Lennard-Jones model - Revision history

86.30.229.2: /* Approximations in simulation: truncation and shifting */

2021-07-30T22:24:00Z

Approximations in simulation: truncation and shifting

← Older revision		Revision as of 00:24, 31 July 2021
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	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)]</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.		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>. See Pártay for the ground state structure <ref>[http://dx.doi.org/10.1039/C7CP02923C Lívia B. Pártay, Christoph Ortner, Albert P. Bartók, Chris J. Pickard and Gábor Csányi "Polytypism in the ground state structure of the Lennard-Jonesium", Physical Chemistry Chemical Physics '''19''' 19369 (2017)]</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>):

77.22.248.29: bug fixes

2021-04-16T23:25:32Z

bug fixes

@@ Line 72: / Line 72: @@
 ==Vapor-liquid equilibrium==
-The vapor-liquid equilibrium of the Lennard-Jones potential has been studied 45 times <ref name="Stephan2019">[https://doi.org/10.1021/acs.jcim.9b00620  Simon Stephan, Monika Thol, Jadran Vrabec, and Hans Hasse "Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment", Journal of Chemical Information and Modeling '''59, 10''' pp. 4248–4265 (2019)]</ref>. Several of these data sets were found to have gross deviations to an entity of data sets, which is both thermodynamically consistent and in good mutual agreement. The mutual agreement of these data sets was found to be approximately <math>\pm 1%</math> for the vapor pressure, <math>\pm 0.75%</math> for the enthalpy of vaporization, <math>\pm 0.2%</math> for the saturated liquid density, and <math>\pm 1%</math> for the saturated vapor density <ref name="Stephan2019">[https://doi.org/10.1021/acs.jcim.9b00620  Simon Stephan, Monika Thol, Jadran Vrabec, and Hans Hasse "Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment", Journal of Chemical Information and Modeling '''59, 10''' pp. 4248–4265 (2019)]</ref>.
+The vapor-liquid equilibrium of the Lennard-Jones potential has been studied more than 45 times in the literature. Several of these data sets were found to have gross deviations to an entity of data sets, which is both thermodynamically consistent and in good mutual agreement <ref name="Stephan2019">[https://doi.org/10.1021/acs.jcim.9b00620  Simon Stephan, Monika Thol, Jadran Vrabec, and Hans Hasse "Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment", Journal of Chemical Information and Modeling '''59, 10''' pp. 4248–4265 (2019)]</ref>. The mutual agreement of these data sets was found to be approximately <math>\pm 1%</math> for the vapor pressure, <math>\pm 0.75%</math> for the enthalpy of vaporization, <math>\pm 0.2%</math> for the saturated liquid density, and <math>\pm 1%</math> for the saturated vapor density <ref name="Stephan2019">[https://doi.org/10.1021/acs.jcim.9b00620  Simon Stephan, Monika Thol, Jadran Vrabec, and Hans Hasse "Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment", Journal of Chemical Information and Modeling '''59, 10''' pp. 4248–4265 (2019)]</ref>.
 ==Melting line==

77.22.248.29 at 22:54, 16 April 2021

2021-04-16T22:54:03Z

← Older revision		Revision as of 00:54, 17 April 2021
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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)]</ref>. Apart from being an important model in itself,		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>. The Lennard-Jones potential is probably the most extensively studied model potential today <ref name="Stephan2019">[https://doi.org/10.1021/acs.jcim.9b00620 Simon Stephan, Monika Thol, Jadran Vrabec, and Hans Hasse "Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment", Journal of Chemical Information and Modeling '''59, 10''' pp. 4248–4265 (2019)]</ref>. 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		the Lennard-Jones potential frequently forms one of 'building blocks' of many [[force fields]] and molecular-based equation of state models <ref name="Stephan">[https://doi.org/10.1016/j.fluid.2020.112772 Simon Stephan, Jens Staubach, Hans Hasse "Review and comparison of equations of state for the Lennard-Jones fluid", Fluid Phase Equilibria '''523''' pp. 112772 (2020)]</ref>. 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]]. Thermophysical property data sampled by molecular simulations since then until 2019 has been compiled, digitalized and assessed by Stephan et al <ref name="Stephan2019">[https://doi.org/10.1021/acs.jcim.9b00620 Simon Stephan, Monika Thol, Jadran Vrabec, and Hans Hasse "Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment", Journal of Chemical Information and Modeling '''59, 10''' pp. 4248–4265 (2019)]</ref>. This database contains about 35,000 datapoints.

	== 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]] for the untruncated potential is		The location of the [[Critical points \|critical point]] for the untruncated potential has been assessed to be at <ref name="Stephan2019">[https://doi.org/10.1021/acs.jcim.9b00620 Simon Stephan, Monika Thol, Jadran Vrabec, and Hans Hasse "Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment", Journal of Chemical Information and Modeling '''59, 10''' pp. 4248–4265 (2019)]</ref>
	<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^* = 1.321 \pm 0.007 </math>
	:<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.~~002~~</math>		:<math>\rho_c^* = 0.316 \pm 0.005 </math>
	~~The~~ critical [[pressure]] ~~is <ref>[http://dx.doi.org/10.1063/1.477787 Jeffrey J. Potoff and Athanassios Z. Panagiotopoulos "Critical point and phase behavior~~ of ~~the pure fluid and a Lennard-Jones mixture", Journal of Chemical Physics '''109''' 10914 (1998)]</ref>~~		and the critical [[pressure]] of
	:<math>p_c^* = 0.~~1279~~ \pm 0.~~0006~~</math>		:<math>p_c^* = 0.129 \pm 0.005. </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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	==Helmholtz energy function==		==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>.		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>.

			==Vapor-liquid equilibrium==
			The vapor-liquid equilibrium of the Lennard-Jones potential has been studied 45 times <ref name="Stephan2019">[https://doi.org/10.1021/acs.jcim.9b00620 Simon Stephan, Monika Thol, Jadran Vrabec, and Hans Hasse "Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment", Journal of Chemical Information and Modeling '''59, 10''' pp. 4248–4265 (2019)]</ref>. Several of these data sets were found to have gross deviations to an entity of data sets, which is both thermodynamically consistent and in good mutual agreement. The mutual agreement of these data sets was found to be approximately <math>\pm 1%</math> for the vapor pressure, <math>\pm 0.75%</math> for the enthalpy of vaporization, <math>\pm 0.2%</math> for the saturated liquid density, and <math>\pm 1%</math> for the saturated vapor density <ref name="Stephan2019">[https://doi.org/10.1021/acs.jcim.9b00620 Simon Stephan, Monika Thol, Jadran Vrabec, and Hans Hasse "Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment", Journal of Chemical Information and Modeling '''59, 10''' pp. 4248–4265 (2019)]</ref>.

			:<math>\ln p^s = n_1 T + \frac{n_2}{T} + \frac{n_3}{T^{n_4}} </math> with <math> n_i =[ 1.156551, -4.431519, -0.423028, 2.638743 ] </math>
			:<math>\Big(\frac{\rho'}{\rho_\mathrm{c}}\Big) = 1 + \sum_{i=1}^{5} n_i\, \Big(1 - \frac{T}{T_\mathrm{c}} \Big)^{t_i} </math>
			:<math>\ln \Big(\frac{\rho''}{\rho_\mathrm{c}}\Big) = \sum_{i=1}^{5} n_i\, \Big(1 - \frac{T}{T_\mathrm{c}} \Big)^{t_i} </math>
			:<math>\Delta h_v = \sum_{i=1}^{4} n_i\, (T_\mathrm{c} - T)^{t_i} </math>

			The numerical values of the parameters ni and $t_i$ are

			<math> {} </math>
			<math> {} </math>
			<math> {} </math>

			1 1.341700 0.327140 −8.135822 1.651685 6.456728 0.411342
			2 2.075332 0.958759 −102.919110 43.469214 2.700099 0.460416
			3 −2.123475 1.645654 −3.037979 0.462877 −3.073573 2.350953
			4 0.328998 17.000001 −44.381841 11.500462 3.149052 5.017010
			5 1.386131 2.400858 −34.55892948 5.394370


	==Melting line==		==Melting line==

Carl McBride: Added a Viscosity section

2018-03-12T13:24:51Z

Added a Viscosity section

← Older revision		Revision as of 15:24, 12 March 2018
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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]].

			==Viscosity==
			[[Viscosity]] <ref>[https://doi.org/10.1063/1.5018483 Kai-Min Tu, Kang Kim, and Nobuyuki Matubayasi "Spatial-decomposition analysis of viscosity with application to Lennard-Jones fluid", Journal of Chemical Physics '''148''' 094501 (2018)]</ref>.

	==Perturbation theory==		==Perturbation theory==

Carl McBride: Corrected Cite error

2017-12-11T15:27:54Z

Corrected Cite error

← Older revision		Revision as of 17:27, 11 December 2017
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	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)]</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.		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>):

Carl McBride: /* Melting line */ Added mention of a recent publication

2017-10-16T12:13:14Z

Melting line: Added mention of a recent publication

← Older revision		Revision as of 14:13, 16 October 2017
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	:<math>\rho_{\mathrm {liquid}} = \beta^{-1/4} \left[ 0.90735 - 0.27120 \beta + 0.91784 \beta^2 -1.16270\beta^3 + 0.68012 \beta^4 -0.15284 \beta^5\right]</math>		:<math>\rho_{\mathrm {liquid}} = \beta^{-1/4} \left[ 0.90735 - 0.27120 \beta + 0.91784 \beta^2 -1.16270\beta^3 + 0.68012 \beta^4 -0.15284 \beta^5\right]</math>


			A study has been performed of the solid-fluid equilibria, and behavior in the high density region <ref>[http://dx.doi.org/10.1063/1.4990667 Andreas Köster, Peter Mausbach, and Jadran Vrabec "Premelting, solid-fluid equilibria, and thermodynamic properties in the high density region based on the Lennard-Jones potential", Journal of Chemical Physics '''147''' 144502 (2017)]</ref>.

	==Zeno line==		==Zeno line==

Carl McBride: Slight re-ordering of sections

2016-04-12T15:25:29Z

Slight re-ordering of sections

← Older revision		Revision as of 17:25, 12 April 2016
Line 90:		Line 90:

	:<math>\rho_{\mathrm {liquid}} = \beta^{-1/4} \left[ 0.90735 - 0.27120 \beta + 0.91784 \beta^2 -1.16270\beta^3 + 0.68012 \beta^4 -0.15284 \beta^5\right]</math>		:<math>\rho_{\mathrm {liquid}} = \beta^{-1/4} \left[ 0.90735 - 0.27120 \beta + 0.91784 \beta^2 -1.16270\beta^3 + 0.68012 \beta^4 -0.15284 \beta^5\right]</math>


	~~==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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	*[[n-6 Lennard-Jones potential]]		*[[n-6 Lennard-Jones potential]]

	==~~Mixtures~~==		==Equation of state==
	*[[~~Binary~~ Lennard-Jones ~~mixtures~~]]		:''Main article: [[Lennard-Jones equation of state]]''
	*[[~~Multicomponent~~ Lennard-Jones ~~mixtures~~]]
			==Virial coefficients==
			:''Main article: [[Lennard-Jones model: virial coefficients]]''

			==Phase diagram==
			:''Main article: [[Phase diagram of the Lennard-Jones model]]''
	==Related models==		==Related models==
	*[[Kihara potential]]		*[[Kihara potential]]
Line 154:		Line 148:
	*[[Soft sphere potential]]		*[[Soft sphere potential]]
	*[[Stockmayer potential]]		*[[Stockmayer potential]]
			;Mixtures
			*[[Binary Lennard-Jones mixtures]]
			*[[Multicomponent Lennard-Jones mixtures]]
	==References==		==References==
	<references />		<references />
	[[Category:Models]]		[[Category:Models]]

Carl McBride: /* Critical point */ Added recent truncated data

2016-04-12T11:01:10Z

Critical point: Added recent truncated data

← Older revision		Revision as of 13:01, 12 April 2016
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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>
			====Truncated at <math>2.5 \sigma</math>====
			For the potential truncated at <math>2.5 \sigma</math> <ref>[http://dx.doi.org/10.1063/1.4944926 Ernesto S. Loscar, C. Gastón Ferrara1, and Tomás S. Grigera "Spinodals and critical point using short-time dynamics for a simple model of liquid", Journal of Chemical Physics '''144''' 134501 (2016)]</ref>
			:<math>T_c^* = 1.1875 (15)</math>
			:<math>p_c^* = 0.1105 (15)</math>

	==Triple point==		==Triple point==

Carl McBride: /* Critical point / added p_c^

2016-04-12T10:49:13Z

Critical point: added p_c^*

← Older revision		Revision as of 12:49, 12 April 2016
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	[[Image:Lennard-Jones.png\|500px]]		[[Image:Lennard-Jones.png\|500px]]
	==Critical point==		==Critical point==
	The location of the [[Critical points \|critical point]] is		The location of the [[Critical points \|critical point]] for the untruncated potential 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>		<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>		:<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 0.002</math>		:<math>\rho_c^* = 0.316 \pm 0.002</math>
			The critical [[pressure]] is <ref>[http://dx.doi.org/10.1063/1.477787 Jeffrey J. Potoff and Athanassios Z. Panagiotopoulos "Critical point and phase behavior of the pure fluid and a Lennard-Jones mixture", Journal of Chemical Physics '''109''' 10914 (1998)]</ref>
			:<math>p_c^* = 0.1279 \pm 0.0006</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>

Carl McBride: Added the Melting line section to this page

2015-12-22T14:37:48Z

Added the Melting line section to this page

@@ Line 64: / Line 64: @@
 ==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==