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arxiv: 2606.09921 · v1 · pith:OHLJOAERnew · submitted 2026-06-07 · ❄️ cond-mat.stat-mech · astro-ph.EP· cond-mat.mtrl-sci

Computing phase diagrams using a convex hull algorithm

C. P. Dullemond , E. D. Young This is my paper

Pith reviewed 2026-06-27 18:14 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech astro-ph.EPcond-mat.mtrl-sci
keywords phase diagramsconvex hullGibbs free energythermodynamicsmineralogycomputational methodsmeltsphase equilibria
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The pith

Convex hull of composition and Gibbs energy points determines all phase diagram features.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that computing the convex hull over points in composition-Gibbs free energy space, followed by classifying the simplices, automatically locates stable phases, tie lines, and all characteristic curves such as solidus, liquidus, solvus, and eutectic points. This replaces manual case-by-case logic with a single geometric operation that works at fixed temperature and pressure. The approach requires only a grid of accurate Gibbs values as input and uses standard library routines for the hull. It is presented as practical and stable for systems with up to four components, such as rock-melt assemblages. The result is an accessible method that turns thermodynamic data into complete phase diagrams without additional geometric coding.

Core claim

All the complexities of determining the stability or separation of phases, the localization and orientation of tie lines, as well as the determination of characteristic points, curves and surfaces such as the solidus, liquidus, solvus, and the eutectic/peritectic points etc, are taken care of by the algorithm that computes the convex hull, supplemented with an algorithm to physically classify the resulting simplices.

What carries the argument

Convex hull computation on points spanning composition and Gibbs free energy, supplemented by physical classification of the resulting simplices.

If this is right

  • The method runs with the publicly available Qhull package inside SciPy.
  • It remains stable and efficient for compositional systems of up to four components.
  • It applies directly to phase diagrams of rocks and their melts at fixed temperature and pressure.
  • Only a grid of Gibbs free energy values is needed as input.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Adaptive or non-uniform sampling of compositions could reduce the number of required Gibbs evaluations while preserving hull accuracy.
  • The same hull-plus-classification structure might locate metastable extensions or spinodal regions if negative-curvature regions of the energy surface are retained.
  • Parallel evaluation of multiple temperatures could generate full T-X sections with minimal extra code.

Load-bearing premise

Accurate Gibbs free energy values can be supplied for a sufficiently dense sampling of compositions at the chosen temperature and pressure so that the discrete convex hull faithfully represents the continuous thermodynamic surface.

What would settle it

For a known ternary or quaternary rock-melt system, supply a dense grid of accurate Gibbs free energies and check whether the hull-derived phase boundaries, tie lines, and invariant points match independent experimental measurements.

Figures

Figures reproduced from arXiv: 2606.09921 by C. P. Dullemond, E. D. Young.

Figure 1
Figure 1. Figure 1: Cartoon illustrating the stability/instability of a substance C consisting of (1 − x)A + xB and Gibbs free energy Gˆ(x) in a binary system consisting of components A and B. Left: Phase C is stable at the composition spec￾ified by x. Right: Phase C is unstable and separates into phases with compositions A (x = 0) and B (x = 1). Green circle is the final state Gibbs free energy. a physical combination of non… view at source ↗
Figure 2
Figure 2. Figure 2: Cartoon illustrating the convex hull of a set of points in 2D space. Right: The set of points. Left: The identification of the convex hull of that set of points. The red points are the vertices of the convex hull (the points ly￾ing on the convex hull), and the red lines are the simplices connecting the vertices. In N-dimensional space, these sim￾plices will become triangles (N = 3), tetrads (N = 4), etc., … view at source ↗
Figure 3
Figure 3. Figure 3: The basic building blocks of phase diagrams using the convex hull method: Simplices. For binary phase dia￾grams, they are line elements (left). For ternary phase di￾agrams, they are triangles (middle). For quaternary phase diagrams they are tetrads (right). 2.8. Application to phase diagrams To illustrate the application of the convex hull algorithm to the computation of phase diagrams, we present here a s… view at source ↗
Figure 5
Figure 5. Figure 5: Cartoon illustrating the convex hull method ap￾plied to a binary system consisting of a set of fixed￾composition phases (red diamonds) and a liquid (blue line and symbols). Left: the true physical continuous case. Right: the discretized approximation used in the convex hull method. Open diamond symbols are unstable phases. Open circles are unstable liquid phase points. Red lines are tie lines between stabl… view at source ↗
Figure 6
Figure 6. Figure 6: Cartoon illustrating the convex hull method ap￾plied to a binary system consisting of a liquid with a non￾convex molar Gibbs free energy curve (left), and a binary system consisting of a liquid and a solid solution (right). In both cases the continua are discretized on a grid, as shown in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: At 1500 K (1227◦C) all phases are solid. The convex hull consists of only four points and three simplices (tie lines). At 2000 K (1727◦C) both quartz/cristobalite and forsterite acquire a liquidus tie line (in green). Also an im￾miscibility region appears in the liquid (in cyan, between x = 0.05 and x = 0.35). The convex hull now consists of five distinct phase assemblages, depending on bulk compo￾sitions.… view at source ↗
Figure 8
Figure 8. Figure 8: Composition-temperature phase diagram of the same system as in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Low-resolution phase diagram of the CaO￾Al2O3-SiO2 ternary system of the Berman (1983) model at T=1400◦C, as computed using the convex hull method, with a base resolution of N0 = 30 and no grid refinement. The convex hull algorithm provides the triangulation. The physical meaning of each triangle (indicated with color) has to be identified in post-processing. See text for sim￾plex classification strategie… view at source ↗
Figure 12
Figure 12. Figure 12: The computation of the binodal curve and the lo [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: Same model as [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Phase diagram of the feldspar ternary system of Elkins & Grove (1990), computed using the convex hull algorithm. Beige color: well-mixed phase. Green/yellow color: miscibility gap. Green: Tie lines. the overall shape of the phase diagram is not known be￾forehand. For the x − T diagram the MgO-SiO2 binary system ( [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Model of an alloy of silver and copper. Left: Gibbs free energy diagram at T = 800◦C, just above the eutectic temperature. In blue is the liquid phase, in red is the solid solution phase. In green the tie lines are shown that connect the liquid and the solid phases. Right: the x-T phase diagram showing the full eutectic behavior of the system. The colors are as follows: Light blue is liquid, red-brown is … view at source ↗
Figure 15
Figure 15. Figure 15: The y-Gˆ diagram of the diopside-enstatite join according to Gasparik (1990). This model consists of five independent solid solutions: high-clino, low-clino, high-p￾clino, proto and ortho. Also plotted are the stable phases as the bottom of the convex hull spanning all five solutions together. In the Berman model of the SiO2, CaO, and MgO sys￾tem (Berman, 1983), diopside and enstatite are only rep￾resente… view at source ↗
Figure 16
Figure 16. Figure 16: The SiO2, CaO, MgO ternary phase diagram at T = 1340 C = 1613 K and P = 1 bar. Left: according to the model of (Berman, 1983) as implemented in PhaseHull. Middle: The same ternary but now with the diopside￾enstatite solid solution of Gasparik (1990) embedded between the diopside and enstatite points. Right: Representation of the embedding of the solid solution (blue) in the ternary. 0.0 0.2 0.4 0.6 0.8 1.… view at source ↗
Figure 17
Figure 17. Figure 17: Left: The grid used for the pyroxene solid solution quadrilateral. Two points are selected for which, in the middle and right panel, the Gibbs free energy as a function of the internal degree of freedom z (the reciprocal reaction) is shown for both solutions. The large dots in the middle and right panels are the points of lowest Gibbs free energy within the range of z. This is then the Gibbs free energy a… view at source ↗
Figure 18
Figure 18. Figure 18: Left and middle panels: The Gibbs free energy Gˆ for the two solid solutions ortho and clino from the model of Saxena et al. (1985). Right panel: The corresponding phase diagram computed with PhaseHull. slice with one of the simplices of the quaternary. Most of these polygons are triangular, but some have four cor￾ner points. Computing the corner points of the polygonal cross sections of an (M − 1)-dimens… view at source ↗
Figure 19
Figure 19. Figure 19: The SiO2 – CaO – Al2O3 – MgO quaternary phase diagram of the Berman 1983 model at T = 1400 C = 1673 K and P = 1 bar. Top-Left: The slices taken. Top-right: the slice at xMgO = 0.25, bottom-left: the slice at xMgO = 0.5, bottom-right: the slice at xMgO = 0.75. Note that the case for xMgO = 0 is not shown here, as it is already shown in [PITH_FULL_IMAGE:figures/full_fig_p016_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The SiO2 – CaO – Al2O3 – MgO quaternary phase diagram of the Berman 1983 model at T = 1400 C = 1673 K and P = 1 bar (same as [PITH_FULL_IMAGE:figures/full_fig_p016_20.png] view at source ↗
read the original abstract

We present a simple universal computational algorithm for computing compositional phase diagrams of rocks and their melts at given temperature and pressure. It makes use of the mathematical concept of the convex hull of a set of points in the space spanned by the composition and the Gibbs free energy. All the complexities of determining the stability or separation of phases, the localization and orientation of tie lines, as well as the determination of characteristic points, curves and surfaces such as the solidus, liquidus, solvus, and the eutectic/peritectic points etc, are taken care of by the algorithm that computes the convex hull, supplemented with an algorithm to physically classify the resulting simplices. For the convex hull computation, the publicly available Qhull package can be used, which is available in SciPy. This makes this method accessible and intuitive for a broad set of scientific and educational applications. Although the method is not practical for systems of a large number of components, it is remarkably stable and efficient for systems of up to four. We present our implementation of the method as a publicly available Python package.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a method to compute compositional phase diagrams at fixed T and P by constructing the lower convex hull of discrete points in composition-Gibbs-energy space using an external algorithm (Qhull), followed by physical classification of the resulting simplices; this is claimed to automatically determine phase stability, tie-line orientations, and all characteristic features (solidus, liquidus, eutectic/peritectic points, etc.) for systems with up to four components, with an accompanying open Python package.

Significance. If the discrete sampling requirement is met and the classification step is robust, the approach supplies a simple, library-based route to phase diagrams that avoids manual equilibrium calculations and is particularly suited to educational use and small-component petrologic or materials problems; the reliance on a publicly available convex-hull routine is a practical strength.

major comments (2)
  1. [Abstract] Abstract: the central claim that 'all the complexities of determining the stability or separation of phases, the localization and orientation of tie lines, as well as the determination of characteristic points, curves and surfaces such as the solidus, liquidus, solvus, and the eutectic/peritectic points etc, are taken care of by the algorithm' is load-bearing yet rests on the unexamined premise that the supplied discrete G values are dense enough for the computed hull to coincide with the true lower convex envelope; no quantitative sampling-density criterion, convergence test, or error estimate is supplied.
  2. [Implementation] Implementation section (or equivalent description of the simplex-classification step): the manuscript provides no analysis of how the classification algorithm behaves when the discrete hull is only an approximation to the continuous G(x) surface, leaving numerical robustness (spurious phases, misoriented tie lines, or shifted invariant points) unquantified.
minor comments (2)
  1. [Abstract] The limitation to four components is stated but not accompanied by a scaling analysis or timing benchmarks that would help readers judge practical boundaries.
  2. [Discussion] No comparison is made to existing phase-diagram codes that also employ convex-hull or Gibbs-energy minimization methods; a brief discussion would clarify the incremental contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight important aspects of the method's assumptions. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'all the complexities of determining the stability or separation of phases, the localization and orientation of tie lines, as well as the determination of characteristic points, curves and surfaces such as the solidus, liquidus, solvus, and the eutectic/peritectic points etc, are taken care of by the algorithm' is load-bearing yet rests on the unexamined premise that the supplied discrete G values are dense enough for the computed hull to coincide with the true lower convex envelope; no quantitative sampling-density criterion, convergence test, or error estimate is supplied.

    Authors: We agree that the method's accuracy depends on the discrete points providing a sufficiently dense sampling of the composition-Gibbs energy space. The algorithm computes the exact convex hull of the supplied points and classifies the resulting simplices; any deviation from the true continuous envelope arises from the input sampling rather than the hull computation itself. The manuscript demonstrates the approach on specific, adequately sampled systems but does not supply a universal quantitative criterion, as the required density depends on the curvature of G(x). In the revised manuscript we will add a dedicated paragraph in the Implementation section with practical guidelines for grid selection and an explicit convergence example for a binary system. revision: yes

  2. Referee: [Implementation] Implementation section (or equivalent description of the simplex-classification step): the manuscript provides no analysis of how the classification algorithm behaves when the discrete hull is only an approximation to the continuous G(x) surface, leaving numerical robustness (spurious phases, misoriented tie lines, or shifted invariant points) unquantified.

    Authors: We acknowledge that the current text does not quantify the robustness of the simplex-classification step under finite sampling. The classification identifies stable phases and tie lines from the lower-hull facets; artifacts can appear only if the input grid is too coarse relative to the curvature of G(x). Our presented examples use grids that avoid such artifacts, but we agree an explicit discussion strengthens the paper. We will revise the Implementation section to include a short analysis of potential numerical issues, together with a practical recommendation that users verify convergence by successively refining the composition grid. revision: yes

Circularity Check

0 steps flagged

No significant circularity; external convex-hull algorithm applied to independent G inputs

full rationale

The paper presents a computational procedure that feeds independently supplied Gibbs free energy values (at discrete compositions, fixed T and P) into the standard Qhull convex-hull routine and then classifies the resulting simplices. No derivation, equation, or central claim reduces to a fitted parameter, a self-definition, or a self-citation chain; the thermodynamic features are asserted to emerge directly from the geometry of the lower convex envelope. The method therefore contains no load-bearing step that is equivalent to its inputs by construction. The only substantive assumption (dense, accurate G sampling) is an external requirement on the data, not a circularity in the algorithm itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard thermodynamic principle that equilibrium states minimize Gibbs free energy and on the geometric property that the lower convex hull identifies those minima; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption The lower convex hull of discrete Gibbs free energy points in composition space identifies the thermodynamically stable phases and their coexistence regions.
    This is the central mapping from geometry to thermodynamics invoked by the method.

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