Editing Lennard-Jones model
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 1: | Line 1: | ||
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]] | 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)] § 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> | <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)] § 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 | The Lennard-Jones [[models |model]] consists of two 'parts'; a steep repulsive term, and | ||
smoother attractive term, representing the London dispersion forces | 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]] | 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. | 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]]. | 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 | 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 == | == Functional form == | ||
The Lennard-Jones potential is given by | 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> | :<math> \Phi_{12}(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] </math> | ||
where | where | ||
Line 22: | Line 17: | ||
* <math> \sigma </math> is the value of <math>r</math> at which <math> \Phi_{12}(r)=0</math> | * <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) | * <math> \epsilon </math> is the well depth (energy) | ||
In reduced units: | In reduced units: | ||
* Density: <math> \rho^* := \rho \sigma^3 </math> | * 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>) | ||
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]] | ||
* 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> | 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]] | [[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== | ==Critical point== | ||
The location of the [[Critical points |critical point]] | The location of the [[Critical points |critical point]] is | ||
:<math>T_c^* = 1. | <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 | at a reduced density of | ||
:<math>\rho_c^* = 0.316 \pm 0. | :<math>\rho_c^* = 0.316 \pm 0.002</math>. | ||
Vliegenthart and Lekkerkerker | Vliegenthart and Lekkerkerker | ||
Line 51: | Line 40: | ||
:<math>B_2 \vert_{T=T_c}= -\pi \sigma^3</math> | :<math>B_2 \vert_{T=T_c}= -\pi \sigma^3</math> | ||
==Triple point== | ==Triple point== | ||
Line 60: | Line 45: | ||
:<math>T_{tp}^* = 0.694</math> | :<math>T_{tp}^* = 0.694</math> | ||
:<math>\rho_{tp}^* = 0.84</math> (liquid); | :<math>\rho_{tp}^* = 0.84</math> (liquid); <math>\rho_{tp}^* = 0.96</math> (solid) | ||
== Approximations in simulation: truncation and shifting == | == 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> | 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>. | ||
== n-m Lennard-Jones potential == | == n-m Lennard-Jones potential == | ||
Line 132: | Line 59: | ||
Such forms are usually referred to as '''n-m Lennard-Jones Potential'''. | 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 | 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. Another example is the [[n-6 Lennard-Jones potential]], | 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. | 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]]. | The potentials form part of the larger class of potentials known as the [[Mie potential]]. | ||
====See also==== | |||
*[[8-6 Lennard-Jones potential]] | *[[8-6 Lennard-Jones potential]] | ||
*[[9-3 Lennard-Jones potential]] | *[[9-3 Lennard-Jones potential]] | ||
*[[9-6 Lennard-Jones potential]] | *[[9-6 Lennard-Jones potential]] | ||
*[[10-4-3 Lennard-Jones potential]] | *[[10-4-3 Lennard-Jones potential]] | ||
*[[n-6 Lennard-Jones potential]] | *[[n-6 Lennard-Jones potential]] | ||
==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.]] | |||
==Equation of state== | ==Equation of state== | ||
:''Main article: [[Lennard-Jones equation of state]]'' | :''Main article: [[Lennard-Jones equation of state]]'' | ||
==Virial coefficients== | ==Virial coefficients== | ||
:''Main article: [[Lennard-Jones model: virial coefficients]]'' | :''Main article: [[Lennard-Jones model: virial coefficients]]'' | ||
==Phase diagram== | ==Phase diagram== | ||
:''Main article: [[Phase diagram of the Lennard-Jones model]]'' | :''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== | ==Related models== | ||
*[[Kihara potential]] | *[[Kihara potential]] | ||
Line 163: | Line 94: | ||
*[[Soft sphere potential]] | *[[Soft sphere potential]] | ||
*[[Stockmayer potential]] | *[[Stockmayer potential]] | ||
==References== | ==References== | ||
<references /> | <references /> | ||
[[Category:Models]] | [[Category:Models]] |