A binary phase diagram maps the equilibrium state of a two-component alloy system as a function of temperature and composition, telling engineers which phases are present, their relative amounts, and their compositions at any given conditions. The most practically important feature is the eutectic point — the unique composition and temperature at which both components solidify simultaneously from a single liquid, producing the lowest possible melting temperature for the system. The Pb-Sn eutectic at 61.9% Sn and 183 °C underpins traditional electrical solder; the Cu-Ag eutectic at 71.9% Ag and 779 °C is used in silver brazing filler metals.
In this simulator you can click or drag any point on the phase diagram to read off the equilibrium phases (liquid, solid α, solid β, or two-phase regions), apply the lever rule to compute the fraction and composition of each phase, and observe the theoretical cooling curve as the alloy passes through the liquidus and solidus boundaries. The simulator displays both the Cu-Ni isomorphous system (complete solid solubility) and the Pb-Sn eutectic system.
What is the lever rule and how do I use it?
The lever rule gives the mass fraction of each phase in a two-phase region. If a point falls in the (α + liquid) two-phase field at overall composition X0, and the phase boundaries at that temperature give the α composition Xα and liquid composition XL, then the fraction of liquid is fL = (X0 − Xα) / (XL − Xα) and the fraction of solid is 1 − fL. Imagine the compositions as distances along a lever; the relative "weight" (fraction) of each phase balances at the overall composition.
What is the eutectic reaction and what microstructure does it produce?
At the eutectic temperature and composition, the liquid transforms simultaneously into two solid phases by the reaction: Liquid → α + β at constant temperature. This invariant reaction (zero degrees of freedom by Gibbs phase rule) produces a characteristic lamellar or rod microstructure of alternating α and β layers, with layer spacing typically 1–10 μm. In Pb-Sn solder, this fine eutectic microstructure gives good strength and ductility; coarser microstructures (hypo- or hypereutectic) form primary dendrites before the eutectic reaction completes.
What is the difference between a liquidus and a solidus?
The liquidus is the upper boundary: above it, the alloy is entirely liquid. Below the liquidus but above the solidus, the alloy is a mixture of liquid and solid. The solidus is the lower boundary: below it, the alloy is completely solid. For a eutectic system, the solidus is horizontal at the eutectic temperature. For a complete solid-solution system like Cu-Ni, both liquidus and solidus are smooth curves that meet at the melting points of the pure components (1085 °C for Cu, 1455 °C for Ni).
The Gibbs phase rule, F = C − P + 2 (or F = C − P + 1 for pressure-fixed diagrams), gives the number of degrees of freedom F: C = number of components, P = number of phases. For a binary system (C = 2) at a eutectic point where three phases coexist (liquid + α + β, so P = 3), F = 2 − 3 + 1 = 0: the temperature and both phase compositions are completely fixed. This zero degrees of freedom is why the eutectic occurs at a single, invariant temperature — the "eutectic invariant".
Hypoeutectic alloys (composition to the left of the eutectic point) begin solidifying by precipitating primary α dendrites as the liquid cools below the liquidus. The remaining liquid becomes progressively enriched toward the eutectic composition, at which point it transforms to lamellar eutectic. Hypereutectic alloys (right of eutectic) precipitate primary β instead. At exactly the eutectic composition, no primary phase forms: the entire melt transforms at once to fine eutectic, giving the finest microstructure and often the best mechanical properties for casting applications.
Pure elements melt at a sharp temperature because they have a single component (C = 1): with liquid and solid in equilibrium, F = 1 − 2 + 1 = 0, fixing both T and composition (trivially). For a binary alloy, C = 2 and with two phases, F = 1: there is one degree of freedom, which is temperature. As temperature changes, both the liquid and solid compositions change (tracking the liquidus and solidus), so the alloy melts gradually over a "pasty range" between solidus and liquidus temperatures.
Many binary systems form intermetallic compounds — ordered crystal structures at specific stoichiometries where the two elements bond with different coordination than in pure metals. In the Fe-C system, cementite Fe₃C (6.67 wt% C) is the key intermetallic. In Al-Cu alloys, θ-CuAl₂ (53 wt% Cu) is the primary strengthening intermetallic in age-hardened aircraft aluminium. These compounds appear as vertical lines (line compounds) or narrow regions in the phase diagram and are often brittle, acting as stress concentrators if they form continuous films at grain boundaries.
Cu and Ni are completely miscible in both liquid and solid states because they have the same crystal structure (FCC), similar atomic radii (Cu 128 pm, Ni 124 pm), similar electronegativities, and the same valence. This makes the Cu-Ni system one of the simplest "isomorphous" phase diagrams — a single lens-shaped two-phase region between a smooth liquidus and solidus. Monel alloy (67% Ni, 33% Cu) has excellent corrosion resistance in seawater and is used in marine hardware and chemical plant equipment precisely because it is a single-phase solid solution throughout the solid range.
Phase diagrams guide the temperature and time cycles used in heat treatments. Solution annealing (heating into the single-phase region to dissolve precipitates), quenching (rapid cooling to freeze a supersaturated solid solution), and ageing (heating to a lower temperature to precipitate fine particles by controlled nucleation) all exploit boundaries on the phase diagram. In Al-Cu alloys, solution treatment at 500 °C (single-phase α), water quench, then age at 150 °C produces GP zones and θ’ precipitates, increasing yield strength from ~100 MPa to over 400 MPa.
A peritectic reaction is the reverse of a eutectic: instead of a liquid forming two solids on cooling, a solid reacts with liquid on cooling to form a different solid: α + Liquid → β. It occurs in many technically important systems including Fe-C near 0.1% C (forming austenite from delta-ferrite and liquid) and Cu-Sn bronzes. Unlike the eutectic, the peritectic reaction is often kinetically sluggish because the newly formed β phase coats the α particles, preventing further reaction and leading to coring (compositional inhomogeneity) in cast alloys.
When an alloy solidifies over a temperature range (mushy zone), the first solid to form is richer in the higher-melting component while the last liquid is enriched in the lower-melting one. This compositional variation is called microsegregation or coring, and it creates local variations in hardness, corrosion resistance, and ductility within each grain. Homogenisation annealing at high temperature (below the solidus) for several hours or days allows solid-state diffusion to equalise composition. Industrial continuous casting uses electromagnetic stirring to reduce segregation in large steel billets.