math> at <math> r = r_{min} </math>;
: <math> \frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ... </math>
==Critical point==
The location of the [[Critical points |critical point]] is
<ref>[http://dx.doi.org/10.1063/1.477099 J. M. Caillol " Critical-point of the Lennard-Jones fluid: A finite-size scaling study", Journal of Chemical Physics '''109''' pp. 4885-4893 (1998)]</ref>
:<math>T_c^* = 1.326 \pm 0.002</math>
at a reduced density of
:<math>\rho_c^* = 0.316 \pm 0.002</math>.

Vliegenthart and Lekkerkerker
<ref>[http://dx.doi.org/10.1063/1.481106 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)]</ref>
have suggested that the critical point is related to the [[second virial coefficient]] via the expression

:<math>B_2 \vert_{T=T_c}= -\pi \sigma^3</math>
==Triple point==
The location of the [[triple point]] as found by Mastny and de Pablo (Ref. 4) is
:<math>T_{tp}^* = 0.694</math>

:<math>\rho_{tp}^* = 0.84</math> (liquid); <math>\rho_{tp}^* = 0.96</math> (solid)

== Approximations in simulation: truncation and shifting ==
The Lennard-Jones model is often used with a cutoff radius of <math>2.5 \sigma</math>, beyond which <math> \Phi_{12}(r)</math> is set to zero. See Mastny and de Pablo
<ref> [http://dx.doi.org/10.1063/1.2753149 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)]</ref>
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:
: <math> \Phi_{12}(r) = c_{m,n} \epsilon \left[ \left( \frac{ \sigma }{r } \right)^m - \left( \frac{\sigma}{r} \right)^n
\right].
</math>
with <math> m </math> and <math> n </math> being positive integers and <math> m > n </math>.
<math> c_{m,n} </math> is chosen such that the minimum value of <math> \Phi_{12}(r) </math> being <math> \Phi_{min} = - \epsilon </math>.
Such forms are usually referred to as '''m-n Lennard-Jones Potential'''.
For example, the [[9-3 Lennard-Jones potential |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 <math>\sigma=3.73 {\mathrm {\AA}}</math> and <math>\epsilon=0.294</math> kcal/mol at a [[temperature]] of 111.06K):
[[Image:LJ_rdf.png|center|450px|Typical radial distribution function for the monatomic Lennard-Jones liquid.]]

#[http://dx.doi.org/10.1063/1.1700653 John G. Kirkwood, Victor A. Lewinson, and Berni J. Alder "Radial Distribution Functions and the Equation of State of Fluids Composed of Molecules Interacting According to the Lennard-Jones Potential", Journal of Chemical Physics '''20''' pp. 929- (1952)]

==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 theory |perturbation theories]], for example see: [[Weeks-Chandler-Anderson perturbation theory]].
==Mixtures==
*[[Binary Lennard-Jones mixtures]]
*[[Multicomponent Lennard-Jones mixtures]]
==Related models==
*[[Lennard-Jones model in 1-dimension]] (rods)
*[[Lennard-Jones disks | 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==
<references />
[[Category:Models]]

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Lennard-Jones model