BIBLIOGRAFIA

Diffusion constants are estimated from calculated time-dependent mean square displacements, MSD, <ra(t)2>, for all ion types a. The quantity

i

(6.2)

where ria(t) is the position of the ith ion of type a at time t. The double summations go over time origins t and ions of type a where Nt , Na are the numbers of time origins and a ions, respectively. This method of extensive

averaging is the same as that adopted by Gillan and Dixon (1980).

As well as calculating the above, F- ion MSD have been calculated separately for directions both perpendicular and parallel to the £ axis. The relations used are:

w h e re xia(t), yja (t) an d Zja (t) are th e resp ective x, y an d z co o rd in ates of the

ith ion of ty p e a at tim e t.

The calculated MSD for lanthanum and fluoride ions at each temperature for a simulation of ~ 10 ps are shown in Figures 6.1 and 6 .2

respectively. The separate curves for F' ion displacements in the x-y and z directions are shown in Figures 6.3 and 6.4 respectively.

These results show very clearly the differences in behaviour between the cations and the anions. The constant long time values for cations in Figure 6.1 confirm that the lanthanum ions are not diffusing, but are vibrating about their regular sites at each temperature. Thus the simulated system is still solid, retaining a rigid cationic lattice even at the highest temperature.

(6.3)

a n d

Conversely, the steep slopes observed in Figure 6.2, which are related to the diffusion constants (discussed later), clearly indicate that F- ions are diffusing even at the lowest temperatures (note the difference in scale between Figures 6.1 and 6.2). It is apparent, however, that a significant increase in diffusion occurs between 1200K and 1500K, as was suggested by the preliminary inspection of ion coordinates mentioned in section 6.3.1. A further point is that the linear behaviour observed in the F- plots is established within a few vibrational periods and resembles the form of liquid diffusion. This is not, however, proof that such behaviour is occurring in LaF3. Figures 6.3 and 6.4 also illustrate that diffusion is greater in the x-y direction (1 to £ axis) than in the z direction (// to £ axis).

Diffusion constants for F- ions are obtained from the mean square displacements using the following equations:

<Ara(t)2> = 6Da|t| + Ba (6.5)

and

<ArCLi(t)2> = 4Da I |t| +

_Bal

_(6-6)

and

<Ara//(t)2> = 2D J t | + Ba,, (6.7)

where Da is the self diffusion constant for species a and Ba is an averaged Debye-W aller factor. Da l and Da// and Ba l and Ba// are the diffusion constants and Debye-W aller factors in the respective directions perpendicular and parallel to the £ axis.

Thus, anion diffusion constants are obtained directly from the gradients of the slopes of the mean square displacement plots. Derived values are given in Table 6.4. The gradients used in the calculations were chosen by eye, without using any special fitting procedure, and so the quoted values are estimated to have an error margin of ~ 10%.

There are at present no experimental values for diffusion constants at these temperatures, but the conductivity has been given as -0.01 Q*1crrr1 at

1000K (Jaroszkiewicz and Strange, 1980).

Conductivity, a , is a macroscopic quantity and is related to the microscopic diffusion constant D, by the Nernst-Einstein equation:

where N is the number density of the conducting species per unit volume, f is the correlation factor, q is the charge per carrier, kg is the Boltzmann constant and T is the operating temperature. The value of f was unknown and so was set to unity. The results for a , obtained from D, are also presented in Table 6.4.

Comparing the calculated conductivity at 1200K with the experimental result at 1000K gives close agreement. This is a useful comparison because, as the mean square displacement plots show, diffusion at 1200K is still minor in comparison to that at higher temperatures. It must, therefore, be representative of an early stage of superionicity, as was the experimental value at 1000K, although a higher value is naturally expected at 1200K.

Thus, this result strongly suggests that the simulation is accurately modelling the diffusion processes of the real system. The results also show that as the temperature is raised the conductivity increases significantly on the 1200K value by factors of 7 and 10 respectively at 1500K and 1700K.

Another result of significance is the ratio of c //c ±. This implies that the conductivity parallel to the £ axis is always slightly less than that perpendicular to the £ axis, although it approaches a 1 :1 ratio at 1500K. This is an interesting result, since although the level of anisotropy would be

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expected to decrease with increasing temperature, the fact that c± is still greater than o//, even at these elevated temperatures, gives credence to the conclusions drawn from the static calculations in Chapter 5. There, results which were applicable for much lower temperatures (theoretically OK) consistently pointed to a preferred motion in the direction perpendicular to the £ axis. These findings were opposite to the experimental conclusions, but perhaps, in view of the supporting MD results, the present interpretation of the experimental data should be reviewed.

Finally, using elementary theory, an estimate for the activation energy for anion diffusion can be obtained from the diffusion constants calculated at each simulation temperature. Using the relations:

loge D = loge D0 - — D 1 e a

' ^ D I = T

V T1

an average value of 0.85 eV is obtained in the superionic region. This is in excellent agreement with the value of 0.84 eV estimated by Roos et al.

(1984) for temperatures > 715K, and again this suggests that the simulation is correctly modelling the real system.