a simulation, run once and kept

Network lab

A cube of glass three and a half nanometers on a side: three thousand atoms of silicon, oxygen and, in one of the two glasses, sodium. They were melted and cooled in a research simulation, and their final positions are kept here so you can turn the block over and slice it. Unlike the disks of the Structure lab, these atoms bond: every silicon holds four oxygens at the corners of a tetrahedron, and the tetrahedra join corner to corner into a network with no repeating pattern.

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What to look for

Silica, a thin slice. Almost every silicon is inked black: it shares all four of its oxygens with neighbors, which chemists write Q⁴ (99% of them in the slower quench). Almost every oxygen is a bridge between two silicons. This is the picture W. H. Zachariasen drew in 1932, in two dimensions, before anyone could check it: a network as connected as a crystal's, with the order taken out. X-ray diffraction soon backed it, and the best of those measurements (Mozzi and Warren, 1969) found the angle at each bridging oxygen spread from 120° all the way to 180°, most often near 144°. In this simulated silica the commonest angle is 143° (to the nearest 5°) and the mean 147°, the spread runs as wide, and 100% of the silicons have exactly four oxygens.

Rings. Follow bridges from one silicon around and back to it and you trace a ring. In cristobalite, the crystal silica forms most easily, every smallest ring has six tetrahedra. Turn on the rings and step through the sizes: the glass has rings from 3 to 9 tetrahedra, most of them six or seven. The variety of rings is what a crystal cannot have, and it is one way to say exactly what "disordered" means for a network whose every atom still has its proper neighbors.

Soda silica. Switch to the glass with sodium, which has the proportions of Na₂O·3SiO₂, a quarter of the oxide units soda. Each Na₂O brings one oxygen and two sodium ions. The extra oxygen cannot join the network as a bridge; instead one Si–O–Si bridge is cut, and the two silicons each keep an oxygen that now hangs free, drawn open here. The sodium ions, dashed rings, sit beside those loose oxygens, each surrounded by about 5 of them, and hold the charge in balance. Cutting the network this way is why soda lowers the melting temperature so far.

Where the breaks fall

Counting is enough to say how many bridges must break: in Na₂O·3SiO₂ there are two loose oxygens for every three silicons. It does not say how they are shared out. The simplest guess, the binary model, puts at most one loose oxygen on each tetrahedron, which makes the glass 67% Q³ and 33% Q⁴. Nuclear magnetic resonance of ²⁹Si can tell the Qⁿ species apart, and measurements of real sodium silicates (Maekawa and colleagues, 1991) find something close to the binary model with a little of the reaction 2Q³ ⇌ Q² + Q⁴ mixed in. The simulations spread the breaks much more widely:

0%25%50%75%100%silica, either quenchQ⁴ 99soda silica, 10¹⁴ K/sQ⁴ 47Q³ 40Q² 12soda silica, 10¹³ K/sQ⁴ 48Q³ 40Q² 10SHIK at 2.25 K/ps (2019)Q⁴ 47Q³ 40Q² 9binary modelQ⁴ 33Q³ 67measured, ²⁹Si NMRQ⁴ 38Q³ 62

Qⁿ is the number of a silicon's four oxygens that bridge to another silicon. The NMR row and the 2.25 K/ps simulation by the authors of the potential are read from their figure 5 (Sundararaman et al., 2019), so they are good to a few percent.

The simulated glass has 12% Q² in the fast quench and 10% in the slower one, close to what the potential's authors found at a rate four times slower still; the measured glass has almost none. The usual explanation, which those authors give, is the cooling rate. A liquid's structure is more disordered the hotter it is, and a glass keeps the structure of the liquid at the temperature where it stopped rearranging. A simulation cools so fast that it stops rearranging hundreds of degrees hotter than a real glass, and keeps that hotter liquid's disorder. Cooling tenfold slower moves it only a little here: the gap to the furnace is a factor of 10¹⁴, and a factor of ten is a small step along it.

Simulation is fast, the furnace is slow

The two quench rates here are 100 and 10 kelvin per picosecond: 10¹⁴ and 10¹³ kelvin per second. A lehr anneals glass at a few kelvin per minute, about 10⁻¹ K/s, so the gap is some fourteen orders of magnitude. Simulations cannot close it: the slower runs here took over two hours each on one processor for a few hundredths of a nanosecond of glass history, and each tenfold slower cooling costs ten times as long. What they can do is show the trend. Vollmayr, Kob and Binder (1996) cooled simulated silica at a range of rates and found the glass transition falling with the logarithm of the rate, and the density, the ring sizes and the bond angles all shifting with it, the atomic structure more than the bulk properties.

50010001500200025003000350040001.81.92.02.12.22.32.42.5temperature (K)density (g/cm³)measured, SiO₂measured, Na₂O·3SiO₂

Density of each run as it was cooled, averaged over every picosecond: silica from 4000 K at one atmosphere, soda silica from 3000 K with the pressure let down from 1,000 atmospheres. The short bars at the left are the measured room-temperature densities: the medians of SciGlass entries for pure silica (2.20 g/cm³, 258 glasses) and for sodium silicates at 25 mol% Na₂O (2.43 g/cm³, 100 glasses).

Read the curves from the right. Liquid silica densifies as it cools until, near 3000 K, the network stops rearranging and the curve flattens: that is the glass transition at simulation speed, far above the real one near 1460 K (1190 °C, the median of 16 SciGlass entries). Below it the glass hardly contracts at all, which is real silica's famous near-zero expansion. The soda silica's curves are harder to read. Its liquid boils at 4000 K and one atmosphere in this model, so, as the potential's authors did, it was melted at the measured glass density and cooled from 3000 K at a pressure let down from 1,000 atmospheres to one along the way. The falling pressure lets the glass expand as it cools and hides the bend; the curves show only that the faster quench ends less dense, 2.37 against 2.43 g/cm³, and that the slower one lands on the measured 2.43. The ring sizes respond to the rate too, and to the sodium:

3456789100.000.350.70tetrahedra in the ringrings per silicon

Rings per silicon atom, by the number of tetrahedra in the ring. A ring is counted for each silicon and each pair of its bridges as the shortest closed path through them (King, 1967), and each distinct ring once. The slower silica has fewer of the strained three- and four-membered rings and more six-membered ones, the direction Vollmayr, Kob and Binder found; in the soda silica, where bridges are cut, rings of six and seven are rarer and rings of nine and ten, and longer, far commoner.

How this was computed

Each glass is a run of LAMMPS, the molecular-dynamics code from Sandia, with the SHIK potential (Sundararaman, Huang, Ispas and Kob, 2018 and 2019). SHIK treats the atoms as charged spheres: silicon +1.7755 e, sodium +0.6018 e and oxygen whatever balances them, smaller than the textbook ionic charges, because the bond is partly covalent. Between each pair it adds a short-range repulsion and a weak attraction (the Buckingham form) plus a steep wall that stops atoms falling into each other at high temperature. Its parameters were fitted so that a simulated liquid matches one computed from quantum mechanics and the glass matches the measured density and stiffness. Nothing in it says "make tetrahedra"; the tetrahedra form because the charges and sizes favor them.

Each run started from atoms scattered at random in a box and melted at 6000 K. Silica was settled at 4000 K and one atmosphere; the soda silica at 3000 K, first at the measured glass density and then at 1,000 atmospheres, because at one atmosphere and 4000 K the model liquid boils away. Each was then cooled in a straight line to 300 K, the soda silica's pressure falling to one atmosphere as it went, and held there for 30 picoseconds. The two runs of each glass start from the same liquid and differ only in the cooling rate. The analysis on this page, which atoms are bonded, the Qⁿ counts and the rings, is computed from the stored positions by this site's own code, with a bond being any Si–O pair closer than 2.2 Å. The input files and the analysis are in the repository, so every number here can be rerun.

What the model leaves out

Each glass is a model: what a particular, published approximation to the forces between atoms produces when cooled impossibly fast. Where it can be checked, against density, the bond angles and NMR, it is close, and it is wrong in the direction its fast quench predicts.

Sources

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