The Hidden Structure of Amorphous Carbon

Twenty-seven simulated carbon structures in five classes: amorphous carbon (ρ 1.5–2.9 g/cm³), carbide-derived carbon (synthesized at 800 and 1200 °C, plus an annealed variant), irradiated graphite (four damage stages), variable-porosity carbon, and a phase-separated phase. Atoms draw as points colored by coordination number; bonds join atoms within 1.8 Å.

Drag to rotate. Shift-drag to pan. Ctrl/⌘ + scroll to zoom. Hover an atom for its coordination, rings, local topology, and coordinates; click it to spotlight it and every ring through it. Reset view restores the camera and clears every selection and filter.

Controls

• Structure — one of the 27 models, grouped by class.
• Theme — four palettes for the same data.
• Color by — coordination number, or bond-network entropy (below).
• Bonds, Rings, Bond strain — toggle bond lines; overlay shortest-path rings (sizes 3–10); color bonds on a compressed-red → stretched-blue ramp about the median length.
• Hide gridlines, Spin, Point size — cell box and axes; slow auto-rotation; dot radius.
• Fly (WASD) — first-person camera inside the cell: W/A/S/D to move, Q/E to descend/climb, Shift for speed, drag to look, Ctrl/⌘ + scroll to dolly, Esc to exit.
• Sequence — ordered series (irradiation damage, CDC annealing, AC densification, VPC density). ◀ ▶ and Play step the stages, the camera holds still between them, and the other stages draw as gray ghost curves in the charts.

Panels

Panel filters compose — every active selection ANDs with the others. The i badge beside each heading holds that panel's full explanation. On phones the g(r) and bond-length panels are hidden.

• Coordination number — click a class to isolate those atoms; shift-click adds more.
• Radial distribution g(r) — brush a range of r to highlight every atom pair at that separation (out to 5 Å); click outside the band to clear.
• Bond lengths — brush to select bonds by length; bars share the strain ramp.
• Rings by size — click a size to isolate those rings; solid bars are rings drawn in the cloud, faded bars cross the cell boundary.

Bond-network entropy

The paper's disorder descriptor. Each atom's local environment — its n nearest atoms and the bonds among them — is classified by ring topology (its H₁ barcode); BNE(n) is the Shannon entropy of that classification over all atoms, and the per-structure number is the growth rate: the mean of BNE(n)/n for n = 14–30. A perfect crystal scores zero; the more distinct local topologies a structure contains, the higher it scores.

Set Color by to bond-network entropy to shade each atom by the rarity (surprisal) of its environment; the average of the shading equals BNE(n) exactly. In the panel, the Environment n slider sets the environment size; the barcode rows list the commonest topologies — click one to isolate its atoms, shift-click to add; the curve plots BNE(n) with the 14–30 band shaded; the tick strip places this structure's growth rate among all 27.

A worked example: Sequence → Irradiation damage — IRG T2→T9, Color by → bond-network entropy, then step the stages. The pristine graphitic class falls from 79% of atoms to 59% while the entropy curve lifts.

Method

Structures from Iwanowski, Csányi & Simoncelli, Bond-network entropy governs heat transport in coordination-disordered solids (Phys. Rev. X 15, 041041 (2025)), relaxed with the GAP potential. Bonds are drawn between atoms within 1.8 Å.

The bond-network entropy is computed with the authors' own reference implementation, the smooth-disorder package, with every atom of every structure catalogued — no sampling. The values here reproduce the paper's (amorphous carbon at 2.9 g/cm³ and 8,000 atoms: 0.240 against Fig. 2c's ≈0.24). Cells of a few hundred atoms are small enough that nearly every neighborhood in them is unique, which caps the entropy and understates their disorder; the panel flags this when it happens.