Lennard-Jones model: Difference between revisions

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The '''Lennard-Jones''' [[intermolecular pair potential]] is a special case of the [[Mie potential]] and takes its name from  [[ Sir John Edward Lennard-Jones KBE, FRS | Sir John Edward Lennard-Jones]] (Ref. 1). The Lennard-Jones [[models |model]] consists of two 'parts'; a steep repulsive term, and
The '''Lennard-Jones''' [[intermolecular pair potential]] is a special case of the [[Mie potential]] and takes its name from  [[ Sir John Edward Lennard-Jones KBE, FRS | Sir John Edward Lennard-Jones]]  
<ref>[http://dx.doi.org/10.1088/0959-5309/43/5/301 J. E. Lennard-Jones, "Cohesion",  Proceedings of the Physical Society, '''43''' pp. 461-482 (1931)] </ref>.
The Lennard-Jones [[models |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,
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]],
the Lennard-Jones potential frequently forms one of 'building blocks' of may [[force fields]],
Line 26: Line 28:
: <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==
The location of the [[Critical points |critical point]] is (Caillol (Ref. 2))
The location of the [[Critical points |critical point]] 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>
:<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>.


Vliegenthart and Lekkerkerker (Ref. 4) have suggested that the critical point is related to the [[second virial coefficient]] via the expression  
Vliegenthart and Lekkerkerker
<ref>[http://dx.doi.org/10.1063/1.481106 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)]</ref>
have suggested that the critical point is related to the [[second virial coefficient]] via the expression  


:<math>B_2 \vert_{T=T_c}= -\pi \sigma^3</math>
:<math>B_2 \vert_{T=T_c}= -\pi \sigma^3</math>
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== 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>. See Mastny and  de Pablo (Ref. 3)
The Lennard-Jones model is often used with a cutoff radius of <math>2.5 \sigma</math>. See Mastny and  de Pablo
<ref> [http://dx.doi.org/10.1063/1.2753149    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)]</ref>
for an analysis of the effect of this cutoff on the melting line.
for an analysis of the effect of this cutoff on the melting line.
== 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:
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*[[Stockmayer potential]]
*[[Stockmayer potential]]
*[[Mie potential]]
*[[Mie potential]]
==References==  
==References==
#[http://dx.doi.org/10.1088/0959-5309/43/5/301 J. E. Lennard-Jones, "Cohesion",  Proceedings of the Physical Society, '''43''' pp. 461-482 (1931)]
<references />
#[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)]
#[http://dx.doi.org/10.1063/1.2753149    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)]
#[http://dx.doi.org/10.1063/1.481106 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)]
[[Category:Models]]
[[Category:Models]]

Revision as of 10:43, 10 March 2009

The Lennard-Jones intermolecular pair potential is a special case of the Mie potential and takes its name from Sir John Edward Lennard-Jones [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

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 or sites
  • 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 which 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 } is the well depth (energy)

In 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^* := \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^* := 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 argon for different 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} } ;
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 \frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ... }

Critical point

The location of the critical point is [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 [3] 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}
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_{tp}^* = 0.84} (liquid); 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_{tp}^* = 0.96} (solid)

Approximations in simulation: truncation and shifting

The Lennard-Jones model is often used with a cutoff radius 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 2.5 \sigma} . See Mastny and de Pablo [4] 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:

Typical radial distribution function for the monatomic Lennard-Jones liquid.
Typical radial distribution function for the monatomic Lennard-Jones liquid.
  1. John G. Kirkwood, Victor A. Lewinson, and Berni J. Alder "Radial Distribution Functions and the Equation of State of Fluids Composed of Molecules Interacting According to the Lennard-Jones Potential", Journal of Chemical Physics 20 pp. 929- (1952)

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

Mixtures

Related models

References

  1. ↑ J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, 43 pp. 461-482 (1931)
  2. ↑ J. M. Caillol " Critical-point of the Lennard-Jones fluid: A finite-size scaling study", Journal of Chemical Physics 109 pp. 4885-4893 (1998)
  3. ↑ 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)
  4. ↑ 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)
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