lindemann

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lindemann is a Python package to calculate the Lindemann index of Molecular Dynamics simulation trajectorys, as well as the progression of the Lindemann index per frame or per atom and frame of temperature ramps for phase transition analysis.

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Installation

It requires python >=3.9,<3.12

pip install lindemann

or install with Poetry

poetry add lindemann

Usage

lindemann [OPTIONS] TRJFILE

or if installed with Poetry:

poetry run [OPTIONS] TRJFILE

Precautions:

Make sure you have enough memory available before you run any flags despite the -t flag. You can check the memory use by using the -m or the --mem_use Option. The high memory use will be fixed as soon as the numpy.nanmean() function with axis is available for numba.

Multiprocessing:

If you don't have a hpc environment available to distribute the workload, I added multiprocessing to parallize the tasks if you are using your local machine. Just add more than one filename with the -t flag. Currently multiprocessing is only implemented for the -t flag, due to the memory issues mentioned above.

Arguments:

• TRJFILE...: The trajectory file(s). If no other option is selected, the lindemann index is calculated for the trajectory. Equivalent to the -t option. If you pass more than one trajectory they will be calculated in parallel. Only works with no flag or -t flag. [required]

Options:

• -t: Calculates the Lindemann-Index for the Trajectory file(s) [default: False]
• -f: Calculates the Lindemann-Index for each frame. [default: False]
• -a: Calculates the Lindemann-Index for each atom for each frame. [default: False]
• -p: Returns a plot Lindemann-Index vs. Frame. [default: False]
• -l: Saves the individual Lindemann-Index of each Atom in a lammpstrj, so it can be viewed in Ovito. [default: False]
• -v, --version: Prints the version of the lindemann package.
• -ti, -timeit: Uses timeit module to show running time [default: False]
• -m, -mem_use: Calculates the memory use. Run it before you use any of the cli functionality despite the -t flag [default: False]
• --help: Show this message and exit.

Demo

Basic usage to calculate the Lindemann Index:

The package has a plotting feature. It will show the a plot Lindemann index vs. the frames. If the trajectory file is a temperature ramp, it is possible to determine the phase transition.

Usage of the of the lammpstrj file output feature to save the progression for each atom per frame into a LAMMPS trajectory file. Afterwards the trajectory can be viewed with ovito for example, here the Lindemann progression was used for the OVITO color coding feature.

Motivation

In my research I have investigated simulated annealing of brass nanoparticles. I had to determine the melting point of many different cluster sizes and zinc compositions and I have chosen the Lindemann index for this purpose. I wanted a cli tool that could be easily used with MPI due to the amount of calculations. I also wanted to get to know numba better and so I combined them in this project. numba does not really like pythonic code and can handle verbose code better.

ybyygu had already worked out a nice solution in cython with the Welford algorithm, which was found to be easy to convert into numba. whashi44 in turn had presented a great Numpy version with a GUI, his version convinced me to use ovito for read write tasks of LAMMPS trajectories.

For the determination of the melting point of nanoparticles you can often find plots in the literature which follow the following scheme. A temperature range is defined where the phase transition is likely to take place. Then step lengths for the temperature are defined and then a trajectory is calculated for the respective temperature. If a phase transition occurs, this can be recognized by a jump of the lindemann index as shown in figure 1

Figure 1.

I realized that the development of the Lindemann index can be observed across the trajectory. The Welford algorithm allows a live calculation of the mean std, which means that the development of the Lindemann index can be observed as the simulations run. Since I worked with heat ramps for simulated annaeling, I was interested in the development of the Lindemann index across the frames of these trajectories.

This could be realized by a modification of the Welford algorithm. With this method you can observe the change of the Lindemann index per frame. Here, too, a characteristic change of the Lindemann index occurs, with which the phase transition can also be determined. The development of the Lindemann index over the frames can then be visualized as a plot for the phase transition determination. If the temperature is plotted against the Lindemann index, the temperature can also be read directly for the phase transition, as seen in figure 2.

Figure 2: Note the noise of the temperature was smoothed with help of numpy.convolve with a box size of 40.

With a further modification of the Welford algorithm you can also see the contribution of each atom to the lindemann index for each frame. This was especially interesting for me because atomic properties can be easily visualized with tools like OVITO. This gives one a detailed view of the phase transition. This is especially interesting for the investigation of nanoparticles during the phase transition, as they often start melting from the surface. This is where the method in combination with OVITO offers a good starting point for further investigations.

Furthermore, computational costs can be reduced by using one heat ramp instead of many trajectories with discrete temperatures. As an example, see figures 1 and 2. In figure 1, 16 trajectories with different temperatures were chosen to determine the phase transition. The step width was reduced near the phase transition for a better determination of the phase transition. Each trajectory has 50k steps, that is 800k in total, whereas in figure 2 the heatramp has only 200k steps.

Background

A key problem with the measurement of the melting point of nanoparticles is that with decreasing size of a given nanoparticle the phase transition, defined as the temperature of a sudden change in the enthalpy, becomes less pronounced. This is caused by the surface effect: for a given cluster the surface area is larger compared to a bulk structure of the same size. Melting does not take place all at once, but is a longer melt transition and no longer really a melting point.

The Lindemann index, stated in the following equation presents a solution for this problem. It describes the root-mean-square (rms) fluctuation of the bonds or interatomic distance in the system over time (or temperature, if the temperature of the system changes as the simulation progresses). The Lindemann index is a more robust method to determine the melting point of nanoparticles, as opposed to enthalpy. Accordingly the Lindemann index is often considered, when the melting point of nano-particles is of interest. The index is defined as,

<a href="https://www.codecogs.com/eqnedit.php?latex={\langle&space;q_{i}&space;\rangle_{\text{atoms}}={\frac&space;{1}{N(N-1)}}\sum&space;_{j\neq&space;i}{\frac&space;{\sqrt&space;{\langle&space;r_{ij}^{2}\rangle&space;-\langle&space;r_{ij}\rangle&space;^{2}}}{\langle&space;r_{ij}\rangle&space;}}}&space;~." target="_blank"><img src="https://latex.codecogs.com/gif.latex?{\langle&space;q_{i}&space;\rangle_{\text{atoms}}={\frac&space;{1}{N(N-1)}}\sum&space;_{j\neq&space;i}{\frac&space;{\sqrt&space;{\langle&space;r_{ij}^{2}\rangle&space;-\langle&space;r_{ij}\rangle&space;^{2}}}{\langle&space;r_{ij}\rangle&space;}}}&space;~." title="{\langle q_{i} \rangle_{\text{atoms}}={\frac {1}{N(N-1)}}\sum _{j\neq i}{\frac {\sqrt {\langle r_{ij}^{2}\rangle -\langle r_{ij}\rangle ^{2}}}{\langle r_{ij}\rangle }}} ~." /></a>

where <a href="https://www.codecogs.com/eqnedit.php?latex=\inline&space;\fn_phv&space;\large&space;N" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\inline&space;\fn_phv&space;\large&space;N" title="\large N" /></a> is the number of atoms in the nano particle. <a href="https://www.codecogs.com/eqnedit.php?latex=\inline&space;\fn_phv&space;\large&space;r_{ij}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\inline&space;\fn_phv&space;\large&space;r_{ij}" title="\large r_{ij}" /></a> is the distance between atom i and atom