Comparison of ion sites and diffusion paths in glasses obtained by molecular dynamics simulations and bond valence analysis

Based on molecular dynamics simulations of a lithium metasilicate glass we study the potential of bond valence sum calculations to identify sites and diffusion pathways of mobile Li ions in a glassy silicate network. We find that the bond valence method is not well suitable to locate the sites, but allows one to estimate the number of sites. Spatial regions of the glass determined as accessible for the Li ions by the bond valence method can capture up to 90% of the diffusion path. These regions however entail a significant fraction that does not belong to the diffusion path. Because of this low specificity, care must be taken to determine the diffusive motion of particles in amorphous systems based on the bond valence method. The best identification of the diffusion path is achieved by using a modified valence mismatch in the BV analysis that takes into account that a Li ion favors equal partial valences to the neighboring oxygen ions. Using this modified valence mismatch it is possible to replace hard geometric constraints formerly applied in the BV method. Further investigations are necessary to better understand the relation between the complex structure of the host network and the ionic diffusion paths.

Figure 1

(a) Pair correlation function of Li-Li, Li-Si and Li-O and (b) time-dependent mean square displacements of Li, O, and Si ions at $T=700\mathrm{K}$.

Figure 2

Number of clusters $N_{\mathrm{cl}}$ in dependence of ${\rho }_{\mathrm{th}}$ for three different grid spacings $\Delta$. Note that the mean number density of Li ions is $144∕{\left(16.68\mathrm{\AA }\right)}^3\cong 0.031{\mathrm{\AA }}^{-3}$ only, which in comparison with the relevant scale of ${\rho }_{\mathrm{th}}$ implies that the Li ions are strongly localized on the clusters.

Figure 3

(a) Mean number density ${\rho }_{\mathrm{cl}}$ of Li ions on the 165 clusters identified by the Hoshen-Kopelman algorithm. (b) Volume $V_{\mathrm{cl}}$ and (c) Li occupation probability $t_{\mathrm{cl}}∕t_{\mathrm{sim}}$ of the clusters. The clusters are sorted according to increasing ${\rho }_{\mathrm{cl}}$.

Figure 4

(a) Total residence and hopping time normalized with respect to the simulation time $t_{\mathrm{sim}}$ and (b) their ratio for the 165 clusters determined by the Hoshen-Kopelman algorithm. The dashed line marks the threshold above which clusters are identified as sites [see Eq. (2)].

Figure 5

(a) Histogram of of bond valence sums for an instantaneous configuration of the equilibrated ${\mathrm{Li}}_2\mathrm{Si}{\mathrm{O}}_3$ glass. (b) Probability density of bond valence sums with respect to the time-averaged structure of network forming O and Si ions.

Figure 6

Volume fraction $v_{\mathrm{BV}}$ of the BV path as a functions of the mismatch threshold $d_{\mathrm{th}}$.

Figure 7

Quality of agreement between (a) the subsets ${\mathcal{P}}_{\text{sites}}$ and ${\mathcal{P}}_{\mathrm{BV}}\left(d_{\mathrm{th}}\right)$ and (b) the subsets ${\mathcal{P}}_{\mathrm{path}}$ and ${\mathcal{P}}_{\mathrm{BV}}\left(d_{\mathrm{th}}\right)$.

Figure 8

Quality of agreement between the percolation path ${\mathcal{P}}_{\mathrm{perc}}$ and ${\mathcal{P}}_{\mathrm{BV}}^{\prime }\left(d_{\mathrm{th}}\right)$ as a function of the BV mismatch threshold $d_{\mathrm{th}}$.

Figure 9

Number of BV clusters as a function of the threshold mismatch $d_{\mathrm{th}}$.

Figure 10

Projection of local maxima of $\rho \left(\mathbf{x}\right)$ (+) and local minima of $d\left(\mathbf{x}\right)$ (×) in a three-dimensional layer with thickness $1.668\mathrm{\AA }$ (corresponding to 10% of the simulation box) onto a reference plane.

Figure 11

Lower panel: Number of possible site centers on the BV path ${\mathcal{P}}_{\mathrm{BV}}^{\prime }\left(d_{\mathrm{th}}\right)$ as determined by the “row-by-row algorithm” described in the text. Upper panel: Cohen’s kappa value for the path ${\mathcal{P}}_{\mathrm{BV}}^{\prime }\left(d_{\mathrm{th}}\right)$ in comparison with ${\mathcal{P}}_{\mathrm{perc}}$ (redrawn from Fig. 8). The number of estimated sites (151) at the optimal $\kappa$ is indicated by the solid lines. The dashed line marks the number of Li ions (144) and the dotted line the number of sites (148) found in Sec. 3.