One alloy, one thermometer — reconstruct the whole map from a temperature log.
HOW TO READ
The stage on the right is driven by one melt state: the macroscopic crucible shows what the alloy is doing, the microscopic atoms explain why, and the symbolic panel switches between the temperature–time curve and the temperature–composition map — the detective’s evidence and verdict.
Current state: All liquid, liquid fraction 100%.
00
In the physical-chemistry lab you are handed a tube of molten alloy — the instructor says it is a "bismuth–cadmium binary" but will not tell you the ratio.
Your only instrument is a thermometer.
Can you reconstruct the complete phase diagram from a single cooling record?
Predict first: what shape will the curve take?
01
If the tube held a pure metal — say pure bismuth — the cooling curve would show a horizontal halt: as the metal freezes, the released latent heat exactly pays for the heat leaking away, and the temperature parks at the melting point until the last drop solidifies.
The halt's height is the melting point; its length is proportional to the latent heat being paid.
But your alloy is not pure. What happens instead?
02
Cool slowly. At some temperature the first crystal appears — bismuth or cadmium? It depends on the ratio: a Bi-rich melt drops Bi first, a Cd-rich melt drops Cd.
The liquidus equation (Schröder–van Laar) says when that first crystal arrives:
And once crystals form, the melt's composition starts to move — pure Bi leaves, the remaining liquid grows Cd-rich, and the liquidus temperature keeps falling. The curve shows a break, then keeps descending on a gentler slope.
03
The liquid slides down the liquidus until it reaches the eutectic point: both crystals now freeze together, weaving lamellae as the last liquid solidifies in one go.
Here the curve shows a horizontal arrest — not because heat transfer slowed, but because three coexisting phases pin the temperature.
The arrest's length is proportional to how much liquid remains on arrival. Remember that line; it earns its keep soon.
04
Now move the evidence onto the map:
Measure several compositions: each gives a liquidus point; every arrest parks at the same temperature; connect the dots — from curves, the diagram. This is exactly what the classic thermal-analysis laboratory has students do.
05
Why does the curve break where it breaks and stop where it stops? One sentence governs everything (condensed system):
The arrest is the phase rule forbidding cooling until the last drop of liquid is gone.
06
Check the map against the world:
Eutectics are everywhere: road salt stops at the brine eutectic of −21.1 °C; 63/37 solder freezes crisply at 183 °C while 50/50 wades through a ~30 °C pasty range; the Wood's-metal link in a fire sprinkler lets go all at once at 70 °C.
07
A caution from the bench: cool too fast and supercooling delays nucleation — the curve dives below the true temperature, then snaps back (recalescence); breaks get rounded, arrests get dragged low.
An ideal cooling curve needs near-equilibrium: the liquid sitting on the liquidus at every instant. The curves on this page are simulated with Newtonian heat loss plus equilibrium freezing — the wiggles and rounded corners on a real recorder are kinetics reminding you it exists.
08
Now the tube is yours: mix any composition, press cool, and watch the curve answer.
Challenges: find the ratio with the longest arrest (the apex of Tammann's triangle); then mix a curve with almost no arrest at all; read a liquidus temperature off a break and check it against the map.
Find the longest arrest; then mix a curve with almost none.
THE TOOLKIT
CONCEPT CHECK · PREDICT, THEN TEST
FREE PLAY
Mix any composition and run a full cool-down; the curve, the crucible, and the map answer together.
Find the ratio with the longest arrest (Tammann's apex); mix a curve with almost none; read a liquidus temperature off a break and check it on the map.
Current state: All liquid, liquid fraction 100%.
MODEL CARD
Bi–Cd simple binary eutectic; terminal solid solubility is negligible (<1 at%), so the freezing solids are pure crystals — which is what makes it the textbook thermal-analysis system (Chinese labs often run Sn-Bi or Pb-Sn; same physics).
Schröder–van Laar ideal-solution equation, analytic in both directions. Pure-component data (CRC Handbook): Bi 271.4 °C, ΔH_fus 11.3 kJ/mol; Cd 321.1 °C, 6.21 kJ/mol.
Model: 135.3 °C at x(Cd)=0.565. Literature: 144–146 °C at ~40 wt% Cd (x(Cd) ≈ 0.55). Composition within one atomic percent; temperature ~10 K low — Bi-Cd mixes nearly ideally (ΔH_mix ≈ +840 J/mol), and the residual bias is declared, not hidden.
Newtonian heat loss plus an equilibrium latent-heat source; in two-phase fields the effective heat capacity is C_p + ΔH·(df_s/dT), and the eutectic arrest pays out the remaining liquid's latent-heat budget step by step. Supercooling and recalescence are not modeled and are discussed in the text.
Brine eutectic −21.1 °C / 23.3 wt% (ice + NaCl·2H₂O); FHWA's practical salting floor near −9.5 °C; Sn-Pb 63/37 vs 50/50 solidus–liquidus; Wood's-metal sprinkler links at 70 °C — all with primary citations in the repository notes.
One temperature log holds the whole map: the breaks surrender the liquidus, the arrest pins the eutectic temperature, and the arrest's length points at the eutectic composition.