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<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 <ref>[http://dx.doi.org/10.1007/BF01421741 F. London "Zur Theorie und Systematik der Molekularkräfte", Zeitschrift für Physik A Hadrons and Nuclei '''63''' pp. 245-279 (1930 | smoother attractive term, representing the London dispersion forces <ref>[http://dx.doi.org/10.1007/BF01421741 F. London "Zur Theorie und Systematik der Molekularkräfte", Zeitschrift für Physik A Hadrons and Nuclei '''63''' pp. 245-279 (1930)]</ref>. 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 Wood and Parker in 1957 <ref>[http://dx.doi.org/10.1063/1.1743822 W. W. Wood and F. R. Parker "Monte Carlo Equation of State of Molecules Interacting with the Lennard‐Jones Potential. I. A Supercritical Isotherm at about Twice the Critical Temperature", Journal of Chemical Physics '''27''' pp. 720- (1957)]</ref> in a study of liquid [[argon]] | One of the first [[Computer simulation techniques |computer simulations]] using the Lennard-Jones model was undertaken by Wood and Parker in 1957 <ref>[http://dx.doi.org/10.1063/1.1743822 W. W. Wood and F. R. Parker "Monte Carlo Equation of State of Molecules Interacting with the Lennard‐Jones Potential. I. A Supercritical Isotherm at about Twice the Critical Temperature", Journal of Chemical Physics '''27''' pp. 720- (1957)]</ref> in a study of liquid [[argon]]. | ||
== Functional form == | == Functional form == | ||
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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]] | ||
==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> | ||
The critical [[compressibility factor]] is given by <ref>[http://dx.doi.org/10.1063/1.4829837 V. L. Kulinskii "The critical compressibility factor of fluids from the global isomorphism approach", Journal of Chemical Physics '''139''' 184119 (2013)]</ref> | The critical [[compressibility factor]] is given by <ref>[http://dx.doi.org/10.1063/1.4829837 V. L. Kulinskii "The critical compressibility factor of fluids from the global isomorphism approach", Journal of Chemical Physics '''139''' 184119 (2013)]</ref> | ||
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:<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== | ||
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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>. | 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== | ==Zeno line== | ||
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==Widom line== | ==Widom line== | ||
It has been shown that the Lennard-Jones model has a [[Widom line]] <ref>[http://dx.doi.org/10.1021/jp2039898 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and E. N. Tsiok "Widom Line for the Liquid–Gas Transition in Lennard-Jones System", Journal of Physical Chemistry B Article ASAP (2011)]</ref> on the [[Phase diagrams: Pressure-temperature plane | pressure-temperature plane]]. | It has been shown that the Lennard-Jones model has a [[Widom line]] <ref>[http://dx.doi.org/10.1021/jp2039898 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and E. N. Tsiok "Widom Line for the Liquid–Gas Transition in Lennard-Jones System", Journal of Physical Chemistry B Article ASAP (2011)]</ref> on the [[Phase diagrams: Pressure-temperature plane | pressure-temperature plane]]. | ||
==Perturbation theory== | ==Perturbation theory== | ||
The Lennard-Jones model is also used in [[Perturbation theory |perturbation theories]], for example see: [[Weeks-Chandler-Andersen perturbation theory]]. | The Lennard-Jones model is also used in [[Perturbation theory |perturbation theories]], for example see: [[Weeks-Chandler-Andersen perturbation theory]]. | ||
== 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 | 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>. In order to avoid any discontinuity, a piecewise continuous version, known as the [[modified Lennard-Jones model]], was developed. | ||
== Cutoff Lennard-Jones potential== | == Cutoff Lennard-Jones potential== | ||
The cutoff Lennard-Jones potential is given by (Eq. 2 in <ref>[http://dx.doi.org/10.1103/PhysRevA.8.1504 Spotswood D. Stoddard and Joseph Ford "Numerical Experiments on the Stochastic Behavior of a Lennard-Jones Gas System", Physical Review A '''8''' pp. 1504-1512 (1973)]</ref>): | The cutoff Lennard-Jones potential is given by (Eq. 2 in <ref>[http://dx.doi.org/10.1103/PhysRevA.8.1504 Spotswood D. Stoddard and Joseph Ford "Numerical Experiments on the Stochastic Behavior of a Lennard-Jones Gas System", Physical Review A '''8''' pp. 1504-1512 (1973)]</ref>): | ||
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*[[n-6 Lennard-Jones potential]] | *[[n-6 Lennard-Jones potential]] | ||
== | ==Mixtures== | ||
*[[Binary Lennard-Jones mixtures]] | |||
*[[Multicomponent Lennard-Jones mixtures]] | |||
==Related models== | ==Related models== | ||
*[[Kihara potential]] | *[[Kihara potential]] | ||
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*[[Soft sphere potential]] | *[[Soft sphere potential]] | ||
*[[Stockmayer potential]] | *[[Stockmayer potential]] | ||
==References== | ==References== | ||
<references /> | <references /> | ||
[[Category:Models]] | [[Category:Models]] |