Reading a Binary Phase Diagram: What the Lever Rule Actually Tells You

One diagram, two questions

A binary phase diagram answers two questions at once for an alloy cooling from the melt: what phases are present at a given temperature, and how much of each. The first question is qualitative — read straight off the diagram by seeing which region a temperature-composition point falls into. The second question is quantitative, and it's where most students stop trusting their intuition and start reaching for the lever rule, because "how much liquid is left at 1350°C" isn't something you can eyeball reliably even when you can see the diagram clearly.

Building the diagram from two melting points

An isomorphous binary system — the simple case, where the two components are fully soluble in each other in both solid and liquid states, like copper-nickel — is bounded by the two pure-component melting points: Component A melting point T_mA and Component B melting point T_mB. Between them sit two curves: the liquidus (above which everything is liquid) and the solidus (below which everything is solid). In between the two curves is the two-phase region, where liquid and solid coexist — and the width of that two-phase region, the gap between liquidus and solidus at a given composition, is controlled by what the tool calls the solidification-gap coefficient κ, which governs how bowed apart the two curves are. A small κ gives a narrow two-phase region and rapid solidification over a narrow temperature window; a larger κ spreads solidification over a wider range, which matters practically because a wide mushy zone tends to promote more pronounced coring (compositional segregation within individual grains) during casting.

For eutectic and peritectic systems, the diagram has a different shape — the liquidus curves slope down from each pure-component melting point to meet at an invariant point (the eutectic point, where liquid transforms directly to two solid phases at a fixed composition and temperature) rather than merging smoothly into a single solidus. The tool supports all three diagram types — isomorphous, eutectic, and peritectic — since the underlying lever-rule math for reading phase fractions at a given point is the same once you know which two phases bound the tie line, but the diagram topology, and therefore which regions exist at which compositions, differs.

The lever rule itself

Pick an Alloy composition C₀ (mol%B) and a Current temperature T (scrub) that lands inside the two-phase region. Draw a horizontal tie line at that temperature; it intersects the solidus at the solid-phase composition and the liquidus at the liquid-phase composition. The lever rule says the fraction of each phase is inversely proportional to how far the overall composition sits from that phase's composition along the tie line — like a literal lever balanced at C₀, with the phase fractions acting as the weights needed at each end to balance it:

f_L    = (C_alpha - C_0) / (C_alpha - C_L)
f_alpha = (C_0 - C_L) / (C_alpha - C_L)

where f_L is liquid fraction, f_alpha is solid fraction, C_L is the liquid-phase composition read off the liquidus, and C_alpha is the solid-phase composition read off the solidus, both at the current temperature. Note the fractions are weighted by the opposite end of the tie line — this is the detail that trips people up on a first pass, since it feels backwards until you picture the literal lever-and-fulcrum analogy the rule is named after.

A worked example

Take a copper-nickel-like isomorphous system with Component A melting point T_mA = 1085°C (pure Cu) and Component B melting point T_mB = 1455°C (pure Ni), an Alloy composition C₀ of 35 mol% Ni, and suppose at Current temperature T = 1280°C the diagram (set by the chosen solidification-gap coefficient κ) reports a liquidus composition C_L = 28 mol% Ni and a solidus composition C_α = 44 mol% Ni — meaning at this temperature, the liquid phase present is leaner in nickel than the bulk alloy, and the solid phase is richer in it, which is the expected direction since nickel has the higher melting point and preferentially partitions into the first solid to form.

Applying the lever rule:

f_L = (44 - 35) / (44 - 28) = 9 / 16 = 0.5625
f_alpha = (35 - 28) / (44 - 28) = 7 / 16 = 0.4375

So at 1280°C this alloy is about 56% liquid and 44% solid by the lever-rule mass fractions — check: 0.5625 + 0.4375 = 1.0000, always a useful sanity check on a lever-rule result before trusting it. The tool reports this directly as Liquid Fraction f_L and Solid Fraction f_α alongside the Present Phase(s) label and the Liquid-phase composition C_L / Solid-phase composition C_α values the fractions were computed from.

Why the coring problem is really a lever-rule problem in disguise

Equilibrium cooling assumes solid-state diffusion has time to homogenize each grain's composition as it forms — an assumption that gets worse the faster you cool. Push the Cooling Rate up and there isn't time for the first-formed, nickel-rich solid to re-equilibrate with the increasingly nickel-lean liquid around it, so each grain ends up with a composition gradient frozen in from the sequence of solid compositions it passed through during freezing (nickel-rich core, copper-rich rim) rather than the single uniform composition the equilibrium lever rule predicts. This is coring, and it's a direct, practical consequence of the same tie-line geometry the lever rule uses — a wider solidus-liquidus gap (larger κ) combined with a higher cooling rate produces worse coring, because there's more compositional range to freeze in and less time to erase it by diffusion.

Where the simple lever rule breaks down

The lever rule as derived assumes complete mixing in the liquid and either complete or partial (Scheil-style, at the extreme) diffusion in the solid; real castings sit somewhere between the equilibrium lever rule and the no-diffusion-in-solid Scheil model depending on cooling rate and diffusivity, and matching a real casting's measured segregation to the equilibrium prediction alone will generally underestimate how cored the actual part is. It also assumes you've correctly identified which two phases bound the tie line — near an invariant point (eutectic or peritectic) in particular, the correct tie line changes character right at the boundary, and using the wrong one gives fractions that look plausible but are wrong.

Eutectic and peritectic systems: the lever rule doesn't change, the geometry does

Everything above assumed an isomorphous system with one continuous two-phase region between a single liquidus and a single solidus. Eutectic systems complicate the picture by adding a second, structurally different two-phase region on each side of the eutectic composition — liquid coexisting with one of the two terminal solid solutions, rather than with a single continuously-varying solid. The lever rule still applies within each of those regions using exactly the same ratio-of-distances logic, but which two phases bound the tie line changes depending on which side of the eutectic point the Alloy composition C₀ sits on, and right at the eutectic temperature itself, three phases coexist at once (liquid plus both solid solutions), which is where the simple two-phase lever rule stops applying directly and a different bookkeeping — comparing the alloy composition against the eutectic composition and the two solid solubility limits — takes over instead. Peritectic systems add a similar complication at the peritectic point, where a liquid and one solid phase react on cooling to form a second, different solid phase. In both cases the practical takeaway is the same: read the Present Phase(s) label the tool reports before trusting a lever-rule fraction, since the correct tie line — and therefore the correct C_L and C_α to plug in — depends on correctly identifying which two-phase field you're actually in.

What the cooling rate slider is really telling you about casting quality

Beyond just widening or narrowing predicted segregation, the interaction between solidification-gap coefficient κ and Cooling Rate maps fairly directly onto real foundry decisions. A casting process with inherently slow, controlled cooling — sand casting a large section, for instance — has more time for solid-state diffusion to smooth out the composition gradients the lever rule glosses over, so equilibrium lever-rule predictions track the actual as-cast structure reasonably well. A process with fast, uneven cooling — die casting, or the outer skin of a large ingot compared to its slow-cooling core — deviates further from the equilibrium picture, and as-cast coring in those regions can be severe enough that a post-cast homogenization heat treatment (holding well below the solidus for an extended time to let diffusion catch up) becomes a necessary processing step rather than an optional one. None of that changes the lever-rule arithmetic itself, but it's the reason the equilibrium diagram is treated as a starting point for alloy design rather than a literal prediction of what a fast-cooled casting will look like.

Try it yourself

Reading a phase diagram is one skill; trusting a lever-rule fraction at an arbitrary scrubbed temperature is another, and it's much faster to build that intuition by dragging the temperature through the two-phase region than by redrawing tie lines by hand. Try the binary phase diagram and lever rule calculator here across isomorphous, eutectic and peritectic systems. If wear and surface behavior in the solidified part matters for your application, the Archard wear tool is a reasonable next stop.