# Molecular Dynamics Simulations of Nanoporous Systems: Dynamic Heterogeneity, Self-Organizationof Voids, and Self-Healing Processes

- Introduction of Dynamic Heterogeneity in Porous Systems
- Summary of the Enhanced Dynamics Found in Porous Lithium Disilicate
- Enhancement Caused by Loosening of Cages
- Explanation of the Mechanism by Geometrical Degrees of Freedom ofCoordination Polyhedra
- Heterogeneous Dynamics of Ions in Porous Lithium Silicates
- Non-Gaussian Parameters of Ions in the Porous System

## Introduction of Dynamic Heterogeneity in Porous Systems

Roles of porous materials are spreading in many fields of science and technologies, including nanoionics, and they are rapidly increasing in importance for both applications and fundamentals [1-3]. In this chapter, two topics concerned with porous systems will be discussed.

The first one is concerned with the dynamic heterogeneity of ionic motions in porous lithium disilicates. Heterogeneous dynamics meaning coexistence of fast and slow ions are commonly found in many systems such as glass-forming materials, ionically conducting glasses, colloidal systems, etc. [4]. To learn changes in heterogeneous dynamics caused by modifying their structures, porous materials are used. As discussed in Chapter 5, the dynamics

*Molecular Dynamics of Nanostructures and Nanoionics: Simulations in Complex Systems *Junko Habasaki

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4800-77-8 (Hardcover), 978-1-003-04490-1 (eBook) www.jennystanford.com of particles (atoms, ions, molecules) show non-Gaussian characters in a certain time and temperature regions and the properties such as diffusion coefficients tend to be dominated by rare events of long-range. This is because the displacement of ion is squared in calculations of mean-squared displacement (MSD), which is connected to the diffusion coefficient through the Einstein equation [5]. Therefore, it seems to be interesting to learn how heterogeneous dynamics are affected by introduced pores and how they are related to the enhanced dynamics of porous systems discussed in Chapter 6. After summarizing the mechanism of the enhanced dynamics discussed in Chapter 6, our attention will be focused on the heterogeneity of the dynamics in ionically conducting porous glasses. The results are useful to learn the role of heterogeneity and how to control properties of materials by the perturbation.

The second one is concerned with the self-organization of larger voids in the *NVE* condition observed by the expansion of the system and healing processes observed in *NPT* runs after that. Results are useful to understand the effects of pressure and stress tensors related to them. Roles of extended conditions in molecular dynamics simulations and relation between structures are important factors to understand the dynamics toward to control nanostructures.

Now, let us start a discussion for the first topic. One may have many questions for the dynamic heterogeneity of porous materials. Are the heterogeneous dynamics also found in porous systems? How is it modified by introducing pores? As shown in Chapter 6, the enhanced dynamics of ions found in porous lithium disilicate in *NVE* conditions are accompanied with the structural changes of cages and paths of ionic motions. First, this system is examined to learn the relationships between changes of structures caused by introducing pores and those for dynamics, especially for exponents characterizing MSD. In the original glass before introducing pores, the dynamics of mobile ions are already heterogeneous. As shown in this chapter, enhanced dynamics found in porous lithium silicates obtained from MD simulations in *NVE* condition are also found to be heterogeneous ones [1]. Because the changes of dynamics are considered to be key features to characterize the slowing down of the dynamics in the glass transition problem, porous systems can provide a new platform to learn the mechanism of the glass transition and related problems.

In this chapter, density is given in g cm'^{3} and time is in ps, unless otherwise stated.

## Summary of the Enhanced Dynamics Found in Porous Lithium Disilicate

Before discussing the heterogeneity of the dynamics, we briefly summarize the essence of the results of Chapter 6 and [1, 2] for the enhanced dynamics found in porous lithium disilicate in *NVE* conditions. If one prepares porous systems started from crystalline ones, the difference between order and disorder in the structures will affect the results considerably. In our approach, possible effects caused by the different ordering in the system were eliminated by using glassy lithium disilicate as a starting material.

In *NVE* conditions, an increase of the diffusion coefficient of Li ions is found with decreasing density (increasing porosity). With a further decrease in density, the diffusion coefficient decreases again. That is, there is a maximum of the diffusion coefficient in the medium density region.

### Enhancement Caused by Loosening of Cages

In Fig. 7.1, the structural changes caused by the expansion of systems are summarized by using mean coordination numbers. In Chapter 6, it was clarified that the increase of the diffusion coefficient of Li ions is caused by the loosening of the cage. This situation is also observed in Fig. 7.1.

The mean coordination number, *N _{v},* is found to decrease and then found to increase. This is consistent with the results observed in distributions of coordination numbers of oxygens around Li ions (Chapter 6). In this figure, wire frame structures are also shown with distinguishing rigidity of structures. Namely, the color of the bonds was changed by the geometrical degree of freedom [5] ofLiO

_{x}structures.

Figure 7.1 Mean number of Li-0 bonds around Li ion, *N _{v},* is plotted against density for the porous disilicate at 600 K. Cut-off length used is 3.0 A. Original system before expansion is found on the right-hand side. The change in the mean coordination number is correlated well with the changes in dynamics. Namely, at around the minimum of the coordination number, the maximum of the diffusivity is found. Wire frame structures for the slice (The thickness of the slice is chosen to be 10 A) are also shown. The number of contact pairs in 0-0 (W

_{b}) is also related to the dynamics as well as the coordination number of 0 around Li ion (Af

_{v}). Colors of bonds change with the geometrical degree of freedom of the polyhedra. That is, the bonds for the structure

*N*(with a floppy mode) is shown in green. Si part of the Si-0 bonds in Si0

_{b}<3Ny-6_{4}unit is shown in blue. Oxygen part of bonds is shown in red. With decreasing density, a loosening of the structure occurred and after that, a tightening of the structures occurred with formation of larger voids.

As discussed in Chapter 6, dynamics of ions are well correlated with such structural changes. With the decrease of the coordination numbers of LiO_{x} structure, diffusion coefficient increases. The loosening of the cages in the small expansion region is also clear from the wire frame structures. With further expansion of the system, coordination number of LiO_{x} structure increases. With this change, diffusion coefficient decreases, and it is related to the tightening of the cage. These tightenings coincide the formation of large voids as shown in this figure. That is, rearrangement of structure occurs with and after the formation of larger voids.

Thus, loosening of the cage causes the enhanced dynamics, while tightening of the cage after the formation of the larger voids causes the slowing down of the dynamics.

### Explanation of the Mechanism by Geometrical Degrees of Freedom ofCoordination Polyhedra

With changes of density, the maximum of the diffusivity appeared near the medium density (~2.0) with the system with enhanced dynamics (see Chapter 6). The trend in dynamics can be well explained by the changes in the coordination number; however, the contribution of other factors is non-negligible.

In general, the geometrical degrees of freedom [6] of the coordination polyhedra is determined by both the coordination number *[N _{v})* and the number of contact pairs (Л/

_{ь}) of oxygen atoms and the latter also changes with density. That is, the change is characterized by both decreasing of coordination number and number of contact ion pairs. Corresponding changes in the time scale of the nearly constant loss (NCL) in susceptibility (i.e. caging) region in the MSD are found and therefore, the caged ion dynamics can be characterized based on the concept.

The concept of the geometrical degrees of freedom has been successfully applied for characterizing glass transition in several systems [7, 8]. These results suggest that explanation for the enhanced dynamics in porous materials is also relevant in discussing the mechanism of the slowing down of the dynamics by caging with decreasing temperature towards the glass transition temperature, T_{g}, in glass-formers.

In the following sections, it will be shown how enhanced dynamics is related to the heterogeneity and/or non-Gaussian parameters of dynamics.

## Heterogeneous Dynamics of Ions in Porous Lithium Silicates

Recently, the importance of space and time heterogeneity in the catalyst was pointed out and several experimental methods begin to apply [9]. Under these circumstances, the knowledge obtained from the molecular dynamics simulation will be useful to understand the nature of dynamics in nanostructured materials. Here we will characterize the heterogeneous dynamics by MD simulations in porous systems [1]. In the original lithium silicate glasses, motions of ions were found to be extremely heterogeneous [3, 4] as shown in Chapter 5, and therefore it is interesting to examine how such heterogeneous dynamics are affected by introducing pores.

When we consider the effect of perturbation by introducing pores, it should be kept in mind that there are remarkable similarities in dynamics between ions in ionically conducting glasses, glass-forming molecules, and/or ionic liquids [3, 4]. This means that changes affected by such modifications of systems are important not only for understanding the ionics in ionically conducting glasses but also for understanding related slow dynamics in molecular and ionic glass-forming materials.

### Non-Gaussian Parameters of Ions in the Porous System

Here the heterogeneity of the dynamics means the coexistence of fast and slow particles (ions) in transport properties [10]. How can we characterize the heterogeneity? One of the indexes to characterize the heterogeneous dynamics is the non-Gaussian (NG) parameters [11]. NG parameters, *a _{2},* are defined by

The parameter characterizes the deviation of G_{s}(r, t) from the Gaussian form. That is, the NG parameter becomes 0 if the dynamics are Gaussian.

In Fig. 7.2, NG parameters of Li ions for the porous lithium disilicate obtained in *NVE* conditions for *p* = 2.47 (purple, solid),

*p* = 2.14 (pink, short-dashed), *p =* 1.98 (red, 2-dot dashed) and 1.58 (blue, dashed) (densities are in g cm"^{3}) at 600 К are shown. In the bottom panel of Fig. 7.2, corresponding MSDs of Li ions are shown. For all porous systems examined, non-Gaussian characters are found at least for the time scales covered in the present work.

From this figure, one can see that atp = 1.98, the non-Gaussianity is the smallest, although the heterogeneity cannot be neglected. As already shown in Chapter 6, at this density, the diffusion coefficient shows the maximum.

Figure 7.2 Top: Non-Gaussian parameters of Li ions for the porous lithium disilicate obtained in *NVE* conditions for *p =* 2.47 (purple, solid) (original system), *p =* 2.14 (pink, short-dashed), *p =* 1.98 (red, 2-dot dashed) and 1.58 (blue, dashed) at 600 K. In the porous system with the highest mobility, the non-Gaussianity decreases but still have a positive value for the longest time. Bottom: Corresponding mean-squared displacements of Li ions. The same kinds of curves and colors as those in the top column are used. The magnitude of the values shows the opposite trend to the Li ions. Reprinted with permission from Habasaki, J., Ngai, K. L. (2018), Heterogeneous dynamics in nano porous materials examined by molecular dynamics simulations-effects of modification of caged ion dynamics,/. *Non-Cryst. Solid,* 498, pp. 364-371. Copyright (2018), Elsevier.

If non-Gaussian parameters for porous systems and original system at 600 К were compared, the maxima of the parameter in porous systems are found to shift toward shorter time in the region with enhancement of dynamics [1]. The maxima are located near [12], which is the beginning of the power law regimes after the jumping out of some of ions from their cages, and hence the value is correlated to the time, t_{xl}, when the caging region ends.

As discussed in Chapter 4, multifractality is another possible measure of the heterogeneity of structures and hence it will be interesting to apply multifractal analysis to the structures and dynamics of porous materials. Examples of such analyses are explained in Chapters 2 and 5.