This simulation displays the equilibrium phase diagram of a two-component (binary) alloy system exhibiting a simple eutectic. You can drag the cursor over the diagram to read the phase region, phase fractions (via the lever rule), and watch a solidification path animate from the liquidus down to room temperature.
Liquidus (A-side): T_L = T_A + (T_E - T_A) * (x / x_E)^k
Liquidus (B-side): T_L = T_B + (T_E - T_B) * ((1-x) / (1-x_E))^k
Lever rule (mushy zone):
f_liquid = (x0 - x_S(T)) / (x_L(T) - x_S(T))
f_solid = 1 - f_liquid
Gibbs phase rule: F = C - P + 2
Single-phase region: F = 2 (T and x free)
Two-phase region: F = 1 (T fixes x_L and x_S)
Eutectic point: F = 0 (invariant)The Pb-Sn eutectic at 63/37 wt% (183 °C) was the backbone of electronics soldering for over a century. Modern lead-free solders (Sn-Ag-Cu, SAC305) shift the eutectic to ~217 °C, which required redesigning the entire PCB assembly industry to handle the higher processing temperature.
What is a binary alloy phase diagram?
A binary alloy phase diagram maps the equilibrium phases of a two-component system as a function of composition (x-axis, mol% or wt% of component B) and temperature (y-axis). It shows which phase — liquid, solid α, solid β, or a mixture — is stable at each (x, T) point.
What is the eutectic point?
The eutectic point is the unique composition and temperature at which a binary alloy melts and solidifies at a single fixed temperature, lower than the melting points of either pure component. At the eutectic, liquid simultaneously transforms into two solid phases (α+β) on cooling.
What are the liquidus and solidus lines?
The liquidus line marks the boundary above which the alloy is entirely liquid. The solidus line marks the boundary below which the alloy is entirely solid. Between them lies the two-phase (mushy) zone where liquid and solid coexist in equilibrium.
The lever rule calculates the fraction of each phase in the two-phase region. If the overall composition is x₀ and the liquidus gives x_L and the solidus gives x_S at that temperature, the fraction of liquid = (x₀ - x_S)/(x_L - x_S) and the fraction of solid = (x_L - x₀)/(x_L - x_S). The rule derives from mass-balance conservation.
A hypoeutectic alloy (composition to the left of the eutectic) first crosses the liquidus on cooling, forming primary α crystals while the remaining liquid becomes enriched toward the eutectic composition. When the temperature reaches the eutectic isotherm, the remaining liquid solidifies as the eutectic mixture of α+β.
A hypoeutectic alloy has a composition lower than the eutectic point (more of component A). During cooling it first precipitates primary α phase. A hypereutectic alloy has a higher composition than the eutectic (more of component B) and first precipitates primary β phase. Both ultimately produce eutectic microstructure at the eutectic temperature.
Mixing two components generally lowers the Gibbs free energy of the liquid phase more than that of either solid, depressing the freezing point. The eutectic is the point where the liquidus curves from both pure components meet, giving the absolute minimum melting temperature for any composition in that binary system.
Classic examples include lead-tin (Pb-Sn, eutectic at ~61.9 wt% Sn, 183 °C, used in solders), silver-copper (Ag-Cu, eutectic at 28.1 wt% Cu, 779 °C), and bismuth-cadmium (Bi-Cd, eutectic at ~40 wt% Cd, 144 °C). The Pb-Sn system is the textbook example taught in most materials-science courses.
Slow (equilibrium) cooling produces coarse lamellar or rod-shaped eutectic microstructures, as atoms have time to diffuse over long distances. Rapid quenching suppresses diffusion, producing fine or even metastable structures such as glass or supersaturated solid solutions. This is why rapid solidification processing can dramatically improve mechanical properties.
The Gibbs phase rule states F = C − P + 2, where F is degrees of freedom, C is the number of components, and P is the number of phases. For a binary alloy (C = 2) at constant pressure: in a single-phase region P = 1 so F = 2 (both T and x can vary freely); in a two-phase region P = 2 so F = 1 (fixing T fixes both phase compositions); at the eutectic point P = 3 so F = 0 (invariant point).
A binary alloy phase diagram maps the equilibrium states of a two-component metallic system as a function of composition (x-axis, mol% of component B) and temperature (y-axis). The diagram displays three key boundaries: the liquidus line above which the alloy is entirely molten, the solidus line below which it is entirely solid, and the eutectic isotherm — the unique temperature at which one special composition melts and freezes at a single fixed temperature lower than either pure component. The eutectic point exists because mixing two metals lowers the Gibbs free energy of the liquid phase, depressing the freezing point; the classic example is lead-tin solder (63/37 wt%, 183 °C), used in electronics for over a century.
Click or drag anywhere on the canvas to move the cursor and read off the phase at that (composition, temperature) point. In the two-phase mushy zone the lever rule is drawn automatically: two coloured markers show the liquidus composition x_L and solidus composition x_S, and the HUD displays the fraction of liquid and solid at a glance. Adjust the melting points of both pure components, the eutectic composition, and the eutectic depression with the HUD sliders to explore a wide range of real and hypothetical alloy systems.
What is the lever rule and how is it applied?
The lever rule is a mass-balance technique for calculating the fraction of each phase in a two-phase region. If the overall composition is x₀, the liquidus composition at the current temperature is x_L, and the solidus composition is x_S, then the fraction of liquid = (x₀ − x_S)/(x_L − x_S) and the fraction of solid = 1 − fraction liquid. The result is exact at thermodynamic equilibrium and follows directly from conservation of mass.
Why is the eutectic temperature the lowest melting point in the system?
Adding a second component to a pure metal generally lowers the chemical potential of the liquid phase more than that of the solid, depressing the freezing point in both directions. The eutectic point is where the liquidus curves from the two pure components intersect, forming the absolute minimum melting temperature for any composition in that binary system — a property exploited in soldering, brazing, and casting alloys.
What is the Gibbs phase rule and what does it say about the eutectic point?
The Gibbs phase rule states F = C − P + 2, where F is degrees of freedom, C is the number of components, and P is the number of phases. For a binary alloy (C = 2) at constant pressure: in a single-phase region P = 1 so F = 2 (T and x are both free); in the two-phase mushy zone P = 2 so F = 1 (fixing T fixes both x_L and x_S); at the eutectic point P = 3 so F = 0 — the system is invariant, meaning T and both compositions are completely fixed.
A hypoeutectic alloy (composition to the left of the eutectic) first crosses the liquidus on cooling, nucleating primary alpha-phase crystals. As temperature falls, the remaining liquid becomes progressively richer in component B, tracking the liquidus curve toward the eutectic composition. When the temperature reaches the eutectic isotherm, the remaining liquid — now at the eutectic composition — solidifies simultaneously into a fine lamellar mixture of alpha and beta phases.
Classic examples include lead-tin (Pb-Sn, eutectic at 61.9 wt% Sn, 183 °C — the backbone of traditional electronics solder), silver-copper (Ag-Cu, eutectic at 28.1 wt% Cu, 779 °C — used in silver brazing filler), and bismuth-tin (Bi-Sn, eutectic at 57 wt% Bi, 139 °C — a low-temperature solder used in thermosensitive applications). Modern lead-free SAC305 solder (Sn-Ag-Cu) is a ternary near-eutectic.
Slow equilibrium cooling gives atoms time to diffuse, producing coarse lamellar microstructure with alternating wide plates of alpha and beta. Rapid quenching suppresses diffusion, producing fine lamellar or even amorphous (glassy) structures. Very fast quenching can create supersaturated solid solutions — the basis of rapidly solidified aluminium alloys used in aircraft, where the fine microstructure dramatically improves strength and corrosion resistance.
A hypereutectic alloy has a composition to the right of the eutectic point (more of component B). On cooling it first precipitates primary beta-phase crystals while the liquid moves toward the eutectic composition. Both hypoeutectic and hypereutectic alloys ultimately produce eutectic microstructure in the remaining liquid — the difference is only in which primary phase precipitates first and in what proportion it appears in the final microstructure.
In an isomorphous system (such as copper-nickel) the two components are completely miscible in both liquid and solid phases, so there is no two-solid region and no eutectic point — only a lens-shaped two-phase region between the liquidus and solidus curves. The eutectic type shown here has limited solid solubility; the two components can only partially dissolve in each other, leading to the characteristic V-shaped liquidus and the eutectic reaction.
Yes. Move the eutectic composition slider (x_E) away from 50% to create an asymmetric system where the eutectic lies closer to one pure component. This mimics most real eutectic systems: the Pb-Sn eutectic at 61.9 wt% Sn is strongly asymmetric, as is the Al-Si eutectic at 12.6 wt% Si that is the basis of cast aluminium alloys used in automotive engine blocks.
The orange dashed line traces how the state of a single alloy (at a composition you can set by clicking) evolves as it cools from above the liquidus. In the liquid region the path descends vertically. On crossing the liquidus, the composition of the remaining liquid follows the liquidus curve (moving toward the eutectic), while the overall composition stays fixed — the path therefore appears to bend. At the eutectic isotherm the remaining liquid solidifies completely at constant temperature.