Defect-mediated viscous flow in vitreous silicon dioxide: flow-defect energetics, the $10^7$ entropic prefactor, and purity-dependent mechanism crossover

N. F. Mott (Phil. Mag. B 56, 257, 1987) proposed that viscous flow in vitreous $\mathrm{SiO_2}$ is mediated by broken Si-O bonds acting as charged point-defect pairs (a threefold-coordinated silicon and a non-bridging oxygen), and analyzed two mechanisms: single-bond ring diffusion with activation energy $W+U$ and two-bond Coulomb-assisted recombination with activation energy $2W+U$, where $W$ is the defect formation energy and $U$ the migration activation energy. The paper left open: (i) the microscopic origin of the very large entropic prefactor $\exp(\Delta S/k_{\mathrm{B}})\approx 10^{7}$ required by fits to high-purity silica data (activation energy $171\ \mathrm{kcal/mol}\approx 7.4\ \mathrm{eV}$), conjecturing but not quantifying configurational multiplicity of multi-defect events; (ii) independent determination of $W$ and $U$ (the paper's $2W+U$ estimate, 5.6-6.6 eV, came from a private communication); (iii) identification of the dominant mechanism in each temperature and composition range, with the prediction that the apparent activation energy drops when dopant-induced non-bridging-oxygen concentration exceeds $\exp(-W/k_{\mathrm{B}}T)$; and (iv) the kinetics of defect formation, mutual capture, and recombination in the three-dimensional continuous random network.

Background

Vitreous silicon dioxide (silica glass) is the archetypal "strong" glass former: above its glass transition temperature $T_{\mathrm{g}}$ (where the viscosity $\eta$ reaches $\sim 10^{12}\ \mathrm{Pa\,s}$) its shear viscosity follows an Arrhenius law $\eta=\eta_\infty\exp(E/RT)$ with an almost constant activation energy $E$, in contrast to "fragile" liquids whose $\eta(T)$ bends strongly upward on cooling. Structurally, vitreous $\mathrm{SiO_2}$ is a continuous random network (CRN) of corner-sharing $\mathrm{SiO_4}$ tetrahedra: a three-dimensional network without periodicity in which nearly every oxygen bridges two silicons. Breaking a Si-O bond in this network creates a pair of coordination defects: a threefold-coordinated silicon and a non-bridging oxygen (NBO), an oxygen bonded to only one silicon; for the charged-defect picture used by Mott these carry positive and negative charge respectively, forming a so-called valence-alternation pair. The Stokes-Einstein relation $\eta=k_{\mathrm{B}}T/(3\pi D a)$ connects viscosity to the diffusion coefficient $D$ of a flow unit of size $a$.

The experimental facts that motivated Mott's paper are classical: careful measurements by Hetherington, Jack and Kennedy (Phys. Chem. Glasses 5, 130, 1964) on high-purity vitreous silica gave an Arrhenius activation energy of about $171\ \mathrm{kcal/mol}\approx 7.4\ \mathrm{eV}$ (for scale, $k_{\mathrm{B}}T\approx 0.17\ \mathrm{eV}$ at 2000 K), several times a single Si-O bond energy ($\sim 3.9\ \mathrm{eV}$), together with a prefactor that, when compared with natural atomic-scale prefactors, implies an entropy factor $\exp(\Delta S/k_{\mathrm{B}})\approx 10^{7}$, i.e. $\Delta S\approx 16\ k_{\mathrm{B}}$ per flow event. The related network oxide $\mathrm{GeO_2}$ shows activation energies consistent with single-bond breaking and no such entropy anomaly. Earlier treatments either used Stokes-Einstein scaling appropriate to simple liquids or Nabarro-Herring creep formulae (diffusional creep of polycrystals) that carry an unphysical specimen- or grain-size dependence.

Mott's 1987 model attributes flow to the thermal generation and motion of the broken-bond defect pairs. In the single-bond mechanism, one broken bond is created ($W$), and the less mobile member of the pair diffuses around an $-\mathrm{Si}-\mathrm{O}-\mathrm{Si}-\mathrm{O}-$ ring ($U$), changing the number of bonds in adjacent rings and producing an irreversible network rearrangement; the activation energy is $W+U$. In the two-bond mechanism, two bonds break close enough (within a Coulomb capture radius $R_0$) that oppositely charged members of different pairs drift together and recombine under Coulomb attraction in the dielectric ($\kappa\approx 4$), giving plastic relaxation with activation energy $2W+U$ and, after the capture radius cancels algebraically, the result

$$\frac{1}{\eta}=\frac{e^{2}}{\kappa a}\,\frac{1}{k_{\mathrm{B}}T}\,\frac{\omega a^{3}}{k_{\mathrm{B}}T}\,\exp\!\left[-\frac{2W+U}{k_{\mathrm{B}}T}\right],$$

with $e$ the elementary charge, $\omega$ an attempt frequency, and $a$ an atomic length. The model predicts a regime switch: when the concentration of non-bridging oxygens introduced by dopants (water, halogens, alkalis) exceeds the thermal concentration $\exp(-W/k_{\mathrm{B}}T)$, the single-bond mechanism takes over and the measured activation energy should drop from $2W+U$ toward $W+U$. Numerically, Mott estimated $W\approx 3$-$4\ \mathrm{eV}$ from the Si-O bond energy, quoted $2W+U=5.6$-$6.6\ \mathrm{eV}$ from a private communication (J. Robertson, 1986), below the experimental $7.4\ \mathrm{eV}$, and found $R_0\approx 40\ \text{Å}$ and a two-bond prefactor of order $40$ at $\sim 1500\ \mathrm{K}$ - far short of the observed $10^{7}$. He conjectured that the simultaneous formation of four nearby defects might supply the missing entropy through configurational multiplicity, but did not quantify this.

Later work established the phenomenological skeleton: the best viscosity data show two Arrhenius regimes, about $515\ \mathrm{kJ/mol}$ above $1400\,^{\circ}\mathrm{C}$ and $720\ \mathrm{kJ/mol}$ below (Doremus 2002); broken-bond ("configuron") fits to viscosity yield effective formation and motion enthalpies and entropies for silica; molecular-dynamics simulations find defect concentrations that are Arrhenius in temperature, activated transport at low temperature, and a fragile-to-strong crossover near $3100$-$3300\ \mathrm{K}$; and systematic measurements on differently doped silica melts confirm that trace OH, Cl, F and alkalis reduce both $T_{\mathrm{g}}$ and the apparent activation energy, with a common crossover near $2200$-$2500\ \mathrm{K}$ attributed to intrinsic defects dominating at high temperature. Alternative descriptions of flow in strong network liquids include Doremus's SiO-molecule line-defect mechanism (flow carried by line defects composed of SiO molecules) and random-first-order-transition theory, a cooperative-rearrangement theory of the glass transition. What is still missing is the microscopic, quantitative closure of Mott's questions.

Problem Statement

Provide a quantitative, microscopic, and testable account of defect-mediated viscous flow in vitreous silicon dioxide that resolves the issues N. F. Mott (1987) left open, namely:

1. Defect energetics. Determine, by independent calculation or measurement rather than by fitting the viscosity itself, the formation energy $W$ (with its entropy) and the migration activation energy $U$ of the flow-mediating defect(s) in the continuous random network of vitreous $\mathrm{SiO_2}$ - including whether the relevant defect is the broken-bond charged pair (threefold-coordinated silicon plus non-bridging oxygen), the over-bonded variants found in ab initio work, or another species - and hence the combinations $W+U$ and $2W+U$ that the two proposed mechanisms predict for the apparent activation energy.
2. Entropic prefactor. Establish microscopically the origin of the large entropy factor $\exp(\Delta S/k_{\mathrm{B}})\approx 10^{7}$ implied by viscosity fits to high-purity silica (activation energy $\approx 7.4\ \mathrm{eV}$): either derive it, or demonstrate that it must be attributed to another source, or refute its attribution to flow events altogether.
3. Mechanism map. Determine which flow mechanism (single-bond ring diffusion with activation $W+U$, two-bond Coulomb-assisted recombination with activation $2W+U$, or higher-order multi-defect or cooperative processes) dominates the measured $\eta(T)$ in each temperature and composition range, and test quantitatively the predicted reduction of the apparent activation energy when the concentration of non-bridging oxygens introduced by dopants (water, halogens, alkalis) exceeds the thermal concentration $\exp(-W/k_{\mathrm{B}}T)$.
4. Network kinetics. Derive the kinetics of defect formation, mutual capture, and recombination in the three-dimensional continuous random network and connect them quantitatively to the measured viscosity of high-purity and doped silica.

All four items are required, and together they define one scientific objective: a validated microscopic model that reproduces the measured $\eta(T)$ of vitreous and doped silica across the experimentally accessible range - including the two-activation-energy structure ($\sim 720\ \mathrm{kJ/mol}$ below $\sim 1400\,^{\circ}\mathrm{C}$, $\sim 515\ \mathrm{kJ/mol}$ above) and the purity-dependent crossover near $2200$-$2500\ \mathrm{K}$ - using defect parameters determined independently of the viscosity data being predicted.

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