Self-Healing Process in NPT Conditions
The system prepared in the constant-volume condition is not necessarily stable under atmospheric pressure. The system will shrink because of negative pressures in the system equilibrated in NVE conditions when the porous system is prepared by expansion.
This process seems to be interesting from the following two points of view; one is controlling the properties of porous materials and another is related to the mechanism of the self-healing process that occurs in porous systems. They are also related to the mechanism of nanofractures. Here, we will examine porous lithium disilicate and metasilicate systems from these points of view.
Self-Healing Process in Porous Lithium Disilicate
If molecular dynamics simulation of porous lithium disilicate (see Chapters 6) obtained in the NVE condition was followed by NPT runs, recovery of the system density was observed in several cases.
In Fig. 7.10, the recovery curves of density starting from the p = 1.98 are shown. During the NPT run, the system gradually shrank to p = 2.44 and the pressure of the system decreased to ~0 к atm within 150 ps at 600 K. At this density, large voids are not formed, and the system can easily return to the original density when the NPT condition was applied. The data of runs at different temperatures are shown in the same figure. Even if the NPT run is performed at 300 К after the NVE run at 600 K, the result is comparable to the run at 600 K. In the case of the run at 300 К after the quench at 100 K, the results are also quite similar. For the data at 600 K, some data points in shorter time scales are shown. As shown in this figure, a large fluctuation of density occurs on a short time scale. These results suggest that the process observed in this region is not controlled by thermally activated processes such as diffusive motions. It seems to be caused by the ductile behavior of the system and/or with fluctuations of structures in a short length scale. The existence of the negative pressure of the system means that the process is related to the stress relaxation and mechanical properties of the system.
Our results are consistent with those by Wang et al.  for sodium silicate. That is, in the small- and medium-density regions, the system of alkali silicate shows an elastic property and nanoductility, respectively. Formation of larger voids observed in the lower the density region (p < 1.58) seems to correspond to the brittle behavior (nanofracture) of the system found in sodium silicate in .
In Fig. 7.11, snapshots of ball-and-stick structures before and after self-healing processes in NPT run of lithium disilicate at 600 K, where the run was starting from p = 1.84, are shown. In this case, recovery is almost completed within ~10 ps. This time scale seems to be quite short, where ions are still in the local region, judging from the MSD of several temperatures shown in Fig. 7.2. Therefore, essentially, this process is considered not accompanied with diffusive motion, although there is a non-zero probability of diffusive motions at this temperature.
Figure 7.10 Self-healing processes observed in porous lithium disilicate systems during NPT runs at 600 К (blue, open diamond) and at 300 К (red, filled diamond) starting from p = 1.98. The result at 300 К after the quench at 100 К is also shown (pale blue, filled diamond). Data are connected for guiding the eyes. When we started the comparable density, the resultant curves are similar even though different temperature conditions are used. That is, the process is not governed by the thermally activated processes in this case.
Figure 7.11 Example of the healing process in structures of porous lithium disilicate at 600 K. Before (left) and after (right) self-healing processes in NPT run. The run was starting from p = 1.84. In this case, almost complete recovery of the density was observed within ~10 ps run. Blue: Si atoms. Red: 0 atoms. Green: Li ions. The size of each figure is nearly proportional to the side length of MD cell of each system.
As shown in Fig. 7.12, when the NPT run is started from the density, p = 1.58, the void remained after ~2 ns. In this case, fractured surfaces are separated from other parts and in each part, tight network structures exist. Initiation of healing probably requires longer lead times to wait for the suitable geometrical conditions of atoms being satisfied. Furthermore, diffusivity the of ions is slower than the medium density region, although the motion of atoms is of a longer length scale required for the recovery of the atoms separated from other regions. This explains why the process is slower than in other cases. Even in this case, once the density becomes larger, the acceleration of the process occurs as found in the change of the slope at around 100 ps.
The recovery of density for the above two cases is compared in Fig. 7.13.
In the case of the starting from p = 1.84, a rapid recovery occurred at around 0.2~1 ps. When it started from p = 1.58, a rapid increase in the density was not observed even at around p~ 1.84.
It is interesting to note that fluctuations are observed in the long-time region in the density curves. This means the initiation and termination of the healing process are alternated, which suggests that the system needs a suitable arrangement for starting of healing processes. It may also be related to the heterogeneity of the structures and dynamics discussed in the first half of this chapter.
Figure 7.12 Example of (a) before and (b) after self-healing processes in NPT run. The system is lithium disilicate at 600 К started from p = 1.58. Complete recovery of the density was not observed after ~2 ns. Pressure of the system becomes ~0 in the earlier stage.
Figure 7.13 Upper curves: examples of self-healing processes observed in porous lithium disilicate systems during NPT runs starting from p = 1.84. In the NPT runs at 600 К (blue, open circles) and at 300 К (blue, filled circles). A solid line is a fitted one with a logarithmic function. Power law functions are also applicable. Lower curves: Examples of self-healing of the same system during NPT runs at 600 К (green, open squares) and at 300 К (green, filled squares) starting from at lower density p = 1.58, where the system shows a larger void in the NVE condition. The healing process is much slower and the difference between at 300 К and 600 К is remarkable at a longer time region than ~1 ps. A large fluctuation of the curves was observed at lower curves especially at 600 K. Data points are connected for guiding the eyes.
Different from the healing at a higher density region, if the run started from the lower density, the process is found to be temperature-dependent. As shown in Fig. 7.13, the difference between the curves at 600 and 300 К is found after 1 ps and becomes larger at longer times. The importance of the activated process such as jump motion of ions for the initiation is suggested by this fact. In porous lithium disilicate systems starting from lower density, some holes tend to remain in the systems even after the longer runs. At the longest time (~4 ns), the density already exceeds 2.15, where the rapid healing process is observed if one started the NPT run from that density after the expansion. Therefore, the slow increase of the density means the stability of the system with these holes after rearrangements of networks. This is caused by the formation of rigid structures of holes being separated from other domains. The healing process is much slower than that observed at the case starting from higher density and the difference between at 300 and 600 К is remarkable at a longer time region than ~1 ps. A large fluctuation of the curves is observed at lower curves especially at 600 K.
Further recovery of the system will not be easy once the ~0 pressure condition was achieved.
In Fig. 7.13, a fitted line using a logarithmic function is shown for data at 600 К starting from the higher density. In this case, the curve in the large change in the middle part can be fitted to
The same data can be fitted to the power law as well:
It is interesting to note that the power law is also observed in the fracture of materials [25,26] in several kinds of plots.
From the lower density curves with large fluctuations of the data, it is not clear what functional form is suitable, although these functions, as well as a linear function, can be applied.
Self-healing Process in Porous Lithium Metasilicate
Porous systems can be similarly prepared for lithium metasilicate as in lithium disilicate . In lithium metasilicate with higher alkali contents, the self-healing process is also observed, and the recovery of the system is found to be almost completed within the relatively short observation time. It is found even after the larger void being formed in the NVE run as shown in Figs. 7.14 and 7.15. Naturally, a larger number of Li ions seem to make the system easier to cause the rearrangement of the structure, although further examinations are required to conclude it. Furthermore, the diffusivity of Li ions in metasilicate is larger than the disilicate system .
Figure 7.14 An upper curve: examples of self-healing processes observed in porous lithium metasilicate systems during NPT runs starting from p = 1.80 in the NPT run at 600 К (red, open circles). Lower curves: Examples of self-healing systems of the same system during NPT runs at 600 К (blue, open squares) and at 300 К (blue, filled squares) starting from at lower density p = 1.53, where the system shows a formation of larger voids in the NVE condition. The healing process is slower than that starting from higher density and the difference between at 300 and 600 К is remarkable in a long-time region ~1 ps. Data points are connected for a guide of eyes and fitted lines to logarithmic functions are shown.
In Fig. 7.14, linear parts can be seen in the plot of density against logarithmic of time in both systems starting from p = 1.53 and 1.80 at 600 K.
Figure 7.15 Self-healing processes observed in porous lithium metasilicates. Example of (a) before and (b) after self-healing processes in NPT run for lithium metasilicate at 600 К starting from p = 1.53. In this case, almost complete recovery of the density was observed within ~2 ns run.
The fitted lines for the former and latter, respectively, are
The result at 300 К starting from the lower density is also shown in the figure. After several ps, the deviation from the curve at 600 К becomes larger and it again confirms that the process is affected by the thermally activated process. In the linear part of the log scale plot, a relation
In all three cases mentioned above, power laws are also good approximations as in the porous lithium disilicate system.
The process depends on temperatures and initial density; however, the existence of the common intersections (1.68-1.75) suggests that systems have both the core part and the flexible part sensitive for repairing.
The process in the final stage after the log or power-law region is slower than early processes. Different from the case of disilicate system, the fluctuation of the density during the healing process in the lithium metasilicate system starting from the low density with large voids (p = 1.53) is not remarkable (see Fig. 7.14). The process seems to proceed rather smoothly, and it seems to be accelerated by a larger number of Li ions. Even in the case with larger voids, the repairing of the systems is almost completed as shown in Fig. 7.15, within ~2 ns.
These processes consist of several regions of different mechanisms. As shown in Fig. 7.16, the difference in the results at higher-density and lower-density regions is observed for the time dependence of the pressure during the NPT runs. Although the process might be affected by the time constant and related weight of the wall of the MD, pressures were controlled by the Nose-Hoover method with the same time constant of 1 ps.
As shown in Fig. 7.16, in the case of the run starting from p = 1.80, pressure gradually increases during the healing process until ~200 ps, which corresponds to the end of the logarithmic (or power law) region. Therefore, in this density region, the process is mainly governed by stress relaxation. On the other hand, when the runs are started from p = 1.53, a rapid increase of the pressure is observed at the beginning of the runs and fluctuation seems to continue during the healing process. In this plot, the pressure at 300 К becomes ~0 in the beginning and therefore clear temperature dependence is not observed for the pressure, in spite of the fact that the changes in density in these regions show temperature dependence. These results mean that the longer time process is affected by the thermally activated process such as jump motions of ions as already mentioned for disilicate systems. The healing process proceeds by the fluctuation of the pressure but not necessarily caused by negative pressures in longer time regions. Since the pressure of the system tends to become ~0 at the beginning of the run as shown in Fig. 7.16, the healing discussed here is not just shrunk by the pressure.
Figure 7.16 Changes of pressures during the self-healing processes observed in porous lithium metasilicate systems during NPT runs at 600 К (open marks), where the pressure of the system is plotted against a logarithmic scale of time. For the lower density system, a result at 300 К (filled square) is also shown. The same marks and colors as in Fig. 7.14 are used here. In the system started from the higher density, negative pressure was observed during the process, while the system started from the lower density, fluctuation of pressure was observed during the process.