Enhanced Dynamics in Lithium Disilicate Systems in NVE Conditions

As discussed in Chapter 2, enhancement of the dynamics (or an opposite case) has been reported for several porous systems and/or related composites as shown, for example, in [24].

For a better understanding of the mechanism of the enhancement of the dynamics in a nanoporous system, molecular dynamics (MD) simulation of porous lithium disilicate has been performed in refs. [6, 7] and is the focus of the present chapter. As shown in detail in the next section, enhancement of diffusion of Li ions is found in the porous lithium disilicate examined in NVE conditions.

6.3.1 Possible Origins of Enhancements

To explain the mechanism of the enhancement in porous ionic conductor, including several composites such as Lil in alumina or silica, several different scenarios have been proposed.

Although the following concepts and/or explanations discussed for nanocomposites (Chapter 3) are not necessarily directly applicable for porous lithium disilicate treated in NVE conditions, it is worth to examine because the existence of some common mechanisms underlying can be expected [8, 25].

So far, enhanced dynamics in several porous systems have been explained by

• 1. a concept of charge transfer and space charge effects [26, 27];
• 2. an effect of epitaxial strain in opening diffusion paths [28];
• 3. the influence of atomic and electronic reconstruction at the interface in stabilizing new phases with increased carrier concentration [29, 30];
• 4. the percolation theory with assuming high conductivity regions on the boundary [31].

Some of the possible origins (like impurity such as water) explained in Chapter 3 can be excluded in well-controlled conditions in MD.

Most of these explanations seem to be related to the characters of the boundaries of the particles of insulator rather than related the porous nature of the components of composites or mixtures. However, a question remains for the role of nanoparticles and/or nanoporous materials.

MD Simulations of Porous Lithium Disilicate

6.4.1 Methods of Simulations for Porous Lithium Disilicate Systems

First, the characteristics of the potential parameters used in MD simulations will be discussed, because the validity of the mechanism discussed in MD simulation depends on the quality of them considerably.

Our potential parameters [32] have been previously developed based on the ab initio MO calculations and the model covers a wide range of potential surfaces. Based on our potential model, lithium disilicate system and related systems have been examined successfully from many points of view by several groups including our groups [8, 12, 32-39] in the molten, glassy, crystalline states. Therefore, it can be used for the comparison of systems with different states, including porous ones although it was not optimized for the study of porous systems. Of course, an examination of applicability is desirable for each purpose.

For the preparation of lithium disilicate systems with several different densities (porosities), the system and the coordinate were expanded gradually. That is, an expansion of the system by scaling of factor 1.005 times (for both the axis length and the coordinate) was repeated until the density to be examined was obtained. Densities p examined are 2.47 (original), 2.30, 2.13, 1.98, 1.84, and 1.58 (density is given in g cm"³). The experimental density of the system in the glassy state was reported to be 2.35 [40]. After the expansion of the system, each system was equilibrated during 300,000 steps run in the NVT condition. Then further 300,000 steps run were done in the NVE condition. If temperature increased during the NVE run (due to non-equilibrium nature of the system), short NVT runs (~30,000 steps using Gaussian thermostat) were inserted between NVE runs. The run for 300,000-4,000,000 steps, which was performed in the NVE condition after the equilibration, was used for further analysis. For temperature dependence of the dynamics (600-1000 K) in porous materials, simulations have been started from the porous systems at 600 К and the same system volume was used at each target temperature. In some cases, additional runs for different density and/or at different temperatures have been performed.

6.4.2 Relation between Porosity and Density in Porous Lithium Disilicate Systems

In refs. [6, 7], the values of the porosity P₀ were determined by geometric considerations using the ionic radius. Here we used

Si: 0.40, О: 1.21, and Li: 0.8 as atomic or ionic radii (in A), which are consistent to the maximum position of the first peak of g[r) for Si-0 (1.63 in the original system at 600 K) and Li-0 (2.04 in the original system at 600 K) pairs. The method used is not for “available space” but the “precise amount of free space.” The method is not necessarily suitable when the overlaps of atoms are non-negligible but seems to be suitable to treat porous systems.

For the systems with p = 2.47, 2.13, 1.98, 1.84, and 1.58 (in g cm"³) at 600 K, porosities are determined to be, 56.62, 62.59, 65.34, 67.80, and 72.29, respectively. The porosity, P₀ (in %), was found to be (negatively) proportional to the density p (in g cm'³) of the system, that is,

Hence, the results related to the different porosities of the systems will be shown simply as the function of the density.

6.4.3 Trajectories of Ions and Atoms in a Porous Systems Compared with theOriginal System

In Figs. 6.2a,b, trajectories (Li ions, Si and 0 atoms) in original and porous lithium disilicates during 100 ps run at 600 К are shown for a slice with a width of 10 A of MD cell, respectively. The density of the porous system is p = 1.98, where the maximum of the diffusion coefficient at 600 К is found as shown later.

In Fig. 6.2b, enhanced dynamics are observed in the bulk part and not necessarily remarkable at the boundary of the clear pores. They are accompanied with the larger motions of Si and 0 atoms than the original system. In Fig. 6.2c, trajectories for longer runs (5 ns) in the original system are shown, where the MSD of Li ions is comparable to the value found in (b). In Figs. 6.2b,c, ion channels are observed by the overlap of trajectories of Li ions with different time scale region. Thus, the enhanced diffusivity in the porous system is found. Some ions are located for a long time, while fast ions cooperatively move in the channels. Therefore, the enhanced ionic motion is heterogeneous one as known for the original system as well as other ionic systems.

Details of the heterogeneous dynamics in porous lithium disilicate systems will be discussed in Chapter 7.

Figure 6.2 (a) Trajectories of ions and atoms during 100 ps (100,000 steps) run of original Li₂Si₂0₅ glass at 600 К for a slice with a thickness of 10 A of MD cell, (b) Enhanced dynamics found in the trajectories of ions in porous lithium disilicate during 100 ps run near the maximum of the mobility at 600 К with p = 1.98. Green: Li ions, Blue: Si atoms and Red: Oxygen atoms. Not only motion of Li ions, mobility of 0 and Si atoms in the porous system are larger than the original system, (c) Trajectories of ions and atoms during 5 ns run of original Li₂Si₂0₅ glass at 600 К for a slice with a thickness of 10 A of MD cell of the system, where the MSD (10 A²) of Li ions is comparable to the Li ions in (b). In each case, data are plotted with interval of each 500 fs. Enhanced dynamics are clearly found in porous lithium disilicates. In these figures, diffusive paths are not necessarily located near the boundaries. Trajectories of Si and 0 are spread in the porous system and it suggests the loosening of cage formed by 0 atoms around Li* ions.

6.4.4 Enhanced Dynamics and Existence of Maximum in NVE Conditions

As shown in the following sections, dynamics of ions in porous lithium silicate in NVE conditions show common characteristics as those found in other porous ionic systems treated in Chapter 3. It includes the existence of the maximum of the enhanced diffusivity in the medium density (or composition) region and smaller slopes in Arrhenius plots of the diffusion coefficients. Since the enhanced dynamics are observed in several composites of lithium silicate and ZnO [41] as well as in the system Lil-Al₂0₃ [42], it can be expected common physics are underlying these behaviors [8, 25]. Common behaviors do not necessarily guarantee that the same explanation holds in all systems. However, the methodology of separating the possible mechanisms and examining the nanostructures and nanoionics in this system will be useful in other composites. In the field of nanocharacterization, where molecular dynamics simulations can be expected to contribute, it seems that new researchers are waited to enter.

Comparison of MSD of Li Ions in the Porous and Original Systems

From the changes in the trajectories, it was suggested that the caging dynamics are affected by the decreased density (and porosity). In this case, we can expect that the MSD of the Li ions, as well as those of Si and О atoms, will change. To separate the possible origins and quantitative assessment of the enhancement of diffusivity (or conductivity) with porosities, time developments of MSD of systems with different density were examined.

Several Characteristic Regions of MSD of Li Ions in Original Lithium Disilicate Glass

Mean-squared displacement (MSD) of ions or atoms can be obtained from the MD simulations data using the following relations.

where r,- is the positional vector

As discussed in Chapter 5, the characterization of the motions of Li ions in each time region has been done with the help of van Hove functions, fractal dimension of random walks, intermediate scattering functions for both lithium metasilicate and lithium disilicate [6, 8, 33, 39, 43]. The result is briefly summarized below because these time regions are compared with each other in porous systems.

Four distinct time regions for MSD of Li ions are found in the original (before expansion) Li₂Si₂0₅ system in the glassy state. MSD of Li ions at 600 and 800 К in this system is shown in Fig. 6.3.

Figure 6.3 Top: MSD of Li ions in original Lithium disilicate glass at 600 К (lower curve) and 800 К (upper curve). Dashed and dotted lines (fitted to power laws) for Li ions at each temperature are for NCL and power law regions, respectively. Bottom: MSD of Si (Blue curves) and 0 (Red curves) at both temperatures. At 600 K, nearly flat (NCL) region is expanded to ~30 ps and the power law region starts at ~3 ns. Flat region of MSD for Si and 0 continues up to ~ns at both temperatures.

Region I (NCL): After the initial increase of MSD due to the local motion of the ion site, a nearly flat region assigned to the nearly constant loss (NCL) is found. This time region is located between approximately 0.2 and 2 ps in the case at 800 К where the MSD increases with time as with a * 0.07. In this time region, ions are still caged by surrounding oxygen atoms and further shells of ions and atoms. Region I is characterized as the caging region, where the anharmonic motion of ions with intermittency was observed [44, 45].

Changes in this characteristic time region are useful to understand the mechanism of the enhancement of dynamics of ions. If the time region becomes longer, this means that the cage is tight, while if the region becomes shorter, it means that the caging is loose and jump motions to the next shell begin. Therefore, the length of this time region is related to the jump rate of the caged ion.

Region II: Primitive jumps (the first jump of each ion after the starting observation for some ions) are found in this intermediate time regime of about t_xl < t < t_x2. The MSD rises more rapidly than t° of the regime (I) in a double logarithmic plot. This increase of MSD is nearly proportional to time as easily found in the linear scale plot.

Region III: Power law behaviors of MSD are found in the region t_x2< t< t_dif. That is, MSD depends on t⁶ with 9 < 1. Both fast and slow ions are found here and the slope 9 is affected by the back-correlated motion of jumps considerably in the glass. That is, the region proportional to time in region II ends by overlaps of jumps returned to original sites and jumps to further shells. In some porous systems with large 9 (>0.8), the linear function to time can be a good approximation for representing MSD.

Depending on the purpose, a correction might be required for the contribution of motions within a cage to MSD for examining the slope in the double logarithmic plot.

Region IV: Diffusive region of ions is found after t_dif, where MSD reaches the square of the typical neighboring distance of ion sites and the MSD shows a linear t dependence in that region.

With decreasing temperature, each time region becomes longer and longer. For example, the diffusive regime is found after 2 ns at 800 К for Li ions, while it is found after ~ 15 ns at 600 K.

From the slope after t_dif, the diffusion coefficient of ions can be determined by the Einstein equation [46]:

where D is the diffusion coefficient and r, is the positional vector.