Draws representative binary phase diagrams, holding the key points (melting points, eutectic and peritectic points, maximum solid solubilities) as literature values. Click or tap the diagram to see the phases at that point, the compositions at the two ends of the tie line, the mass and mole fraction of each phase from the lever rule, and the working. "Cooling path" tabulates and plots how the phase fractions and compositions change as an alloy cools from the liquid, and sketches the microstructure, including the fractions of primary phase and eutectic and the segregation (coring). "Phase rule & reactions" counts degrees of freedom and explains the invariant reaction types, and "Practice" asks you to read points off the diagram. The default is Callister's example: a Pb–40 wt% Sn alloy at 150 °C.
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Diagram and results
A phase diagram shows equilibrium at 1 atm. The key points are literature values, but the shapes of the lines between them are schematic (for Cu–Ni they are calculated from a solution model), so compositions read from the diagram are approximate. The lever rule applies only in two-phase regions; on a wt% diagram it gives mass fractions, and on an at% diagram it gives mole fractions (fractions by number of atoms).
See it as a graph
Other inputs stay at their current values while the chosen input varies from 1/4 to 4 times its current value. The red dot is the current value.
How to read a binary phase diagram
The horizontal axis is composition (wt% or at% of the right-hand component), the vertical axis is temperature, and the pressure is fixed at 1 atm. Lines divide the diagram into single-phase regions (coloured here) and two-phase regions where two phases coexist (white). The lower boundary of the all-liquid region is the liquidus, the upper boundary of an all-solid region is the solidus, and the line giving the limit of how much can dissolve in a solid solution is the solvus. For a point inside a two-phase region, draw a horizontal line (tie line) at that temperature; the compositions where it meets the single-phase regions on each side are the compositions of the two coexisting phases. The key to reading the diagram is that, at a fixed temperature, changing the alloy composition does not change the compositions of the two phases, only their proportions. Neighbouring regions differ by one phase, and two single-phase regions touch only at points such as melting points. Three phases coexist only on horizontal lines, which mark the temperatures of invariant reactions such as eutectics and peritectics.
Deriving the lever rule
Suppose an alloy of overall composition C₀ has separated into an α phase of composition Cα and a β phase of composition Cβ. Taking the total mass as 1, with mass fractions Wα and Wβ, conservation of total mass gives Wα + Wβ = 1, and conservation of the right-hand component (the solute) gives Wα·Cα + Wβ·Cβ = C₀. Solving gives Wα = (Cβ − C₀)/(Cβ − Cα) and Wβ = (C₀ − Cα)/(Cβ − Cα). If the tie line is a balance beam with its fulcrum at C₀, each phase's amount is proportional to the length of the opposite arm, hence the name. Applying the same formula on an at% diagram solves a balance of numbers of atoms instead of mass, and gives mole fractions. Mass fractions from wt% and mole fractions from at% differ because the two phases have different average atomic weights (this tool lists both). Area fractions in a micrograph are close to volume fractions, which differ again when the two phases have different densities; convert with Vα = (Wα/ρα)/(Wα/ρα + Wβ/ρβ).
The Gibbs phase rule and cooling curves
The phase rule F = C − P + 2 gives the number F of intensive variables (temperature, pressure and phase compositions) that can be varied freely in a system of C components and P phases. Fixing the pressure at 1 atm, as in metallic phase diagrams, removes one, giving F = C − P + 1. In a binary system (C = 2) a single-phase region has F = 2 (temperature and composition can vary independently), a two-phase region has F = 1 (fixing the temperature fixes the two phase compositions at the ends of the tie line), and three coexisting phases give F = 0, fixing the temperature and all three compositions. This is why pure metals and eutectic alloys solidify at a single temperature and their cooling curves show a temperature arrest. While eutectic solidification proceeds, latent heat is released but the temperature stays constant; it starts to fall only once one of the three phases is used up. Passing through a two-phase region (F = 1), the temperature keeps falling but more slowly because of the latent heat, so the cooling curve bends at the liquidus temperature. Thermal analysis measures liquidus, solidus and eutectic temperatures using exactly these features.
Types of invariant reaction
Three-phase horizontal lines in binary systems are named by the reaction on cooling. A eutectic reaction splits a liquid into two solids at once, L → α + β (183 °C in Pb–Sn, 577 °C in Al–Si); a eutectoid reaction does the same from a solid, γ → α + β (pearlite forms at 727 °C in Fe–C); a peritectic reaction combines a liquid with a previously formed solid into another solid, L + α → β (1186 °C in Pt–Ag, 1495 °C in Fe–C); a peritectoid reaction is the all-solid version, α + β → γ; and a monotectic reaction splits a liquid into a solid and a second liquid, L₁ → α + L₂. On the diagram, if the phase at the middle composition of the horizontal line lies above the line, the reaction is of the eutectic type (two lines descend in a V and meet there); if it lies below, it is of the peritectic type. In a eutectic reaction the two solids grow side by side at the same time, giving a fine lamellar or rod-like eutectic structure. In a peritectic reaction the new β forms as a shell on the surface of α, after which the reaction can proceed only by diffusion through that shell, so in real cooling it often stops short and α remains.
Calculating a phase diagram from solution models (Cu–Ni)
The isomorphous Cu–Ni system is the standard example whose liquidus and solidus can be calculated from a model. For each component, the chemical potentials in the solid and the liquid must be equal. For an ideal solution (no heat of mixing, only the entropy of mixing), each component obeys ln(x_L/x_S) = −(ΔH_f/R)(1/T − 1/T_m), and the two equations fix the solid and liquid compositions at each temperature. Only the melting points and heats of fusion of the two ends are needed. The "ideal solution" lines in this tool use 13.26 kJ/mol for Cu and 17.48 kJ/mol for Ni; both lines lie on the Ni-rich side of Callister's diagram, and the 1250 °C tie line comes out at about 38 / 45 wt% Ni (the figure gives 31.5 / 42.5). A regular-solution model adds a heat of mixing Ω·x(1 − x), and giving the solid and the liquid separate Ω values corrects this offset. This tool chooses Ω_s ≈ 9.2 and Ω_l ≈ 10.0 kJ/mol so that the model passes through the 1250 °C tie line. Cooling a 35 wt% Ni alloy in this model, the first solid contains about 46 wt% Ni and the last liquid about 25 wt% Ni, in good agreement with the readings from Callister's lecture figure (46 and 24). A positive Ω_s means that Cu and Ni mix slightly reluctantly in the solid (they tend to separate into two phases at low temperature). Thermodynamic databases used in practice (the CALPHAD method) make such coefficients functions of temperature and composition and fit them to many measurements.
Equilibrium versus real cooling (segregation and the Scheil equation)
Phase diagrams and the lever rule assume equilibrium, with diffusion complete even inside the solid at every temperature. In real casting, diffusion in the liquid is fast but diffusion in the solid is much slower, so the composition of each layer that solidified earlier is frozen in. Cu–Ni dendrites therefore show segregation (coring): the centres keep the Ni-rich composition that solidified first, and the outer parts are progressively richer in Cu. The limiting case of no diffusion at all in the solid is the Scheil equation, C_s = k·C₀·(1 − f_s)^(k−1), where k is the partition coefficient C_s/C_L; the "Cooling path" mode of this tool solves the general case with k not constant by integrating numerically along the liquidus and solidus. In Scheil solidification the average composition of the solid lags behind the equilibrium solidus, so solidification ends below the equilibrium solidus, and eutectic alloys keep more eutectic than at equilibrium. Even an alloy such as Al–4 wt% Cu, which forms no eutectic at equilibrium, ends up with a few per cent of non-equilibrium eutectic when its last liquid reaches the eutectic composition (about 8 % in this calculation). Real solidification lies between equilibrium and Scheil, and segregation is reduced by homogenization annealing (holding for a long time a little below the solidus). Non-equilibrium eutectic melts below the equilibrium solidus, however, so overheating can cause local melting.
Fractions of primary phase and eutectic
In eutectic alloys, the most common mistake is confusing phase fractions with constituent (microstructure) fractions. When a hypoeutectic alloy (left of the eutectic composition) cools, primary α crystallizes once the liquidus is crossed, and just above the eutectic temperature primary α coexists with liquid of the eutectic composition. All this liquid becomes eutectic structure (layers of α and β) at the eutectic temperature, so the primary α fraction equals the α fraction just above the eutectic temperature, (C_E − C₀)/(C_E − Cα), and the eutectic fraction equals the liquid fraction at that moment. To find the fractions of all α phase and all β phase just below the eutectic temperature, put the α and β compositions at the ends of the tie line instead. For Callister's Pb–40 wt% Sn example, primary α is about 50 %, eutectic about 50 %, all α phase about 73 % and all β phase about 27 %; of the 73 % α, about 22 % is the α inside the eutectic. In a hypereutectic alloy (to the right) the primary phase is β, found the same way. Below the eutectic temperature the solubility limits fall, so a second phase precipitates within the primary crystals and the phase fractions shift slightly, but the primary and eutectic fractions do not change.
Common alloys and their diagrams
Solder: Sn–37Pb (63/37) solder is close to the Pb–Sn eutectic composition of 61.9 wt% Sn and melts and freezes at essentially a single temperature, 183 °C, which made it the long-standing choice for joining electronic components. Alloys away from the eutectic composition have a range between liquidus and solidus where they are partly molten. Aluminium castings: the Al–Si system has a eutectic at 577 °C and 12.6 wt% Si, and hypoeutectic alloys around 7 wt% Si (such as A356) and near-eutectic alloys flow well and are widely used for castings. Eutectic Si tends to form coarse plates and is refined by modification with small additions of Na or Sr (one source gives the Na-modified eutectic as about 564 °C and 13 % Si). In hypereutectic alloys, hard primary Si gives wear resistance. Precipitation hardening: on the Al-rich side of Al–Cu, the solubility of Cu drops sharply from 5.65 wt% at the 548 °C eutectic temperature. Solution-treating Al–4 wt% Cu above the solvus but below the eutectic temperature and quenching gives a supersaturated solid solution; subsequent ageing at low temperature produces a fine dispersion of GP zones and metastable precipitates such as θ″ and θ′, which strengthen the alloy (2000-series alloys, duralumin). The diagram shows only the equilibrium θ (Al₂Cu); the metastable phases that do the strengthening do not appear on the equilibrium diagram. Brazing alloys: Ag–Cu alloys near the Cu–Ag eutectic composition (71.9 wt% Ag) are used as brazing alloys that melt at 779 °C.
Where this calculation does not apply (limitations)
A phase diagram represents equilibrium at 1 atm, so after rapid cooling or short heat treatments both the phases present and their fractions differ from the diagram (martensite and the metastable precipitates in Al–Cu do not appear on equilibrium diagrams). The key points are literature values, but the line shapes between them in this tool are schematic (a fitted model for Cu–Ni), so in places they differ from published diagrams by a few wt%. When precise values are needed, use assessed diagrams such as the ASM Alloy Phase Diagram Database or CALPHAD thermodynamic calculations. Impurities and third elements shift liquidus and eutectic temperatures (Na and Sr in Al–Si, Ag and Cu in solders, for example). In ternary and higher systems, the direction of two-phase tie lines within an isothermal section changes with composition, so the phase compositions cannot be found just by drawing a horizontal line as in a binary diagram. Remember also that on a three-phase horizontal line the lever rule alone cannot give the fractions (use the limits just above and just below). Finally, lever-rule fractions are mass fractions and differ from area fractions in a micrograph by the effect of density differences.
With the lever rule, the fraction of a phase is the distance from the alloy composition to the composition of the opposite phase, divided by the total length of the tie line. The liquid fraction is W_L = (Cα − C₀)/(Cα − C_L) = (42.5 − 35)/(42.5 − 31.5) = 7.5/11.0 ≈ 0.68. About 0.32 is the mistake of taking the distance on the same side, (35 − 31.5)/11.0, which is the fraction of α. About 0.82 is the ratio of the alloy composition to the solid composition, 35/42.5, which is not the lever rule. The fractions of the two phases always sum to 1, which you can use as a check once you have found one of them.
The eutectic structure is what the liquid (61.9 wt% Sn) remaining just above the eutectic temperature turns into through the eutectic reaction. So apply the lever rule in the α + L region just above the eutectic temperature: W_eutectic = W_L = (30 − 18.3)/(61.9 − 18.3) = 11.7/43.6 ≈ 0.27. About 0.73 is the fraction of the remaining primary α, the mistake of taking the opposite distance. About 0.15 is the fraction of all β phase just below the eutectic temperature, (30 − 18.3)/(97.8 − 18.3). All the β is inside the eutectic structure, but the eutectic structure also contains α, so the two do not match. The key to this kind of problem is distinguishing microconstituents (primary and eutectic) from phases (α and β).
Gibbs' phase rule is F = C − P + 2, but for condensed systems at constant pressure we use F = C − P + 1. At the eutectic point of a binary system (C = 2), three phases, liquid L, α and β, coexist (P = 3), so F = 2 − 3 + 1 = 0: an invariant point at which the temperature and the composition of every phase are fixed. That is why an alloy of eutectic composition solidifies at a single temperature and its cooling curve shows a thermal arrest. One results from applying F = C − P + 2, which includes pressure as a variable, to a constant-pressure phase diagram. Two is the number of degrees of freedom in a single-phase region (P = 1), where temperature and composition can be varied independently.
BRicher in Cu than the average alloy composition, with the outer parts richer in Ni
CAlways equal to the average composition, because diffusion in the solid is fast enough
Show answer and explanation
Answer: A. Richer in Ni than the average alloy composition
The solid that forms early in solidification has the composition at the end of the higher-temperature tie line, so it is rich in Ni. With rapid cooling, diffusion in the solid cannot keep up, and the outer layers that solidify later are progressively richer in Cu, leaving segregation in which the composition changes from the center of the dendrite outward (a cored structure, or coring). The center is therefore richer in Ni than the average composition. Diffusion in the solid is far slower than in the liquid, so the equilibrium composition is not necessarily reached. The last regions to solidify have a lower melting point and may melt locally on reheating, so a homogenizing anneal is used to reduce the segregation.
Just above the eutectic temperature, primary Si coexists with liquid of the eutectic composition (12.6 wt% Si). From the lever rule, W_primary Si = (18 − 12.6)/(100 − 12.6) = 5.4/87.4 ≈ 0.062, or about 6.2%. About 5.4% takes the composition difference 18 − 12.6 directly as a percentage without dividing by the length of the tie line. About 16.6% is the fraction of all Si phase just below the eutectic temperature, (18 − 1.65)/(100 − 1.65), which includes the Si inside the eutectic structure. Hypereutectic Al-Si alloys are used for pistons and similar parts because the hard primary Si provides wear resistance, and treatments are applied to refine the primary Si.
FAQ
Should I use the wt% or the at% diagram?
They are the same phase diagram with a different horizontal scale. The lever rule on a wt% diagram gives mass fractions, and on an at% diagram it gives mole fractions (fractions by number of atoms). Mass fractions are convenient for weighing out an alloy or thinking about the weight of a casting; mole fractions are convenient for atomic arrangements in crystals and for diffusion and thermodynamic calculations. Always use the same unit for all three compositions (C₀ and the two phases).
Are the phase fractions undefined at exactly the eutectic temperature?
Yes. At the eutectic temperature L, α and β coexist and there are zero degrees of freedom. While an alloy sits at this temperature during cooling, the liquid is being consumed and α and β are growing, so the fractions depend on how far solidification has gone. Use the lever rule in the limits just above (α and L) and just below (α and β) the eutectic temperature instead. Setting the temperature to exactly the eutectic temperature in this tool shows both.
What does it mean that the lines are schematic?
Key points such as melting points, eutectic points and maximum solubilities are literature values, but the lines between them are smooth curves (monotone cubic interpolation) through the key points and through points read from textbook figures and examples. Readings near the key points agree well with the literature, but far from them they can be off by a few wt%. The Cu–Ni lines are calculated from a regular-solution model fitted to the tie line in Callister's worked example.
How can coring (segregation) be removed?
Homogenization annealing, a long hold a little below the solidus, lets diffusion in the solid even out the composition differences. Diffusion distances scale roughly as √(Dt), so finer dendrite arm spacings need shorter times. If the last liquid from Scheil-like solidification has left low-melting regions (such as non-equilibrium eutectic), however, melting starts below the equilibrium solidus, so the temperature must not be set too high.
Does the lever rule work for ternary phase diagrams?
In two-phase regions the same form of equation applies along a tie line in an isothermal section. The direction of the tie line, however, is fixed by experiment or calculation; unlike in a binary diagram, it is not simply horizontal. Inside a three-phase triangle, the fractions of the three phases are found with a centre-of-gravity construction (the triangular lever rule).
Are the fractions in the cooling-path table the same as in a real casting?
The equilibrium column is the limit of very slow cooling, and the Scheil column is the limit of no diffusion in the solid. A real casting lies between them, depending on the cooling rate and how fast the alloying elements diffuse. Undercooling and delayed nucleation can also make solidification start below the liquidus, so the changes do not follow the table temperatures exactly.