Convex Hull of Stability
Adapted from Bartel (2022). The convex hull of stability represents the lowest energy surface in composition space for a given chemical system. It is constructed by connecting the energies of the most thermodynamically stable compounds at each composition. For visual clarity, only ground-state phases at each composition are shown. Legend:
- Blue circles: Stable phases lying on the convex hull (blue line)
- Red squares: Unstable phases above the hull
- Orange dashed line: Extension of the AX–A₂X₅ hull facet to x = 0. Its intercept and the energy of A bound the chemical potential range of A (orange arrow) where AX is stable
- ΔE_d: Decomposition enthalpy, the energy above the hull for unstable phases (A₂X), or the negative distance below the hull constructed without it for stable phases (A₂X₅)
Materials on the hull are stable against decomposition into other phases. The energy difference between a material and the hull (hull distance) quantifies its thermodynamic stability, with larger distances meaning less stable and harder to synthesize, making the convex hull a powerful tool for guiding experimental synthesis efforts.
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Code
#import "@preview/cetz:0.5.3": canvas, draw
#import draw: circle, content, intersections, line, rect
#set page(width: auto, height: auto, margin: 3pt, fill: none)
#set text(size: 12pt)
#canvas({
// Diagram dimensions and styles
let width = 12
let height = 8
let point-radius = 0.15
let line-thickness = 1.5pt
let arrow-style = (mark: (end: "stealth"), stroke: line-thickness, fill: black)
let hull-color = blue.darken(20%)
let green = rgb("#4d8000")
// Darker than Typst's red/orange/gray so labels stay legible on light and inverted dark pages
let red = rgb("#c42a20")
let orange = rgb("#b35500")
let hypothetical = luma(110)
// Draw axes first to establish named positions
line((0, 0), (0, height), ..arrow-style, name: "y-axis-left")
line((0, 0), (width, 0), stroke: line-thickness, name: "x-axis")
line((width, 0), (width, height), ..arrow-style, name: "y-axis-right")
// Stable phases at x = 0, 1/2, 5/7, 1. Both elemental references sit at Delta E_f = 0.
for (pos, label, anchor, padding, name) in (
((0, height - 1), "A", "west", (left: 10pt), "a"),
((width / 2, 2), "AX", "north", (top: 10pt), "ax"),
((width * 5 / 7, 1.5), $A_2X_5$, "north", (top: 10pt), "a2x5"),
((width, height - 1), "X", "east", (right: 10pt), "x"),
) {
circle(pos, radius: point-radius, fill: hull-color, name: name)
content(pos, label, anchor: anchor, padding: padding)
}
// Unstable phases at x = 1/3, 7/9
for (pos, label, name, anchor) in (
((width / 3, height - 1.5), $A_2X$, "a2x", "south"),
// Upper-left label keeps clear of the hypothetical hull leaving A₂X₇ to the upper right
((width * 7 / 9, 3.7), $A_2X_7$, "a2x7", "south-east"),
) {
rect((rel: (-0.15, -0.15), to: pos), (rel: (0.3, 0.3)), fill: red, stroke: .5pt, name: name)
content(pos, label, anchor: anchor, padding: (bottom: 8pt))
}
for (start, end) in (("a", "ax"), ("ax", "a2x5"), ("a2x5", "x")) {
line(start, end, stroke: hull-color + 2.5pt, name: "hull-" + start + "-" + end)
}
content(
(rel: (-1.8, -.8), to: "ax"),
text(fill: hull-color)[convex hull\ of stability],
frame: "rect",
stroke: none,
padding: (left: 5pt),
fill: rgb("#cdd3da"),
name: "hull-label",
)
line(
"hull-label.north",
"hull-a-ax.90%",
stroke: (paint: hull-color, thickness: .5pt),
padding: 1pt,
)
for (start, end) in (("ax", "a2x7"), ("a2x7", "x")) {
line(
start,
end,
stroke: (paint: hypothetical, thickness: line-thickness, dash: "dashed"),
name: "hyp-hull-" + start + "-" + end,
)
}
content(
(rel: (0, 0.3), to: "hyp-hull-a2x7-x.mid"),
text(fill: hypothetical, size: 13pt)[hypothetical hull for\ evaluating $A_2X_5$],
anchor: "east",
)
// Vertical arrow from the (hypothetical) hull down or up to a phase: its decomposition energy
let decomposition-arrow(phase, hull-segments, color) = {
// Intersection helpers must not expand the visible diagram's bounds.
draw.hide(line((rel: (0, 3), to: phase), (rel: (0, -3), to: phase), name: phase + "-vertical"))
intersections(phase + "-isect", phase + "-vertical", ..hull-segments)
line(
phase + "-isect.0",
phase,
mark: (end: "stealth", fill: color),
stroke: color + line-thickness,
name: "arrow-" + phase,
)
}
let reaction-box = (frame: "rect", padding: (1pt, 3pt), anchor: "west")
decomposition-arrow("a2x", ("hull-a-ax", "hull-ax-a2x5"), red)
content((rel: (-0.5, 0), to: "arrow-a2x.60%"), text(fill: red)[$Delta E_d$])
content(
(rel: (.2, 0), to: "arrow-a2x.30%"),
text(fill: red)[A + AX → A₂X],
stroke: red + .3pt,
fill: red.lighten(90%),
..reaction-box,
)
decomposition-arrow("a2x5", ("hyp-hull-ax-a2x7", "hyp-hull-a2x7-x"), green)
content("arrow-a2x5.mid", text(fill: green)[$Delta E_d$], anchor: "east", padding: (right: 3pt))
content(
(rel: (0.1, 0), to: "arrow-a2x5.mid"),
text(fill: green)[4/5 AX + 3/5 A₂X₇ → A₂X₅],
stroke: green + .3pt,
fill: green.lighten(90%),
..reaction-box,
)
// The AX–A₂X₅ hull facet extended to x = 0 bounds μ_A from below; the A–AX facet from above.
line(
("a2x5", 1000% / 3, "ax"),
"ax",
stroke: (paint: orange, thickness: line-thickness, dash: "dashed"),
name: "mu-line",
)
content((rel: (2.5, 0.15), to: "mu-line.start"), rotate(11deg, text(
fill: orange,
size: 13pt,
)[$μ_A$ range\ where AX is stable]))
line(
"hull-a-ax.2%",
"mu-line.4%",
mark: (symbol: "stealth", fill: orange, offset: 0.1, scale: 0.6),
stroke: orange + line-thickness,
)
// Legend: the unstable entry hangs below the stable marker
let legend-label = (anchor: "west", padding: (left: 5pt))
circle((0.5, 1), radius: point-radius, fill: hull-color, name: "legend-stable")
content("legend-stable.east", "stable", ..legend-label)
rect((rel: (-0.15, -0.6), to: "legend-stable"), (rel: (0.3, 0.3)), fill: red, stroke: .5pt)
content((rel: (0.15, -0.45), to: "legend-stable"), "unstable", ..legend-label)
content((rel: (-0.5, 0), to: "y-axis-left.mid"), rotate(
-90deg,
reflow: true,
)[$Delta E_f$ (energy/atom)])
content((rel: (0, -0.5), to: "x-axis.mid"), $x "in" A_(1-x)X_x$)
})