Difference between revisions of "Lennard-Jones model"

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The location of the [[triple point]] as found by Mastny and  de Pablo (Ref. 3) is
 
The location of the [[triple point]] as found by Mastny and  de Pablo (Ref. 3) is
 
:<math>T_{tp}^* = 0.694</math>
 
:<math>T_{tp}^* = 0.694</math>
 +
 +
:<math>\rho_{tp}^* = 0.84</math> (liquid); <math>\rho_{tp}^* = 0.96</math> (solid)
  
 
== Approximations in simulation: truncation and shifting ==
 
== Approximations in simulation: truncation and shifting ==

Revision as of 15:49, 29 January 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 (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

 \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|
  •  \Phi_{12}(r) is the intermolecular pair potential between two particles or sites
  •  \sigma is the diameter (length), i.e. the value of r at which  \Phi_{12}(r)=0
  •  \epsilon is the well depth (energy)

In reduced units:

The following is a plot of the Lennard-Jones model for the parameters \epsilon/k_B \approx 120 K and \sigma \approx 0.34 nm. See argon for different parameter sets.

Lennard-Jones.png

This figure was produced using gnuplot with the command:

plot (4*120*((0.34/x)**12-(0.34/x)**6))

Special points

  •  \Phi_{12}(\sigma) = 0
  • Minimum value of  \Phi_{12}(r) at  r = r_{min} ;
 \frac{r_{min}}{\sigma} = 2^{1/6} \simeq   1.12246 ...

Critical point

The location of the critical point is (Caillol (Ref. 2))

T_c^* = 1.326 \pm 0.002

at a reduced density of

\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

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

T_{tp}^* = 0.694
\rho_{tp}^* = 0.84 (liquid); \rho_{tp}^* = 0.96 (solid)

Approximations in simulation: truncation and shifting

The Lennard-Jones model is often used with a cutoff radius of 2.5 \sigma. 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:

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

with  m and  n being positive integers and  m > n .  c_{m,n} is chosen such that the minimum value of  \Phi_{12}(r) being  \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 \sigma=3.73 {\mathrm {\AA}} and \epsilon=0.294 kcal/mol at a temperature of 111.06K:

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. 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)
  4. 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)