# Characterization of Complex Systems

The meaning of complex systems discussed is sometimes confusing. The systems having different scales, components are complex. On the other hand, the word "complex system” can be used to mean the existence of deterministic Chaos, which shows complex behaviors even though the system, is represented by quite simple equations. In this book, "complex” is simply concerned with the first meaning, where the multifractal mixing of length scales resulted in heterogeneous structures, although the second meaning is important to consider the origin of behaviors of heterogeneous motions. Here the word "complex” also means the modifications caused by nanostructures.

**4.9.1 Complexity of the Systems and Vitrification of Systems**

From the structural point of view, complexity of systems due to coexistence and/or mixing of different kinds of structural units, length scales, different sizes of domains are often found. One of the causes of the complexity of the structures (and dynamics related to them) of systems is the existence of a plural number of components.

Such complexity prevents the rapid crystallization of the system and that's why usually binary systems are used to study the glass transition, although even in the simple soft sphere model with r" type potential, the existence of substructures (such as fee like and bcc like structures) makes the system complicated one to allow it vitrifying.

**4.9.2: Structures of Silica and Silicates**

As well known in mineralogy, almost of crystalline silica systems consist of Si0_{4} units. Networks formed by tetrahedral units of

Si0_{4} are commonly found in silica and silicates. Structures formed by them are quite complex ones. Even in silica, which consists of Si and О only, a variety of structures with different kinds of connections of units by vertices or by faces are found.

To characterize silica and the silicate systems, the concept of "fragility” is proposed by Angell to characterize the glassforming liquids [45]. The fragile system shows super-Arrhenius behavior of the dynamics, while the strong system shows temperature dependence closer to the Arrhenius behavior.

Simple silicates are binary mixtures of silica and oxides such as M_{2}0 (M=Li, Na, К—) and fragility of the system is known to be affected by the mixing of another component. Usually, silicate with higher contents of M_{2}0 is known to be more fragile. Therefore, first, we will consider the changes caused by the additional component in structure and dynamics.

The addition of M_{2}0 cut the structures of silica part and it resulted in the decrease of the dimension of networks of Si0_{4} units with the formation of different substructures as shown by different *Q _{n}* structures (see Section 4.11) as well as different statistics of rings. The distribution of coordination numbers in MO

_{x}units also brings some complexity.

Specific conditions used in MD simulations of porous lithium silicate systems including the original glassy systems can be found in Chapters 6 and 7 and their appendixes. Similar conditions can be used in many cases.

# Existence of Different Length Scales in Ionic Systems

From the dynamical point of view, the coexistence of slow and fast particles is related to that of different length scale motions. Their different temperature dependences resulted in the curvature of the diffusivity against 1/T plot [4]. That is, super-Arrhenius behavior is caused by the coexistence of different length scales. This view is previously obtained from the changes in the dynamics observed in ionic liquid, EMIM-N0_{3} [46]. In that paper, it was concluded that the large "fragility,” shown by the curvature of diffusion coefficients, can be explained by the changes in the slopes in both temperature regions rather than the suddenly caused rapid decrease of dynamics near *T _{0}.* Above the inflection point, long-ranged motions with cooperative motions are dominant, while below it, the localized motions become dominant. Then, the acceleration of the dynamics with heterogeneity near the inflection points, which resulted in the gentle slope in the high temperature region, ensures the large changes in the slopes. As found in the above discussions, the fragility of glass-forming liquids is closely related to the concept of heterogeneity of dynamics. Accordingly, it is suggested that multifractality [46-48] due to the coexistence of the different length scales is a good measure of the fragility as well as the heterogeneity. The concept of multifractal can be seen in the Cantor set shown in Fig. 4.4. As depicted in this figure, the mixing of different exponents resulted in the formation of complex heterogeneous structures.

Figure 4.4 An example of the multifractal Cantor set. When different exponents are mixed as in this figure, the final structures are heterogeneous ones. The length 1/5 and 1/2 remained in the first stage, where the measure is 3/5 and 2/5, respectively. This procedure was repeated. Resultant distribution after *n* stages is heterogeneous. That is, multifractal analysis gives a spectrum characterizing the heterogeneity. See [3, 21-23] for more detail. Reprinted with permission from Habasaki, J., Leon, C., Ngai, K. L. (2017), *Dynamics of Glassy, Crystalline and Liquid Ionic Conductors: Experiments, Theories, Simulations,* Springer International.

In the following sections, multifractal analyses have been applied to characterize the paths or sites in ionic systems using density maps [43, 46-48]. It is noteworthy that such maps are obtained by the cumulative positions of particles during MD runs so that it reflects both the mean structures and dynamics governing them.

## Multifractality of Jump Path and Sites-Singularity Spectra,f(α)

Structures of systems formed such different length scales are complicated ones with multifractal nature. In lithium metasilicate, singularity spectra (/(a)) [27, 28] has been calculated by the direct method [27] from cumulative positions of Li^{+} ions during MD run as shown below.

In Fig. 4.5a, the contour map for the cumulative density of Li ions in lithium metasilicate glass at 700 К is shown as an example. Several kinds of substructures such as local ion sites and a pair of ion sites, jump paths connecting several ion sites and each sub structure, have different fractal dimensions. Such a complicated pattern was formed by the coexistence of fast and slow particles (ions). The following characteristics can be seen from the figure. As shown by the color code bar, gradual changes of the color mean those in the density. That is, red color means localized ion sites, while dark blue means the region separated by frameworks made of Si0_{4 }units. Naturally, the structure formed by localized motion has a high density, while the delocalized motion has a lower density. This heterogeneous character is also observed in depths of ion sites. As found in this figure, structures formed by ions are heterogeneous one with many substructures, such as ion sites and ion channels.

It may be worth to mention that this map is obtained for the run during 1 ns and gradual changes of the pattern was found in the following 1 ns and therefore, these paths and sites are not fixed ones at least in the lithium metasilicate at 700 K. The system with high alkali metal ions has a small number of free oxygen atoms, which can modify the network structures gradually, while the network structures in low alkali content glasses are more stable.

Figure 4.5 (a) Example of a contour map of accumulated density for Li ions in lithium metasilicate at 700 K. Dense part is due to localization of ions. Such ion sites and jump paths have different fractal dimensions, (b) Example of the corresponding singularity (/(a)) spectrum obtained from a 3-D structure of accumulated density. A convex curve means the multifractality. The *a _{max}* and a

_{min}correspond to the most rarefied part (delocalization of ions) and the densest part of the density map (localization of ions), respectively, so that the multifractal spectra can represent the heterogeneity of the structures and dynamics. The value at

*q*= 0 corresponds to the capacity dimension, while the value at

*q =*1 corresponds to the information dimension. Structures with different fractal dimensions as shown in (a) can be represented by a single curve. Reprinted with permission from Habasaki, J., Leon, C., Ngai, K. L. (2017),

*Dynamics of Glassy, Crystalline and Liquid Ionic Conductors: Experiments, Theories, Simulations,*Springer International.

For characterizing the spatial heterogeneity of structures, a multifractal analysis by using a singularity (/(«)) spectrum is a useful method, where many fractal dimensions of substructures can be represented as a single curve. By using the spectra, localization and delocalization shown in the density map can be quantitatively represented.

In Fig. 4.5b, the singularity spectrum obtained by the multifractal analysis for this density map is shown. The convex shape of the spectrum means the multifractality. Generally, the value at a_{max} corresponds to the rarefied part of the density map and hence meaning delocalization, while a_{min} corresponds to the densest part of the density map meaning the localization. Therefore, the difference of these values can be used as a measure of the strength of the heterogeneity. The value at *q =* 1 corresponds to the capacity dimension and when the value is larger than the percolation threshold value for 3-D (=2.53 [49]), it means that the jump path is percolated. One should note that the value larger than the percolation threshold is the essential requirement for the occurrence of the diffusion or conduction within an observation time.

## Calculation of the Singularity Spectrum

Here the method of how to calculate the singularity spectrum [27] is summarized. Singularity spectrum, /(or), is obtained from the probability /q(<5).

The procedure to obtain the function from the (accumulated) density profile of Li ions is described as follows.

A pattern (a density map) was divided into cells with size <5.

Probability /<,(<5) (=n,/2n,) of the cell is measured. In this case, the number "how many times ions visit the /-th cell during the run” is regarded as *rij.*

The measure /г,(<5) is represented as

where *a* is the strength of the singularity. The probability of *a *lying between or' and *a'* + *da'* is

Thus, the multifractal spectrum, /(a) corresponds to the fractal dimension of the set of small cells with a singularity strength *a.*

Each spectrum is obtained by using the qr-th moment.

The concentrated region is emphasized in [/u(d)]/ when *q* is positive, while the rarefied region is emphasized when *q* is negative.

When *q* = 0, *f[a[q*)) value corresponds to the capacity dimension, *D _{0},* while when

*q*= 1, the value corresponds to the information dimension

*D*Correlation dimension is given by

_{v}*D*with

_{2 }*q = 2.*

Using the normalized *q*-th moment, /q(q, d), the numerical values of *a(q)* and /(or(q)) are obtained for a certain range of *q* (-40 < *q <* 30 were used in our analysis) for several d values using the following relations:

where d) is defined by

Care must be taken for the numerical errors in calculations because wide ranges of *q* values are treated.

The obtained spectra are useful for understanding the dynamic heterogeneity forming the density profile. A convex shape of the curve in the /(or) spectrum means the multifractality (mixing of more than one exponent). That is, the structure is formed by the mixing of localized and accelerated motions having different exponents.

Using the singularity spectra, one can compare the heterogeneity of different systems on the same basis. This is because these relations have a similar structure as the thermodynamic formalism of equilibrium thermodynamics, which can be a general framework to study the dynamic and structural heterogeneity.

In Fig. 4.6, a density map for EMIM^{+} ion in the ionic liquid, l-ethyl-3-methylimidazolium nitrate (EMIM-N0_{3}) obtained by MD simulations and corresponding /(or) spectrum are shown.

Figure 4.6 (a) An example of a density map of EMIM* ions of EMIM-N0_{3 }at 370 К projected on a plane. Colors are changed by a logarithmic scale, (b) An example of the singularity spectrum for the cumulative density of EMIM* ions in the same system obtained from a 3-D structure of accumulated density. Convex shape of the curve means the multifractality. The maximum position with *q* = 0 corresponds to the capacity dimension. The value 3 means the three-dimensional connections of the paths in the liquid state. The a_{max} and *a _{mm}* correspond to the most rarefied part and the densest part of the density profile, respectively. Reprinted with permission from Habasaki, J., Leon, C., Ngai, K. L. (2017),

*Dynamics of Glassy, Crystalline and Liquid Ionic Conductors: Experiments, Theories, Simulations,*Springer International.

In ILs, ions are charge carriers and at the same time glass formers. Therefore, ion sites for cations are separated by moving anions. The sites seem to be connected and show complicated fractal shapes. Even though it has a quite different appearance from that in ionically conducting glasses, quite a similar spectrum in the shape is obtained. While both a_{min} and a_{max} values in the spectrum in IL are larger than the lithium silicate glasses and the capacity dimension, *D _{0},* is also larger than those of lithium silicate glasses. These differences are reasonable to consider the IL at 370 К is in the liquid state.

This situation with multifractal is not limited to the ionic systems. Generally, multifractal analysis can treat complex systems represented by different shades. For example, it was used to characterize fractures [50, 51], turbulence [52], reactions over a rough surface [53, 54], genomes [55], and crystalline aggregates formed in a drying gel medium [56].

A porous system is expected to take a role as a storage of gases as well as liquids, and the distributions of oil and gases in porous systems is known to be heterogeneous. Recently, multifractal analyses were used to characterize the pore structure of tight sandstone [57].

Strong correlations were found to exist between the porosity and the multifractal parameters, such as a_{min}, a_{max}, Да_{к} (=a_{0}-a_{min})

^max^{-}AI (= [Ac^l ^{—} , and A*f* (-/amax ^{—} /.-min)■

Here *a _{0}* is a position of the maximum.

Peng and coworkers proposed a classification of types of pore structures based on these parameters.

Thus, the multifractal analysis seems to be a promising method for systematic comparison of complex systems, the classification of the materials, and characterization of surfaces as well as their porosity.

It will be interesting to use the classificaion of multifractal materials suggested in Chapter 2 for these systems.

The method is also applied to the structures obtained by simulations of non-ionic systems. As demonstrated by Sakikawa and Narikiyo [58], the method was successfully applied to examine the heterogeneity in two-dimensional supercooled liquids of the soft sphere model and found that the cooperative rearrangement regions (CRR) [59, 60] can be identified as the weakly bonded regions of the multifractal structure of (fictive) bonds.

Besides the /(a) spectra, the generalized dimension, Z)_{q}, can be used for the analyses of multifractality, because the dimension is related to by a Legendre transformation, r(q) *= [q -* 1 *)D _{q},*

where *i[q)* and *q* are conjugate thermodynamic variables to /(a) and *a.*

## Relationship between “Fractal Dimension of Random Walks,” and Power Law Exponent of MSD

In 1967, the concept of fractal dimension of the random walks, *d _{w}* was introduced by Mandelbrot to measure the complexity of the coast [20].

One can imagine the situation that one measures the length of the coast on foot with a fixed stride, so that the value for an uneven coastline will be longer compared with the value measured on the small map by a divider. This is because, details of the curves are coarse-grained in the latter. If the coast is more complex, the difference of these values will be larger. Therefore, if one measured a total length and/or total frequency to use the ruler to cover the coast by using several fixed lengths of rulers, one can determine how the coastline is complex.

Of course, coast or trajectory is not an exact fractal in the sense of the self-similarity but has the characteristic of statistical fractal and trajectories obtained by MD directly reflect the motion of particles (atoms, ions, and molecules) and its complexity is a good measure of the dynamics.

Some examples of trajectories of ions found in molecular dynamics simulations of the ionic system are shown in Fig. 4.7.

As shown in Fig. 4.8, using divider (i.e., a straight ruler) of length *L,* one can count how many times, *N,* are required to cover the trajectory.

From the slope in the double logarithm plots, one can determine the fractal dimension of random walk, *d _{w},* defined by

Normal random walk like in Brownian diffusion has *d _{w}* = 2. The value increases if the trajectories are complex and localized, while the value decreases if the trajectories are like straight lines and delocalized. Curves on the trajectory can overlap each other and therefore the obtained fractal dimension is a latent one, which can be larger than the dimensionality of space and this situation is different from the coast.

Figure 4.7 Example of trajectory of a particle obtained by MD for lithium ion in lithium metasilicate at 1000 K, (a) a localized ion and (b) an ion accelerated by forward correlated motions, where the motion is highly cooperative.

Figure 4.8 Procedure how to determine the fractal dimension of random walks from the trajectory. Using the divider of length *L,* counts required to cover the trajectory are determined. This procedure has been repeated by changing length *L.* The value *N{L)* for the sum of all trajectories is plotted against *L* in a double. From the slope, one can determine the fractal dimension of the random walks. If more than one region is found in the plot, the motion is multifractal rather than monofractal.

Previously, we have reported the existence of two different length scales in trajectories of Li ions in lithium silicate systems [43] because more than one region was found in the (double logarithmic) plots of *N* against *L.* Similar behaviors are also found in other ionic systems, that is typical ionic liquid (IL), EMIM-NO3 [46].

Due to the existence of two different length scale regions showing different exponents, such walks are characterized as "multifractal (random) walks” [61, 62]. Two length scales are thus represented by two different values of *d _{w}.* In general, the dynamics of ions in disordered materials show large dynamical heterogeneity both temporally (waiting-time distribution of jumps) and spatially (geometrical correlation among successive motions). The fractal dimension analysis of the trajectories is concerned with the spatial term.

Here the residence time (temporal term) in each site does not affect the result. That is, *d _{w}* is determined solely by the geometrical correlations. The slope with

*9*(=2/d

_{w}) is expected [25] for the fractional power-law region of the MSD (before the onset of the diffusive regime) in the theory of simple fractal. In this case, a temporal term might be included in

*d*

_{w}If the ionic site is visited by the walk with *d _{w}* in a monofractal manner, the mean squared displacement in the power-law regime is given by

Naturally, the "power law” dependence of the MSD explains the large non-linear effect of the slowing down with a change of these values.

At first sight, an explanation for the mono-fractal case seems to hold in many systems; however, we have observed two different length scales in *d _{w},* in lithium metasilicate or IL, EMIM-NO3, and hence the observed exponent

*9*is for mean behavior of the

*d*and

_{wl}*d*The former is for the short- length-scale motion while the latter is for the longer-length- scale one. The coexistence of two length scales in trajectories means that the walk of ions has a multifractal nature. It seems to be a natural result obtained from the multifractal nature of the density profile. That is, if dynamics are localized, density in localized regions becomes large and if dynamics are delocalized, low-density regions are formed. In Fig. 4.9, an example of the motion of ion (EMIM

_{w2}.^{+}ion in EMIM-N0

_{3}) at 370 К is shown with the corresponding singularity spectrum,

*D[h).*The spectrum for the trajectories can be obtained similarly to the /(a) spectrum of the density profile. Multifractality of the walks can be characterized by the singularity spectrum using Hurst exponent [63, 64],

*h,*defined by the following equation in the case of one-dimensional motion:

Figure 4.9 (a) An example of the changes in the position of an EMIM* ion along X axis at 370 K. The motion consists of fast and slow motions, (b) A singularity spectrum *D(h)* for the motion shown in (a). A convex shape of the spectrum means the multifractality of the dynamics. Reprinted with permission from Habasaki, J., Leon, C., Ngai, K. L. (2017), *Dynamics of Glassy, Crystalline and Liquid Ionic Conductors: Experiments, Theories, Simulations,* Springer International.

In the power-law region of the MSD, (r^{2}^)) ~ with *в* = 2*H.*

Here *H* is the Hurst exponent for the mean behavior of particles. Since the fractal dimension of the random walk *d _{w}* is related to

*9*by

*9 = 2/d*the value of

_{w},*d*for the mean behavior corresponds to the inverse of the Hurst exponent. That is,

_{w}