Modified Lennard-Jones model: Difference between revisions
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Carl McBride (talk | contribs) (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 ...") |
Carl McBride (talk | contribs) m (Corrected lexing error) |
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\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 | 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. | ||
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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>. | 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> <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 = | <math>C_4 = -0.0833111261 \epsilon </math> | ||
and <math>C_5 = 0.746882273 \epsilon</math>. | and <math>C_5 = 0.746882273 \epsilon</math>. | ||
==References== | ==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>. 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/> | ||
[[category: models]] | [[category: models]] | ||
Latest revision as of 16:00, 14 September 2016
The modified Lennard-Jones model is given by (Eq. 2 [1]):
![{\displaystyle \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 {r}{\sigma }}\right)^{2}+C_{5}&2.3\sigma <r<2.5\sigma \\0&2.5\sigma \leq r\end{array}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acfc3386bc36e4115e126d6104de0d86d1f7e714)
where , and . These parametrs are chosen so that the function , as well as the first derivative, is continuous both at and . These parameters have recently been recalculated with greater precision by Asano and Fuchizaki [2], leading to , [3], and .
Virial coefficients[edit]
The virial coefficients up to the seventh order have been calculated for the range of temperatures [4]. See also [5].
References[edit]
- ↑ 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)
- ↑ Yuta Asano and Kazuhiro Fuchizaki "Phase diagram of the modified Lennard-Jones system", Journal of Chemical Physics 137 174502 (2012)
- ↑ Note: due to a typographical error in the original ms, in Table I appears to be positive.
- ↑ M. V. Ushcats "Modified Lennard-Jones model: Virial coefficients to the 7th order", Journal of Chemical Physics 140 234309 (2014)
- ↑ 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)
