Multi-Component Phase Stability

Composition geometry · convex hulls · finite temperature

Nico Unglert

22.04.2026

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Outline

From composition geometry to finite-T stability

1. Composition geometry

  • Binary — Fe–O
  • Ternary — triangle
  • Quaternary — LiMnOF
  • Quinary — (Li,Mn,Ti)(O,F)

2. Stability at T = 0

  • Convex-hull LP (theory)
  • Binary hull — Fe–O
  • Ternary hull — Li–Mn–O

3. Stability at T > 0

  • Free-energy hull (theory)
  • Entropy softens the hull
  • Sampling F(x, T) with MC

Appendix — Constrained Gibbs triangle (Li,Mn,Ti)(O,F)

Binary Composition Space

The FeO system

2 species = 2D plane (xFe, xO)

Thermodynamical stability at T = 0

Convex hull of formation energies

Formation energy (per config \(\sigma\))

$$\Delta E_f(\sigma) = E(\sigma) - \sum_i x_i(\sigma)\, E_i^{\text{ref}}$$

Convex hull (function of \(\mathbf{x}\))

$$E_{\text{hull}}(\mathbf{x}) = \min_{\substack{\lambda_i \,\ge\, 0 \\[2pt] \sum_i \lambda_i\, \mathbf{x}(\sigma_i) \,=\, \mathbf{x} \\[2pt] \sum_i \lambda_i \,=\, 1}} \; \sum_i \lambda_i\, \Delta E_f(\sigma_i)$$

A linear program (LP) in \(\{\lambda_i\}\): optimum sits on a facet with \(\le d{+}1\) non-zero \(\lambda_i\) — the decomposition products.

Energy above hull

$$E_{\text{above}}(\sigma) = \Delta E_f(\sigma) - E_{\text{hull}}\!\big(\mathbf{x}(\sigma)\big) \;\ge\; 0$$

Stable ⇔ \(E_{\text{above}}(\sigma)=0\) — \(\sigma\) sits on a hull vertex.

Workflow

  • Enumerate configurations \(\sigma\) at each composition
  • Predict \(E(\sigma)\) with fitted CE
  • Construct formation-energy hull \(E_{\text{hull}}(\mathbf{x})\)
  • Hull vertices = thermodynamically stable phases

Binary convex hull at T = 0

Convex hull of formation energies — which compounds are stable?

Formation Energy

$$\Delta E_f = E - x_A E_A^{\text{ref}} - (1-x_A)\, E_B^{\text{ref}}$$

Points on the hull → stable at T=0. Above → decompose.

Decomposition

Hover a compound or drag the slider for details.

Ternary → 2D hull on Gibbs triangle (next slide)

Ternary Composition Space

The LiMnO system

3 elements → 2D triangle (xLi + xMn + xO = 1)

Ternary Convex Hull at T = 0

The LiMnO system

Drag to rotate (3D) · hover a point for formation energy & decomposition

Quaternary Composition Space

The LiMnOF system

4 elements → 3D tetrahedron (xLi + xMn + xO + xF = 1)

Quinary Composition Space

The (LiMnTi)(OF) system

5 elements − 2 constraints = 3 DOF — triangular prism

Thermodynamical stability at T > 0

Convex hull of free energies — rebuilt at each temperature

Helmholtz free energy (function of \(\mathbf{x}, T\))

$$F(\mathbf{x}, T) = U(\mathbf{x}, T) - T\, S_{\text{conf}}(\mathbf{x}, T)$$

Free-energy hull (at fixed \(T\))

$$F_{\text{hull}}(\mathbf{x}, T) = \min_{\substack{\lambda_i \,\ge\, 0 \\[2pt] \sum_i \lambda_i\, \mathbf{x}_i \,=\, \mathbf{x} \\[2pt] \sum_i \lambda_i \,=\, 1}} \; \sum_i \lambda_i\, F_i(T)$$

Same linear program as \(T = 0\) with \(F\) in place of \(\Delta E_f\); supporting facet = common-tangent line (binary) / plane (ternary).

Coexistence condition

$$\mu_i(\mathbf{x}^{(\alpha)}, T) = \mu_i(\mathbf{x}^{(\beta)}, T) \quad \forall i$$

Chemical potentials \(\mu_i = \partial F / \partial x_i\) equal across coexisting phases ⇒ common tangent.

Workflow

  • Sample \(U(\mathbf{x},T)\) with canonical MC at many \(T\)
  • Thermodynamic integration \(\to S_{\text{conf}} \to F(\mathbf{x},T)\)
  • Build convex hull of \(F(\cdot,T)\) at each \(T\)
  • Entropy can stabilise phases above the \(T=0\) hull

Finite Temperature: Entropy Softens the Hull

Sampling F(x, T) with Monte Carlo

Canonical vs semi-grand canonical — and the miscibility gap

Drag the red μ-line or use sliders · green = stable branch, red = spinodally unstable, orange band = gap

Summary

Composition geometry
Each constraint removes one DOF. Sum-to-1 → simplex · sublattice → product of sub-simplices · charge balance → flat facet.
Stability at T = 0
\(E_{\text{hull}}(\mathbf{x}) = \min_{\boldsymbol{\lambda}} \sum_i \lambda_i\, \Delta E_f^{(i)}\) · stable ⇔ hull vertex; decomposition = optimal \(\lambda_i\).
Stability at T > 0
\(F(\mathbf{x}, T) = U - T\, S_{\text{conf}}\) · same LP with \(F_i(T)\) · entropy softens the hull.
Common tangent
≡ hull facet. Supporting hyperplane touching \(F\) at coexisting compositions; \(\mu_i = \partial F/\partial x_i\) equal across phases.
In practice
MC (canonical / semi-grand canonical) + thermodynamic integration deliver \(F(\mathbf{x}, T)\); the hull is re-built at each temperature.

Appendix — Constrained Gibbs Triangle: (Li,Mn,Ti)(O,F)

5 elements − 3 constraints = 2 DOF → Gibbs triangle with adjustable basis

Basis: Same grid, different reference corners