Common Behaviors of Porous Lithium Disilicate and Composites Showing Enhanced Dynamics
The next step is to treat the transport coefficients quantitatively. For this purpose, diffusion coefficients of ions and atoms are plotted against the density of the system (Fig. 6.7).
In Fig. 6.7a, diffusion coefficients of Li ions determined by the Einstein equation are plotted against the density of the systems at 600, 700, and 800 K. The maximum of the diffusion coefficient is predicted at around p = 1.98, where the diffusion coefficient is more than one order higher than the original system at 600 K.
Figure 6.7 (a) Diffusion coefficients of Li ions in porous lithium disilicate in the NVE condition. The largest density (the right end point) corresponds to the original system before the expansion. Blue triangles: 600 K, purple diamonds: 700 К and red open squares: 800 K. The maxima are found at around p = 1.98 (in g cm"3) for all temperatures, where the diffusion coefficients are predicted to be one order higher than the original systems. The enhancement is larger at lower temperatures in the log-scale plot, (b) Temperature dependence of diffusion coefficients of Li ions for each p value. Inverse triangles (red): p = 2.47, system in the glassy state without expansion. Filled circles (orange): p = 2.13, open circles (green): p = 1.98, open squares (blue): p = 1.84 and filled squares (purple): p = 1.58. The solid black line indicates the approximate position of experimental values for lithium disilicate glass gathered in Ref. . Smaller negative slopes are found as in other porous composites. Reprinted from Habasaki, J. (2016), Molecular dynamics study of nano-porous materials—Enhancement of mobility of Li ions in lithium disilicate in NVE conditions, J. Chem. Phys., 145, 204503(1-11), with the permission of A1P Publishing.
As already shown qualitatively by MSD, the maximum of the transport properties was found for the diffusion coefficient of Li ions in porous lithium disilicate systems simulated in the NVE condition.
The shape of curves connecting different densities (porosities) slightly changes with temperature. This trend is characterized mainly by the larger temperature dependence of the diffusivity in the large density region. The existence of the maximum is usually found in the composition dependence of the composites showing enhanced dynamics as shown in Chapter 2 and/or in [41,42].
Temperature dependence of the diffusion coefficients in these systems is shown in Fig. 6.7b. The similarity with other porous composites is also found in the temperature-dependent behavior. The result of the original system shown by the dotted line is comparable to the trend shown by the solid one, which represents the approximate positions of experimental values of Li ions in the original glass . Interestingly, smaller negative slopes in the temperature dependence in porous systems are found in this figure. As in porous composites showing enhanced diffusions and/or conductivities, the £-(1/7) plot shows the smaller negative slope than the original system. For example, see Ref.  for common trends for several systems. This fact is promising to use the porous system as high conductivity materials in the lower temperature region.
These common trends also suggest the possibility of the existence of the common physics for the origin of the enhancement and these characteristics. That is, we can expect that porous materials act as supports of mobile ions in composites and they will cause a modification of caging dynamics, although the mechanism should not be the same. A porous system will also modify jump paths as shown later. Of course, the difference in the conditions used in MD simulations and experimental ones also should be considered, since experimental results tend to be obtained in the NPT condition instead of the fixed volume in the NVE condition in the present system for further comparison.
Comparable behaviors of the conductivity to the diffusion coefficient for the existence of the maxima and/or similar trend in the slopes in Arrhenius plots are naturally expected.
6.6.1 Contribution of Other Factors: Percolation Theory, Fractal Dimension of RandomWalks and Cooperative Jumps
How can we distinguish the possible mechanisms, which affect the MSD and hence the diffusion coefficient? Here we will examine the characteristic time scales defined for MSD to clarify the origin of the enhancement and the cause of the maximum in the porous lithium disilicate. Causes of the asymmetry of the curve for the diffusion coefficient against density plot (Fig. 6.7a) are clearer at lower temperatures, and dynamics are more enhanced in a longer time scale in some cases. These findings suggest the existence of another factor (or factors) playing roles to determine diffusivity. One of the possible explanations concerns the geometrical changes of the trajectories, related to the structure of the jump paths and cooperative jump motions.
This is because the exponent 9 is a function of the fractal dimension of the random walk , dw which can be used as an index to represent the complexity of trajectories [53, 54]. Here, the exponent contains only the geometrical term. For a free random walk, one can expect dw = 2, while for more localized motions the value becomes larger. In the trajectory of fast ions, the trajectory becomes more linear, and the value becomes less than 2 . Larger dw value means that the trajectory is more complex and the ion tends to be localized and so that the small value of в (larger value of dw) results in the retardation of the time scale of tdif.
6.6.2 Fractal Dimension of Random Walks
The fractal dimension of the random walk, dw was determined from the trajectories of each ion by using a divider method . This dimension represents the complexity of the trajectory and is defined by the following equation:
In this analysis, trajectories were obtained by the MD simulation during a run of time t,.un. For the determination of dw for heterogeneous dynamics, the long run should be used for sampling of rare events. If the time scale is long enough, exchanges between fast and slow ions will also be observed.
6.6.3 Multifractal Random Walks Found in Lithium Silicates
In the lithium metasilicate, the slope of the NL-Lr plot was found to change at around 3 A at lower temperatures. The dw was named dwl for a shorter length scale (<3 A) and dw2 for a longer length scale (3 < Lr < 10 (A)). From the theories of random walks, 9 = 2/dwis expected for a simple random walk as already mentioned . However, the existence of different length scales found here means that the value is for the average of slow and fast ions forming multifractal random walks [55-59]. Since the fractal dimensions of the walks are closely related to the structures of jump paths, it is interesting to consider the relation with percolation theories [60-62].
For the porous media, the effects of the enhancement of transport by the contribution of low-frequency elastic waves also have been pointed out in .
6.6.4 Percolative Aspect of Jump Paths
Generally, the concept of percolation is quite useful. However, in the case of the enhancement of porous lithium disilicate system examined in the NVE condition, the main cause of the enhancement, as well as the existence of the maximum of the conductivity, cannot be explained by this kind of theory alone, although the effect concerning the path still exists. Instead, it can be mainly explained by the changes in the caging dynamics at least in the present system in the NVE condition. Since the slow dynamics in the densely packed systems are affected by the back correlated motions of ions, the geometrical characteristics of paths are the non-negligible factors to control dynamics, as well. Here we consider the percolative aspect of the jump path of lithium ions in the porous lithium disilicate system. To learn the effect of geometrical factors such as correlation among jump motions of ions with separating from the changes in caged ion dynamics, characteristic times of MSD are examined. Effects of caging dynamics are shown by the changes in txl, while the changes in the power law exponent 9 of MSD is shown by the time scale separation between tx2 and tdif, based on the physical meaning of each stage of ionic motion (see Chapter 5). In the following section, we will make a quantitative assessment of these factors and relations between them.
6.6.5 Characteristic Times in Porous Systems
To understand how changes in dynamics are brought, to specify time scales of motions in porous lithium disilicate is useful and they can be quantitatively examined by characteristic times txl, tx2 and tdif (in ps), which separate different regions found in MSD for Li ions.
Figure 6.8 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 /txl. Circles, blue: 1 /tx2. Filled circles, dark blue: l/tdif. Maxima are found even in the shortest time scale. Each error of datum in this figure is estimated within the size of marks, except for the value at the lowest temperature for l/tdif, where the value is for the largest limit, due to the overlap of an aging process. Thus, the shortening of the time scales contributes to the enhancement of the diffusive motion. See also Chapter 7 for further explanation of this figure. Reprinted with permission from Habasaki, J., Ngai, K. L. (2018), Heterogeneous dynamics in nanoporous materials examined by molecular dynamics simulations—effects of modification of caged ion dynamics, /. Non-Cryst. Solid, 498, pp. 364-371. Copyright (2018), Elsevier.
As shown in Fig. 6.8, 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 at p = 1.98 (g cm'3) at 600 K. The shortening and lengthening of the characteristic time txl found to correspond to the loosening and tightening of cages, respectively. For the latter behavior, more details will be shown later. Here our attention is turned toward the maxima. Changes in dynamics are more enhanced on a longer time scale. Especially, the change of l/tdif between p = 2.15 and original system is larger compared with those for l/tx2 and l/txl at high-density region. Asymmetry of the curve similar to this situation was clearly found in the diffusion coefficient plotted against density as shown in Fig. 6.8. This result is consistent with the observation that the activation energy of the tdif is comparable to that of the diffusive motions .
Similar behaviors with maxima were observed at 700 and 800 К as shown in Fig. 6.9, The maxima were found even for l/txl at higher temperatures, although the changes in short time scale at 800 К are small.
Figure 6.9 Inverse of the characteristic times (in ps'1) found in MSD of Li ions plotted against density of the system for lithium disilicate at (a) 800 К and (b) 700 K. Upper, medium and bottom are for l/txl, l/tx2 and 1/tdif, respectively. The maximum of the values has been already found in the shortest time scale even at higher temperatures. The change near the dense part seems to be weakened at the higher temperature.