Bonding in Large(r) Assemblies

Two particles that obey a Lennard-Jones potential form a bond if they are close enough to each other and their kinetic energy isn't too high. Now let's think about what would happen if we have three particles (we'll call them atoms now - that's what we're simulating) in close proximity to each other. For this section we'll use the NetLogo model in NetLogo model 5.3.1 to simulate various structures that arise when more and more atoms are added to the simulation.

Complete Exercise 5.3.1 while referring to NetLogo model 5.3.1 to explore what happens in this situation.

Exercise 5.3.1: The Lennard-Jones Potential - More Atoms!
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Let's expand our Lennard-Jones potential and apply it to group of atoms, which are much more interesting to materials scientists than pairs of atoms. Complete the questions below, taking about 15 minutes complete the work.

Note, these models are incredibly powerful! While there are some basic observations you can make about what happens when atoms get together, you'll also start to observe various emergent behavior - phenomenon we don't explicitly model, but that emerge from our computation. These can mean many things: they may be actual behaviors that also occur in real materials, or they may be artifacts of the assumptions and simplicity of our models that do not have real, physical analogs. We have to adapt and interrogate the models to find and - and often, do real experiments.

  1. Don't play around with NetLogo model 5.3.1 yet! Instead, create a mental model of three atoms (represent them with circles) occupying a two-dimensional plane. Imagine that these atoms interact with each other according to a Lennard-Jones potential. (See Eq. 4.5.1). First draw the atoms at an "initial" state in which they're positioned at the vertices of an equilateral triangle with sides of length $\ell = 4r_0$, where $r_0$ is the equilibrium bond distance.

    Then, sketch how you think the atoms will be positioned over some time $\Delta t$ later at which they've reached "equilibrium". Sketch the atoms in their final equilibrium position. Upload your sketches with a short explanation of your reasoning.

  2. Use the NetLogo model 5.3.1 to validate your sketch. Make sure that settings are on the default:

    • num-atoms is set to 3
    • initial-config is set to "Random"
    • Constant temp is enabled.
    • temp is set to 0.1
    • First, set go-mode is set to "drag atoms" and press go. Arrange the atoms as you'd like.
    • Then, go-mode to "simulate" and press go.

    Watch the system equilibrate. Describe the results. Do your results agree with your predictions above? (If they don't, explain why you see what you see now.)

    Try this a few times with different initial configurations. Do you ever get any variation in your configuration - or is it always the same?

  3. Repeat the previous question for 4, 6, and 30 atoms. For simulations of 4 and 6, you'll want to drag atoms so that they are close enough to "feel" each other. For 30 atoms, you can just run the simulation. What arrangements do you observe for each? Include the images and describe. For simulations of 3, 4, and 6 atoms, count the total number of bonds in each atomic assembly.

    Upload a single image showing these configurations (you can screenshot each and put them into a single image), with all the bonds indicated.

  4. Comment on the trends you observed in the previous question. What patterns arise from these simulations? What can you say about the number of bonds formed in each configuration? Did you observe anything for very large assembles that differed from smaller ones?