Modeling the Atomic Hypothesis with Interatomic Potentials

When building a model, you first have to decide what the entities of your model will be. In this course, we will typically model atoms as Newtonian objects (i.e. point objects with position and velocity) that exert forces on each other. While this is clearly "wrong", as atoms (and therefore many of their properties) are quantum mechanical in nature, the Newtonian approximation is powerful enables us to build many illuminating models.

Sketching A Potential Energy Function for Interatomic Interactions

In this section we'll introduce a NetLogo molecular dynamics model of two atoms interacting with each other. NetLogo is a programming language designed specifically to model the interaction and behaviors of multiple entities, or "agents". In materials science, these are often atoms or molecules. A molecular dynamics model treats molecules and atoms as Newtonian objects with position, velocity and mass. They accelerate due to forces they exert on one another (and sometimes external forces like electric or magnetic fields) and move according to Newton’s laws.

The potential energy that results from the interaction of the atoms, and therefore the force the atoms feel from each other, is modeled using a function called an interatomic potential. These interatomic potentials come in lots of different varieties - we’ll first explore how these potentials work in a computational model that we provide, and then we’ll later show you some mathematical models that we can manipulate to derive important metrics such as equilibrium bond energies and distances.

In our first NetLogo interatomic potential model, you will use the interactive graphic to draw the interatomic potential function and see how the atoms react. The video demo in Video 4.4.1 demonstrates how to use the model. We'll use it first to draw an interatomic potential function similar to the potential energy function for an ideal spring that we reviewed in the previous section. For simplicity, we'll use the reference from of the left (green) atom - holding it "fixed" while only the right atom moves. From an outsider's perspective, both atoms would be moving. The force the right atom feels will be proportional to the slope of the interatomic potential function at its current location, as $F = -\frac{dU}{dx}$.

Your job is to draw an interatomic potential function to model the way you think atoms behave, using the embedded model in NetLogo model 4.4.1. We have already stated in the atomic hypothesis that atoms repel each other when squeezed together and attract each other when they are further apart. We can also deduce from previous models that atoms must not attract each other very much when they are very far apart, because otherwise everything everything in the universe would quickly condense into one giant blob.

Spend a few minutes exploring the construction of interatomic potentials that will align with the atomic hypothesis and answer the questions below the NetLogo model and complete the the tasks in Exercise 4.4.1. In the next section we will introduce a formal mathematical model of the interatomic potential function based on what we observe here.

Exercise 4.4.1: Sketching Interatomic Potentials
Not Currently Assigned

Let's try to model some potentials. This exercise should take about 10 minutes. As a reminder: do your best and submit your own work first before peeking at the answer.

  1. First, construct a spring potential like that on the previous page (parabola). When you have one you like, click the export-sketch button and upload your sketch here (you can drag a file onto the "choose file" button or click it and find the file).

    Make some observations about how the atoms behaves in this potential. Which features does the spring model have that fits atomic hypothesis (Section 3.4)? What is it missing? Would this be a good model for simulating atomic bonding?

  2. Now, construct an interatomic potential that more accurately models the atomic hypothesis. When you have one you like, click the export-sketch button and upload your sketch here. Explain what the main difference is between your new potential and the spring potential? Make some observations about the atomic pair in this potential.