The Lennard-Jones Potential

Now that you have worked to figure out what sort of shape an interatomic potential must have to satisfy the atomic hypothesis, we will look at one of the simplest mathematical functions used to model interatomic potentials. There are hundreds of these potentials that serve to model all sorts of phenomena - not just materials. These potentials are used by chemists, physicists, biologists, and engineers to better understand and predict all sorts of physical behavior!

The Lennard-Jones (LJ) potential, named after Sir John Edward Lennard-Jones, specifies the potential energy between two atoms, $U_{\text{LJ}}$, as function of the distance between them, $r$ (we use $r$ to specify the distance between the two atoms regardless of where they are as opposed to $x$ which usually implies distance from some fixed origin).

$$U_{\text{LJ}}(r) = 4 \epsilon \left[\Big(\frac{\sigma}{r}\Big)^{12}-\Big(\frac{\sigma}{r}\Big)^6\right] \tag{4.5.1}$$

The two positive constants, $\sigma$ and $\epsilon$, can be adjusted to model different behaviors and yield different depths and widths to the potential. We'll practice manipulating these in a bit. These constants are not (usually) derived from more fundamental theory. They are parameters of the model that can be adjusted to fit experimental data. What they represent will be discussed below, although you may be able to determine this yourself from looking at Eq. 4.5.1.

Where do the exponents 12 and 6 in Eq. 4.5.1 come from? The 6th power term can be derived from quantum mechanical considerations due to the effect of fluctuating electron clouds. Interestingly, the 12th power term cannot actually be derived from more fundamental theory. It was selected for mathematical convenience and efficiency, but it serves well enough to model the repulsion of atoms at short distances.

Figure 4.5.1 below shows an interactive graph of the LJ potential along with each of the terms graphed separately. Here, we plot $+\Big(\frac{\sigma}{r}\Big)^6$ not $-\Big(\frac{\sigma}{r}\Big)^6$) so you can better see which term is larger as well as when the cross-over of the two terms occurs. Use the graph to answer Exercise 4.5.1 below about the potential. We suggest you spend a bit of time wrestling with the question by yourself, and then discuss it with a friend or a teaching assistant. Engaging with the questions in this way will allow you to explore the model more deeply and help to improve your understanding. These questions aren’t intended to bog you down, however. If you feel stuck after a few minutes, we recommend submitting your best effort and then reviewing the solution.

Figure 4.5.1 The Lennard-Jones potential (red line). The absolute values of the attractive (dashed black) and repulsive terms (solid blue) are also graphed individually.

Exercise 4.5.1: Analytical Exploration of the LJ Model
Not Currently Assigned

Let's first just explore the math using the dynamic plots in Figure 4.5.1. Take about 15 minutes on this exercise.

  1. What happens to $U_{\text{LJ}}$ when you change $\epsilon$? What do you think we might be able to model in a two-atom system by changing $\epsilon$?

  2. What happens to $U_{\text{LJ}}$ when you change $\sigma$? What do you think we might be able to model in a two-atom system by changing $\sigma$?

  3. What happens to the interatomic potential energy as the distance between the atoms gets very large ($r \rightarrow \infty$)? Show this mathematically. What does this value at $U(r \rightarrow \infty)$ mean?

  4. What happens to the interatomic potential energy $U_{\text{LJ}}$ as the distance between the atoms gets very small ($r \rightarrow 0$)? What does this mean?

  5. The LJ potential (and indeed other interatomic potentials) are constructed such that the potential energy is positive at short distances at very short interatomic separation distances $r$, becomes negative as $r$ increases and reaches a minimum. Then, it increases and approaches zero at $r \rightarrow \infty$.

    What does the value of the negative potential energy minimum $U_0 = |U(r=r_0)|$ represent?

  6. Find the general expression for the distance at which the potential energy is minimized. We call this the equilibrium bond distance $r_0$, and it is the point of at which net force in the system is zero ($F = 0$).

    Hint: how are potential energy and force related? And how would you manipulate Eq. 4.5.1 to find the values of $r_0$ and $U_0$?