Relation between Glass Transition and Changes of Slow Dynamics by Introducing Pores

As shown in Chapter 6 and Section 7.5, enhancement of dynamics in porous systems seems to be closely related to the mechanism of the glass transition, where both the path of particles and caging dynamics are changed. These results mean that one needs to consider both changes in percolation of jump motions related to the power law exponent, 9, of MSD and those in caging dynamics for the enhancement, although the contribution of the former to the diffusion coefficient is larger in the case of the glass transition.

In the original system, fast ionic motion tends to be suppressed at lower temperatures, before ions being tightly caged. In the case of ionically conducting glasses, the behavior of these processes is not so deviated from the Arrhenius type temperature dependence, since the ionic motion is fairly decoupling from the motion of the matrix (that is, silica part, where packing of oxygen atoms play roles.) On the other hand, in the case of ionic liquids, ionic motion is directly related to the glass transition of the system and the fragile behavior, which can be represented as changes in slopes in the Arrhenius plot [20]. In both cases, a weakened cage is expected by introducing pores and therefore, the fast ionic motion will continue to lower temperatures.

The large effect of в near the original glass means a remarkable contribution of geometrical correlations in slowing down of the dynamics in the original glass at the lower temperature. This conclusion is consistent with those in the study of the original system using fractal dimension analysis [3].

Self-Organization of Larger Voids

Heterogeneous dynamics mentioned in previous subchapters can be explained by the distribution of different structures in the environment of Li ions. In a short time scale, it is governed by the coordination polyhedra (and further shells) as already shown in Chapter 6. In the lowest density examined, gradually larger voids are formed, and it is interesting to examine this self-organization process of the void with the structure of LiOx.

First, the process of larger voids in the NVE condition will be re-examined from this point of view. In Fig. 7.6, time development of large void is shown by the wire frames and polyhedral structures of the system starting from the smallest density system at 600 K. In this figure, time development of structures of porous systems is shown along this formation process with the interval of 400 ps, where the polyhedra of LiO_x are colored by its coordination number. On the left, many small pores are brought in the system. Different sizes of pores are found in systems. In the middle, the region containing the polyhedra with a small coordination number of Li developed and many small pores exist. In this stage, the system contains some structures having large coordination numbers, ~7. In the right panel, a larger void was formed with the rearrangement of polyhedra and networks. After the formation of the large void, contribution of large coordination numbers becomes negligibly small and the coordination number of 4-6 becomes dominant.

This structure with the larger void was essentially kept after further 400 ps, while the further rearrangement of coordination polyhedra is found. The porosity of the system is a linear function of the density (see, Eq. (6.1) of Chapter 6 or Appendix of this chapter) even after the formation of larger voids, and this means that it accompanied with the decrease of free space

(small pores) in other parts. Due to rearrangements of the rest part, the smaller pores found in the left and middle panels disappeared. That is, a rearrangement of the structure occurs with keeping total free volume.

Figure 7.6 Development of large void in the system at p = 1.58 with an elapse of time from left to right, which is shown in the interval of 400 ps (100,000 steps) at 600 К for the slice of 10 A. Each polyhedron (Li ions surrounded by oxygen atoms) is colored by its coordination number, N_v, shown in the figure. Blue spheres: Si. Red spheres: 0. Pale blue spheres: Li. Li ions colored purple are for the structure with N_v = 2.

The size of the larger void discussed here is about ~2 nm in diameter for the case observed in Fig. 7.6. Based on the recommendation by IUPAC for the field of physisorption [22], the classification of porous systems is mentioned as follows. "In the context of physisorption, it is expedient to classify pores according to their sizes: (i) pores with widths exceeding about 50 nm are called macropores; (ii) pores of widths between 2 and 50 nm are called mesopores; (iii) pores with widths not exceeding about 2 nm are called micropores." Even the large void found in the present simulations is still between microporous and mesoporous regions. Because the porous system in the present work was prepared by the expansion of the original system in NVE conditions, observed enhancement is correlated with the residual stress in the system.

These effects of the residual stress in the system are comparable to that found in the stress-strain curve [2] because pressure gradually changes during this process.

Effect of Stress Tensor on the Enhancement

In Fig. 7.7, pressures of the system obtained in NVE conditions at 600 К in the quasi-equilibrate states are shown as a function of the density.

The minimum is found near the maximum of the diffusion coefficient. This means that the loosening of the cage in the porous lithium disilicate is accompanied with the large negative pressure of the system in NVE conditions. In other words, the enhancement is suggested to be caused by the changes in the stress tensor in the system.

Figure 7.7 The pressure of the system under NVE conditions plotted against the density of the system. At the lowest density in the plot, the negative pressure is found but it increased gradually (see arrow) with the formation of a larger void observed in Fig. 7.6.

In the case of the lowest density (p = 1.58), an increase of the pressure was observed during the formation of larger voids in NVE condition as shown by the arrow in the figure. This means that the further rearrangement of the structure was brought by the stress relaxation in the system. In the following NPT run, shrinks of the size of voids are observed with increase of the pressure (to ~0 atm). However, voids formed do not disappear even after 1 ns (1,000,000 steps) run. The result of a longer length scale will be shown later.

It is useful to see how the pressure of the system is related to this behavior. The pressure of the system is defined as follows from the virial theorem [23]:

where г_ц= |r, - r_; |.

Accordingly, the pressure of the system, as well as each component of the pressure tensor of the system, is a function of the position of the particles on the potential surface.

To consider the mechanism of the enhancement in other systems such as nanocomposites discussed in Chapter 3, it seems to be important to examine the stress tensor and related structures. It will be interesting to examine if the a similar structure and/or mechanical changes occurred in other systems showing enhanced dynamics and if similar structural changes without negative pressure can cause the enhancement or not.

Thus, the mechanism of the enhancement is closely related to the mechanical properties found in the stress-strain curves and nanofracture of the system. In the stress tensor, its components are stresses exerted across surfaces perpendicular to the coordinate directions. The observed relation between pressure and volume in Fig. 7.7 is comparable to the relationship between stress and strain if we considered the difference of plus and minus sign of values and dimensionality.

Recently, Wang et al. [24] examined the stress-strain curve of sodium silicate and silica using MD simulations. As shown in Fig. 7.8, the silicate system shows ductile behaviors and after that, nanofracture is observed. The result is compared with that of silica, where brittle behavior is observed. In the sodium silicate, the following observation was reported [24]. "As shown in the inset of Fig. 7.8, as the strain increases, the relative deformation of Si-0 bonds presents the same shape as that of the stress-strain curve and eventually goes back to its initial zero-stress value.

This means that Si-0 bonds essentially deform in a reversible elastic fashion under stress. On the other hand, Na-0 bonds behave in a significantly different way. Indeed, the maximum relative elongation of the Na-0 bonds is much lower than that of the Si-0 bonds, which shows that, after a short elastic regime, these bonds yield at low stress and, thereby, initiate the fracture through some plastic deformations prior to the failure of any Si-0 bond.”

Figure 7.8 Stress-strain response of the simulated (Na₂O)₂₀(SiO₂)₈₀ glass during fracture. The snapshots illustrate the volume of the cavities, with a radius larger than 5 A, that are formed at various strains. The inset shows the relative variation of the average Si-0 and Na-0 bond lengths (Д///₀) with respect to the strain. The cut-offs used to identify Si-0 and Na-0 bonds are 1.974 and 3.311 A, respectively, as determined from the position of the first minimum after the peak associated to the first coordination shell in the pair distribution functions. Reprinted from Wang, B., Yu, Y., Wang M., Mauro, J. C., Bauchy, M., Phys. Rev. B, 93, 064202 (2016) with permission.

From these results, alkali metal ions seem to take important roles in the mechanical properties of the system. In the case of porous silicates, the distances between pairs of atoms (or ions) are shown as the function of the density in Fig. 7.9a. For the distance in each pair, a position of the maximum of the first peak in g[r) is used. In this plot, there are no remarkable changes in Li-0 and Si-0 pairs. Instead, changes in the Li-Li distance seem to be large and correlated well for the changes in the pressure of the system.

The position of the maximum for the Li-Li distance is also found near the maximum of the diffusivity. A small maximum is also observed for Li-Si and Si-Si pairs. These results mean that the changes in the structures are mainly for spaces without direct bonding. Therefore, structural changes related to these parts should not be neglected for the arguments of mechanical behaviors.

At first sight, this result is not the same as those in the inset of Fig. 7.8. However, this result is not necessarily against the result concerned with the local structures, because the changes shown in the inset are in an order of ~1% in length. To examine the behaviors of this order, in Fig. 7.9b, the values for Si-0 and Li-0 are replotted in an enlarged vertical axis. Then, similar trends as in Na-0 and Si-0 bonds in the inset of Fig. 7.8 are found for Li-0 and Si-O. As shown in this figure, the changes in Li-0 occurred at the beginning of the expansion and those for Si-0 occurred later. The trend observed is consistent with that in sodium silicate.

Figure 7.9 (a) Density dependence of the distance between neighboring atoms for all pairs. The distance for Li-Li ions shows the largest change with density in our model. It becomes longer with decreasing density. With the formation of voids, it returns to be short. For Li-Si and Si-Si pairs, maxima are also observed, (b) Density dependence of the distance for Si-0 and Li-0 pairs in enlarged scales. The distance for Li-0 increases with decreasing density and it decreases with a further decrease of density. The maximum for Si-0 is not so clear but seems to be located in a smaller density region than in Li-O. Trends are consistent with the results in the inset of Fig. 7.8.

From their results, it is clear that alkali metal ions take important roles in the mechanical properties of the system.

That is, they prevent the brittle behavior of the system and make rearrangement of the system easier.

This finding is important to consider the self-healing process of the system in the following sections.