Modified Lennard-Jones model - Revision history

Carl McBride: Corrected lexing error

2016-09-14T14:00:34Z

Corrected lexing error

← Older revision		Revision as of 16:00, 14 September 2016
Line 18:		Line 18:
	<math>C_1 = 0.0163169237\epsilon</math>,		<math>C_1 = 0.0163169237\epsilon</math>,
	<math>C_2 = 3136.5686 \epsilon</math>		<math>C_2 = 3136.5686 \epsilon</math>
	<math>C_3 = -68.069 \epsilon</math> <ref>Note: due to a typographical error in the original ms, in Table I <math>C_3</math> appears to be positive. </ref>.		<math>C_3 = -68.069 \epsilon</math> <ref>Note: due to a typographical error in the original ms, in Table I <math>C_3</math> appears to be positive. </ref>,
	<math>C_4 = −0.0833111261\epsilon</math>		<math>C_4 = -0.0833111261 \epsilon </math>
	and <math>C_5 = 0.746882273 \epsilon</math>.		and <math>C_5 = 0.746882273 \epsilon</math>.
	==Virial coefficients==		==Virial coefficients==

Carl McBride: /* Virial coefficients */ Added a recent publication

2014-09-26T10:02:42Z

Virial coefficients: Added a recent publication

← Older revision		Revision as of 12:02, 26 September 2014
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	and <math>C_5 = 0.746882273 \epsilon</math>.		and <math>C_5 = 0.746882273 \epsilon</math>.
	==Virial coefficients==		==Virial coefficients==
	The [[Virial equation of state\|virial coefficients]] up to the seventh order have been calculated for the range of temperatures <math>k_BT/\epsilon = 0.3-70</math> <ref>[http://dx.doi.org/10.1063/1.4882896 M. V. Ushcats "Modified Lennard-Jones model: Virial coefficients to the 7th order", Journal of Chemical Physics '''140''' 234309 (2014)]</ref>.		The [[Virial equation of state\|virial coefficients]] up to the seventh order have been calculated for the range of temperatures <math>k_BT/\epsilon = 0.3-70</math> <ref>[http://dx.doi.org/10.1063/1.4882896 M. V. Ushcats "Modified Lennard-Jones model: Virial coefficients to the 7th order", Journal of Chemical Physics '''140''' 234309 (2014)]</ref>. See also <ref>[http://dx.doi.org/10.1063/1.4895126 M. V. Ushcats "Communication: Low-temperature approximation of the virial series for the Lennard-Jones and modified Lennard-Jones models", Journal of Chemical Physics '''141''' 101103 (2014)]</ref>.

	==References==		==References==
	<references/>		<references/>

	[[category: models]]		[[category: models]]

Carl McBride: Added a recent publication (Virial coefficients)

2014-06-23T11:42:52Z

Added a recent publication (Virial coefficients)

← Older revision		Revision as of 13:42, 23 June 2014
Line 20:		Line 20:
	<math>C_3 = -68.069 \epsilon</math> <ref>Note: due to a typographical error in the original ms, in Table I <math>C_3</math> appears to be positive. </ref>.		<math>C_3 = -68.069 \epsilon</math> <ref>Note: due to a typographical error in the original ms, in Table I <math>C_3</math> appears to be positive. </ref>.
	<math>C_4 = −0.0833111261\epsilon</math>		<math>C_4 = −0.0833111261\epsilon</math>
	and <math>C_5 = 0.746882273 \epsilon</math>.		and <math>C_5 = 0.746882273 \epsilon</math>.
			==Virial coefficients==
			The [[Virial equation of state\|virial coefficients]] up to the seventh order have been calculated for the range of temperatures <math>k_BT/\epsilon = 0.3-70</math> <ref>[http://dx.doi.org/10.1063/1.4882896 M. V. Ushcats "Modified Lennard-Jones model: Virial coefficients to the 7th order", Journal of Chemical Physics '''140''' 234309 (2014)]</ref>.
	==References==		==References==
	<references/>		<references/>

	[[category: models]]		[[category: models]]

Carl McBride: Typo confirmed.

2012-11-12T14:33:04Z

Typo confirmed.

← Older revision		Revision as of 16:33, 12 November 2012
Line 18:		Line 18:
	<math>C_1 = 0.0163169237\epsilon</math>,		<math>C_1 = 0.0163169237\epsilon</math>,
	<math>C_2 = 3136.5686 \epsilon</math>		<math>C_2 = 3136.5686 \epsilon</math>
	<math>C_3 = 68.069 \epsilon</math>		<math>C_3 = -68.069 \epsilon</math> <ref>Note: due to a typographical error in the original ms, in Table I <math>C_3</math> appears to be positive. </ref>.
	<math>C_4 = −0.0833111261\epsilon</math>		<math>C_4 = −0.0833111261\epsilon</math>
	and <math>C_5 = 0.746882273 \epsilon</math>~~. Note however, that it seems that '''both''' <math>C_3</math> and <math>C_4</math> should be negative for these parameters to work~~.		and <math>C_5 = 0.746882273 \epsilon</math>.
	==References==		==References==
	<references/>		<references/>

	[[category: models]]		[[category: models]]

Carl McBride: Added note that both C3 and C4 should be negative.

2012-11-07T17:18:02Z

Added note that both C3 and C4 should be negative.

← Older revision		Revision as of 19:18, 7 November 2012
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	<math>C_4 = 0.083312 \epsilon</math>		<math>C_4 = 0.083312 \epsilon</math>
	and <math>C_5 = 0.74689\epsilon</math>. These parametrs are chosen so that the function <math>\Phi_{12}(r)</math>, as well as the first derivative, is continuous both at <math>r = 2.3\sigma</math> and <math>r = 2.5\sigma</math>.		and <math>C_5 = 0.74689\epsilon</math>. These parametrs are chosen so that the function <math>\Phi_{12}(r)</math>, as well as the first derivative, is continuous both at <math>r = 2.3\sigma</math> and <math>r = 2.5\sigma</math>.
	These parameters have recently been recalculated by Asano and Fuchizaki <ref>[http://dx.doi.org/10.1063/1.4764855 Yuta Asano and Kazuhiro Fuchizaki "Phase diagram of the modified Lennard-Jones system", Journal of Chemical Physics '''137''' 174502 (2012)]</ref>, leading to		These parameters have recently been recalculated with greater precision by Asano and Fuchizaki <ref>[http://dx.doi.org/10.1063/1.4764855 Yuta Asano and Kazuhiro Fuchizaki "Phase diagram of the modified Lennard-Jones system", Journal of Chemical Physics '''137''' 174502 (2012)]</ref>, leading to
	<math>C_1 = 0.0163169237\epsilon</math>,		<math>C_1 = 0.0163169237\epsilon</math>,
	<math>C_2 = 3136.5686 \epsilon</math>		<math>C_2 = 3136.5686 \epsilon</math>
	<math>C_3 = 68.069 \epsilon</math>		<math>C_3 = 68.069 \epsilon</math>
	<math>C_4 = −0.0833111261\epsilon</math>		<math>C_4 = −0.0833111261\epsilon</math>
	and <math>C_5 = 0.746882273 \epsilon</math>.		and <math>C_5 = 0.746882273 \epsilon</math>. Note however, that it seems that '''both''' <math>C_3</math> and <math>C_4</math> should be negative for these parameters to work.
	==References==		==References==
	<references/>		<references/>

	[[category: models]]		[[category: models]]

Carl McBride: typo

2012-11-07T16:36:05Z

typo

← Older revision		Revision as of 18:36, 7 November 2012
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	<math>C_3 = -68.069 \epsilon</math>		<math>C_3 = -68.069 \epsilon</math>
	<math>C_4 = 0.083312 \epsilon</math>		<math>C_4 = 0.083312 \epsilon</math>
	and <math>C_5 = 0.74689\epsilon</math>. ~~THese~~ parametrs are chosen so that the function <math>\Phi_{12}(r)</math>, as well as the first derivative, is continuous both at <math>r = 2.3\sigma</math> and <math>r = 2.5\sigma</math>.		and <math>C_5 = 0.74689\epsilon</math>. These parametrs are chosen so that the function <math>\Phi_{12}(r)</math>, as well as the first derivative, is continuous both at <math>r = 2.3\sigma</math> and <math>r = 2.5\sigma</math>.
	These parameters have recently been recalculated by Asano and Fuchizaki <ref>[http://dx.doi.org/10.1063/1.4764855 Yuta Asano and Kazuhiro Fuchizaki "Phase diagram of the modified Lennard-Jones system", Journal of Chemical Physics '''137''' 174502 (2012)]</ref>, leading to		These parameters have recently been recalculated by Asano and Fuchizaki <ref>[http://dx.doi.org/10.1063/1.4764855 Yuta Asano and Kazuhiro Fuchizaki "Phase diagram of the modified Lennard-Jones system", Journal of Chemical Physics '''137''' 174502 (2012)]</ref>, leading to
	<math>C_1 = 0.0163169237\epsilon</math>,		<math>C_1 = 0.0163169237\epsilon</math>,

Carl McBride: Corrected typo, the C4 term should be r/sigma

2012-11-07T16:33:18Z

Corrected typo, the C4 term should be r/sigma

← Older revision		Revision as of 18:33, 7 November 2012
Line 5:		Line 5:
	\begin{array}{ll}		\begin{array}{ll}
	4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] + C_1 & r \leq 2.3 \sigma \\		4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] + C_1 & r \leq 2.3 \sigma \\
	C_2 \left(\frac{\sigma}{r} \right)^{12} + C_3 \left(\frac{\sigma}{r} \right)^{6} + C_4 \left(\frac{\sigma~~}{r~~} \right)^{2} + C_5 & 2.3 \sigma < r < 2.5 \sigma\\		C_2 \left(\frac{\sigma}{r} \right)^{12} + C_3 \left(\frac{\sigma}{r} \right)^{6} + C_4 \left(\frac{r}{\sigma} \right)^{2} + C_5 & 2.3 \sigma < r < 2.5 \sigma\\
	0 & 2.5 \sigma \leq r		0 & 2.5 \sigma \leq r
	\end{array} \right.		\end{array} \right.

Carl McBride: Created page with "The '''modified Lennard-Jones model''' is given by (Eq. 2 [http://dx.doi.org/10.1063/1.445633 Jeremy Q. Broughton and George H. Gilmer "Molecular dynamics investigation ..."

2012-11-07T16:08:18Z

Created page with "The '''modified Lennard-Jones model''' is given by (Eq. 2 <ref>[http://dx.doi.org/10.1063/1.445633 Jeremy Q. Broughton and George H. Gilmer "Molecular dynamics investigation ..."

New page

The '''modified Lennard-Jones model''' is given by (Eq. 2 <ref>[http://dx.doi.org/10.1063/1.445633 Jeremy Q. Broughton and George H. Gilmer "Molecular dynamics investigation of the crystal–fluid interface. I. Bulk properties", Journal of Chemical Physics '''79''' pp. 5095-5104 (1983)]</ref>):

:<math>
\Phi_{12}(r) = \left\{
\begin{array}{ll}
4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] + C_1 & r \leq 2.3 \sigma \\
C_2 \left(\frac{\sigma}{r} \right)^{12} + C_3 \left(\frac{\sigma}{r} \right)^{6} + C_4 \left(\frac{\sigma}{r} \right)^{2} + C_5 & 2.3 \sigma < r < 2.5 \sigma\\
0 & 2.5 \sigma \leq r
\end{array} \right.
</math>

where <math>C_1 = 0.016132 \epsilon</math>,
<math>C_2 = 3136.6 \epsilon</math>
<math>C_3 = -68.069 \epsilon</math>
<math>C_4 = 0.083312 \epsilon</math>
and <math>C_5 = 0.74689\epsilon</math>. THese parametrs are chosen so that the function <math>\Phi_{12}(r)</math>, as well as the first derivative, is continuous both at <math>r = 2.3\sigma</math> and <math>r = 2.5\sigma</math>.
These parameters have recently been recalculated by Asano and Fuchizaki <ref>[http://dx.doi.org/10.1063/1.4764855 Yuta Asano and Kazuhiro Fuchizaki "Phase diagram of the modified Lennard-Jones system", Journal of Chemical Physics '''137''' 174502 (2012)]</ref>, leading to
<math>C_1 = 0.0163169237\epsilon</math>,
<math>C_2 = 3136.5686 \epsilon</math>
<math>C_3 = 68.069 \epsilon</math>
<math>C_4 = −0.0833111261\epsilon</math>
and <math>C_5 = 0.746882273 \epsilon</math>.
==References==
<references/>

[[category: models]]