MAT-05

Lever rule

Wα = (Cβ − C0)/(Cβ − Cα) for a two-phase alloy.

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Phase diagramsGibbs phase rule

Governing equation

W_\alpha=\dfrac{C_\beta-C_0}{C_\beta-C_\alpha},\quad W_\beta=1-W_\alpha

$C_0 \in \left[0, 100\right]$

where

C_0: Alloy composition (wt%)
C_\alpha: α composition (wt%)
C_\beta: β composition (wt%)
W_\alpha: α fraction (—)
W_\beta: β fraction (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

W_\alpha=\dfrac{C_\beta-C_0}{C_\beta-C_\alpha},\quad W_\beta=1-W_\alpha

$C_0 \in \left[0, 100\right]$

$C_0 = 0$ $W_\alpha = 1.143$

Along the curve

②

Domain of the variable — blue

C_0_{\min} = 0,\quad C_0_{\max} = 100

$C_0 \in \left[0, 100\right]$

$C_0 = 0$ $C_0 = 0$

Along the curve

③

Response curve — green

W_\alpha=\dfrac{C_\beta-C_0}{C_\beta-C_\alpha},\quad W_\beta=1-W_\alpha

$C_0 \in \left[0, 100\right]$

$C_0 = 0$ $W_\alpha = 1.143$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}W_\alpha}{\mathrm{d}C_0}

$C_0 \in \left[0, 100\right]$

$C_0 = 0.629$ $W_\alpha^{\prime} = -0.014$

Along the curve

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-05 — Lever rule) is the form associated with Gibbs phase rule. Working symbols: $C_0$ , $C_\alpha$ , $C_\beta$ $\rightarrow$ $W_\alpha$ , $W_\beta$ . Mass conservation on a tie-line of a binary phase diagram is a mechanical lever: the phase fractions are inverse length ratios.

Purpose

Purpose: compute

W_\alpha

W_\beta

from

C_0

C_\alpha

C_\beta

in Materials via

W_\alpha=\dfrac{C_\beta-C_0}{C_\beta-C_\alpha},\quad W_\beta=1-W_\alpha

Wα = (Cβ − C0)/(Cβ − Cα) for a two-phase alloy. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

C_0 = 40.000\,\mathrm{wt%}

C_\alpha = 10.000\,\mathrm{wt%}

C_\beta = 80.000\,\mathrm{wt%}

, the governing relation

W_\alpha=\dfrac{C_\beta-C_0}{C_\beta-C_\alpha},\quad W_\beta=1-W_\alpha

yields

W_\alpha = 0.571\,\mathrm{—}

W_\beta = 0.429\,\mathrm{—}

. Binary isomorphous or eutectic tie-line, mass %. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Alloy composition C_040 wt%α composition C_\alpha10 wt%β composition C_\beta80 wt%

Outputs

α fraction W_\alpha0.571 —
β fraction W_\beta0.429 —

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Narration of this film

Binary isomorphous or eutectic tie-line, mass %.

Mass conservation on a tie-line of a binary phase diagram is a mechanical lever: the phase fractions are inverse length ratios.

Reading speed

Watch on YouTube