Structure Lab

Three runs worth making

All alike, quenched. Set the size difference to zero and cool at the fastest rate. The atoms crystallize anyway, almost at once. Identical spheres in a plane pack so easily into triangles that no cooling rate a computer can reach outruns them. The same holds in the laboratory: whether a liquid forms a glass is a race between cooling and crystal growth (Turnbull, 1969), and simple liquids win it only at extreme rates. Pure metals have been made glassy, but only in nanometer-sized samples quenched at about 10¹⁴ kelvin per second (Zhong et al., 2014). The pure substances that form glass easily, such as silica, owe it to the connected network their bonds build rather than to speed.

Mixed, slow then fast. Put the size difference back to 20%, cool slowly, then melt and quench. The slow run ends mostly in ordered patches; the quench leaves far more of the box disordered. On the curve, the slow run sinks lower: the crystal packs tighter and holds less energy. Each run is random, so two runs at the same rate can differ by twenty points or so; the trend holds over repeated runs.

Very mixed, any rate. At 30% or more the two sizes no longer fit one lattice between them. Every run ends as a glass, and the volume curve bends rather than drops: the liquid stiffens until its atoms stop trading places, and below that bend the volume shrinks only as fast as a solid's would. The bend is the glass transition. Compare g(r), the count of neighbors at each distance: the glass keeps the liquid's broad humps, only sharper, while a crystal splits them into narrow peaks at the lattice spacings.

Why mixing makes glass

A crystal has to sort its atoms: every site in the lattice expects a particular size. When the melt holds two sizes in equal numbers, a growing crystal must either make room for both or push one out and wait for it to diffuse away. Either way it slows down, and if the cooling is fast enough the melt is too stiff to move before the crystal has grown. Metallurgists call this the confusion principle, after A. L. Greer (1993), and it is why the metallic glasses that can be cast thick are alloys of at least three elements with very different atomic sizes (one of Inoue's empirical rules); Vitreloy 1, a widely studied one, has five (Peker and Johnson, 1993).

The glass in a window works on the same principle in a different way. Silica alone forms a glass because its tetrahedra lock into a network that is very slow to rearrange (stage 00 on the front page draws it). Soda and lime cut that network so it melts in a studio furnace, and a soda–lime batch is then a mixture of several oxides that would have to separate to crystallize. When it does crystallize, glassmakers call it devitrification, and they avoid it by keeping a melt out of the temperature range where the crystal grows fastest. (This paragraph is the general account, not taken from a cited source here.)

The real network, counted

The atoms above have no bonds to break. A silicate glass does: each silicon sits in a tetrahedron of four oxygens, and in silica every one of those oxygens bridges to a second silicon. Chemists name a silicon by how many bridges it keeps. Q⁴ has all four, Q³ three, down to Q⁰, a lone SiO₄ ion. Each Na₂O or K₂O added brings one oxygen, which cuts one bridge into two non-bridging oxygens, so the average count is fixed by the recipe: at x mol% alkali oxide there are 2x/(100 − x) cuts per silicon. What the recipe does not fix is how the cuts are shared out, and that can be measured. The ²⁹Si nucleus resonates at a slightly different frequency for each Qⁿ, and magic-angle-spinning NMR separates the peaks well enough to count them.

Two simple answers bracket the truth. If every bridge were equally likely to break, the shares would follow a binomial law, and some silicons would lose two or three bridges almost at once. If each silicon had to lose one bridge before any lost a second, the shares would change in straight lines: all Q³ at the disilicate, M₂O·2SiO₂, and all Q² at the metasilicate, M₂O·SiO₂. The NMR points sit close to the second, with the corners rounded.

soda, Na₂O potash, K₂O

statistical mechanics random binary

the model chosen²⁹Si NMR, Maekawa et al. 1991

The rounding is what the statistical-mechanics model describes. Each new cut chooses a silicon species with a probability proportional to how many of that species are left and to a Boltzmann factor, exp(−H/RT_f), where H is how much less favorable it is to cut that species than a Q⁴. When the liquid is hot the Boltzmann factors flatten and the cuts drift toward the random pattern; as it cools the preference sharpens toward the binary order. The structure stops changing at the fictive temperature T_f, roughly the glass transition, so a glass quenched faster keeps a hotter, more random network. The model was proposed by John Mauro in 2013 and fitted to silicates by Bødker and colleagues (2019); the curves here are reimplemented from their equations and their Table 1 enthalpies, and each glass is frozen at its own Tg from the oxide bench's viscosity model. Potash holds the order more strictly than soda: its fitted enthalpies are larger.

This is the bridge between the two halves of the site. On the oxide bench in stage 01, every slider moves the same count for a real batch, after alumina and boron have taken their share of the modifier, and the viscosity model beside it shows what fewer bridges do: the network rearranges at a lower temperature and the glass melts, works and anneals cooler.

How this was computed. Binary glasses from 0 to 60 mol% alkali in steps of half a percent, each frozen at its own 10¹² Pa·s temperature from the bench's viscosity model. Modifier is added a hundredth of a non-bridging oxygen per silicon at a time, and each step is shared among Q⁴…Q¹ by the weights above. The miss is the root mean square over every species of every measured glass. Code: qn.js; enthalpies and NMR rows: qn.json, read from the data files by tools/qn/extract_nmr.mjs; the quoted figures, including the 800 K check: tools/qn/validate.mjs.

Simulation is fast, the furnace is slow

Taken as argon, the slowest rate here is about 10¹¹ kelvin per second. A lehr cools glass at a few kelvin per minute, about twelve orders of magnitude slower. Molecular-dynamics glasses are therefore always quenched far harder than any made in a studio, and their glass transitions sit correspondingly higher. On the front page, stage 00 follows the transition of a real soda–lime glass across the rates a studio can reach, using its measured viscosity instead of atoms. The two pictures agree on the shape: cool faster and the liquid falls out of equilibrium sooner, hotter and less dense.

What the model leaves out

• The chemistry. These atoms attract and repel with no preferred direction, as argon does. Silicon and oxygen bond covalently at fixed angles, which is what builds the network. A model of silica needs a three-body or charged potential, such as the BKS potential (van Beest, Kramer and van Santen, 1990).
• The third dimension. The atoms live in a plane. Two-dimensional crystallization differs in detail from three, and two-dimensional crystals can melt through an intermediate, hexatic phase that three-dimensional ones lack (predicted by Halperin and Nelson in 1978, and for hard disks shown by Bernard and Krauth in 2011). Two-dimensional glass also exists: in 2012 a sheet of silica glass two atoms thick was imaged atom by atom on graphene, and it looks much like Zachariasen's 1932 drawing.
• Size. Four hundred atoms in a box that wraps around at its edges, so a grain the size of the box can meet itself. Real grains hold billions of atoms.
• The bath. Temperature and pressure are held by the Berendsen thermostat and barostat, which rescale velocities and the box each step. They steer the averages well but do not give exactly the right fluctuations. The run uses velocity Verlet steps of 0.005 τ.
• The order detector. An atom is drawn solid when its neighbors sit in a near-perfect hexagon and at least five of them are oriented the same way, by the sixfold bond-order test of ten Wolde, Ruiz-Montero and Frenkel adapted to a plane. The threshold is a choice; a glass in two dimensions always has a few small ordered clusters.

For the real thing in a browser, Atomify runs LAMMPS, the research code, with silica and other potentials, in three dimensions. The Network lab keeps four such runs, silica and a soda silica, to turn over and slice.

Sources

• W. Kob and H. C. Andersen, Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture, Phys. Rev. E 51, 4626 (1995). The standard model glass former; the engine keeps it as an option.
• R. Brüning, D. A. St-Onge, S. Patterson and W. Kob, Glass transitions in one-, two-, three- and four-dimensional binary Lennard-Jones systems (2009): in two dimensions the 80:20 Kob–Andersen mixture forms hexagonal crystals, and a 65:35 mix is needed to suppress them.
• A. L. Greer, Confusion by design, Nature 366, 303–304 (1993): the confusion principle for glass-forming alloys.
• A. Inoue, Stabilization of metallic supercooled liquid and bulk amorphous alloys, Acta Mater. 48, 279–306 (2000): the empirical rules for bulk metallic glasses.
• A. Peker and W. L. Johnson, A highly processable metallic glass: Zr41.2Ti13.8Cu12.5Ni10.0Be22.5, Appl. Phys. Lett. 63, 2342 (1993).
• D. Turnbull, Under what conditions can a glass be formed?, Contemp. Phys. 10, 473–488 (1969).
• L. Zhong, J. Wang, H. Sheng, Z. Zhang and S. X. Mao, Formation of monatomic metallic glasses through ultrafast liquid quenching, Nature 512, 177–180 (2014).
• B. I. Halperin and D. R. Nelson, Theory of two-dimensional melting, Phys. Rev. Lett. 41, 121 (1978); E. P. Bernard and W. Krauth, Two-step melting in two dimensions: first-order liquid-hexatic transition, Phys. Rev. Lett. 107, 155704 (2011).
• B. W. H. van Beest, G. J. Kramer and R. A. van Santen, Force fields for silicas and aluminophosphates based on ab initio calculations, Phys. Rev. Lett. 64, 1955 (1990): the BKS potential.
• P. R. ten Wolde, M. J. Ruiz-Montero and D. Frenkel, Numerical evidence for bcc ordering at the surface of a critical fcc nucleus, Phys. Rev. Lett. 75, 2714 (1995): the bond-order test for which atoms are crystalline.
• P. Y. Huang et al., Direct imaging of a two-dimensional silica glass on graphene, Nano Lett. 12, 1081–1086 (2012); see the Cornell summary.
• M. S. Bødker, S. S. Sørensen, J. C. Mauro and M. M. Smedskjaer, Predicting composition–structure relations in alkali borosilicate glasses using statistical mechanics, Front. Mater. 6, 175 (2019), open access (CC BY): the model's equations and the silicate enthalpies of Table 1. Mauro's original proposal is cited there.
• M. S. Bødker, C. J. Wilkinson, J. C. Mauro and M. M. Smedskjaer, StatMechGlass: Python based software for composition–structure prediction in oxide glasses using statistical mechanics, SoftwareX 17, 100913 (2022): mixing modifiers (eq. 3). The software is GPL and is not used here; its data files supplied the NMR table below.
• H. Maekawa, T. Maekawa, K. Kawamura and T. Yokokawa, The structural groups of alkali silicate glasses determined from ²⁹Si MAS-NMR, J. Non-Cryst. Solids 127, 53–64 (1991): the measured points, as tabulated in StatMechGlass's data files (some potassium points are from Schroeder and colleagues, 1973, per Bødker et al.).
• A. Rahman, Correlations in the motion of atoms in liquid argon, Phys. Rev. 136, A405 (1964): the first molecular dynamics with a Lennard-Jones potential. The argon scale (ε/k ≈ 120 K, σ ≈ 0.34 nm, τ ≈ 2.2 ps) is the textbook one, quoted for scale and not checked against this paper.

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