How is the convex hull (phase diagram) constructed?

The convex hull is a geometric construction used to determine thermodynamic phase stability across a composition space.

Construction Process

1. Collect formation energies: Gather the DFT-computed formation energy for every known phase in a chemical system (e.g., all Li-Fe-O compounds)
2. Plot in composition-energy space: Each phase is a point with coordinates (composition, formation energy per atom)
3. Compute the convex hull: The lower convex envelope of these points defines the set of thermodynamically stable phases
4. Determine stability: Phases on the hull are stable; phases above the hull are metastable or unstable

Energy Above Hull

For any phase not on the hull, the energy above hull (Ehull) is the vertical distance from that phase to the hull at the same composition. This represents the thermodynamic driving force for decomposition.

Limitations

• The hull is only as complete as the set of known phases. If a stable phase is missing from the database, the hull may be incorrect.
• DFT energies have uncertainties of 0.02-0.05 eV/atom, which can affect stability predictions near the hull.
• Temperature effects (entropy, vibrations) are not included in standard 0 K convex hulls.
• Kinetic barriers to decomposition are not captured.

In MatCraft

MatCraft computes convex hulls on demand for any chemical system using the full database of known phases. The phase diagram endpoint returns both stable and unstable phases with their hull distances.