The Atomic-Scale Structure of Network Glass-Forming Materials

Abstract

A prerequisite for understanding the physico-chemical properties of network glass-forming materials is knowledge about their atomic-scale structure. The desired information is not, however, easy to obtain because structural disorder in a liquid or glass leads to complexity. It is therefore important to design experiments to give site-specific information on the structure of a given material in order to test the validity of different molecular dynamics models. In turn, once a molecular dynamics scheme contains the correct theoretical ingredients, it can be used both to enrich the information obtained from experiment and to predict the composition and temperature/pressure dependence of a material’s properties, a first step in using the principles of rational design to prepare glasses with novel functional properties. In this chapter the symbiotic relationship between experiment and simulation is explored by focussing on the structures of liquid and glassy ZnCl$_2$ and GeSe$_2$, and on the structure of glassy GeO$_2$ under pressure. Issues to be addressed include extended range ordering on a nanometre scale, the formation of homopolar (like-atom) bonds, and the density-driven mechanisms of network collapse.

Appendix: Concentration of Defects in GeSe\(_2\) Glass from the Law of Mass Action

Following Feltz [127, 128], consider the reversible reaction

$$\begin{aligned} \mathrm{2Ge}{-}\mathrm{Se} \rightleftharpoons \mathrm{Ge}{-}\mathrm{Ge} + \mathrm{Se}{-}\mathrm{Se} \end{aligned}$$

(1.18)

where homopolar or defect bonds are formed in pairs, and for which the law of mass action gives an equilibrium constant

$$\begin{aligned} K = \frac{\left[ \mathrm{Ge}{-}\mathrm{Ge}\right] \left[ \mathrm{Se}{-}\mathrm{Se}\right] }{\left[ \mathrm{Ge}{-}\mathrm{Se}\right] ^2} = \exp \left( -\frac{\varDelta G}{\mathrm{R}T}\right) \end{aligned}$$

(1.19)

where $\left[ \mathrm{A}{-}\mathrm{B}\right]$ represents the concentration of A$-$B bonds, $\varDelta G$ is the standard reaction Gibbs energy, R is the molar gas constant, and $T$ is the absolute temperature. From (1.18) it follows that the concentration of Ge$-$Ge or Se$-$Se defect bonds $n_d$ = $\left[ \mathrm{Ge}{-}\mathrm{Ge}\right]$ = $\left[ \mathrm{Se}{-}\mathrm{Se}\right]$ where the Ge$-$Ge homopolar bonds might be in ethane-like Se$_{3/2}$Ge$-$GeSe$_{3/2}$ units as suggested by $^{119}$Sn Mössbauer spectroscopy experiments [83, 129] and the Se$-$Se homopolar bonds might be in dimers linking Ge-centred tetrahedra. Equation (1.19) can therefore be re-written as

$$\begin{aligned} \frac{n_d}{n_0} = \exp \left( -\frac{\varDelta G}{2\mathrm{R}T}\right) \end{aligned}$$

(1.20)

where $n_0 \equiv \left[ \mathrm{Ge}{-}\mathrm{Se}\right]$. If the concentration of defects is small such that $n_d \ll n_0$ then the latter is approximately equal to the concentration of Ge$-$Se bonds in a non-defected system.

$\varDelta G$ can be estimated from the difference between the Ge$-$Se, Ge$-$Ge and Se$-$Se bond enthalpies which take values of 225, 188 and 227 kJ mol$^{-1}$, respectively, at 298 K i.e. $\varDelta G \simeq \varDelta H$ = 2$\times$225 $-$ 188 $-$ 227 = 35 kJ mol$^{-1}$ [127]. Hence, an estimate for the fraction of defects in the melt at the glass transition temperature ($T_g$ = 665 K) is given by $n_d/n_0 \simeq$ 0.042. Alternatively, if $n_d \equiv N_d/V$ and $n_0 \equiv N_\mathrm{bond}/V$, where $N_d$ is the number of Ge$-$Ge or Se$-$Se homopolar bonds and $N_\mathrm{bond}$ is the total number of bonds, it follows that $N_d/N_\mathrm{bond} \simeq$ 0.042. This ratio is probably a lower limit because the value of $\varDelta G$ used in the calculation is likely to decrease when the entropy term $\varDelta S$ is taken into account ($\varDelta G = \varDelta H - T\varDelta S$ if the absolute temperature $T$ is constant), and the reaction enthalpy $\varDelta H$ is likely to be smaller at $T_g$ as compared to room temperature [127, 128].

Let the total number of atoms in the system be denoted by $N = N_\mathrm{Ge} + N_\mathrm{Se}$ where $N_\mathrm{Ge}$ and $N_\mathrm{Se}$ are the numbers of Ge and Se atoms, respectively, such that the atomic fractions are given by $c_\mathrm{Ge} = N_\mathrm{Ge}/N$ and $c_\mathrm{Se} = N_\mathrm{Se}/N$. From the NDIS results on GeSe$_2$ glass [22, 23], the coordination number for Ge$-$Ge homopolar bonds $\bar{n}_\mathrm{Ge}^\mathrm{Ge}$ = 0.25(5). If these bonds form only in pairs then $\bar{n}_\mathrm{Ge}^\mathrm{Ge} = \left( N_\mathrm{Ge, homo} \times 1\right) /N_\mathrm{Ge}$ such that the number of Ge$-$Ge bonds is given by $N_\mathrm{Ge{-}Ge} = \left( \bar{n}_\mathrm{Ge}^\mathrm{Ge} \times N_\mathrm{Ge}\right) /2$ where the factor of two avoids double counting and $N_\mathrm{Ge} = N/3$. It follows that $N_\mathrm{Ge{-}Ge}$ = 0.042(8)$N$. Similarly, from the NDIS results the coordination number for Se$-$Se homopolar bonds $\bar{n}_\mathrm{Se}^\mathrm{Se}$ = 0.20(5). If these bonds form only in pairs then $\bar{n}_\mathrm{Se}^\mathrm{Se} = \left( N_\mathrm{Se, homo} \times 1\right) /N_\mathrm{Se}$ such that the number of Se$-$Se bonds is given by $N_\mathrm{Se{-}Se} = \left( \bar{n}_\mathrm{Se}^\mathrm{Se} \times N_\mathrm{Se}\right) /2$ where the factor of two avoids double counting and $N_\mathrm{Se} = 2N/3$. It follows that $N_\mathrm{Se{-}Se}$ = 0.067(17)$N$. Thus, within the experimental error, $N_\mathrm{Ge{-}Ge} \sim N_\mathrm{Se{-}Se}$ as in the model of Feltz [127] such that $N_d \simeq \left( N_\mathrm{Ge{-}Ge} + N_\mathrm{Se{-}Se}\right) /2$ = 0.05(2)$N$.

For GeSe$_2$, the number of Ge$-$Se bonds in a non-defected system $N_\mathrm{bond}$ $=$ $(N_\mathrm{Ge}Z_\mathrm{Ge}$ $+$ $N_\mathrm{Se}Z_\mathrm{Se})/2$ = $\left( c_\mathrm{Ge}Z_\mathrm{Ge} + c_\mathrm{Se}Z_\mathrm{Se}\right) N/2$ where $Z_\alpha$ is the number of bonds formed by chemical species $\alpha$. Since $Z_\mathrm{Ge}$ = 4, $Z_\mathrm{Se}$ = 2, $c_\mathrm{Ge}$ = 1/3, $c_\mathrm{Se}$ = 2/3 it follows that $N_\mathrm{bond}$ = $4N/3$. Thus $N_d/N_\mathrm{bond} \simeq$ 0.04(2) for the NDIS results, which is in agreement with the value $N_d/N_\mathrm{bond} \simeq$ 0.042 estimated by using the law of mass action.