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 (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,

[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 at a distance r;
$σ$ is the diameter (length), i.e. the value of $r$ at $Φ 12 (r) = 0$ ;
$ε$ : well depth (energy)

Reduced units:

Density, $\rho^* \equiv \rho \sigma^3$ , where $ρ = N / V$ (number of particles $N$ divided by the volume $V$ .)
Temperature; $T^* \equiv k_B T/\epsilon$ , 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 parameters $\epsilon/k_B \approx$ 120 K and $\sigma \approx$ 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))

[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 (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

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

[edit] Triple point

The location of the triple point as found by Mastny and de Pablo (Ref. 3) is

$T_{tp}^* = 0.694$

[edit] Approximations in simulation: truncation and shifting

The Lennard-Jones model is often used with a cutoff radius of $2.5σ$ . See Mastny and de Pablo (Ref. 3) for an analysis of the effect of this cutoff on the melting line.

[edit] 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 $Φ 12 (r)$ being $Φ m i n = - ε$ . 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.

[edit] 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 $ε = 0.294$ kcal/mol at a temperature of 111.06K: