**3.1 Position of sodium ions in c-axis without applied field**

After ten runs have been carried out with different initial velocities for the individual sodium ions with their average value being set by the defined temperature, the average positions for each of the sodium ions in the c-axis are then calculated. Figure 5a shows the average position for the first six sodium ions along the c-axis with the initial conditions of 273K, 100000 intervals and time step=10-15s. This gives a good comparison of the positions of the sodium ions throughout the 100000 intervals. The sodium ions only vibrate in their equilibrium positions depending upon where their equilibrium positions were. For example, the sixth sodium ion (pink) only vibrates around the cavity it belongs to.

Fig. 5. (a) The average position along the c-axis for the first six sodium ions (b) The arrangement for the first six sodium ions in the tunnel. The arrows show the alignment of the cavities in the tunnel and the graphs.

#### **3.2 Position of sodium ions in c-axis with applied field**

An electric field in the range of 7.43MV/m to 74.3GV/m was applied along the c-axis to the hollandite model at the 5001th time interval. The initial conditions for the results shown below were temperature=273K, time step=10-15s, 100,000 intervals and electric field=743MV/m. Figure 6 shows the positions of the first six sodium ions as a function of time. Over the initial 5000 intervals, which was in the absence of the electric field, the sodium ions just vibrate around their equilibrium positions. Starting from 5001th intervals, the positions of the sodium ions change dramatically. For example, the sixth sodium ion (pink) moved to the next cavity at some points and then back to the original cavity and then to the next cavity again.

Molecular Dynamics Simulation and Conductivity

results of interest are those under the effect of electric field.

over 100 intervals eliminates vibration periods at 10-13 s and shorter.

Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16 379

intervals, the polarisation tends to fluctuate around zero. Once the electric field is applied at 5001th interval, the polarisation increases rapidly to around 0.7 C/m2 and after that sudden increase, the polarisation remains at an average of that value for the rest of the time intervals. The results under different electric field are similar, but a smaller field gives a smaller polarisation. A running average of the polarisation was taken over 100 intervals and only the results starting from 5001th intervals are taken into account. This is because the

Figure 8 gives the continuous average for the polarisation as a function of time. The plot obtained was not as noisy as the polarisation plot shown in figure 7, as the running average

Fig. 8. Continuous average for the polarisation with an applied field of 743MV/m as a

The Fast Fourier Transform (FFT) is performed in order to calculate the behaviour of the hollandite model in the frequency domain. The continuous average of the polarisation is imported to "Origin" program. A graph of polarisation versus time is plotted. Only the data between the 5001th and 100000th gives the polarisation since the electric field is switched on at the 5000th interval. The time derivative of the polarisation is then obtained via the program, and is plotted. The one-sided Fourier Transform of / / <sup>0</sup> *dP dt Ee* gives the frequency dependent dielectric susceptibility for comparison with experiment. The easier way to carry out the FFT in "Origin" software is by performing FFT on the *dP dt* / , the

**3.4 Fast Fourier Transform (FFT) – Susceptibility** 

function of time.

Fig. 6. Position of the first six sodium ions in c-axis when electric field of 743MV/m was applied along the c-axis to the hollandite model at 5001th time interval.
