# Diffusion: MSD and VACF¶

In this tutorial we will compute the diffusion coefficients of Lithium atoms in a Li_{0.4}S cathode by means of molecular dynamics simulations with the ReaxFF engine.

The systems and workflows presented here are originally described in the publication **ReaxFF molecular dynamics simulations on lithiated sulfur cathode materials** Phys. Chem. Chem. Phys. 17, 3383-3393 (2015).

The tutorial consists of the following steps:

- Importing a CIF file
- Manipulating the structure (e.g. inserting particles) and equilibrating the system
- Creating an amorphous structure by simulated annealing
- Calculating diffusion coefficients from MD trajectories

If you wish, you can download the already-relaxed amorphous Li_{0.4}S structure `Li04S_amorphous.xyz`

and jump directly to the diffusion coefficient calculation

## Importing the Sulfur(α) crystal structure¶

To speed up the calculations, we will use a smaller system than the one described in the publication. The crystal structure can be directly imported from a CIF file:

**1.**Download the`CIF file by clicking here`

**2.**In AMSinput:**File → Import Coordinates****3.**Select the CIF file you just downloaded using the file dialog window

## Generating the Li_{0.4}S system¶

We first need an `xyz`

file of a single Lithium atom:

**1.**Open a new AMSinput:**SCM → New Input****2.**Draw a single Li atom**3.****File → Export coordinates → .xyz**and give it the name “Li.xyz”**4.**Close the AMSinput window without saving

We use the builder functionality of AMSinput to randomly insert 51 Li-atoms into our Sulfur system.

See also

A better way of building the Li_{0.4}S system is to use Gran Canonical Monte Carlo (GCMC). See also the Voltage profile from Grand-Canonical Monte Carlo tutorial.

**1.**Click on**Edit → Builder****2.**Next to “Fill box with”, set the number to**51****3.**Click on the folder and select the “Li.xyz” you just made**4.**Set**Distance**to 1.0 Å**5.**Click on**Generate Molecules****6.**Click on**Close**

We now need to relax the geometry by running a geometry optimization **including lattice relaxation**:

**1.**In the main panel, select**Task → Geometry Optimization****2.**Click on the folder next to**Force Field**and select**LiS.ff****3.****Details → Geometry Optimization****4.**Tick the**Optimize lattice**checkbox

We are now ready to run the Li_{0.4}S relaxation calculation:

**1.****File → Save As…**and give it an appropriate name (e.g. “LiS_optimization”)**2.****File → Run****3.**Click**yes**when asked if the structure in AMSinput should be updated

The volume of the unit cell should have increased significantly during the optimization (from about 3300 Å^{3} to about 4400 Å^{3}):

**1.**Open**AMSmovie**by clicking on**SCM → Movie****2.**In AMSmovie, click on**Graph → Cell volume****3.**Click on play to see the movie of the geometry optimization

## Creating the amorphous systems by simulated annealing¶

Amorphous systems can be created with a Molecular Dynamics simulation by slowly heating up the system followed by a rapid cool-down.

As in the publication, we will anneal up to 1600 K followed by a rapid cool-down to room temperature. In order to speed up the calculation, only 30000 steps are calculated here.

We will now set up the following temperature profile:

- From start until step 5000: T = 300 K (constant)
- From step 5000 to step 25000: heating up from 300 K to 1600 K
- From step 25000 to step 30000: cooling down from 1600 K to 300 K

For more details on temperatures and pressure regimes, see the AMS manual on MD.

We are now ready to run the Li_{0.4}S simulated annealing calculation:

**1.****File → Save As…**and give it an appropriate name (e.g. “LiS_simulated_annealing”)**2.****File → Run****3.**Click**yes**when asked if the structure in AMSinput should be updated

In AMSmovie you can follow the progress of the MD simulation, and you’ll be able to see the three temperature regimes:

**1.**Open**AMSmovie**by clicking on**SCM → Movie****2.**In AMSmovie, click on**MD Properties → Temperature****3.**Click on play to see the movie of the MD simulation

We now need to relax the geometry of our new amorphous system by running a geometry optimization **including lattice relaxation** (as we did before):

**1.**In the main panel, select**Task → Geometry Optimization****2.****Details → Geometry Optimization****3.**Tick the**Optimize lattice**checkbox**4.****File → Save As…**and give it an appropriate name (e.g. “LiS_amorphous_optimization”)**5.**Run the calculation**6.**Click**yes**when asked if the structure in AMSinput should be updated

## Calculating the diffusion coefficients¶

We are now ready to run the final MD simulation to compute the Lithium diffusion coefficient D at T=1600 K.

You can calculate the diffusion coefficient in two different ways:

- Through the slope of the mean squared displacement (MSD,
**recommended**):

- Through the integration of the velocity autocorrelation function (VACF, this requires setting
**sampling frequency**to a small number):

Important

Because of finite-size effects, the diffusion coefficient depends on the size of the supercell (unless the supercell is very large). Typically, you would perform simulations for progressively larger supercells and extrapolate the calculated diffusion coefficients to the “infinite supercell” limit.

### Set up and run the production simulation¶

We will use a minimal proof-of-principle setup of 10000 equilibration steps followed by only 100000 production steps:

**1.**In the main panel, select**Task → Molecular Dynamics****3.**Set the**number of steps**to**110000****4.**Set the**Sample frequency**to**5**(this writes both the atomic positions and velocities to disk every 5 steps)**6.**Set the**Type**to**Berendsen****6.**Set the**Temperature**to**1600**K**7.**Clear the**Duration**field**8.**Set the**damping constant**to**100**fs

Tip

If you use the MSD to calculate the diffusion coefficient, you can set **Sample frequency** to a higher number (giving a smaller trajectory file).

Note

The time between two steps on the trajectory file will be sample_frequency * time_step = 5 * 0.25 fs = 1.25 fs.

We are now ready to run the calculation:

**1.****File → Save As…**and give it an appropriate name (e.g. “LiS_MD_1600K”)**2.****File → Run****3.**You can follow the progress of the calculation in**AMSMovie**and**AMStail**(**SCM → Logfile**)

### Diffusion coefficient through MSD (recommended)¶

After the calculation has finished, open it in AMSmovie:

**1.**In**AMSMovie**select**MD Properties → MSD****2.**In**Steps**set**2000 - 22001****3.**Set**Atoms**to**Li****4.**Set**Max MSD Step**to**5000**(corresponding 5000*1.25 fs = 6250 fs).

**5.**Click on**Generate MSD**

Your results may look a bit different because of the short MD simulation and random initial velocities.

**The MSD is the red straight line**. If the MSD line is not straight, it means
that you need to run a longer simulation to gather more statistics. It is
calculated for times up to 6250 fs (controlled by the **Max MSD Step** option
in the MSD window).

The blue curve shows the slope of the MSD divided by 6, calculated by performing a linear fit on the MSD from the “Start Time Slope” (that can be set in the MSD window) to each point on the MSD. The blue curve thus corresponds to the diffusion coefficient D. It should ideally be perfectly horizontal.

Here, the blue curve seems to converge to a value around 3.73 × 10⁻⁸ m² s⁻¹.

Thus, **D = 3.73 × 10⁻⁸ m² s⁻¹**.

### Diffusion coefficient through VACF¶

After the calculation is finished, we can obtain the Li diffusion coefficient by computing the **velocity autocorrelation function** for the Li atoms:

**1.**In**AMSMovie**select**MD Properties → Autocorrelation function****2.**In**Steps**set**2000 - 22001****3.**Select**Property → Diffusion Coefficient (D)****3.**Set**Atoms**to**Li****4.**Set**Max ACF Step**to**5000**(corresponding 5000*1.25 fs = 6250 fs).

**6.**Click on**Generate ACF**

This plots

- the velocity autocorrelation function,
- the Fourier transform of the velocity autocorrelation function (power spectrum),
- the integral of the velocity autocorrelation function divided by 3 (diffusion coefficient)

In this case, only the bottom plot is of interest to us. It should ideally become perfectly horizontal (converge) for large enough times.

Here, the value of the diffusion coefficient **D = 3.73 × 10⁻⁸ m² s⁻¹**, which is the same value that was obtained using the MSD.

### Extrapolate to lower temperatures¶

Calculating the diffusion coefficient at 300K would require a very long trajectory. However, it is possible to provide an upper bound to the Li diffusion by means of extrapolation from elevated temperatures using the Arrhenius equation:

where \(D_0\) is the pre-exponential factor, \(E_a\) is the activation energy, \(k_B\) is the Boltzmann constant,
and \(T\) is the temperature. The activation energy and pre-exponential factors can then be obtained from an Arrhenius plot of \(\ln{(D(T))}\) against \(1/T\). In order to extrapolate the diffusion coefficients for Li_{0.4}S we calculate trajectories
for at least four different temperatures (600 K, 800 K, 1200 K, 1600 K) for each system. One can then extrapolate the diffusion coefficient to lower temperature.