Diffusion coefficient¶

Importing the Sulfur(α) crystal structure¶

1. Open a new AMSinput window

2. Switch to ReaxFF: →

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_0.4S 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.

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_0.4S 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 ų to about 4400 ų):

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.

1. In the main panel, select Task → Molecular Dynamics

2. Click on next to Task: Molecular Dynamics to go to the MD details

3. Set the number of steps to 30000

We will now set up the following temperature profile:

1. From start until step 5000: T = 300 K (constant)
2. From step 5000 to step 25000: heating up from 300 K to 1600 K
3. 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.

1. Click on next to Thermostat to go to the thermostat details

2. Click on to add a new thermostat

3. Select Thermostat → Berendsen

4. In Temperature(s), set the values 300 300 1600 300

5. In Duration(s), set the values 5000 20000 5000

6. Set the damping constant to 100 fs

We are now ready to run the Li_0.4S 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):

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.