softmatter:Lennard-Jones Potential

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The Lennard-Jones Potential.
The Lennard-Jones Potential.
The Lennard-Jones potential (LJ) is used to model the excluded volume interactions and van der Waals interaction attraction of neutral atoms. The commonly used 6-12 form of the potential is as follows:

U_{LJ}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]

Where ε is the well depth, σ is the characteristic diameter (typically the diameter of the smallest particle), and r is the radial separation of the two atoms.


van der Waals Attraction

In theory, the van der Waals interaction for atoms with similar ionization frequencies and where the dispersion (London) interactions are dominant is proportional to  - (\frac{\alpha_{01}\alpha_{02}}{r^{6}}) where α01 and α02 are the polarizabilities of atom 1 and atom 2 respectively. Again, this assumes that dispersion (London) forces are dominant and that there are no permanent dipoles (Keesom forces) or induced dipoles (Debye forces). In the LJ construction, the term 
is used to describe this attractive van der Waals interaction [1].

Excluded Volume Interaction

As the seperation distance between atoms decreases, the electron clouds will eventually overlap, resulting in a very strong repulsion that rapidly increases as interatomic spacing is further decreased. In the LJ construction, the term 
describes this repulsive interaction. The 12th power is used for two main reasons: it is very steep, rapidly becoming dominant as r is small and it is also a multiple of the 6th power allowing for efficient computation [1].

LJ in Simulation

Typical implementation

Typically, the LJ potential is truncated at a distance of 2.5σ. Computationally, this drastically reduces the number of force calculations needed at each time step because the potential interaction is only calculated on neighbors within a distance of 2.5σ away from a particle, rather than over the entire box. At distance of 2.5σ, the potential has acquired a value of less than 1/60th of the well depth. In dynamical simulation methods (such as Molecular Dynamics and Brownian Dynamics), it is not favorable to have a potential that has a discontinous jump at the tail. As such, the potential is typically shifted by the value at 2.5σ such that the potential has a value of zero at the tail. Mathematically this is represented by [2]:

U_{LJ}(r) = \begin{cases}
4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right] - 4\varepsilon \left[ \left(\frac{\sigma}{r_c}\right)^{12} - \left(\frac{\sigma}{r_c}\right)^{6} \right] &r<r_c \\
0 & r \geq r_c 

where rc is the cutoff. While less common, a cutoff value of rc = 2(2(1/6)) ~ 2.245 is also used, where 2(1/6) corresponds to minimum of the potential well.

Truncation and long range corrections

When one uses a truncated pair potential, it would be assumed that the contributions of the long tails would be negligible, however, in practice this is not true. By assuming that at large r values the density is equal to the average density, ρ, we can integrate and arrive at an expression that gives a corrective term for the overall potential energy[2]:

 PE^{tail} = \frac{8}{3}\pi\rho\epsilon\sigma^{3}     \left[\frac{1}{3}      \left(\frac{\sigma}{r_c}\right)^9  -\left(\frac{\sigma}{r_c}\right)^3\right]



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