Lennard-Jones model

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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 ^[1] ^[2] 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 itself, the Lennard-Jones potential frequently forms one of 'building blocks' of many force fields. It is worth mentioning that the 12-6 Lennard-Jones model is not the most faithful representation of the potential energy surface, but rather its use is widespread due to its computational expediency. One of the first computer simulations using the Lennard-Jones model was undertaken by Rahman in 1964 ^[3] in a study of liquid argon.

1 Functional form
2 Special points
3 Critical point
4 Triple point
5 Approximations in simulation: truncation and shifting
6 n-m Lennard-Jones potential
- 6.1 See also
7 Radial distribution function
8 Equation of state
9 Virial coefficients
10 Phase diagram
11 Perturbation theory
12 Mixtures
13 Related models
14 References

[edit] Functional form

The Lennard-Jones potential is given by

$\Phi_{12}(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right]$

where

$r := |\mathbf{r}_1 - \mathbf{r}_2|$
$Φ 12 (r)$ is the intermolecular pair potential between two particles or sites
$σ$ is the value of $r$ at which $Φ 12 (r) = 0$
$ε$ is the well depth (energy)

In reduced units:

Density: $ρ * : = ρσ 3$ , where $ρ: = N / V$ (number of particles $N$ divided by the volume $V$ )
Temperature: $T * : = k B T / ε$ , where $T$ is the absolute temperature and $k B$ is the Boltzmann constant

The following is a plot of the Lennard-Jones model for the Rowley, Nicholson and Parsonage parameter set ^[4] ( $ε / k B =$ 119.8 K and $σ =$ 0.3405 nm). See argon for other parameter sets.

[edit] Special points

$Φ 12 (σ) = 0$
Minimum value of $Φ 12 (r)$ at $r = r m i n$ ;

$\frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ...$

[edit] Critical point

The location of the critical point is ^[5]

$T_c^* = 1.326 \pm 0.002$

at a reduced density of

$\rho_c^* = 0.316 \pm 0.002$ .

Vliegenthart and Lekkerkerker ^[6] have suggested that the critical point is related to the second virial coefficient via the expression

$B_2 \vert_{T=T_c}= -\pi \sigma^3$

[edit] Triple point

The location of the triple point as found by Mastny and de Pablo ^[7] is

$T_{tp}^* = 0.694$

$\rho_{tp}^* = 0.84$ (liquid); $\rho_{tp}^* = 0.96$ (solid)

[edit] Approximations in simulation: truncation and shifting

The Lennard-Jones model is often used with a cutoff radius of $2.5σ$ , beyond which $Φ 12 (r)$ is set to zero. Setting the well depth $ε$ to be 1 in the potential on arrives at $\Phi_{12}(r)\simeq -0.0163$ , i.e. at this distance the potential is at less than 2% of the well depth. See Mastny and de Pablo ^[7] for an analysis of the effect of this cutoff on the melting line. See Panagiotopoulos for critical parameters ^[8].

[edit] n-m Lennard-Jones potential

It is relatively common to encounter potential functions given by:

$\Phi_{12}(r) = c_{n,m} \epsilon \left[ \left( \frac{ \sigma }{r } \right)^n - \left( \frac{\sigma}{r} \right)^m \right].$

with $n$ and $m$ being positive integers and $n > m$ . $c n, m$ is chosen such that the minimum value of $Φ 12 (r)$ being $Φ m i n = - ε$ . Such forms are usually referred to as n-m 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. Another example is the n-6 Lennard-Jones potential, where $m$ is fixed at 6, and $n$ is free to adopt a range of integer values. The potentials form part of the larger class of potentials known as the Mie potential.

[edit] See also

[edit] Radial distribution function

The following plot is of a typical radial distribution function for the monatomic Lennard-Jones liquid^[9] (here with $\sigma=3.73 {\mathrm {\AA}}$ and $ε = 0.294$ kcal/mol at a temperature of 111.06K):

[edit] Equation of state

Main article: Lennard-Jones equation of state

[edit] Virial coefficients

Main article: Lennard-Jones model: virial coefficients

[edit] Phase diagram

Main article: Phase diagram of the Lennard-Jones model

[edit] Perturbation theory

The Lennard-Jones model is also used in perturbation theories, for example see: Weeks-Chandler-Anderson perturbation theory.

[edit] Mixtures

[edit] Related models

[edit] References

[0] John Edward Lennard-Jones "On the Determination of Molecular Fields. I. From the Variation of the Viscosity of a Gas with Temperature", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 106 pp. 441-462 (1924) § 8 (ii)

[1] John Edward Lennard-Jones "On the Determination of Molecular Fields. II. From the Equation of State of a Gas", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 106 pp. 463-477 (1924) Eq. 2.05

[2] A. Rahman "Correlations in the Motion of Atoms in Liquid Argon", Physical Review 136 pp. A405–A411 (1964)

[3] L. A. Rowley, D. Nicholson and N. G. Parsonage "Monte Carlo grand canonical ensemble calculation in a gas-liquid transition region for 12-6 Argon", Journal of Computational Physics 17 pp. 401-414 (1975)

[4] J. M. Caillol " Critical-point of the Lennard-Jones fluid: A finite-size scaling study", Journal of Chemical Physics 109 pp. 4885-4893 (1998)

[5] 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)

[Mastny-6] 7.0 ^7.1 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)

[7] 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)

[8] 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)

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Lennard-Jones model

From SklogWiki

Contents

[edit] Functional form

[edit] Special points

[edit] Critical point

[edit] Triple point

[edit] Approximations in simulation: truncation and shifting

[edit] n-m Lennard-Jones potential

[edit] See also

[edit] Radial distribution function

[edit] Equation of state

[edit] Virial coefficients

[edit] Phase diagram

[edit] Perturbation theory

[edit] Mixtures

[edit] Related models

[edit] References

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