MIT - Massachusetts Institute of Technology



22.091 Fall 2008

Understanding Lennard Jones Potential with Matlab

The Lennard Jones (LJ) potential functions are often used in the Molecular Dynamics simulation of materials. The functions normally have two parameters Epsilon [pic] and Sigma [pic] with the following 6-12 form:

[pic] --------- (1)

where [pic]are the well-depth and distance of closest approach, respectively. For the noble gas element argon, they have the values, [pic]=120oK and [pic]=3.405A. A sketch of this potential is shown in Fig. 2. It is also useful to plot the force [pic] and note that F vanishes at the minimum of V(r). At separations less than this minimum the force is repulsive, whereas at greater separations the force is attractive. The feature of repulsive force at close range and attractive force far away is the essence of all interatomic potentials. The repulsion means that any such potential will give the system of atoms a certain size, equivalently, a certain excluded volume or effectively density, while attraction means that the system has a certain cohesion (glue) that holds it together. These are the two fundamental characteristics that all atomic systems must have. It is not surprising that a minimum description, like the Lennard-Jones potential, should have two parameters. In Fig. 2 one sees why [pic] and [pic]have the physical meaning that we have mentioned.

Figure 1 below shows a Matlab generated plot for Argon:

The following table gives you LJ parameters for few materials:

|S.N |Material | [pic] (K) |[pic] (A) |

|1 |He |10.22 |2.58 |

|2 |Ne |35.7 |2.79 |

|3 |Ar |120 |3.405 |

|4 |Xe |229 |4.06 |

Using Matlab create scripts to do following tasks:

1. Plot the LJ potential for each material separately

2. Save data files for potential energy and distance r in datafile

3. Use datafile to find the distance r at the minimum potential energy for each substance and create a table based on your datafile.

4. Now plot all LJ potentials on the same graph

Please provide interpretation of potential function curve and the physical properties of the substances.

Note: Submit solution to this problem to your class TA.

Sidney Yip

Nish Sonwalkar

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[pic]

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