Lennard-Jones model: Difference between revisions
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where: | where: | ||
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math> | * <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math> | ||
* <math> \ | * <math> \Phi_{12}(r) </math> is the [[intermolecular pair potential]] between two particles at a distance r; | ||
* <math> \sigma </math> is the diameter (length), i.e. the value of <math>r</math> at <math> \ | * <math> \sigma </math> is the diameter (length), i.e. the value of <math>r</math> at <math> \Phi_{12}(r)=0</math> ; | ||
* <math> \epsilon </math> : well depth (energy) | * <math> \epsilon </math> : well depth (energy) | ||
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==Special points== | ==Special points== | ||
* <math> \ | * <math> \Phi_{12}(\sigma) = 0 </math> | ||
* Minimum value of <math> \ | * Minimum value of <math> \Phi_{12}(r) </math> at <math> r = r_{min} </math>; | ||
: <math> \frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ... </math> | : <math> \frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ... </math> | ||
==Critical point== | ==Critical point== | ||
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== m-n Lennard-Jones potential == | == m-n Lennard-Jones potential == | ||
It is relatively common to encounter potential functions given by: | It is relatively common to encounter potential functions given by: | ||
: <math> \ | : <math> \Phi_{12}(r) = c_{m,n} \epsilon \left[ \left( \frac{ \sigma }{r } \right)^m - \left( \frac{\sigma}{r} \right)^n | ||
\right]. | \right]. | ||
</math> | </math> | ||
with <math> m </math> and <math> n </math> being positive integers and <math> m > n </math>. | with <math> m </math> and <math> n </math> being positive integers and <math> m > n </math>. | ||
<math> c_{m,n} </math> is chosen such that the minimum value of <math> \ | <math> c_{m,n} </math> is chosen such that the minimum value of <math> \Phi_{12}(r) </math> being <math> \Phi_{min} = - \epsilon </math>. | ||
Such forms are usually referred to as '''m-n Lennard-Jones Potential'''. | Such forms are usually referred to as '''m-n Lennard-Jones Potential'''. | ||
For example, the [[9-3 Lennard-Jones potential |9-3 Lennard-Jones interaction potential]] is often used to model the interaction between | For example, the [[9-3 Lennard-Jones potential |9-3 Lennard-Jones interaction potential]] is often used to model the interaction between | ||
Revision as of 17:44, 24 July 2008
The Lennard-Jones intermolecular pair potential was developed by Sir John Edward Lennard-Jones in 1931 (Ref. 1). The Lennard-Jones model consists of two 'parts'; a steep repulsive term, and smoother attractive term, representing the London dispersion forces. Apart from being an important model in its-self, the Lennard-Jones potential frequently forms one of 'building blocks' of may force fields,
Functional form
The Lennard-Jones potential is given by:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] }
where:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r := |\mathbf{r}_1 - \mathbf{r}_2|}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r) } is the intermolecular pair potential between two particles at a distance r;
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma } is the diameter (length), i.e. the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r)=0} ;
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon } : well depth (energy)
Reduced units:
- Density, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^* \equiv \rho \sigma^3 } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = N/V } (number of particles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } divided by the volume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } .)
- Temperature; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^* \equiv k_B T/\epsilon } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T } is the absolute temperature and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B } is the Boltzmann constant
The following is a plot of the Lennard-Jones model for the parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon/k_B \approx}
120 K and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma \approx}
0.34 nm. See also argon for appropriate parameter sets.
This figure was produced using gnuplot with the command:
plot (4*120*((0.34/x)**12-(0.34/x)**6))
Special points
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(\sigma) = 0 }
- Minimum value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r) } at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = r_{min} } ;
Critical point
The location of the critical point is (Caillol (Ref. 2))
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c^* = 1.326 \pm 0.002}
at a reduced density of
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_c^* = 0.316 \pm 0.002} .
Vliegenthart and Lekkerkerker (Ref. 4) have suggested that the critical point is related to the second virial coefficient via the expression
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_2 \vert_{T=T_c}= -\pi \sigma^3}
Triple point
The location of the triple point as found by Mastny and de Pablo (Ref. 3) is
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{tp}^* = 0.694}
Approximations in simulation: truncation and shifting
The Lennard-Jones model is often used with a cutoff radius of . See Mastny and de Pablo (Ref. 3) for an analysis of the effect of this cutoff on the melting line.
m-n Lennard-Jones potential
It is relatively common to encounter potential functions given by:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r) = c_{m,n} \epsilon \left[ \left( \frac{ \sigma }{r } \right)^m - \left( \frac{\sigma}{r} \right)^n \right]. }
with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } being positive integers and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m > n } . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{m,n} } is chosen such that the minimum value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r) } being Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{min} = - \epsilon } . Such forms are usually referred to as m-n Lennard-Jones Potential. For example, the 9-3 Lennard-Jones interaction potential is often used to model the interaction between the atoms/molecules of a fluid and a continuous solid wall. On the '9-3 Lennard-Jones potential' page a justification of this use is presented.
Radial distribution function
The following plot is of a typical radial distribution function for the monatomic Lennard-Jones liquid (here with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma=3.73 {\mathrm {\AA}}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon=0.294} kcal/mol at a temperature of 111.06K:
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
Related models
- Lennard-Jones sticks
- Lennard-Jones disks
- 9-3 Lennard-Jones potential
- 10-4-3 Lennard-Jones potential
- Stockmayer potential
- Mie potential
References
- J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, 43 pp. 461-482 (1931)
- J. M. Caillol " Critical-point of the Lennard-Jones fluid: A finite-size scaling study", Journal of Chemical Physics 109 pp. 4885-4893 (1998)
- Ethan A. Mastny and Juan J. de Pablo "Melting line of the Lennard-Jones system, infinite size, and full potential", Journal of Chemical Physics 127 104504 (2007)
- G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics 112 pp. 5364-5369 (2000)