# Dynamic Heterogeneity: Coexistence of Slow and Fast Ions

Here, the term "dynamic heterogeneity” is used to imply the coexistence of both fast and slow diffusive dynamics of particles (ions, molecules). In this section, dynamic heterogeneity found in ionically conducting glasses is discussed, although the phenomena are common with other non-ionic glass-forming materials. Often, heterogeneity is regarded as the important properties to characterize glass transitions.

The concept is not necessarily the same as the fast and slow relaxations, because fast relaxation is usually accompanied with the local and/or rotational motions. One should keep in mind that the beginning of the researches of dynamic heterogeneity, the same or similar concepts were sometimes described by different terminology.

Self-part of the van Hove functions at 1673 К (in the molten state) and 800 К (near the glass transition temperature) for Li ions in lithium metasilicate, Li_{2}Si0_{3} [60] are shown in Fig. 5.10.

In the molten state at 1673 K, the function is well represented by the Gaussian-type diffusive motion, while the large deviation is observed at lower temperatures.

In the power law region of MSD, the distribution of the displacement or corresponding behavior in the self-part of the van Hove functions shows an inverse-power law tail of the following form for the large *X* (=d| r |/dt or r) region,

with an exponential tail. This part of the van Hove functions is related to the Levy (stable alpha) distribution (when *a <* 2) as found in molten and glassy lithium metasilicate [51].

The broadening of the peak found in a short-time region represents the motion within a cage and the fluctuations of the cage itself, while the decreased area of the first peak represents the motion to the next shell. With the decay of the first peak, the tail part of the function develops with elapse of time. The long tail with large r is related to the long-ranged motion as found in the trajectory, i.e., by fast ions with successive jump-like motions.

Figure 5.10 (a) Self-part of the van Hove function of Li ions in lithium metasilicate at 1673 К (for t = 16, 32, 64 and 128 ps from left to right in the curves of long ranged region). Smooth curves are for the Gaussian- type distribution having the same diffusion coefficient as the observed function. Same colors are used when the time is the same. Deviation from Gaussian form is observed only at short-time region, (b) Self-part of the van Hove function of Li ions in lithium metasilicate at 800 К (for *t =* 50, 100, 200, 400, 800 ps from left to right in the in the curves of long ranged region) in lithium metasilicate, respectively. Both non-decaying part and jumps to the long length coexist. Reprinted with permission from Habasaki,). (2007), On the nature of the heterogeneous dynamics of ions in ionic conducting glasses, *J. Non-Cryst. Solids,* 353, pp. 3956-3968. Copyright (2007), Elsevier.

As already mentioned, the fractal dimension of the random walk with *d _{w} =* 2 corresponds to the free random walk (Brownian diffusion). Fast ions with accelerated motion show smaller

*d*value (

_{w }*d*2), which means the trajectories are more linear like and this also means the acceleration is achieved by the geometrical correlation among successive jumps.

_{w}<Localization of particles (ions, molecules) is governed by the complexity of paths and motions of them. Localized motion tends to have larger *d _{w}* (> 2) values, which means the trajectories are folded many times. On the path with small dimensionality, such motion can be understood as "fracton” [61], coming from the complicated motion within the path with small dimensionality. The concept of "fracton” can be explained as follows.

When the motion of ions is examined along the trajectory, the numbers of sites, *V(N],* visited during the *N* steps is represented by

where *d _{{}* is the density dimension of the cluster formed by the visited sites and

*R*is a linear size of it. Here, "fracton” dimension,

*d*is defined by

_{s},

That is, the vibrational state on the fractal structure depends on both the fractal dimension of the visited sites and the walk.

Then we consider a back-correlation probability, < P_{0}(t) >, of finding a walker. The walker is located at an origin initially at time *t _{0}* and back to the initial position at time

*t.*

The probability is proportional to an inverse of the numbers of the site, which is a function of the fracton dimension.

See ref. [7] for more details of dynamic heterogeneity in lithium silicates and ionic liquids.

## Heterogeneity Found in Individual Ionic Motions

One fundamental question for the phenomena is if the fast and slow ions are part of the one distribution or mixture of different distributions. At least in the single alkali silicates, they can be regarded as parts of the one distribution (Levy distribution with a truncation) [7, 51, 62] if the motions are averaged adequately, because of exchanges among fast and slow motions exist.

Figure 5.11 Squared displacement of arbitrarily chosen 6 ions in lithium metasilicate at 700 K. Initial values are arbitrarily shifted to reduce overlaps. The lowest one is arrested during 1 ns run. Several ions cooperatively jump to the nearly same direction, as shown by arrows. Some localized jumps are also shown. Reprinted with permission from Habasaki, J., Hiwatari, Y. (2002), Dynamical fluctuations in ion conducting glasses: Slow and fast components in lithium metasilicate, *Phys. Rev. E,* 65, 021604, pp. 1-8. Copyright (2002) by the American Physical Society.

To understand the characteristics of dynamic heterogeneity, motions of ions including individual ones have been examined in several MD works [51, 63-67]. In Fig. 5.11, squared displacement of several arbitrarily chosen ions is shown. Both fast ions showing forward correlated jumps and slow ions showing a long waiting time of jumps and/or localized jumps are observed. Within ~1 ns, they tend to keep their characteristics; however, exchanges among these behaviors are observed. Trajectories of 7 ions with large mobility found in lithium metasilicate at 700 К are shown in Fig. 5.12. These ions tend to move together and show common paths. Such kind of paths (ion channels) formed by Si0_{4} networks can change in position gradually at high temperature [68], especially at high alkali content glasses, it can also change due to network tends to be cut because "free oxygens" exist in the system [69-71].

Figure 5.12 Trajectories of several Li ions with large motions at 700 К in lithium metasilicate glass during 1 ns run after averaging the local motions. There are common paths for motions of ions, which means the existence of ion channels. Reprinted with permission from Habasaki, J., Hiwatari, Y. (1999). Characteristics of slow and fast ion dynamics in a lithium metasilicate glass, *Phys. Rev. E,* 59, pp. 6962-6966. Copyright (1999) by the American Physical Society.

That is, motions of ions are highly cooperative and dominated by the accelerated motions of a small number of ions.

## Fractal Dimension Analyses of Jump Paths and Walks

The changes of the characteristic times in the temperature dependence have been remarkable at longer time scales. To explain the slowing down of the dynamics at longer time scales more quantitatively, the description of jump paths and walks by fractal dimension analyses is useful. It describes how the trajectory of ions spread over the space through the jump motions.

The fractal dimension of the path can be calculated in the cumulative number of the same kinds of ions within *r* when one ionic species is concerned. From the slope of the power law region, one can estimate the dimension for the possible connections of the path during a certain time of MD runs.

Multifractal analyses of density profiles of ions [68] are also useful to judge the percolation of paths as shown in Fig. 4.5 in Chapter 4. This analysis was applied for the accumulated position of ions during a long-time run so that the result can cover both the localized motions and diffusive motions. In our case, several times longer time scales than t_{dif} are necessary for accumulation to judge the percolation of the path [21].

## Relation between Temporal and Spatial Terms in Dynamics

The dimension of the path is not the same as that for the motions of ions. One can easily imagine a walker coming and going with hanging around on the way, which requires a long time to reach the destination. This means that the structure of the pathway alone cannot determine the behavior of the walker. This example also shows that how geometrical correlations with strong back correlation result in the temporal slowing down of the motion. This relation will be useful to consider the mechanism of the enhanced dynamics as well.

That is, analysis along the trajectory of the walker is required. The analysis of the topological nature of paths and motions is useful to understand dynamics [63, 64] and fractal dimension analysis of the random walk of trajectories of ions is applied successfully to understand the mixed alkali effect in mixed alkali silicates [21, 24]. Such an analysis is also useful to characterize the fast and slow components of diffusive motions, that is, dynamic heterogeneity [72]; that is, analysis of the coexistence of two different regions can be done as follows.

The fractal dimension of the random walks [35], that is the complexity of the trajectory obtained by simulation during the certain run time t_{run}, can be measured by using a divider method. We examined all the trajectories of the ions, obtained during long runs [43]. (For example, t_{run} = 4 ns at 700 К for x = 0, which is about 8 times longer than t_{dif}.) From the trajectories, we determine the fractal dimension of random walk, *d _{w}* defined by

### Heterogeneous dynamics and mixing of them in longer time scales

The *d _{w}/2* value corresponds to the power law exponent

*в*when the system is monofractal. A similar situation might be found in the fractal time random walk [73-75] when we used the effective localization time below t

_{dif}. In that theory, waiting time distribution with a long tail and change in the distribution can be an origin of the dynamic slowing down. The role of back correlation for slowing down of the dynamics discussed in the previous subsection might be mathematically similar to that caused by the waiting time distribution and sometimes they are not distinguished from each other. However, particles with localized motions are not the same as immobile particles. That is, spatial term and the temporal term can be distinguished in the molecular dynamics if necessary.

First, we need to discuss which term is the origin of the power law of MSD. It can be clarified we have separated the contribution of temporal and spatial terms in MSD [7, 76-78]. In the case of lithium metasilicate glasses, it was clarified that the power law of MSD for motions of Li ions is due to the back-correlation and not by the waiting time distribution. It was found that the change of the temporal term can contribute to the slowing down of the dynamics through the smaller jump rate but not to the exponent of the power law directly.

There are several works reported for the role of the fractal structures on random walks [79-84] related to the slowing down of the dynamics. The importance of the fractal geometry has been pointed out [82] and that on cluster-cluster aggregates and multicenter diffusion limited aggregates [83] have been discussed based on the trajectories of particles obtained by using Monte Carlo simulations.

The scenario for mono-fractal behavior of ions does not necessarily hold in single alkali silicate systems [43] as well as ionic liquids [72]. In the fractal dimension analyses of trajectories in lithium metasilicate, the slope of *N* against *L* plot was found to change at around 3 A at lower temperatures. Since two regions of slopes are found, the *d _{w}* for a shorter length scale (<3 A) is named

*d*and a longer length scale (3 <

_{wl}*L*< 10) is named

*d*Thus, the walks of Li ions in the lithium metasilicate have a multifractal nature. On the other hand, the difference between

_{w2}.*2/d*and 2

_{wl}*/d*is negligibly small for the mixtures as found in lithium-potassium metasilicate. It is also clear that

_{w2 }*2/d*value of trajectories with the geometrical origin is responsible for

_{w}*в*in the power law region of MSD, although generally, the fractal dimension of the random walk can include the temporal factor.

The role of the spatial term can be proved by the fractal structure of jumps related to the wavenumber dependence or by several works. Our definition of fast and slow ions is related to the power law distribution of length scales, which is related to the wavenumber dependence of the т in *F _{s}[k, t).*

In refs. [1] and [84], *F _{s}[k, t)_{s}* for wide

*к*range for this system were examined.

The *F _{s}[k, t)* for intermediate wavenumber (at around

*к = 2n/2*~ 2tt/3) is concerned with the back motion to the site occupied at

*t*= 0. The back correlation is governed by the fracton dimension rather than the fractal dimension of the random walks. The effect of fast ions is larger in the small wavenumber for MSD due to squared distance. The fast ions contribute to the fast decay of

*F*t) at small wavenumber as well. For the discussion of dynamic heterogeneity, thus the wavenumber dependence of it is useful.

_{s}(k,To distinguish the role of the geometrical factor and the temporal factor is interesting for describing the dynamics of ionics as well as in examining glass-forming materials.

After t_{dif}, the distinction of fast and slow particles might be coarse-grained in observations but still exists. It may be found in the effective length scale and time scale of events on average, which are affected by the coexistence of fast and slow ions. In the diffusive regime, the geometrical correlation between successive jumps determines the effective length and time scales. These insights might be lost or substituted by the effects of the temporal term if the random walk scenarios without spatial terms are applied.

## Exchanges between Fast and Slow Dynamics

It may be interesting to examine how fast and slow motions are mixing and change each other and how temporal and spatial terms are related to this. The data obtained by molecular dynamics simulations are adopted to examine further details of the dynamic heterogeneity.

Fast ions tend to have a short interval of jumps and show the forward correlated jumps, while slow ions tend to have longer intervals of jumps and/or strong back correlated jumps [64]. Therefore, it is interesting to examine the relation between temporal and spatial characters and its mixing of longer time scales than t_{dif}.

Examples of such results [63] will be shown here. In Fig. 5.13, the number of jumps, *N _{it}* for 10 arbitrarily chosen Li ions in each time-region 0-1, 1-2, 2-3 and 3-4 ns of lithium metasilicate at 700 К is shown. Some ions keep their mobility in jump rate for a long time, while other ions tend to show changes of their mobility.

Resultant squared displacements of these ions are shown in Fig. 5.14.

For example, for Li-ions of No. 2 and No. 7, both the number of jumps and squared displacement in each time interval are small. In the case of Li-ion of No. 6, numbers of jumps are relatively large, while the squared displacement is small. This means the jumps are localized ones.

For the Li-ion of No. 1, the largest squared displacement is found in the third time region (2-3 ns); however, the jump rate here is not the largest. That is, a forward correlation among jumps contributes to the large squared displacement in this region.

Figure 5.13 Number of jumps, *Nj,* for 10 arbitrarily chosen Li-ions in each time-region of 0-1, 1-2, 2-3 and 3-4 ns of lithium metasilicate at 700 K. Reprinted with permission from Habasaki, J., Hiwatari, Y. (2002), Dynamical fluctuations in ion conducting glasses: Slow and fast components in lithium metasilicate, *Phys. Rev. E,* 65, 021604, pp. 1-8. Copyright (2002) by the American Physical Society.

Figure 5.14 Squared displacements r*^{2} for 10 arbitrarily chosen Li-ions in each time-region (the number of Li-ion corresponds to that in Fig. 5.13). Reprinted with permission from Habasaki, J., Hiwatari, Y., (2002), Dynamical fluctuations in ion conducting glasses: Slow and fast components in lithium metasilicate, *Phys. Rev. E,* 65, 021604, pp. 1-8. Copyright (2002) by the American Physical Society.

For Li-ions of No. 8 and No. 9, jump rates after 1 ns are larger than that for the 0-1 ns. However, squared displacements for these ions are kept small for 4 ns. Therefore, motions tend to be localized and the increase in the jump rate does not directly cause an increase in diffusivity. Displacements become large after 3 ns for the Li-ions of No. 3 and No. 4. A corresponding increase in the jump rate is found in the former case, while remarkable changes are not found in the latter case. In the former case, the change can be correlated with the increased jump rate. However, in the latter case, the change is mainly concerned with geometrical correlations rather than changing the jump rate. As found in these Figs. 5.13 and 5.14, changes in temporal characteristics related to the jump rate and spatial characteristics related to the geometrical correlations are not simple nor necessarily directly linked. Since heterogeneity in structures and/or dynamics is widely found in nanostructured materials such as catalysts [85], colloidal systems [86], it will be interesting to examine the relationship with the functions of systems and heterogeneity of systems.