Convex Hull of Stability
Reproduced 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: Chemical potential range where AX is stable
- ΔE_d: Decomposition enthalpy, the energy above the hull for unstable phases
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 - 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
convex-hull-of-stability.typ (195 lines)
#import "@preview/cetz:0.5.2": 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-style = (stroke: blue.darken(20%) + 2.5pt)
let hyp-hull-style = (
stroke: (paint: gray, thickness: line-thickness, dash: "dashed"),
)
// 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")
let stable-point(pos, label, anchor: "north", padding: none, ..rest) = {
circle(pos, radius: point-radius, fill: blue.darken(20%), ..rest)
content(pos, label, anchor: anchor, padding: padding)
}
for spec in (
(pos: (0, height - 1), label: "A", args: (anchor: "west", padding: (left: 10pt), name: "a")),
(pos: (width / 2, 2), label: "AX", args: (anchor: "north", padding: (top: 10pt), name: "ax")),
(
pos: (width * 5 / 7, 1.5),
label: $A_2X_5$,
args: (anchor: "north", padding: (top: 10pt), name: "a2x5"),
),
(
pos: (width, height - 1.5),
label: "X",
args: (anchor: "east", padding: (right: 10pt), name: "x"),
),
) { stable-point(spec.pos, spec.label, ..spec.args) }
let unstable-point(pos, label, ..rest) = {
let (x, y) = pos
rect((x, y - 0.15), (x + 0.3, y + 0.15), fill: red, stroke: .5pt, ..rest)
content(pos, label, anchor: "south", padding: (bottom: 8pt))
}
for (pos, label, name) in (
((width / 3, height - 1.5), $A_2X$, "a2x"),
((width * 7 / 9, 3.7), $A_2X_7$, "a2x7"),
) { unstable-point(pos, label, name: name) }
for (start, end, name) in (
("a", "ax", "hull-a-ax"),
("ax", "a2x5", "hull-ax-a2x5"),
("a2x5", "x", "hull-a2x5-x"),
) { line(start, end, ..hull-style, name: name) }
content(
(rel: (-1.8, -.8), to: "ax"),
text(fill: blue.darken(20%), size: 12pt)[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: blue.darken(20%), thickness: .5pt),
padding: 1pt,
name: "hull-label-line",
)
for (start, end, name) in (
("ax", "a2x7", "hyp-hull-ax-a2x7"),
("a2x7", "x", "hyp-hull-a2x7-x"),
) { line(start, end, ..hyp-hull-style, name: name) }
content(
(rel: (0, 0.3), to: "hyp-hull-a2x7-x.mid"),
text(fill: gray, size: 13pt)[hypothetical hull for\ evaluating $A_2X_5$],
anchor: "east",
)
// Intersection helpers must not expand the visible diagram's bounds.
draw.hide(line(
(rel: (0, 3), to: "a2x"),
(rel: (0, -3), to: "a2x"),
stroke: none,
name: "a2x-vertical",
))
intersections("a2x-isect", "a2x-vertical", "hull-a-ax", "hull-ax-a2x5")
line(
"a2x-isect.0",
"a2x",
mark: (end: "stealth", fill: red),
stroke: red + line-thickness,
name: "arrow-a2x",
)
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, size: 12pt)[A + AX → A₂X],
frame: "rect",
padding: (1pt, 3pt),
stroke: red + .3pt,
name: "box1",
anchor: "west",
fill: red.lighten(90%),
)
// Second arrow - find intersections first
draw.hide(line(
(rel: (0, 3), to: "a2x5"),
(rel: (0, -3), to: "a2x5"),
stroke: none,
name: "a2x5-vertical",
))
intersections("a2x5-isect", "a2x5-vertical", "hyp-hull-ax-a2x7", "hyp-hull-a2x7-x")
line(
"a2x5-isect.0",
"a2x5",
mark: (end: "stealth", fill: rgb("#4d8000")),
stroke: rgb("#4d8000") + line-thickness,
name: "arrow-a2x5",
)
content(
"arrow-a2x5.mid",
text(fill: rgb("#4d8000"))[$Delta E_d$],
anchor: "east",
padding: (right: 3pt),
)
content(
(rel: (0.1, 0), to: "arrow-a2x5.mid"),
text(fill: rgb("#4d8000"), size: 12pt)[4/5 AX + 3/5 A₂X₇ → A₂X₅],
frame: "rect",
padding: (1pt, 3pt),
stroke: rgb("#4d8000") + .3pt,
name: "box2-label",
anchor: "west",
fill: rgb("#4d8000").lighten(90%),
)
line(
(0, height - 4.5),
"ax",
stroke: (paint: orange, thickness: line-thickness, dash: "dashed"),
name: "mu-line",
)
content((rel: (2.4, 0), to: "mu-line.start"), rotate(14deg)[#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,
name: "mu-arrow",
)
circle(
(0.5, 1),
radius: point-radius,
fill: blue.darken(20%),
name: "legend-stable",
)
content("legend-stable.east", "stable", anchor: "west", padding: (left: 5pt))
rect(
(0.5 - 0.15, 1 - 0.6),
(0.5 + 0.15, 1 - 0.3),
fill: red,
stroke: red,
name: "legend-unstable",
)
content((0.5 + 0.15, 1 - 0.45), "unstable", anchor: "west", padding: (
left: 5pt,
))
content((rel: (-0.5, 0), to: "y-axis-left.mid"), [#rotate(
-90deg,
reflow: true,
)[$Delta E_f$ (energy/atom)]])
content((width / 2, -0.5), $x "in" A_(1-x)X_x$)
})