# softmatter:Lennard-Jones Potential

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.

## Contents |

## 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 where α_{01} and α_{02} 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]:

where *r _{c}* is the cutoff. While less common, a cutoff value of

*r*= 2(2

_{c}^{(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]:

## Examples

- Radial Distribution Function for a 500 particle Lennard-Jones gas simulated with Molecular dynamics at a number density = 0.01, T=1.0.

- Radial Distribution Function for a 500 particle Lennard-Jones fluid simulated with Molecular dynamics at a number density = 0.5 and Temperature = 1.0.

- Radial Distribution Function for a 2000 particle Lennard-Jones crystal simulated with Molecular dynamics at a number density = 1.0 and Temperature = 0.5.

- Mean squared displacement of 1000 particle Lennard-Jones fluid simulated using Molecular dynamics with number density = 0.85 and Temperature =2.0.

## References

- [1] The Structure and Rheology of Complex Fluids
- [2] Understanding molecular simulation : from algorithms to applications

## Additional Sources

- van der Waals interactions are addressed in depth in chapter 2.3 of the first edition of The Structure and Rheology of Complex Fluids starting on page 78.
- Excluded volume interaction are addressed in chapter 2.2 of the first edition of The Structure and Rheology of Complex Fluids starting on page 61.
- van der Waals interaction and the development of semi-emperical expressions can be found in chapter 1 of the second edition of Intermolecular and Surface Forces starting on page 8.
- The Lennard-Jones potential and 2-D phase diagram [1].
- Potential truncation and long range corrections are discussed in detail starting on page 35 of the second edition of Understanding molecular simulation : from algorithms to applications