Glass Fragility and Atomic Ordering on the Intermediate and Extended Range

The relation between the fragility of glass-forming systems, a parameter which describes many of their key physical characteristics, and atomic scale structure is investigated by using neutron diffraction to measure the topological and chemical ordering for germania, or ${\mathrm{GeO}}_2$, which is an archetypal strong glass former. We find that the ordering for this and other tetrahedral network-forming glasses at distances greater than the nearest neighbor can be rationalized in terms of an interplay between the relative importance of two length scales. One of these is associated with an intermediate range, the other with an extended range and, with increasing glass fragility, it is the extended range ordering which dominates.

Figure 1

The measured Bhatia-Thornton partial structure factors, $S_{IJ}^{\mathrm{BT}}(k)$, for glassy ${\mathrm{GeO}}_2$ (solid curves with vertical error bars—the latter are comparable to the curve thickness at most $k$ values) and ${\mathrm{ZnCl}}_2$ [broken curves (red online)] plotted as a function of the scaled scattering vector $k{r}_{AX}$ where $r_{AX}$ is the separation of unlike nearest neighbors. In the case of ${\mathrm{GeO}}_2$ each function has a FSDP at $k\approx 1.53{\AA }^{-1}$ or $k{r}_{AX}\approx 2.65$.

Figure 2

The measured partial pair-distribution functions $g_{\mathrm{GeO}}(r)$ (solid black curve), $g_{\mathrm{GeGe}}(r)$ [light solid curve (red online)] and $g_{\mathrm{OO}}(r)$ [broken curve (blue online)] for glassy ${\mathrm{GeO}}_2$. The inset shows the Bhatia-Thornton partial pair-distribution functions, $g_{IJ}^{\mathrm{BT}}(r)$, for ${\mathrm{GeO}}_2$ (solid black curves) and ${\mathrm{ZnCl}}_2$ [broken curves (red online)] plotted as a function of the scaled distance, $r/r_{AX}$, which ensures that the first peak positions are superimposed. The $g_{IJ}^{\mathrm{BT}}(r)$ were obtained from the $g_{\alpha \beta }(r)$ by using the relations [26] $g_{\mathrm{NN}}^{\mathrm{BT}}(r)=c_A^2{g}_{AA}(r)+c_X^2{g}_{XX}(r)+2c_A{c}_X{g}_{AX}(r)$, $g_{\mathrm{CC}}^{\mathrm{BT}}(r)=c_A{c}_X[g_{AA}(r)+g_{XX}(r)-2g_{AX}(r)]$ and $g_{\mathrm{NC}}^{\mathrm{BT}}(r)=c_A[g_{AA}(r)-g_{AX}(r)]-c_X[g_{XX}(r)-g_{AX}(r)]$ where $c_A=1/3$ and $c_X=2/3$.

Figure 3

Decay of the Bhatia-Thornton partial pair-distribution functions for glassy ${\mathrm{GeO}}_2$ and ${\mathrm{ZnCl}}_2$ as shown by plotting $\ln |r^{\prime }{h}_{IJ}^{\mathrm{BT}}(r^{\prime })|$ vs $r^{\prime }$ (dark solid black curves for ${\mathrm{GeO}}_2$) or $\ln |r{h}_{IJ}^{\mathrm{BT}}(r)|$ vs $r$ [light solid curves (red online) for ${\mathrm{ZnCl}}_2$] [24] where the scaled distance, $r^{\prime }$, is related to the actual distance by $r^{\prime }=1.267r$. The decay length for the N-N, N-C, and C-C functions is 4.9(7), 4.9(2), 5.1(4) Å for ${\mathrm{GeO}}_2$ and 5.0(4), 5.4(2), and 5.0(1) Å for ${\mathrm{ZnCl}}_2$, respectively, and was obtained from the fitted straight lines given by the broken blue (${\mathrm{GeO}}_2$) or chained green (${\mathrm{ZnCl}}_2$) curves. The quoted $r_{\mathrm{decay}}^{\prime }$ values for ${\mathrm{GeO}}_2$ are related to the actual decay lengths by $r_{\mathrm{decay}}^{\prime }=1.267r_{\mathrm{decay}}$.

Figure 4

The measured total-structure factor for ${\mathrm{SiO}}_2$, $F_{\mathrm{Si}}(k)$, (solid curve—the error bars are smaller than the curve thickness) is compared with its reconstruction, $F_{\mathrm{Si}}^{\mathrm{rec}}(k)$, (broken curve) from the measured partial structure factors for ${\mathrm{GeO}}_2$ (see Fig. 1) where $F_{\mathrm{Si}}^{\mathrm{rec}}(k)/\mathrm{barn}=0.2758(3)[S_{\mathrm{NN}}^{\mathrm{BT}}(k)-1]+0.00608(3)[S_{\mathrm{CC}}^{\mathrm{BT}}(k)/c_A{c}_X-1]-0.1737(4)S_{\mathrm{NC}}^{\mathrm{BT}}(k)$ and the weighting factors are calculated from the scattering lengths and atomic fractions of Si and O [26]. Both data sets are plotted as a function of the scaled scattering vector $k{r}_{AX}$. The atomic number density of ${\mathrm{SiO}}_2$ is $0.0665(2){\AA }^{-3}$ and the coherent scattering length $b(\mathrm{Si})=4.1491(10)\mathrm{fm}$.