Practice Problems

W3 Miniproject

Interatomic potentials are mathematical functions that we use to model the behaviors of a wide variety of materials. Researchers from physics to biology formulate and fine-tune these potentials so that can do computational experiments to learn about the structure and behavior of matter.

These potentials are usually 1. empirical and 2. parametric meaning that researchers adapt the functions so that they match experimental (or other types) of data that tell us how atoms and molecules interact with each other. For example, a researcher might measure a bond distance using X-ray diffraction or a bond strength using melting temperature, and fit the potential accordingly.

If a researcher can find a strong match between their interatomic potential and the experimental data, they can have more confidence in the results of their simulation.

Your job today is to try something similar: we'll challenge you to explore a physical behavior in NetLogo (thermal expansion) and engineer an interatomic potential to change the thermal expansion behavior. We'll use a simple NetLogo model (NetLogo model 4.9.1) and explore interatomic potentials in Desmos to see if we can simulate materials with different thermal expansion behaviors.

Exercise 4.9.1: Engineering Interatomic Potentials
Not Currently Assigned

  1. The NetLogo model in NetLogo model 4.9.1 simulates the behavior of a string (or rod) of atoms, with one side fixed, and all the atoms free to move only along the lateral direction. All atoms are interacting with each other according to the Lennard-Jones potential:

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

    The white bar indicates the length of the string of atoms at 0 K. Now, increase the temperate by intervals of 0.5 by clicking on the slide. Report what happens. Is this expected? Why or why not? Note, the light blue bars on the graph shows the range of lengths of the bar (it is wiggling in the lateral direction), while the black line connects time-averaged positions.

  2. Now sketch three images on paper or your notepad and upload them (in a single image) here:

    1. A schematic interatomic potential like what you'd expect underlays the simulation in NetLogo model 4.9.1.
    2. A schematic interatomic potential that has a larger thermal expansion coefficient compared to the original LJ potential.
    3. A schematic interatomic potential that has a smaller thermal expansion coefficient compared to the original LJ potential.

  3. The Lennard-Jones potential, while simple and powerful, is sometimes difficult to use because it has relatively few parameters (only $\epsilon$ and $\sigma$. Spend a few minutes to try to invent your own potential that would allow you more freedom to tune the thermal expansion behavior in your model. There're many ways to do this, of course. Chat with your group members about general approaches and then submit an expression for your proposed potential.

    You can use LaTeX in your solution by copying and pasting the code snippet for the Lennard-Jones potential below and adapting it as you see fit.

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

    After you submit, view the solution for a new potential that you can use!

  4. Use the potential provided in the previous solution to derive parameters ($\epsilon$, $sigma$, $n$, and $m$ that give the same equilibrium bond length and equilibrium bond energy as the LJ potential, but either a larger or smaller thermal expansion coefficient. Report the parameters of the potential.

    Alternatively, you can use a potential that you invented yourself, report the potential and the parameters. Use LaTeX for effective communication.

  5. Finally (!), verify that your new potential and the parameters you found do indeed yield a larger/smaller thermal expansion coefficient, as expected. Calculate the force of your potential by differentiating, and replace Line186 in the NetLogo code in NetLogo model 4.9.1. Can you confirm that thermal expansion is larger/smaller? That is, what are the lengths of the the atomic strings for the LJ and your potential at $T = 0.1$ and $T = 3.6$?