leverrule

The lever rule is a principle used in phase diagrams to determine the relative amounts of two coexisting phases in a two-phase region. It is commonly applied to binary systems, where the compositions of the two phases at a given temperature are fixed by the phase boundaries, and the overall composition lies on a horizontal tie-line between them.

In a binary phase diagram, within a two-phase field such as α + β, the compositions of the two phases are denoted Cα and Cβ. The overall alloy composition C0 lies on the tie-line between Cα and Cβ. The fractions of the two phases are given by the lever rule: Fα = (Cβ − C0) / (Cβ − Cα) and Fβ = (C0 − Cα) / (Cβ − Cα). These fractions are mass (weight) fractions; converting to volume fractions requires accounting for the phase densities.

The lever rule derives from mass conservation along the tie-line and the geometry of the diagram, with the ratios corresponding to the lengths of the lever segments between the overall composition and the phase boundaries. It is widely used to interpret solidification and phase separation in eutectic, peritectic, and miscibility gap systems, and applies broadly to two-phase equilibria in binary diagrams. While the concept extends to more complex systems, the simple lever rule strictly applies only when exactly two phases are present in equilibrium.

Note that the rule yields mass fractions. If volume fractions are needed, densities of the phases must be considered to convert between mass and volume.