3.4 Bulk Dynamics
3.4.1 Mean-Squared Displacement
Pioneering work on alkali diffusion in MD simulations was done by Huang and Cormack, though due to computer resource constraints, they collected data after artificially increasing the kinetic energy of the alkali while keeping the silicate network at room temperature in order to calculate reasonable alkali diffusion coefficients.45,74,174 A more recent MD diffusion study of the Na2O·2SiO2
composition using a Vessal potential model found that simulation temperatures had to be increased to 1000 K to show significant diffusion of sodium ions.181 For the glasses in this study, a simulation temperature of 1000 K is well below the estimated simulation Tg (discussed below), and should be sufficient to detect alkali motion without disruption of the network structure by the long-range migration of network ions.
Figure 3.28 shows mean-squared displacement (MSD) plots for all ions in each glass at 1000 K over the saved configurations of the simulation (timesteps 20,000—500,000). The P and Ca plots do not show long-range diffusive behavior, but the Na plot shows that by the end of the 1 ns simulation, the average Na ion is over 10 Å away from its original position (Section 3.4.4 below shows that there is a considerable distribution of diffusive behavior for Na ions).
Figure 3.28. MSD at 1000 K for 500,000 timestep (1 ns) simulations: a) P, b) Ca, and c) Na. Ca and P do not show long range transport behavior.
From the structural data and the reaction kinetics of bioactive glasses in the body, we would expect MSD plots to show greater ion motion in the 45S5 glass, but in the Na and Ca plots we see greatest motion in the 55S4.3 glass. The lower diffusion for 45S5 may not be necessary for rapid creation of the HCA layer, as the fragmented structure may be the dominant factor. On the other hand, the greater Na and Ca migration in the 55S4.3 glass may be a factor in determining its greater bioactivity as compared to 60S3.8 glass, as their structures are similar based on the structural analysis above.
c
a b
No long-range diffusion is observed for phosphorus, but only local displacements. Little explanation for this observation is necessary for the phosphorus ions that are bonded to the network (non-P0). The lack of movement for independent PO4 groups is likely due to the constraints of the bulk simulation cell (however, see Section 3.4.4 below). The PO4 specie is relatively large, so diffusion is likely prohibitive except when the glass surface has been reacted with water, as in the first two steps of the reaction sequence for creation of the HCA layer. These act to open up the glass structure, presumably opening avenues for ion migration. The hydrolysis step would also act to create more PO4 groups which may also increase the diffusive nature of this specie to the surface, where it can participate, as proposed8,23,25, in Stage 4 of Hench’s reaction model.
Nonlinearity in the phosphorus MSD plot for phosphorus is likely due to poor statistics; contributions include the low number of P ions in the simulation and the certainty that we have not run the simulations long enough to get a proper look at long-time migration. Both of these factors are due to the limitations of available computing power. However, similar MSD behavior was observed for a silica surface in which the MSD was found to decrease at longer times (as short as 50 ps), presumably due to the averaging of only few ions (i.e., poor statistics).179 It may also be due to the presence of multiple diffusion regimes182 or the observation that individual ions of a given specie can move at drastically different rates.45,77,183 These may help to explain the nonlinearity and/or multiple regions of linearity in the Na and Ca plots.
It is understandable that phosphorus should exhibit no long-range diffusion, as it would migrate as relatively large PO4 groups.8,23 However, there must be another explanation for the low diffusivity of calcium, which is actually “smaller” (i.e., has a smaller ionic radius) than sodium (even though they are both point charges in the current simulations). The most likely explanation is that the greater ionic strength of calcium serves to attach it to the relatively immobile network, even if some sort of migratory path exists (i.e., due to nearby Ca sites). This is born out in the deconvolution of the Ca-O correlation plots from Section 3.3.1.9 above, as Ca has a greater contribution of NBO than Na for its first peak. This implies a greater ability
of Ca to compensate the negative charge of NBO (as expected due to its greater cationic charge), and thus a lower probability that it would leave this type of configuration (i.e., migrate to a different site). In sum, with twice the formal valence charge of a sodium ion, a calcium ion is expected to better compensate the negatively-charged NBO (for a given separation distance) by a factor of two.
Figure 3.29. MSD at 1000 K, including 1 and 5 million timestep simulation results:
a) P, b) Ca, and c) Na.
The length of simulations is a significant issue in MD diffusion studies.
While large simulation cells have been developed for structural studies, in many cases, small cells are used for dynamical studies, ignoring possible simulation size
a b
c
effects on structure or dynamics, in order to increase simulation length.183 One possible effect of small simulation size is the reduced number of sites available for diffusion. Simulation length was tested by running a 45S5 1000 K simulation for 5 million steps (5 ns), which took approximately four weeks to complete, making it prohibitive for a complete dynamical study, but useful for comparison with simulations an order of magnitude shorter. Figure 3.29 shows the results of this simulation on the MSD of P, Ca, and Na, with the first 2 ns of the 5 ns simulation used as an intermediate comparison.
Clearly, the P MSD is drastically different in shape and slope among the three simulation lengths. The slope has actually decreased, showing that over the time range investigated, little long-range diffusion is still present. It seems that the local vibrational motions have been averaged out for the longest simulation. The calcium plot also shows a decreasing slope of greater linearity with simulation length, but there is less of an effect on the Na plot, likely due to the greater number of Na ion in the simulation. This is similar to the results of a previous MD simulation of a sodium trisilicate glass, where the Na diffusion coefficient was found to decrease by a factor of 5-8 in going from a 30 ps simulation to 1 ns.184 In sum, it has been shown that increasing the simulation length has a significant effect on MSD plots, confirming that simulations were not run for sufficient time to observe long-range diffusion for Ca, but that long-range diffusive motion is still only observed for Na in the time regime of the current study.