Self-Part of the Van Hove Functions in Porous Lithium Disilicates
The self-part of the van Hove function is defined as follows :
where r, is the positional vector.
In Fig. 7.3a, the self-part of the van Hove functions at 600 К for p = 1.98 is shown. As one can see, the functional form in Fig. 7.3a is different from the Gaussian form and has an inverse power law tails with exponential truncation. It can be regarded as the truncated Levy distribution , although the bumps due to the jump characters of motions make the functional form more complicated. Both localizations within the site and motions with several forward correlated jumps still coexist for Li ions in porous systems, albeit with weaker caging.
If the function, Gs(r, t), (r = |r|) obeys Gaussian form, the solution of the diffusion equation should be
as shown in the example shown in Fig. 7.3b. In this figure, the maximum of the peak moves toward larger r with time, while in Fig. 7.3a, the initial peak remains for a long time even when the fast ions reached the third and/or fourth shells. Although the width of the region of the tail part is rather limited in this case, the inverse power law tail (Levy distribution)  with truncation [15-17] in an exponential form seems to be a good representation as found in other systems. The enhanced dynamics are dominated by long-ranged motions with higher jump rates and it is still extremely heterogeneous, although the NG parameter decreases considerably with the shortened time scales of jumps.
Figure 7.3 (a) Self-part of the van Hove function of Li ions at 600 К for porous lithium disilicate at p = 1.98, where the maximum diffusion coefficient was found. Curves are for 0.1, 0.2, 0.4, 0.6, 0.8 and 1.1 ns from left to right. (b) Example of a behavior expected from Gaussian dynamics, where D = 2.0E'11 (m2 s'1), t = 0.08, 0.16, 0.32, 0.64, 1.28 and 2.56 (ns) were used. Reprinted with permission from Habasaki, J., Ngai, K. L. (2018), Heterogeneous dynamics in nano porous materials examined by molecular dynamics simulations-effects of modification of caged ion dynamics, J. Non-Cryst. Solid., 498, pp. 364-371. Copyright (2018), Elsevier.
Percolative and Cooperative Aspect of the Dynamics
Time scales in porous lithium disilicate can be quantitatively examined by characteristic times of MSD for Li ions. As shown in Fig. 7.4 (Fig. 6.8 in Chapter 6 is modified slightly), the inverse of the characteristic times (in ps"1) of Li ions in the porous lithium disilicate system obtained by the NVE condition shows the maximum atp = 1.98 at 600 К and changes were already found to be at short time scales (in NCL region). In this chapter, the relation between the short-time-scale and long-time-scale motion will be discussed. The changes in caged ion dynamics explain the main features of the dynamics of ionics in porous systems at least in NVE conditions. However, can it explain all features? In the plot of Fig. 7.4, an asymmetry is found in a longer-time-scale behavior and larger change is found in the dense region as shown by a line with arrows. This means that the dynamics in porous systems cannot be explained by the changes in the caging alone. The time scale of this region is characterized by the power law exponent 9 of the MSD before the diffusive regime. Changes in log (tdif/txl) against density at 600, 700, and 800 К are shown in Fig. 7.5a. These values are related to the time scale separation of caged ion dynamics and diffusive dynamics, which is governed by the geometrical correlations among jump motions after the caging region. In this figure, it was found that the minimum is clearer at higher temperatures. Thus, the changes in the trajectories of ions also contribute to the changes of the dynamics with density (and porosity). Another index to represent this separation of the time scale is the slope of the power law region of MSD, 9. The values of 9 plotted against density in porous lithium disilicate systems are shown in Fig. 7.5b for the same three temperatures. The maximum of 9, which corresponds to the maximum of the diffusivity is found at 800 K, but clear maxima were not found for lower temperatures.
This situation can be explained as follows. Results in Figs. 7.5a,b are consistent with each other. The slope of the power law region of MSD is affected by the complexity of the trajectories, as explained in Chapters 4 and 5. In the power law region of MSD, both fast and slow ions are usually found in the original glass, and the slope 9 is affected by the forward or back-correlated motion of jump motions of ions . For the mean behavior of fast and slow ions, 9 = 2/dw is expected, where the exponent 9 is a function of the fractal dimension of the random walk, dw [18, 19], which is an index to represent the complexity of trajectories.
Figure 7.4 Inverse of the characteristic times (in ps"1) found in MSD of Li ions plotted against density of the system for lithium disilicate at 600 K. Filled squares, green: 1 /tyl. Circles, blue: l/tv2. Filled circles, dark blue: l/tdif. Maxima are found even in the shortest time scale. The shape of the plot is asymmetrical especially for l/tdif, because of the large difference of 1 /tx2 and l/tdif in the right part of the plot (shown by a line with arrows). This means the large contribution of the change in the power law exponent, 9, for the slowing down of the dynamics in the original system. Reprinted by permission from Habasaki, J., Ngai, K. L. (2018), Heterogeneous dynamics in nano porous materials examined by molecular dynamics simulations-effects of modification of caged ion dynamics,/. Non-Cryst. Solid, 498, pp. 364-371. Copyright (2018), Elsevier.
If the dw value is large, it means that the trajectory is a more complex one and ion tends to be localized. Therefore, the small value of p (larger value of dw) resulted in the retard of the time scale of tdi[. Thus, the large change in l/tdif was attributed to the geometrical changes of the trajectories, related to the structure of the jump paths and cooperative jumps. The result mentioned above for porous systems is useful to understand the dynamics in glasses, where ions (atoms, molecules) tend to be located near the bottom of the potential surfaces.
Figure 7.5 (a) Temperature dependence of the value log(tdjf/txi) plotted against density for three temperatures. 600 K: Blue, filled circles. 700 K: black, squares. 800 K: red, filled diamonds, (b) Temperature dependence of the power law exponent 9 of MSD plotted against density. The same colors and marks as in (a) are used. The change in 9 also contributes to the enhanced dynamics in the porous system and asymmetry of the behaviors in the diffusion coefficient plotted against density, but the contribution is smaller than the changes in the caging region. The maximum is found at high temperatures but not at lower temperatures. Reprinted with permission from Habasaki, J., Ngai, K. L. (2018), Heterogeneous dynamics in nano porous materials examined by molecular dynamics simulations effects of modification of caged ion dynamics, J. Non-Cryst. Solid, 498, pp. 364-371. Copyright (2018), Elsevier.
For the dynamics of ions in the glassy lithium disilicate system, change in the characteristic times related to the slope of the MSD in the power law region contributes to the slowing down of the dynamics considerably and this situation is changed by introducing pores. Thus, both changes in caged ion dynamics and power law exponent 9 contribute to those in the dynamics in porous systems but differently.
The result suggests that power law exponent plays roles in the existence of the maximum at a high-temperature region as well as caged ion dynamics, while at low temperatures, changes in the caging dynamics are dominant for this. Results for the separation of different contributions of caging and power law exponent 9 examined in the present work clarify the difference of dynamics of high-temperature and low-temperature regions.
The result discussed above is consistent with the previous results  found in an ionic liquid, EMIM (l-ethyl-3-methyl imidazolium)-N03. In this IL, diffusivity at a high-temperature region is found to be dominated by changes in the long-ranged motion, while that at low-temperature region, it is slowed down by the short-ranged motion (caged ion dynamics). This situation is comparable to the porous systems, although these contributions are mixing in some extents. In the porous systems, caging becomes looser and hence, fast dynamics motion observed at high- temperature region tends to continue at lower temperatures. As a result, the slope of the Arrhenius plot at the high temperatures is extended to lower temperatures. In the case of porous systems obtained by NVE conditions, the potential surface related to the longer r region affects the motion of particles.
7.5.1 Explanation of Weakened Slowing Down of Ionic Motion with Decreasing Temperature in Porous Systems
The large decrease of 9 was found in the original system with decreasing temperature (~0.55 at 600 K), and it resulted in a considerable increase in the time scale separation at low temperatures. The result means that the remarkable contribution of geometrical correlations in slowing down of the dynamics in the original glass.
The weaker temperature dependence of 9 explains why the smaller absolute slopes in the Arrhenius plot of diffusion coefficients were found in porous systems . That is, slowing down by complexity of trajectories is weakened in porous materials. The trend observed for 9 in porous lithium disilicate can be explained by the following mechanism. In the lithium disilicate and related systems in the dense region, cooperative motions of several Li ions with long length scale with both forward correlated motion and back-correlated motions are found and the latter causes the slowing down of the dynamics at low temperatures. In porous systems, availability of the next vacant sites for jumps of ions increases and therefore it will decrease the back-correlation probability of ionic motions. The mechanism seems to be just opposite to the slowing down of the dynamics near Tg but is related to the longer r region of the potential functions. For an explanation of enhanced dynamics, both the changes in percolation of jump motions related to 9 and those in caging dynamics for changes in dynamics should be considered.