Editing 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 KBE, FRS | Sir John Edward Lennard-Jones]] 
<ref>[http://dx.doi.org/10.1098/rspa.1924.0081  John Edward Lennard-Jones "On the Determination of Molecular Fields. I. From the Variation of the Viscosity of a Gas with Temperature", 	Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character '''106''' pp. 	441-462 (1924)] &sect; 8 (ii)</ref> <ref>[http://dx.doi.org/10.1098/rspa.1924.0082 John Edward Lennard-Jones "On the Determination of Molecular Fields. II. From the Equation of State of a Gas",  Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character '''106''' pp. 463-477 (1924)] Eq. 2.05</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 itself,
the Lennard-Jones potential frequently forms one of 'building blocks' of many [[force fields]]. It is worth mentioning that the 12-6 Lennard-Jones model is not the 
most faithful representation of the potential energy surface, but rather its use is widespread due to its computational expediency.
For example, the repulsive term is maybe better described with the [[exp-6 potential]].
One of the first [[Computer simulation techniques |computer simulations]] using the Lennard-Jones model was undertaken by Rahman in 1964 <ref>[http://dx.doi.org/10.1103/PhysRev.136.A405 A. Rahman "Correlations in the Motion of Atoms in Liquid Argon", Physical Review '''136''' pp. A405–A411 (1964)]</ref> in a  study of liquid [[argon]].
== 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 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 Rowley, Nicholson and Parsonage parameter set <ref>[http://dx.doi.org/10.1016/0021-9991(75)90042-X   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)]</ref> (<math>\epsilon/k_B = </math> 119.8 K and <math>\sigma=</math> 0.3405 nm). See [[argon]] for other parameter sets.<br>
[[Image:Lennard-Jones.png|500px]]

==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>
<ref>[http://dx.doi.org/10.1063/1.3496468  L. A. Bulavin  and V. L. Kulinskii "Generalized principle of corresponding states and the scale invariant mean-field approach", Journal of Chemical Physics ''''133''' 134101 (2010)]</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 name="Mastny"> [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> is
:<math>T_{tp}^* = 0.694</math>

:<math>\rho_{tp}^* = 0.84</math> (liquid); <math>\rho_{tp}^* = 0.96</math> (solid)
==Radial distribution function==
The following plot is of a typical [[radial distribution function]] for the monatomic Lennard-Jones liquid<ref>[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)]</ref> (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.]]
==Helmholtz energy function==
An expression for the [[Helmholtz energy function]] of the [[Building up a face centered cubic lattice | face centred cubic]] solid has been given by van der Hoef <ref>[http://dx.doi.org/10.1063/1.1314342 Martin A. van der Hoef "Free energy of the Lennard-Jones solid", Journal of Chemical Physics '''113''' pp. 8142-8148 (2000)]</ref>, applicable within the density range <math>0.94 \le \rho^* \le 1.20</math> and the temperature range <math>0.1 \le T^* \le 2.0</math>. For the liquid state see the work of Johnson, Zollweg and Gubbins <ref>[http://dx.doi.org/10.1080/00268979300100411 J. Karl Johnson, John A. Zollweg and Keith E. Gubbins "The Lennard-Jones equation of state revisited", Molecular Physics '''78''' pp. 591-618 (1993)]</ref>.

==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]]''
==Zeno line==
It has been shown that the Lennard-Jones model has a straight [[zeno line]] <ref>[http://dx.doi.org/10.1021/jp802999z E. M. Apfelbaum, V. S. Vorob’ev and G. A. Martynov "Regarding the Theory of the Zeno Line", Journal of Physical Chemistry A '''112''' pp. 6042-6044 (2008)]</ref> on the [[Phase diagrams: Density-temperature plane |density−temperature plane]].

==Perturbation theory==
The Lennard-Jones model is also used in [[Perturbation theory |perturbation theories]], for example see: [[Weeks-Chandler-Anderson perturbation theory]].
== 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. Setting the well depth <math> \epsilon </math> to be 1 in the potential on arrives at <math> \Phi_{12}(r)\simeq  -0.0163</math>, i.e. at this distance the potential is at less than 2% of the well depth. For an analysis of the effect of this cutoff on the melting line see the work of Mastny and  de Pablo  <ref name="Mastny"> </ref> and of Ahmed and Sadus <ref>[http://dx.doi.org/10.1063/1.3481102  Alauddin Ahmed  and Richard J. Sadus "Effect of potential truncations and shifts on the solid-liquid phase coexistence of Lennard-Jones fluids", Journal of Chemical Physics '''133''' 124515 (2010)]</ref>. See Panagiotopoulos for critical parameters <ref>[http://dx.doi.org/10.1007/BF01458815 A. Z. Panagiotopoulos "Molecular simulation of phase coexistence: Finite-size effects and determination of critical parameters for two- and three-dimensional Lennard-Jones fluids", International Journal of Thermophysics '''15''' pp. 1057-1072 (1994)]</ref>. It has recently been suggested that a truncated and shifted force cutoff of <math>1.5 \sigma</math> can be used under certain conditions <ref>[http://dx.doi.org/10.1063/1.3558787 Søren Toxvaerd and Jeppe C. Dyre "Communication: Shifted forces in molecular dynamics", Journal of Chemical Physics '''134''' 081102 (2011)]</ref>-

== n-m Lennard-Jones potential ==
It is relatively common to encounter potential functions given by:
: <math> \Phi_{12}(r) = c_{n,m} \epsilon   \left[ \left( \frac{ \sigma }{r } \right)^n - \left( \frac{\sigma}{r} \right)^m 
\right].
</math>
with <math> n </math> and <math> m </math> being positive integers and <math> n > m </math>.
<math> c_{n,m} </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 '''n-m 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
a continuous solid wall and the atoms/molecules of a liquid.
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 <math>m</math> is fixed at 6, and <math>n</math> is free to adopt a range of integer values.
The potentials form part of the larger class of potentials known as the [[Mie potential]].
<br>
Examples:
*[[8-6 Lennard-Jones potential]]
*[[9-3 Lennard-Jones potential]]
*[[9-6 Lennard-Jones potential]]
*[[10-4-3 Lennard-Jones potential]]
*[[200-100 Lennard-Jones potential]]
*[[n-6 Lennard-Jones potential]]

==Mixtures==
*[[Binary Lennard-Jones mixtures]]
*[[Multicomponent Lennard-Jones mixtures]]
==Related models==
*[[Kihara potential]]
*[[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]]
*[[Mie potential]]
*[[Soft-core Lennard-Jones model]]
*[[Soft sphere potential]]
*[[Stockmayer potential]]

==References==
<references />
[[Category:Models]]