Lever Rule

The lever rule has no discoverer and needs none. It is a mass balance, and it says only this: the solute in the alloy has to be somewhere. If the alloy sits in a two-phase field, and the two phases have compositions read off the ends of a tie line, then the fractions of the two phases are fixed by the requirement that their weighted average composition equals the alloy's own. Two equations — total mass, and total solute — three unknowns, one of which is known. There is nothing to approximate and nothing to fit.

The mechanical analogy that names it is exact. Put the alloy composition C₀ at the fulcrum of a lever whose ends are at C_α and C_β, and hang the phase fractions from the ends. The lever balances. And that is where the classic mistake lives: the fraction of α is proportional to the arm running from C₀ to the OPPOSITE end, not to the short arm on α's own side, exactly as the longer arm of a physical lever carries the smaller weight. The sanity check that catches it every time is that the phase whose composition is closer to the alloy composition must be the more abundant one.

Two conditions bound the whole exercise. The tie line belongs to ONE TEMPERATURE. Move the temperature and all three compositions move, so a phase fraction quoted without the temperature it was read at is not an answer. Read all three numbers off the same horizontal line on the diagram, and read them off the phase boundaries rather than estimating. And the alloy composition must lie BETWEEN the two phase compositions; if it does not, the alloy is single-phase at that temperature and the rule does not apply — which is usually a sign the tie line was taken at the wrong temperature.

The larger limitation is that the lever rule knows nothing about time. It is a statement of equilibrium, so it tells you what the alloy would contain if it were held long enough for diffusion to finish everywhere. Real cooling is almost never long enough. That gap is not a small correction to be waved away: it is why a cast alloy has cored dendrites with a composition gradient across every arm, and why it contains eutectic that the phase diagram says should not exist at that composition at all. The Scheil–Gulliver equation on this site handles the opposite extreme — no diffusion in the solid whatever — and the truth of any real casting lies between the two, nearer Scheil for substitutional solutes and nearer equilibrium for interstitials like carbon that diffuse fast enough in the solid to homogenise as they go.

The rule does more than count phases in a binary. Applied at the eutectoid in the iron–carbon system it gives the classic first-year results: a 0.4 % carbon steel just below the eutectoid is about half proeutectoid ferrite and half pearlite, and the pearlite itself is about 88 % ferrite and 12 % cementite by the same arithmetic applied to a second tie line. Applied to the liquid and solid in a freezing range, it gives the equilibrium fraction solid at any temperature between liquidus and solidus, which is the starting point for casting simulation. It works identically in three-phase and multicomponent systems, though the tie line becomes a tie triangle or a tie simplex and the arithmetic gets heavier — which is what CALPHAD software is for.

One quiet caveat for people reading micrographs. Point counting on a random section gives an AREA fraction, which by Delesse's principle equals the volume fraction. The lever rule as usually written is a MASS balance. If the two phases differ appreciably in density — cementite against ferrite, an intermetallic against an aluminium matrix — convert before comparing, or the comparison is biased towards the heavier phase.