Caged Ion Dynamics in NCL Region
Localized motions of ions in NCL regions
The characteristics of the caged ion dynamics are clearer at lower temperatures. The motion within a cage tends to continue for a long time. This situation is clear when individual motions of ions are examined. Squared displacements of localized motions of Li ions in NCL regions for arbitrarily chosen 6 ions in lithium disilicate are shown in Fig. 5.4 . Each ion has a different color in each figure. The impression gathered from the figure may be different depending on the time intervals of data points used. Figures 5.4a,b show motions at 300 К during the time scale of NCL for the same ions but at different time intervals. In both figures, the active and quiet states intermittently appeared for the two ions shown in red and blue. Other ions are more localized. Strong heterogeneity of the motion is also observed at 500 К during the time scale of NCL (10 ps) in Fig. 5.4c. Here strong heterogeneity means the coexistence of ions with large displacements and immobile ions, although almost ions are still within cages. There are some remarkable characteristics in the motion of ions in the caging region.
Figure 5.4 Squared displacements of arbitrarily chosen 6 Li ions during the time scale of the NCL region, (a) At 300 K, (1250 data points for each ion), (b) 300 К (125 data points for each ion.) (c) 500 К (125 data points for each ion). Large displacements are found intermittently. The motion is not a harmonic vibration. Some ions show large squared displacements, while other ions are strongly localized. Adapted with permission from Habasaki, J., Ngai, K. L. (2017), Nearly constant loss (NCL) in lithium metasilicate glass at low temperatures—anisotropic and dynamical caging from molecular dynamics simulations, J. Phys. Chem. C, 121, pp. 13729-13737. Copyright (2017) American Chemical Society.
- 1. Ionics in the glassy state are dominated by jump motion of ions and it has intermittent nature related to chaos , where both laminae and turbulent states appeared at different intervals. Similar intermittent^ is found in many systems including the behavior of nature such as rainfalls . The motion is also sporadic [39, 40] so that rare events such as cooperative motions of several ions occur. Even though low probability, such cooperative and intermittent motions tend to show a remarkable contribution to the mean value of dynamics because of the long-range scale of motion with forward correlated motions. In the low- temperature regions, pre-jump motions contribute to MSD, while jump motions become rarer.
- 2. The motion of ions is rather deterministic as explained later. A large deviation from the harmonicity in the motion of ions in cages exists as shown in the following section and anharmonicity is suggested to be the cause of the NCL.
Anharmonic and intermittent characters of caged ion dynamicsin nearly constant loss region:shapes of the density profile oflocalized ions
Here we will examine what determines the motion of ions in cages.
The local motion of ions in the NCL region is affected by the positions of surrounding atoms and hence the motion is governed by the geometry of cages and their deformations. That is, the shape of the set of points in each potential well (site) in these figures is not necessarily spherical but is ellipsoid-like with some distortions as shown in Fig. 5.5, where an example of positions visited by Li ions at 300 К in the NCL regime is shown. Deviation from the spherical shape is observed and it was correlated well for the structure of cages formed by surrounding oxygen atoms [8, 41, 42]. As a natural consequence, the transformation of cages is governed by the geometric rules of coordination polyhedra.
Characteristics of the anharmonic motions are summarized as follows.
1. Within their cages, Li ions near the edge of the site tend to be sharply drawn back within the central part of the site. These motions correspond to the intermittent spike-like motions found in Fig. 5.4. Therefore, on a longer time scale than vibrations, the ion exhibited anharmonic motion. Jumps, including trial ones, are related to deterministic chaos as shown by the phase-space plot. It is previously clarified for each ionic motion in lithium metasilicate and also in ionic liquid. [6, 43]. That is, anharmonic motions of ions in cages are followed by jumps having the deterministic nature.
Figure 5.5 An example of the three-dimensional plot of the positions of a Li ion during the MD run at 300 K, which exhibited the non-spherical shape. It means a contribution of the higher order terms of the multipole expansion. Positions are accumulated for 500 ps, that is, during the NCL region. The point is plotted at every 0.4 ps.
- 2. This non-spherical shape of density of ions means contributions of dipole and quadrupole of the ionic interactions, and therefore the charge-charge interaction is not enough to represent the motion within the cage. Then, the multipole expansion of the potential field will be useful to represent such situations. In other words, anharmonicity of the potential was considered as an origin of the NCL [6, 7, 10], because the contents of the potential in the NCL regime can be represented by the multipoles.
- 3. As shown in several works for the lithium silicates [10, 42, 43] as well as an ionic liquid [44, 45], the local field in the caging regime is dynamically fluctuating, where the large displacement of Li ions to the next site is accompanied with the changes in the coordination number of surrounding oxygen atoms. A similar result is also observed in a colloidal system  in spite of the fact that the system is quite different from the ionic conductor examined here.
Lower than 500 К in lithium disilicate systems, it is not easy to observe the diffusive motions within ~ns order simulations. However, the comparison with the temperature dependence of dynamics obtained by MD and that of experimental values is successful [6, 47], and this fact means the caged ion motions observed in MD are comparable to experimental ones. The conductivity is still measurable down to ~300 К or below in experiments.
As shown in Chapters 6 and 7, the enhancement of the dynamics of Li ions is found by introducing pores to the system in the NVE conditions. The enhancement is found to be accompanied with the structural changes of cages (coordination polyhedra), which causes a shortening of the time scale of the caging region. Thus, the importance of the caged ion dynamics is found in such cases. In some theories or models in ionics or glass transition, the importance of the NCL region tends to be neglected, while it is pointed out in the Coupling Model by Ngai [3, 6].
As found in several properties in each region, ions escaping from ion sites show jump (hopping) motions to neighboring sites and after power law regions, finally they contribute to diffusive motions. Therefore, the role of motions in the NCL region is determining jump rates, and motions in cages are regarded as precursors of the following processes.
Non-decaying part obtained from the Van Hove function found in the NCL region at low temperatures
The van Hove function can be used for further characterization of the time-regions found in MSD. The characteristics of the MSD in the NCL region at lower temperature regions are examined by the self-part of the van Hove function defined as follows [48, 49]:
In Fig. 5.6, the self-part of the van Hove function at 300 К is shown, where r = |r|. At temperatures ~500 К and above, development of the second and further shells by jump motion is clear within a time scale of observation (~several ns), while at 300 К contribution at around r = 3 A is found only at the longest time. Therefore, at low temperatures, Li ions are almost trapped within original sites during the observation time. This time-region becomes considerably longer at low temperatures than those at higher temperatures.
Figure 5.6 Self-part of the van Hove function of Li ions in lithium disilicate at 300 К is shown, where r = | r | in a log-log plot. Curves from inner to outer are for 0.8, 8, 39.2, 80.0, 560, and 1520 ps, respectively. Time development is small but is clear. Contribution at around 3 A by the first jumps is found at the longest time in the plot.
Further details of the motion in the NCL region are discussed here for the lithium metasilicate system using both MSD and cage decay function. Another evidence of lengthening of the NCL region at low temperature will be shown as follows. In ref. , the temperature dependence of MSD of Li ions, including lower temperature data (at 100 K) and corresponding cage decay functions Ncage(t) [1, 46] were examined. These results are shown in Fig. 5.7a,b.
In Fig. 5.7a, the temperature dependence of the MSD of Li ions is shown. As expected, with decreasing temperature, each time-region becomes longer and longer. At the low-temperature regions, ions are localized for a long time because of the long lifetime of the cage.
Figure 5.7 (a) Mean squared displacement of Li ions in Li2Si03 at 7 temperatures, (b) Corresponding cage decay function measured by the area under the first peak of the self-part of the van Hove function (normalized to 1 at the initial time) for the same temperatures. The function decays by the net jump to the neighbor or further shells. The data of the same temperature are represented by the same color. Adapted with permission from Habasaki, J., Ngai, K. L. (2017), Nearly constant loss (NCL) in lithium metasilicate glass at low temperatures— anisotropic and dynamical caging from molecular dynamics simulations, /. Phys. Chem. C, 121, pp. 13729-13737. Copyright (2017) American Chemical Society.
The cage decay functions are defined as follows:
Here, the summation is taken within a distance rc of the initial cage (less than 1.7 A). The cut-off value corresponds to the first minimum of the self-part of the van Hove function when the clear second peak exists (500-800 K). The area under the curve of the first peak of the self-part of the van Hove function is proportional to the number of particles in cages. In this figure, the area is normalized to 1 at the initial time.
During the NCL region, essentially no ion can jump out of the cage, although in the case of ionically conducting glasses near room temperature, there is still a low probability for jumps.
As shown in Fig. 5.7b, Ncage(t) at each low temperature does do not decay for a long time. At the lowest temperature, the NCL region has covered the entire observation time during the MD runs. It is worth to mention that the structural arrest observed in molecular dynamics often is concerned with the lengthened NCL (the caging region in non-ionic systems) and not necessarily mean the loss of diffusivity.
Therefore, careful treatment and serious consideration of this region found by molecular dynamics (MD) simulations in glassforming materials are essential for a fundamental understanding of the glass transition.
For calculations of transport coefficients such as diffusion coefficient and corresponding DC conductivity, one needs to use the long-time region after tdif, although it is not necessarily easy due to a limited simulation time up to ~ns in many cases. The covered time scale may be restricted by the real-time required to do such simulations.