CHEN E4880 – Atomistic Simulations

Project 1: Properties of an Elemental Transition Metal

(Due on February 27, 2025 at 11:59 PM)

Transition metals are of great technological relevance for the chemical industry due to their unique properties and versatility in various applications, including catalysts, magnetic materials, and electronics. In this project, you will explore the properties of an elemental transition metal, for now, in the absence of temperature or pressure.

The technical objective of this project is to familiarize yourself with the LAMMPS software and its  input/output file formats. Scientific objectives are to find the ground-state energy, lattice parameter, vacancy formation energy, and one surface energy of a face-centered cubic (FCC) transition  metal. As you work through the project, you will also learn about the importance of convergence parameters and the limitations of interatomic potentials used in atomic-scale simulations.

Geometry optimizations (relaxations)

In nature, a crystal in equilibrium is automatically in the lowest energy (ground-state) configuration. The ground-state lattice parameters and atomic positions define the ground-state lattice geometry that minimizes the lattice energy. The ground-state lattice parameters are generally not known a priori. Given a reasonable guess, we can perform. a geometry optimization, also sometimes called relaxation, that finds the lowest energy configuration by adjusting the atomic positions and lattice  parameter(s) until an energy minimum is found.

Vacancy formation energy

The vacancy formation energy Ev is defined as the energetic cost to remove an atom from a lattice site and reinsert it into the bulk of the material. This leads to the energy difference

where Ebulk and EN - 1 are the energies of the perfect bulk structure with Nbulk atoms and the defect structure with one vacancy containing (N − 1) atoms, respectively. By evaluating the bulk energy per atom (Ebulk /Nbulk) and multiplying by the number of atoms in the vacancy structure (N − 1), we obtain the bulk energy of (N − 1) atoms and can compare the two structures.

Surface energy

The surface energy is the energy required to truncate an infinitely extended crystal along a specific lattice  plane.  Calculating  surface  energies follows a similar overall approach to calculating vacancy formation energies, but different convergence parameters must be considered.

Surface relaxations and pair potentials

The following equation gives the Lennard-Jones potential

Pair potentials, such as the Lennard-Jones potential, show outward surface relaxations, which disagrees with experimental observation.

1.   Lattice constant of elemental platinum

a.   Calculate the lattice constant (in Å) and total energy (in eV) using the supplied LJ potential. First, use LAMMPS’ built-in minimizer (input file lmp-in.1a-relax), then find the lattice constant by manually adjusting the lattice parameter (input file lmp-in.1a-single). Start with a lattice parameter close to the relaxed lattice parameter. Plot the energy as a function of the lattice parameter from above to below the optimized value (Figure 1) and identify the equilibrium lattice constant.

Hint: See the project guide for an explanation of the LAMMPS input format.

b.   Repeat the calculations, both automated and manual (Figure 2), for the supplied embedded-

atom model (EAM) potential (input file: lmp-in.1b-single). Hint: Don’t forget to upload the EAM potential file to nanoHUB.

c.   How do the calculated lattice constants compare to the experimental value?

d.  Which potential agrees better with experiment? Is this result expected? Explain your answer. Hint: Consider how the different potentials were constructed (see table below).

2. Vacancy formation energy of Pt

a.   Compute the vacancy formation energy (in eV) with the provided LJ potential as a function of the supercell size. Do not relax the atomic positions after taking out an atom. Perform a convergence test and plot the energy against the convergence parameter (Figure 3).

b.   Calculate  the  ratio  of  the  vacancy  formation  energy  to  the  cohesive  energy  per  atom. Document how you calculated the cohesive energy. Hints: The (conventional) FCC unit cell contains 4 atoms. Use the optimized LJ lattice parameter from problem 1.

c.   Repeat your calculation, but relax the atomic positions after creating the vacancy. Perform another convergence test and add results to Figure 3. Describe how the vacancy formation energy changes compared to the unrelaxed calculations. Hint: Adjust your LAMMPS input file for relaxations (see problem 1).

d.   Now  compute the vacancy formation energy using the provided  EAM potential.  Do  it as accurately as you can. Report your convergence test (Figure 4). Use what you learned in parts a. and b. Hint: Use the optimal EAM lattice parameter from problem 1 as the initial value.

e.  Without relaxation, the absolute value of the ratio of the vacancy formation energy to the cohesive energy per atom equals 1 for the Lennard-Jones potential. Explain why.

f.   Why does the vacancy formation energy decrease when the atoms are allowed to relax?

3.   Surface energy of the Pt(100) facet

a.  Which two convergence parameters need to be considered for surface slab calculations?

b.   Compute the surface energy of the Pt(100) surface using the LJ potential. Document your approach. Perform. and plot convergence tests (Figure 5). Report your result in meV/Ǻ2 .

c.   Repeat with the EAM potential, including convergence tests (Figure 6). Please do not perform. relaxations for problem 3.

4.   Conceptual understanding

a.   Calculate the distance r/  where the LJ potential reaches its minimum (Derivation 1). Express r/  in terms of ε and σ , and evaluate using the values from your calculations.

b.   Determine  the  nearest-neighbor  distance  dNN    in  the  optimized  structure  of  problem   1 (Derivation 2).

c.   Why is dNN  different from r/ ?

d.  Why are numerical simulations needed even for simple models such as the LJ potential?

5.   Short answers

a.   For what class of compounds are Lennard-Jones potentials most suitable (name 1 example)?

b.   For what class of compounds/materials are EAM potentials most suitable (name 1 example)?

c.   Name 1 example of materials/interactions for which neither LJ nor EAM are appropriate.

d.

Assignment

1.   Perform. the simulations described on the previous page.

2.   Document your work in a two-page memo.

a.  Address all questions.

b.   Highlight key results (lattice parameters, energies, etc.).

c.   Refer to all figures and derivations.

d.   Keyword style. is fine if the presentation is clear.

3.   Include Figures 1–6 and Derivations 1 and 2 in the appendix. Pictures of hand-written notes are fine for derivations.

4.   For problems 1–3, include examples of your input files in the appendix.

Make sure to follow the formatting guidelines. Limit the text portion of your report to no more than two pages.

Provided Files

The Lennard-Jones potential parameters (ε = 0.200 eV; σ = 2.540 Å) were determined by a fit to the lattice constant and the vacancy formation energy. The EAM potential was fitted to sublimation energies, elastic constants, and vacancy formation energies.

Problems 2 and 3:

We do not provide separate files for Problems 2 and 3. Use the files provided for problem 1 and start from there. See the project guide for hints regarding creating vacancies and surface slab models.