# Relationships among Multifractal Random Walks, Heterogeneous Dynamics, and Power Law Exponent 9, of MSD

Although the heterogeneous dynamics are considered as the multifractal random walks [55-58] having different length scale (and time scale) regions [8], the relation mentioned in 6.6.3 between *в* and *d _{w}* is expected to hold for mean behaviors of heterogeneous dynamics, although large fluctuations are found frequently due to coexistence of fast and slow motions of particles [33, 64].

The concept of the multifractal random walk is useful to understand the complex behaviors, not only for ionics but also for other time series in many systems found in geophysics, biology and economics, *etc.* Therefore, it seems to be useful to understand the possible origins of such behaviors. Recently one molecular motion in several systems [65, 66] is examined and they are interesting examples to be examined concerning the multifractality. In ionic systems, the complex motions of ions, where exchanges among fast and slow dynamics construct the multifractal walks [8, 70-72], have been attributed to the convolution of truncated Levy distribution of displacements [67-69] and geometrically correlated jumps.

**6.7.1 Percolative and Cooperative Aspect of Dynamics**

In Fig. 6.8, an asymmetry of the plot is found in a longer time scale behavior and larger change is found in the dense region marked by line with arrows. This asymmetry found in the dynamics in porous systems cannot be explained by the changes in the caging dynamics alone. Instead, the large change in l/t_{dif} was attributed to the geometrical changes of the trajectories, related to the structure of the jump paths and correlation among successive jump motions. This conclusion was drawn from the following characteristics of t_{dif} found in MSD.

As discussed in Chapter 5 and in [8], in the power law region of MSD, both fast and slow ions are usually found in the original glass and the slope *в* is affected by the foiward or back-correlated motion of jump motions of ions. For the mean behavior of fast and slow ions, *9 = 2/d _{w}* is expected, where the exponent

*9*is a function of the fractal dimension of the random walk [53],

*d*The value can be used as an index to represent the complexity of trajectories [8]. Larger

_{w }*d*value means that the trajectory is more complex one and ion tends to be localized and so that the small value of

_{w}*9*(larger value of d

_{w}) resulted in the delay of t

_{dif}.

The result is useful to understand the dynamics in the glass, where the ions tend to be located near the bottom of the potential surfaces. For the dynamics of ions in the glassy lithium disilicate system, change in the characteristic times related to the slope of the MSD in the power law region contributes to the slowing down of the dynamics considerably and this situation is changed by introducing pores.

Changes in *9* were determined from the power law region of MSD and plots against density will be shown in Chapter 7. As discussed there, separation of different contributions of caging and the power law exponent *9* examined in the present work is consistent with the previous results [70] found in an ionic liquid, EMIM (l-ethyl-3-methyl imidazolium)-N0_{3}.

**6.7.2 Structural Changes Related to Enhancement of Dynamics**

Hereafter, we consider what kind of structural changes are found at the atomistic level by introducing pores. To consider the origin of fast ion dynamics, the concentration of Li ions is one of the important factors to be considered. This is because, for composition dependence of M_{2}0-Si0_{2} (M=Li) glasses, it is known that the system shows higher diffusivity and corresponding conductivity with increasing contents of alkali metal ions. In the high-alkali- content system, the contribution of fast ions with cooperative jumps increases. In Fig. 6.10, a pair correlation function £f(r) of Li-Li pairs of the porous system *(p* = 1.98) at 600 К is compared with that in the original system. The peak positions of the first and second peaks shift toward the right and peak heights become smaller by decreasing density. Usually, the formation of ion channels is observed in the region with a high density of ions in the original system. Therefore, one may expect that the formation of the high-density region of Li ions as a cause of the enhancement. However, this is not the case. Instead, enhancement is mainly caused by the loosening of the cage. Li ions in the porous system are more separated by decreasing density as shown in Fig. 6.10.

Figure 6.10 Pair correlation function *g[r)* for Li—Li pair in original Li2Si20s glass (Blue, solid curve) and porous system (red dashed curve) *{p* = 1.98) both at 600 K. In the latter, maximum diffusivity was observed.

# The Role of the Caged Ion Dynamics in the Enhancement of Dynamics

## The Concept of Geometrical Degree of Freedom

Structure of cage in lithium silicate system can be well characterized by the structure of LiO_{x} [73] as well as structures of silica part using the concept of geometrical degree of freedom [74], which is schematically shown in Fig. 6.11.

The role of the cage in the ionic system is rather general. A quite similar discussion is applicable for ionic liquids. In our previous works [75] for ionic liquid, l-ethyl-3-methyl imidazolium nitrate (EMIM-N0_{3}), we have shown that the mobility of ions is closely related to both the change of the geometrical degree of freedom of the coordination polyhedra and total number of bonds (including fictive ones for neighboring (contact) ions or atoms). The combination of these concepts can explain both characteristic temperatures, *T _{B}* and T

_{g}, of the system. A loss of the geometrical degrees of freedom occurs with the closed packing of coordination polyhedra in a shell, while the saturation of the number of the constraints (bonds, contact ion pairs) is found near Tg, which is related to the concept of the rigidity percolation of (fictive) bonds [74].

Figure 6.11 Examples of LiO_{v} polyhedra and their characteristics for the coordination number (number of vertices), *N _{v} =* 5 (left) and 6 (right). Structures are taken from lithium disilicate systems obtained by MD simulations. Other Li ions and Si atoms within 2.8 A from each central Li ion are also shown. The number of contact О pairs (fictive bonds),

*N*are counted. Geometrical degree of freedom [74] is lost when (3/V

_{b},_{v}-6)-/V

_{b}= 0 as shown in upper figures. In the lower figures, there are floppy modes. In other words, when the coordination number is

*N*bonds are required to fix the shape of the polyhedron. Existence of the floppy part also means the location of free space in the structure. With the decreasing density of the system, a decrease of the coordination number occurs (Fig. 6.5a) and flexible-type structures also increase as shown in Fig. 6.12.

_{v}, N_{b}= 3N_{v}-6In the lithium silicate systems, LiO_{x} polyhedra and networks formed by Si0_{4} polyhedra are mixing. For the structures of porous lithium disilicate atp = 1.98, the coordination number of 0 around Si is 4 for ~100% of Si as found in the original disilicate glass at temperatures 600-800 K. That is, Si0_{4} units are kept unchanged while the *N _{v}* value of LiO

_{x}structure changes gradually. Therefore, structural changes in the porous systems are characterized well by the latter. The length of Si-0 bonds (in A) is only slightly changed with the density of the system. That is, the distance 1.493-1.846 is found for the original system, while 1.498-1.843 is found for

*p*= 1.98 both at 600 K.

## Changes in Coordination Polyhedra in Porous Systems

The concepts for the geometrical degrees of freedom in structures are applied to characterize the porous system as follows. As already shown in Fig. 6.5, each distribution of the coordination number *N _{v}* (number of О atoms around Li ion) for each density of system at 800K was obtained for an instantaneous structure after the quasi-equilibration of the system. In this figure, the position of the maximum of the peak is found at 5 before the expansion of the system. The findings are comparable to the situation in lithium metasilicate, where the saturation of the total number of Li-0 bonds and a comparable distribution were found [73] near T

_{g}, when the temperature of the system was decreased. The increase of the coordination number with decreasing temperature means that the coordination polyhedra more overlapped each other at lower temperatures. In the case of LiO

_{x }polyhedra, the sum of

*N*values of all polyhedra corresponds to the total number of Li-0 bonds. Therefore, this saturation means that further increase of overlap of polyhedra is not allowed. The peak position of the distribution shifts to 4 with a decreasing density as shown in Fig. 6.5a. The deviation naturally resulted in the enhanced diffusion of Li ions and this trend is following the relation between the slowing down of the dynamics near T

_{v}_{g}and the number of constraints discussed in [73]. The distribution patterns for 600 К (not shown) and 800 К are found to be almost the same and it means that the distribution mainly depends on density.

Original structure at 600 К is shown in the top panel of Fig. 6.12 and the porous system with *p* = 1.98 (in g cm"^{3}) at the same temperature is shown in the bottom panel.

Figure 6.12 Topological change of the coordination polyhedra (0 atoms around Li ion) by introducing pores with decreasing density. Changes of the geometrical degree of freedom of the coordination polyhedra are shown for instantaneous structures at 600 K. Upper panel: Structure before expansion. The structure with *N _{b} = 3N_{v}-6* is shown by yellow. The structure

*N*(structure with a floppy mode) is shown in green. SiC>4 units are shown in blue. Lower panel: The structure after expansion at 600 K, for which the largest diffusion coefficient was observed (p = 1.98). Colors are the same as those in the upper panel. By expansion, coordination polyhedra with

_{bc}< 3N_{v}-6*N*decreased from 318 to 232 and the floppy structure with

_{b}= 3N_{v}-6*N*increased from 370 to 420. Here, total number of Li ions (and hence coordination polyhedra around Li ions) is 768. The size of each figure in these panels is proportional to the system size, approximately. Reprinted from Habasaki, J. (2016), Molecular dynamics study of nano-porous materials—Enhancement of mobility of Li ions in lithium disilicate in NVE conditions,

_{b}< 3N_{v}-6*J. Chem. Phys.,*145, 204503(1-11), with the permission of AIP Publishing.

In this figure, polyhedra with *N _{b} = 3N_{v}-6* are colored yellow, while those with

*N*are colored green. The deformation of the coordination polyhedra is also affected by the

_{b}< 3N_{v}-6*N*explained below. The geometrical degree of freedom, F

_{b}_{po}i

_{y}hedron. changes by introducing pores. The degree is defined by F

_{po}i

_{y}hedron = [(3iV

_{v}- 6) - Al

_{b}], where

*N*is the number of vertices, that is, the coordination number, and

_{v}*N*is the number of bonds (contact ion pairs for ionic liquid and contact oxygen pairs for LiO

_{b}_{x}structures). In the concept of deformation characterized by the f

_{po}i

_{yhe}dron' the effects of bond-bending constraints are included. The F

_{po}i

_{y}hedron is 0 when

*N*where 6 means the translational and rotational degrees of freedom of the polyhedron. In other words,

_{b}= 3N_{v}-6,*N*braces characterized by this equality are required to fix the shape of the polyhedron. Thus, one needs to consider the change in

_{b}*N*which is relating to the mobility of the caged ions.

_{b},Coordination polyhedra for the network part (Si0_{4} units) are colored blue. With expansion of the original system to this density, coordination polyhedra (yellow) with *N _{b} = 3Ny6* decreased from 318 to 232 (0.73 times larger) and the floppy structure (green) with

*N*increased from 370 to 420 (1.14 times larger), while the number of the structure with over-constraint

_{b}< 3N_{v}-6*[N*was almost unchanged. The number of polyhedra with a floppy mode

_{b}> 3N_{v}-6]*[N*< 3/Vy-6) having a looser caging increases by the expansion of the system. This change, as well as that found in the coordination number of the cage of ions, are responsible for the changes in MSD. Thus, the porous systems are characterized by the increase of the structure with a floppy mode as well as smaller coordination numbers. Therefore, similar concepts as used to characterize the glass transition in the ionic liquids are found to be applicable to characterize the conduction, that is, "caged ion dynamics” in silicate glasses.

_{b}## Effects of Tightening of Cages and Formation of Larger Voids

A decrease of the coordination number in LiO_{A}. polyhedra is found before the maximum of the diffusivity, while an increase of it is observed again after that. The latter change is accompanied with the rearrangement of networks after the formation of larger voids. Therefore, the maximum is explained by the change of the caging as a first approximation and we found that the existence of the larger void is not necessarily effective for the enhancement of the dynamics. These behaviors are closely related to the residual stress in the system and further study of the structures and stress in related composites will be interesting.