Editing Lennard-Jones model
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==Vapor-liquid equilibrium== | ==Vapor-liquid equilibrium== | ||
The vapor-liquid equilibrium of the Lennard-Jones potential has been studied | 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== | ||
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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 | 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>): |