Open almost any CVD paper on monolayer WS₂ and the optical micrographs look the same: sharp equilateral triangles scattered across the substrate. MoS₂ does it too. Hexagonal boron nitride usually doesn't, and graphene tends to grow into hexagons. So why does this particular crystal keep ending up with three sides?

The short answer is that the triangle is a kinetic outcome rather than an equilibrium shape. A WS₂ island nucleates with six zigzag edges, and only three of them survive growth. The shape you see under the microscope is a record of which edges grew slowest.

The interactive model below is a kinetic Wulff construction, the classical geometric rule that a faceted crystal's steady-state outline is bounded by its slowest facets. Drag the anisotropy slider from balanced (hexagon) to S-rich CVD (triangle) and watch the fast edges erase themselves.

Two edges that look alike but aren't

A TMD monolayer is a honeycomb of metal and chalcogen atoms. From a distance the lattice looks six-fold symmetric, but up close it isn't. Tungsten sits on one sublattice and sulfur occupies the other (really a pair of S atoms, one above and one below the W plane). A 60° rotation swaps those two sublattices, so it is not a symmetry of the crystal. The true rotational symmetry is three-fold.

That broken equivalence shows up at the island perimeter. The six zigzag edges of a hexagonal nucleus alternate between two chemically distinct terminations:

• S-zigzag: the outermost row is sulfur. Under sulfur-rich CVD these edges are relatively passivated and stable.
• W-zigzag: the outermost row is tungsten, with dangling bonds that readily bind incoming chalcogen. These edges are reactive.

Same geometry, different chemistry. There is no reason for them to advance at the same speed.

Fast edges eliminate themselves

In faceted growth, a fast-moving facet shrinks in length as its slow neighbors lag behind, until it grows itself out of the crystal entirely. This is what the kinetic Wulff construction formalizes, and it is exactly what the animation above computes: six half-planes advancing at two different speeds, with the intersection redrawn at every frame.

Under typical S-rich CVD conditions the W-zigzag edges are the fast ones, so they vanish. The three slow S-zigzag edges remain, and the island becomes an equilateral triangle bounded by sulfur-terminated zigzag edges.

The argument in three steps:

1. Two edge types. Because W and S occupy different sublattices, the hexagonal nucleus alternates between S-terminated and W-terminated zigzag edges, with different atoms, different dangling bonds, and different edge energies.
2. Different speeds. Under S-rich CVD the W-zigzag edges are reactive and advance fast, while the sulfur-passivated S-zigzag edges advance slowly.
3. Fast edges vanish. Three fast edges disappear, three slow edges remain. What's left is a triangle.

The shape is tunable

Because the triangle is set by kinetics, you can change it by tuning the S:W flux ratio, which changes which edge family is slow:

• Sulfur-rich gives S-zigzag triangles, the familiar orientation.
• Near-balanced gives hexagons, since both edge types advance at comparable speed.
• W-rich gives triangles pointing the opposite way, with the W-zigzag edges now the survivors.

Truncated triangles, hexagons with three short sides and three long ones, are the intermediate you see when the chemical potentials are close but not equal. That continuum is why the slider above matters: the shape is a continuous function of edge-speed anisotropy, not a discrete switch.

You can push the same idea further with atomistic kinetics. Matter42's kinetic Monte Carlo growth simulator (run_kmc) evolves a lattice under explicit temperature and S:metal flux, and reproduces the same hexagon-to-truncated-to-triangle family when you sweep the flux ratio. The Wulff cartoon above is the geometric skeleton of that simulation.

Why it matters for characterization

Domain shape is one of the first things a growth team reads off an optical image, and it is also one of the easiest things to misread. A triangle does not mean "thermodynamically preferred." It means these three edges won the kinetic race under this particular precursor chemistry. Flip the flux and the race flips with it.

That reading matters when you later map Raman or PL across the flake. Edge regions, grain boundaries between co-nucleated triangles, and the interior of a single domain are chemically different environments, and they show up differently in defect-sensitive metrics. Knowing why the domain is triangular is part of knowing which spatial features are growth artefacts and which reflect intrinsic material quality.

If you want the calibrated side of that story, how a Raman linewidth becomes a vacancy percentage, see How a Raman linewidth becomes a defect density. For the broader product shape around growth simulation and spectroscopy in one workflow, see AI-native characterization for 2D materials.