Abstract

@mainpage Accurate relaxation of a 2D crystal interacting with a quasicrystalline surface [TOC]

Abstract

We investigate the energetic properties of a hexagonal 2D monolayer of colloidal particles interacting with a rigid decagonal quasicrystalline corrugation potential. Using classical molecular dynamics, we study the effectiveness and applicability of numerical optimization techniques to investigate the presence of an optimal angular orientation, following an extension of the Novaco-McTague theory.

Dependencies

A C compiler
Make
Gnuplot
LAMMPS
Doxygen for documentation
Coreutils (only tested with GNU coreutils, so the scripts may be reliant on non-standard flags)

Compiling

The code complies with the C99 standard, it has been tested with GCC, but Clang (or any other C compiler for that matter) should also work.

$ make # to compile
$ make check # to compile and run tests

Some of the scripts depend on small utilities stored in the bin/ directory, so add it to your path, to ease this you can simply source environment.sh:

$ . ./environment.sh

Documentation

To generate HTML documentation use Doxygen:

$ doxygen

This will populate the html directory with docs, open html/index.html in any web browser to browse them.

Debugging

If you want to change the code, or you find a bug, compile it like this to include debug symbols:

$ make DEBUG=1

and run the programs with the highest log level (i.e. with the option -l VERBOSE). This will let you use a debugger (e.g. gdb), and will add source file and line information to all of the logging. This will also compile the code with the Undefined Behavior Sanitizer. If you want to enable the Address Sanitizer also add the variable ASAN=1.

Generating simulations

To generate a simulation use the simulation program:

$ simulation rotation simulation-directory

This will create subdirectories and create all of the necessary files, it should also generate an sbatch-compatible shell script, called run.sh, to run the simulation and collect the results. Note that you might want to modify the sbatch parameters. If you don't need to run under SLURM, then you can just run LAMMPS with the *.in files as inputs, like so:

$ lmp -i lattice_r0.in

By default in rotation mode it will generate 60 simulation inputs, one for each angle from 0.0° to 6.0° every 0.1°. The program also collects some of the parameters in a file called METADATA, this might be useful when reviewing the results later. To explore the options run

$ simulation -h

Analyzing results

Once simulations are over, you can analyze the results with the analyze program:

$ analyze simulation-directory

Analyzing effectively means extracting the energies from the LAMMPS outputs and subtracting the initial values from the relaxed ones, the result of this is stored in 4 files:

angles.energies or translation.energies: relaxed - rigid (zJ)
normalized.energies: relaxed - rigid / Ntot (zJ)
normalized_movable.energies: relaxed - rigid / N (zJ) (excluding the fixed border from the normalization)
rigid.energies rigid energies (zJ)

One of the most important invariants of the simulations is that it should be 2D, this requires multiple LAMMPS commands to achieve and is prone to error, therefore the analyze command accepts the -e switch to check the final results for out of plane z components i.e.

$ ./analyze -e some-wrong-simulation
[WARNING] some-wrong-simulation/conf_finale_r0_t0.data: non zero z component
44523 1 1 -104.40039562276402 -1155.2665943350287 0.04082829989134281 0 0 0
                                              ~~~~^
Errors occurred, the analysis might not be complete. Good luck.

Expected energy profile

To get the expected energy profile, as a function of the mismatch angle, run novaco_v1p0.py, this will also store all of the q vectors (see Novaco Notes) used for the energy calculations in a file called file_q.dat so whenever gnuplot scripts refer to some file_q-SOME_NAME.dat, this is the file they expect.

Gnuplot scripts

Under thesis/ there are a lot of gnuplot scripts I wrote to produce the plots I used in my thesis, a couple of them seem to be important enough to be worth explaining here.

q vectors

To plot the q vectors in the first Brillouin zone you can use the scripts under thesis/qs. Take thesis/qs/qs-5.4.gp as an example to make your own representation of the q vectors, this requires two files:

file_q.dat: which is the q vectors file generated by novaco_v0p5.py,
tauvectors.dat: which is a file containing the 6 shortest vectors of the triangular lattice, used to plot the first Brillouin zone.

Moiré pattern

Scripts under thesis/moire are used to produce the colorful moiré patterns, see Fig. 12 in my thesis for an example. The most important script in that directory is thesis/moire/generate-polar-displacements.sh. In order to plot the moiré pattern you will need to setup a simulation that stores particle configurations both before and after the minization. It is important to have LAMMPS do this because it will reorder particles at the beginning of the simulation, but to be able to calculate displacements we need to rely on the fact that both files will have the same particle ordering. To setup a simulation e.g. at the optimal NM angle for apot = 5.4 µm, which is 5.3°, you may do something like this:

$ . environment.sh # If you didn't already do this
$ simulation r -f 5.3 -t 5.3 -m fire -d data/bigger.lmpdat -p 5.4 moire-5.3-apot-5.4 # 5.3 is ≈ the optimal angle

this will generate a single LAMMPs input under moire-5.3-apot-5.4/lattice_r53.in which you need to edit in order to have LAMMPS save the initial configuration, e.g. add this line before the minimize command:

write_data initial_something_something.data

Then run the simulation:

$ cd moire-5.3-apot-5.4
$ lmp -i lattice-r53.in

Then copy the relevant files under thesis/moire and run thesis/moire/generate-polar-displacement.sh like this:

$ cd thesis/moire
$ ./generate-polar-displacements.sh initial_something_something.data conf_finale.data novaco.dat > some-moire.dat

where novaco.dat refers to the output of novaco_v1p0.py for the same values of the parameters. some-moire.dat will contain the following columns:

# particle-type x y ux uy |u| angle

where x and y are the particle positions, ux and uy are the components of the displacement, |u| is the length of the displacement, angle is the angle with respect to the horizontal axis. The last step is to create a copy of one of the preexisting moire scripts, e.g. thesis/moire/moire-5.4.gp, changing the variables that specify the input data; these scripts just include thesis/moire/polar-color-dots.gp which does all of the heavy lifting, i.e. mapping displacements and angles to colors, finding the shortest q vector with thesis/moire/min.sh, and plotting a nice figure with a legend.

All of this process assumes you want to look at the displacement pattern for the NM angle, if you don't, you will have to modify the scripts so that they accept the angle as a parameter rather than relying on the output of novaco_v1p0.py to find the NM angle.

License

All of the programs I wrote are released under the terms of the GNU General Public License version 3, or any later version, see the LICENSE file for information. If the paternity of any of the files is unclear, contact me.