Binary Phase Diagram & Lever Rule Calculator

The lever rule calculates the mass or mole fractions of two coexisting phases (α and Liquid) at equilibrium for a given overall composition $x_0$ and temperature T. It's a mass balance derived from the phase diagram.

$f_\alpha, f_L$: Fraction of α (solid) and Liquid phase.
$x_0$: Overall composition of the alloy (mol% B).
$x_L, x_\alpha$: Composition of the Liquid and α phase at the temperature T, read from the liquidus and solidus lines.

The Scheil equation models non-equilibrium solidification, where diffusion in the solid is negligible. It predicts how solute builds up in the remaining liquid, leading to segregation.

$C_S$: Composition of the solid forming at fraction $f_S$.
$C_0$: Initial liquid composition ($x_0$).
$k$: Partition coefficient. If $k\lt 1$, solute is rejected into the liquid, enriching it as solidification proceeds.

Alloy Design & Processing: Engineers use these diagrams to select heat treatment temperatures (like for annealing or homogenization) and to predict the melting range of an alloy, which is critical for casting and welding processes.

Casting & Solidification Simulation: The lever rule and Scheil equation are foundational inputs for CAE software that simulates casting. They help predict where shrinkage porosity or severe segregation (like "centerline segregation" in steel ingots) might occur, guiding the design of risers and chills.

Welding Metallurgy: The rapid heating and cooling in a weld's Heat-Affected Zone (HAZ) can be approximated using phase diagrams. Predicting the phases that form helps assess the risk of cracking or corrosion, crucial for welding aerospace or pipeline components.

Foundation for CALPHAD: Binary systems are the building blocks for the CALPHAD method used in sophisticated materials design software. It computationally extrapolates these principles to complex, multi-component alloys used in jet engines or battery materials.

When you start using this tool, there are a few common pitfalls to watch out for. First, be mindful of whether the overall composition is given in weight percent (wt%) or atomic percent (at%). The simulator input is typically mole fraction (similar to at%), but material specification sheets in practice are often written in wt%. For example, the eutectic point of an Al-Si alloy is about 12.6 wt% Si, but it's about 18.6 at% Si. Confusing these units will cause you to completely misplace the point on the phase diagram, so be careful.

Second, remember that the lever rule calculates for "equilibrium conditions". In actual casting, the cooling rate is fast, so the composition of the liquid and solid phases does not become uniform as calculated. Treat the equilibrium calculation result from the tool as an "ideal limit value." The trick is to compare it with the Scheil equation shown as a reference to intuitively grasp how much reality deviates. For instance, for an alloy with a partition coefficient k=0.5 at a solid fraction $f_S$ of 0.5, the solid composition $C_S$ returns to $C_0$ under equilibrium. However, under non-equilibrium (Scheil equation), it remains significantly lower at $C_S = 0.5 C_0 (1-0.5)^{-0.5} \approx 0.707 C_0$. This is the essence of segregation.

Third, understand the limitations of "binary systems". Most practical alloys, like Fe-Cr-Ni, are ternary or more complex. The principles you learn with this tool are foundational, but in reality, the presence of a third element can significantly shift liquidus lines and eutectic temperatures. It's risky to directly apply binary system calculation results to practical work; think of this strictly as a "training tool for understanding phase equilibrium concepts."