A phase diagram is a single map of “at this composition and this temperature, which phases coexist, and in what proportions.” In ①, move freely over the diagram and see the microstructure at that point (primary crystals, eutectic structure, peritectic shells, and precipitates, drawn the same way as Figs. 2-7 and 2-8 of Understanding Inorganic Materials Chemistry, in Japanese) together with the phase fractions from the lever rule. In ②, confirm that the phase diagram follows from the common tangent of the free-energy curves, and move the interaction parameter Ω to watch the complete-solid-solution, eutectic, peritectic, and monotectic types appear (Fig. 3 of Tanaka 2008). In ③, cool from the melt, switch between equilibrium cooling and rapid cooling (Scheil), and follow how cooling stops at the eutectic and peritectic temperatures (thermal arrests), as well as microsegregation, nonequilibrium eutectics, and even vitrification by quenching. All curves are computed on the spot from the regular-solution model, and the melting, eutectic, and peritectic points are matched to real materials (6 metal systems, 1 semiconductor system, and 6 ceramic systems). You can also choose schematic systems with the same shapes as the textbook figures (eutectic and peritectic) and a schematic monotectic system (two-liquid separation type). Composition can be switched between mol% and mass%. In ④, schematic pictures of the atomic arrangement show a melt turning into glass when quenched and into crystals when cooled slowly; in ⑤, they show a glass or solid solution that was uniform at high temperature separating into two phases by spinodal decomposition on cooling.
This is a “phase diagram drawn from a model.”
The free energy per mole of each phase is taken to be
G = (1−X)gA + X gB
+ RT[(1−X)ln(1−X) + XlnX]
+ X(1−X)[L0 + L1(1−2X)
+ L2(1−2X)2]
(a Redlich–Kister expansion truncated at three terms),
and the phase diagram is obtained by finding the lower convex hull (= common tangents) numerically.
The melting points and enthalpies of fusion of the pure components are measured values, and the interaction parameters
L0, L1, L2 are
values chosen so that the temperatures and compositions of the eutectic and peritectic points and the solubility limits agree with experiment
— not a CALPHAD thermodynamic assessment.
All compositions in the internal calculation are mole fractions (the display can be switched between mol% and mass%).
The three schematic systems (the eutectic and peritectic systems shaped like textbook Figs. 2-7 and 2-8, and the monotectic system) are fictitious systems whose parameters were chosen to reproduce the shapes in the figures.
The monotectic system and the “Vary Ω” mode of ② follow Figs. 1 and 3 of Toshihiro Tanaka, “Alloys and Equilibrium Phase Diagrams” (in Japanese), Journal of the Japan Welding Society 77 (6), 576–581 (2008).
Sources of the data: the melting points and enthalpies of fusion are from the
CRC Handbook of Chemistry and Physics and
Kubaschewski, Materials Thermochemistry;
the eutectic and peritectic points are from ASM Handbook Vol. 3, Alloy Phase Diagrams and
“Phase Diagrams for Ceramists” (The American Ceramic Society).
Rapid cooling uses the Scheil–Gulliver equation
(CL − CS)dfS =
(1 − fS)dCL
integrated numerically along the liquidus and solidus,
which is the limit in which diffusion in the solid is ignored entirely (real cooling falls somewhere in between).
For the detailed derivation, see From Free Energy to Phase Diagrams and the Phase-Field Method (in Japanese).
See also Gibbs free energy.