LJ-Potential Atoms in a Molecular Dynamics Simulation

Summary of LJ-potential

Now that you have spent some time exploring the LJ-potential:

$$U_{\text{LJ}}(r) = 4 \epsilon \left[\Big(\frac{\sigma}{r}\Big)^{12}-\Big(\frac{\sigma}{r}\Big)^6\right] \tag{1.2.1}$$

Let’s summarize.

  • The negative 6th-term models the attraction of atoms.
  • The positive 12th-term models the repulsion of atoms.
  • Both potential energy terms approach zero as $r \rightarrow \infty$. The 12th-term goes to zero faster as $r \rightarrow \infty$ which gives the potential its attractive "tail".
  • Attractive and repulsive potential energy terms approach $-\infty$ and $+\infty$, respectively, as $r \rightarrow 0$. The repulsive (12th power term) goes to infinity faster as $r \rightarrow 0$ which dominates the energy at small values of $r$.
  • At intermediate distances (typically on the order of a few Å in most systems), the attractive and repulsive terms cancel out, achieving equilibrium (a minimum where $F = 0$) at $r_0$.
  • At values larger than $r_0$, the atoms are attracted (although very weakly as $r$ increases).
  • At values smaller than $r_0$, the atoms are repelled.

In all, we have an interatomic potential function for which atoms repel each other when squeezed very close together, attract each other at moderate distances and essentially do not interact at large distances.

Next we will use the LJ-potential in a molecular dynamics (MD) simulation with two atoms and observe some properties of materials that can be explained by the interatomic potential model. After that, we will use the LJ-potential in a MD simulation with many atoms.

Using LJ Potential in an MD simulation.

The NetLogo model in below is similar to the one you used earlier (NetLogo model 1.2.1), but now, instead of you drawing an interatomic potential that satisfies the atomic hypothesis, it is modeled by the Lennard-Jones potential within the code itself. You will see the two atoms with the LJ potential drawn between them from the perspective of the blue atom. The red atom will move simply according to Newton's laws - this is a very simple molecular dynamics (MD) simulation.

Again, in reality, both atoms would feel a force from the interatomic potential and both would move, but for simplicity, we are holding the blue atom fixed and observing the red atom's behavior from the blue's reference frame. The red atom starts at $r_0$, the distance which minimizes the potential energy, and has an initial velocity in the $+x$ direction determined by the initial-KE term. Use the model to answer Exercise 1.2.1.