LJ-Potential Atoms in a Molecular Dynamics Simulation

Summary of LJ-potential

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

$$U_{\text{LJ}}(r) = 4 \epsilon \left[\Big(\frac{\sigma}{r}\Big)^{12}-\Big(\frac{\sigma}{r}\Big)^6\right] \tag{4.6.1}$$

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 4.4.1), 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 4.6.1.

In previous sections, we've modeled interatomic potentials with a parabolic spring potential and a sketched potential. In this section, we'll use the Lennard-Jones equation Eq. 4.5.1 to model the interatomic interactions.

Take about 10 minutes on this exercise.

To change our model in NetLogo we will of course need to change the code! Take a look at Line 131 in NetLogo model 4.6.1 under NetLogo Code. This line asks NetLogo to report the force acting between that atoms at the so the code will accelerate the atom appropriately during the next tick.

Does this line of code make sense based on your understanding of the interatomic force from the Lennard-Jones potential? Explain.

Using the Graphical User Interface (not the code), set the initial kinetic energy initial-KE to zero and click the setup and go buttons. The red atom is originally positioned at equilibrium. What does it do? Why?

Now, notice that you have a few plots to the right: interatomic distance v time (where $r$ is the current interatomic distance and $r_0$ is the equilibrium interatomic distance) and energy (which shows potential energy $\text{PE}$, kinetic energy $\text{KE}$, and total energy $\text{Total}$. Keep your eye on these as you drag the red atom a small distance from equilibrium in the $+x$ direction.

What happens to the energy curves (they may be overlaid - careful) when you move the atom?

Now, let the atom go. Explain why the atom moves the way it does. Explain why the energy plots show what they show. You may want to slow the model speed down to observe this better.

Interatomic Force and Interatomic Energy

One thing we want to make clear is that we can show both interatomic potential curves and interatomic force curves. The potential is useful because it tells us about the energy of the system, but the force curves are useful because they give us intuition about what net forces the atoms are experiences at some position. One can, for example, sketch a interatomic force curve from an interatomic energy curve! Give it a try in Exercise 4.6.2.

The figure below shows an interaomic energy curve (solid, red curve). Use your understanding of the force-energy relationship $F = -\frac{dU}{dr}$ to sketch the corresponding interatomic force curve in the axes above it.

Explain the trends in the force curve from small $r$ to large $r$.

Take 3-5 minutes on this activity.