Relationships between Structure and Dynamics
Thus, the loosening of the cage is considered as the main factor to explain the enhancement of the dynamics observed in the porous materials in the NVE condition. Concerning these findings, some quantitative relations between the dynamics of ions and the changes in the coordination polyhedra were shown in previous works for lithium metasilicate glass. It was also shown that the spectra obtained for the fluctuation of Nv and Nb in the coordination polyhedra are closely related to those for the motion of caged ions in EMIM(N03) .
In Fig. 6.13, partial MSD of Li ions is plotted for each Ny value at initial times t0 for the original and porous lithium disilicate systems (p = 1.98) at 700 and 600 K. Here the partial MSD value for Li ions was obtained for 500 initial times covering 500 ps to 1 ns of time windows in each case. As shown in Figs. 6.13a,b, for the structure with smaller Nv in the original system, the larger squared displacement of Li ions was found at around 1-4 ps. Jump motions of ions accompanied with the fluctuation of the coordination number started there. The initial increase of partial MSD depends on the coordination number Ny and it decreases in the order of [Ny = 2) > [Ny = 3) > [Ny = 4) > [Ny = 5) > [Ny = 6) > [Ny = 7) both at 600 and 700 K. Li ions surrounded by two or three oxygen atoms show large mobility during several ps.
After the changing coordination number with time, behaviors are gradually averaged. While the partial MSDs in the case of Ny larger than 4 are small and this trend continued for a long time. For the structure with Ny = 6, the value in the short time scale was larger than that for Nv = 5 and it decreased later. The value for Ny = 3 also shows the decreases at longer time scales. The difference by the coordination number is larger at lower temperatures in the original system.
As shown in Figs. 6.13c,d, in the case of the porous system with p = 1.98, the initial increases of the partial MSD with the fluctuation of the coordination number are observed much earlier time (~1 ps) as already discussed. The relation between the coordination number and the partial MSD is lost rapidly in the porous system. Nevertheless, a clear relationship between the coordination number and MSD exists but the coordination number alone is not enough to explain the behavior of ions in porous materials. One needs to consider the changes in Nb values and changes in the time scale of the NCL region at the same time. A total number of bonds also play a role in the dynamics of the system.
Figure 6.13 Relation between partial MSDs of Li ions and the coordination number (1VV) determined at an initial time. These values are averaged for 500-1000 initial times, (a) The original system at 700 K. The Nv values distribute from 2 to 6 (from upper to lower at the right part). Red: Nv = 2, Pink: Nv = 3, Blue: Nv = 4, Green: Nv = 5, Orange: Nv = 6. The displacement of the system with a larger coordination number tends to be smaller. The difference by the initial coordination number is kept for a long time even after its gradual changes, (b) Original system at 600 K. Nv changes from 2 to 6. Long-time localization was found especially for the polyhedra with the large coordination number, (c) Porous system with p = 1.98 at 700 K. Nvchanges from 3 to 6 (from lower to upper at the beginning). The difference of mobility accompanied with the coordination number rapidly dissipated (by a change of the coordination number with the first jump). The result for Ny= 7 (brown) is included here, although the numbers of events are not large enough for the statistical treatment, (d) Porous system with p = 1.98 at 600 K. Nv changes as similar manner as in (c) at the beginning. In porous systems, the structures for LiOvare rapidly changed by the jump to the next site. The same colors as in (a) are used in (b-d).
How to Separate Different Origins of Enhancement?
The methods used in the previous sections to separate the possible mechanisms of enhancement are summarized here.
Some of the procedures and concepts are applicable to examine other systems.
- 1. The systems with different porosity were prepared at 600 K. First, the effect of the difference of disorder and order in the structures was eliminated by using glass as a starting material as already mentioned.
- (Please note that the difference by ordering is quite large and should be considered on many occasions).
- 2. The systems at different temperatures with keeping the volume at each density (porosity).
Thus, the volume change with temperature can be neglected here.
3. Mean-squared displacements of Li ions are examined for different densities.
The behaviors in each time-region in porous systems are compared to clarify how events of each time region contribute to the enhancement.
4. Diffusion coefficients are plotted against density (porosity) and/or temperature.
The existence of the maximum in the former and slower slopes in the latter was confirmed.
To learn temporal aspects of dynamics, the following analyses were used.
- 1. To clarify the changes in time scale, each characteristic time in the mean-squared displacements of Li ions was examined [6, 7]. The contribution of the different mechanisms becomes clearer by it.
- 2. Since the enhancement already started in the early time (NCL or caging) region, this means the changes in caged ion dynamics being dominant for the enhancement. Then structures of coordination polyhedra were examined to learn the changes in the cage.
To specify the region characterizing the enhancement, the following analyses were used.
1. Direct observation of trajectories was done to determine spatial region where enhancement occurred.
In the case of porous lithium disilicate examined for some slices, it was found that the enhancement is not necessarily along the boundary, although it does not exclude the possibility of the enhancement in layers with a larger thickness.
2. Structures of coordination polyhedra of Li ion were characterized by using the concept of the geometrical degrees of freedom of the polyhedra because the fluctuation of the cage and its decay are dominated by the degrees of freedom.
Enhancements of the dynamics are known to depend on the compositions, pore sizes, preparation methods and so on, and therefore the examination of mechanism in different conditions will be required, although one can expect some common mechanism underlying [8, 24].
In the present work, our attention was mainly focused on the single-phase system simulated in NVE conditions. However, the methods used to separate and characterize the enhanced dynamics would be expected to be useful for other systems with and without some modifications.
Comparison with Porous Lithium Disilicate and Other Porous Composites
In several nanoporous or mesoporous composite systems, the maximum of the transport properties observed in experiments tend to be explained by the percolation theory [31, 59-61].
For example, in Ref. , the existence of the maximum conductivity has been explained by the percolation theory with two critical points. In this (three-resistor) model, existence of enhanced interface conductivity is assumed. With increasing contents of insulator particles, the conduction paths percolate and further increase causes immersing of conductive materials in the insulator, which resulted in the decrease of conductivity (see Chapter 3 for more details).
On the other hand, in our results for porous the lithium disilicate systems in NVE conditions, the importance of the caged ion dynamics is found.
There are some differences between our systems and experimental nanocomposites discussed in Chapter 3, where enhanced dynamics have been observed. In several composites, conductivity maxima were obtained with increasing concentration of porous media (or insulator), while our observation of the maximum is diffusion coefficient plotted against density, so that only the effect of introducing pore is considered, without remarkable changing Li contents in NVE condition.
In the enhancement observed in composites, both changes of porosity and the content of Li ions occur at the same time. Therefore, the enhancement effect found in the experiment in such conditions exceeds the effect of decreasing the contents of mobile ions.
The difference in the thermo-dynamical conditions should also be taken into account because, in experiments, systems tend to be measured in NPT conditions.
Due to these differences, our results do not necessarily exclude other explanations of the enhancement for the composite immediately. For example, a layered structure of ions is worth to consider for effective conduction.
In spite of the limitation of the direct comparison, the methods used for the separation of the possible factors as shown in this chapter will be useful to understand the mechanism and applicable to other systems with more complex structures.
We also note here that insulator introduced in composites might have a role like pores introduced in NVE conditions. That is, it can modify the coordination polyhedra of ions.
Some cautions are required for the separation of possible origins of enhancement. Sometimes, limited numbers of factors and/or mechanisms are assumed from the beginning and one may overlook other possible causes. Furthermore, these possible factors of the mechanism are not necessarily separable completely.
Therefore, careful treatment of each problem is required, although the treatment of the system in this chapter is applicable to other systems.
In the porous lithium disilicate system, enhancement of the diffusion coefficient having the maximum in the medium density region is predicted by the molecular dynamics simulation in the NVE condition. In this chapter, procedures and concepts to separate the possible factors contributing to the enhancement were discussed and exemplified for the study of porous lithium silicate systems. The porous model system for MD is prepared by the expansion of the system started from the glassy lithium disilicate, so that the effect of disorder (difference of ionics in crystal and glass) was separated beforehand. The enhancement of the diffusive motion of Li ions is found to occur not only at the boundary of pores but also in the bulk part of the systems. The enhancement of the diffusion coefficient (and hence that of the conductivity) starts from an early time (NCL) region of the mean-squared displacement and this means that the loosening of the cage contributes to the enhancement. Changes found in the distribution of the coordination number and geometrical degrees of freedom of coordination polyhedra (formed by the oxygen atoms around Li ions) are consistent with this view.
However, our system is not a composite and the ratio of Li ions and Si atoms is kept unchanged, when the pore was introduced. Furthermore, in this chapter we discussed the simulation in NVE conditions not in NPT conditions. Therefore, the systems are not directly comparable to the Liang effect. Nevertheless, it is interesting to point out that the observed maximum in the medium density region of Li ion and smaller absolute slopes in Arrhenius plots are quite similar to those observed in composites of ionic conductors diluted by silica or alumina. We would like to suggest that the structural changes of MOx (M is mobile ions) or related polyhedra should be considered even in the composites and that the role of alumina or silica is not only the separator of paths, but also the modifier of the density of Li ions and geometrical structures of the cage.
As found in this chapter, nanoionics in porous materials is an interesting problem not only for nanotechnology but also for the fundamental research to understand the complex ion-dynamics and the relation to the glass transition problem. It provides a new platform to understand the heterogeneous dynamics in supercooled liquids and glasses.