Lever rule

The lever rule is a rule used to determine the mole fraction (x_i) or the mass fraction (w_i) of each phase of a binary equilibrium phase diagram. It can be used to determine the fraction of liquid and solid phases for a given binary composition and temperature that is between the liquidus and solidus line.[1]

In an alloy or a mixture with two phases, α and β, which themselves contain two elements, A and B, the lever rule states that the mass fraction of the α phase is

${\displaystyle w^{\alpha }={\frac {w_{\rm {B}}-w_{\rm {B}}^{\beta }}{w_{\rm {B}}^{\alpha }-w_{\rm {B}}^{\beta }}}}$

where

• ${\displaystyle w_{\rm {B}}^{\alpha }}$ is the mass fraction of element B in the α phase
• ${\displaystyle w_{\rm {B}}^{\beta }}$ is the mass fraction of element B in the β phase
• ${\displaystyle w_{\rm {B}}}$ is the mass fraction of element B in the entire alloy or mixture

all at some fixed temperature or pressure.

Derivation

Suppose an alloy at an equilibrium temperature T consists of ${\displaystyle w_{\rm {B}}}$ mass fraction of element B. Suppose also that at temperature T the alloy consists of two phases, α and β, for which the α consists of ${\displaystyle w_{\rm {B}}^{\alpha }}$, and β consists of ${\displaystyle w_{\rm {B}}^{\beta }}$. Let the mass of the α phase in the alloy be ${\displaystyle m_{\alpha }}$ so that the mass of the β phase is ${\displaystyle m^{\beta }=m-m^{\alpha }}$, where ${\displaystyle m}$ is the total mass of the alloy.

By definition, then, the mass of element B in the α phase is ${\displaystyle m_{\rm {B}}^{\alpha }=w_{\rm {B}}^{\alpha }m^{\alpha }}$, while the mass of element B in the β phase is ${\displaystyle m_{\rm {B}}^{\beta }=w_{\rm {B}}^{\beta }\left(m-m^{\alpha }\right)}$. Together these two quantities sum to the total mass of element B in the alloy, which is given by ${\displaystyle m_{\rm {B}}=w_{\rm {B}}m}$. Therefore,

${\displaystyle w_{\rm {B}}m=m_{\rm {B}}=m_{\rm {B}}^{\alpha }+m_{\rm {B}}^{\beta }=w_{\rm {B}}^{\alpha }m^{\alpha }+w_{\rm {B}}^{\beta }\left(m-m^{\alpha }\right)}$

By rearranging, one finds that

${\displaystyle w^{\alpha }\equiv {\frac {m^{\alpha }}{m}}={\frac {w_{\rm {B}}-w_{\rm {B}}^{\beta }}{w_{\rm {B}}^{\alpha }-w_{\rm {B}}^{\beta }}}}$

This final fraction is the mass fraction of the α phase in the alloy.

Calculations

A phase diagram with tie line (LS) for a binary isomorphous system. The x dimension defines the mass fraction of elements A and B.

Binary phase diagrams

Before any calculations can be made, a tie line is drawn on the phase diagram to determine the mass fraction of each element; on the phase diagram to the right it is line segment LS. This tie line is drawn horizontally at the composition's temperature from one phase to another (here the liquid to the solid). The mass fraction of element B at the liquidus is given by w_B^l (represented as w_l in this diagram) and the mass fraction of element B at the solidus is given by w_B^s (represented as w_s in this diagram). The mass fraction of solid and liquid can then be calculated using the following lever rule equations:[1]

${\displaystyle w^{\rm {s}}={\frac {w_{\rm {B}}-w_{\rm {B}}^{\rm {l}}}{w_{\rm {B}}^{\rm {s}}-w_{\rm {B}}^{\rm {l}}}}}$

${\displaystyle w^{\rm {l}}={\frac {w_{\rm {B}}^{\rm {s}}-w_{\rm {B}}}{w_{\rm {B}}^{\rm {s}}-w_{\rm {B}}^{\rm {l}}}}}$

where w_B is the mass fraction of element B for the given composition (represented as w_o in this diagram).

The numerator of each equation is the original composition that we are interested in is +/- the opposite lever arm. That is if you want the mass fraction of solid then take the difference between the liquid composition and the original composition. And then the denominator is the overall length of the arm so the difference between the solid and liquid compositions. If you're having difficulty realising why this is so, try visualising the composition when w_o approaches w_l. Then the liquid concentration will start increasing.

Eutectic phase diagrams

Tie line in the Alpha plus Liquid two phase region

There is now more than one two-phase region. The tie line drawn is from the solid alpha to the liquid and by dropping a vertical line down at these points the mass fraction of each phase is directly read off the graph, that is the mass fraction in the x axis element. The same equations can be used to find the mass fraction of alloy in each of the phases, i.e. w^l is the mass fraction of the whole sample in the liquid phase.[2]

References

1. Smith.h, William F. halkias; Hashemi, Javad (2006), Foundations of Materials Science and Engineering (4th ed.), McGraw-Hill, pp. 318–320, ISBN 0-07-295358-6.
2. Callister, William D.; Rethwisch, David (2009), Materials Science and Engineering An Introduction (8th ed.), Wiley, pp. 298–303, ISBN 978-0-470-41997-7.