3.3 Bulk Analysis
3.3.1 Bulk Structure
3.3.1.4 Ring Size Distribution
Ring size distribution is an important means to describing medium range structure in glasses. In this work, only the primitive ring, defined as a ring that cannot be divided into two smaller rings, is considered.124,125 A four-membered ring, for example, consists of four network formers cations and four bridging oxygen ions (Figure 3.5). Ring statistics have found use in the study of silicates and borates, for which the presence of small rings (2-8 members) is documented.126-129 However, due to disruption of the network structure in the presence of modifier cations, large rings (>30 members) can be present in the modified random networks of modified silicates, as they surround modifier-rich regions.45,99 Rings with more than six members are generally not distinguishable using spectroscopic techniques, so molecular dynamics provides a unique method of counting large rings.71,97,98,130,131
Previous MD models have shown that pure silica has a nearly symmetrical ring size distribution in the range from 3 to 12-membered rings, with 5, 6, and 7- membered rings in the greatest proportion.97,132-135 Previous studies have shown that as modifier content is increased, the symmetrical shape gradually disappears and large-membered rings are created at the expense of small-membered rings.97,99,124 This is due to the ability of the structure to reduce the strain inherent in small- membered rings by incorporation of NBO due to the presence of modifier. This reduces O-Si-O bond angles, for example. The presence of large-membered rings also suggests evidence for Greaves’ MRN36,117, as there may be large-membered rings
which surround high concentrations of modifier ions. For example, Du found 38- membered rings for an MD-simulated sodium silicate glass of composition 50Na2O·50SiO2.99
Figure 3.5. Four-membered ring structure from 45S5.
It should be noted that by the definition of a ring, only network formers with two or more BO (Q2, Q3, Q4) may participate in a ring, so the low connectivity of these glasses provides even greater reason to expect relatively few rings, and that the number of rings may decrease with decreasing silica content. However, based on the low connectivity, larger rings are expected because the local connectivity has decreased, making the presence of small rings less likely.
The ring-size distribution up to 49 members was calculated from the final configuration and without discrimination between phosphorus and silicon as network formers using an algorithm developed by Yuan and Cormack.124 As in previous studies, a maximum cutoff of 3.40 Å between network formers was applied to define linkages in the structure (i.e., instead of explicitly considering each Si-O or P-O bond as a linkage), to speed calculation.97,99,124 A second, independent bulk simulation was
1
4
2
3
Si
BO
NBO
created from a different random configuration for comparison. Also, a simulation of three times the “normal” size was created. Plots of the number of rings per network former (Si or P) are presented in the figures below. A value of unity indicates that, on average, one network former is involved in a ring of a given size.
For 45S5, there are far fewer rings overall than for the other two composition (Table 3.V), which is indicative of a more disrupted structure rather than an extensively connected network structure, as more evident in the 55S4.3 and 60S3.8 simulations (Figure 3.6). However, the 45S5 composition also contains the largest rings, which may be due to large alkali regions surrounded by a continuous network ring.97 There are some differences between the configurations of the same size, but the above comments hold true in general.
Overall, there are no 2-rings and very few 3-rings found in these simulations, while these exist in pure silica.50,126,135 The 60S3.8 composition shows perhaps a remnant of the silica ring distribution from 4-10 membered rings, but that is even less clear for the lower silica content glasses. This lack of small rings is expected, based on results for the 50Na2O·50SiO2 simulated glass97 noted above, because of the high modifier content and depolymerized networks of bioactive glasses. And while there are differences for configurations of the same size, the general shape of the distribution for small rings is similar.
Table 3.V. Ring Statistics
Glass # Nodes Total # rings # Rings/node
45S5 513 741 1.44
55S4.3 603 5258 8.72
60S3.8 653 5001 7.66
45S5+ 513 635 1.24
55S4.3+ 603 4287 7.11 60S3.8+ 653 6170 9.45
45S5-3X 1539 2090 1.36
55S4.3-3X 1809 12934 7.15 60S3.8-3X 1959 15570 7.95
+ - second (independent) bulk simulation of the original size
The 45S5 and 55S4.3 compositions show a concentrations of large rings (>30 members). The existence of these large rings for the lower silica content glasses may be explained by the presence of relatively large alkali-rich regions that are surrounded by a continuous part of the network. It is proposed that the rather high values for large-membered rings (the high peaks in Figure 3.6 and Figure 3.8) may be due to the presence of large rings that share many linkages. For example, the 22-membered ring shown in Figure 3.10 for 55S4.3 has on its periphery a four-membered ring; this affords two equal-length paths and so two 22-membered rings are counted.
Therefore, in interpreting Figure 3.6, it is not necessary to conclude, for example, that there is a large number of 21-membered rings for 60S3.8, but perhaps only many paths with a number of common linkages. Also, an odd-numbered peripheral ring may contribute to increasing values in adjacent peaks in the ring-size distribution.
For example, if the four-membered ring in Figure 3.10 was instead a five-membered ring, there may be a 22-membered path and a 23-membered path. This may explain in part the large values of 32, 33, and 34-membered rings for 55S4.3 (Figure 3.6).
A previous study has shown that the simulation box size can have a significant effect on ring size distribution in modified silicates.124 In that study, when the simulation box size was increased 8-fold for a sodium-silicate glass, larger rings (27- 28-membered) appeared at the expense of some smaller (13-15-membered) rings that were apparently restricted by the periodic boundary conditions of the smaller box.
One proposed explanation was that larger boxes have larger modifier-rich regions and thus larger rings surrounding these regions.124
The simulation box size effect was tested in this study by creating bulk simulations with three times the number of ions. Because of the differences in RSD observed for simulations of the same size (Figure 3.9 and Table 3.V), it is not clear that a simulation-size dependence exists for these glasses, at least between simulations that differ in size by a factor of three. By comparing the number of rings per node, we can only say that there is a difference between the 45S5 composition and the other two compositions, but not that simulationsize matters.
Figure 3.6. Bulk simulation ring size distributions.
Figure 3.7. Bulk simulation ring size distributions for simulations three times regular size.
Figure 3.8. Bulk simulation ring size distributions for two different simulation sizes.
a
b
c
Figure 3.9. Bulk simulation ring size distributions for two simulations of the same size.
a
b
c
Figure 3.10. Ring structure from 55S4.3 showing small and large-membered rings.