Lever-Rule

Stefan Bringuier

March 2, 2014

The essence of the lever-rule is that the fraction of each phase at a givencomposition is the ratio of the differences in composition weight percent (oratomic). Let us take for example a complete solubility system such as Si-Gewith two phases α and L, shown in Fig. 1. The total number of moles isgiven by:

nT = nL + nα (1)

Let us now select a composition corresponding to the weight fraction atXoGe, the total number of moles of Ge at this composition is:

XoGenT (2)

So that the that the number of moles of Ge is the sum of the number ofmoles in the α and L phases.

XoGenT = XL

GenL +XαGenα (3)

Normalizing Eq. 3 with respect to the total number of moles, yieldsEq. 4a which allows us to determine the fraction of each phase anywhere onthe phase diagram.

Lets look at the Si-Ge phase diagram again, specifically at the compo-sition with weight fraction XSi = 0.3 at 1100 ◦C (note the axis goes fromSi to Ge). Notice how this point falls in the two phase region, we thus caninvoke the lever-rule (Eq. 4a).

nTnT

=nL + nαnT

1 = ZL + Zα

XoGe = XL

GeZL +XαGeZα (4a)

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Figure 1: Si-Ge phase diagram1.

Before getting the values for the composition of the L and α from thephase diagram, let us rearrange Eq. 4a using Si weight fraction, to solve forZα:

ZL = 1 − Zα

XoSi = XL

Si (1 − Zα) +XαSiZα

Zα =XoSi −XL

Si

XαSi −XL

Si

(5a)

On the phase diagram we use the tie-line construction method to measurethe composition of the two phases. The tie-lines correspond to the composi-tion at which the the free energies are a minimum and satisfy the conditionsfor thermodynamic equilibrium. The tie-lines are always constructed be-tween phase-boundaries. This is shown by the red line, the tie-line willdepend on initial starting overall composition. For a value of Xo

Si = 0.3 thecorresponding XL

Si and XαSi are 0.18 and 0.49, respectively. Plugging these

numbers into Eq. 5a gives:

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Zα = 0.2926

The reader who is not familiar with the lever-rule will quickly notice thatthe fraction of α is not the same as the composition of Si. It maybe the casethat by incorrectly understanding the derivation of the lever-rule one mightinitially mistake or guess that then fraction of α is equal to composition ofSi.

References

[1] R.W. Olesinski and G.J. Abbaschian. The gesi (germanium-silicon) sys-tem. Bulletin of Alloy Phase Diagrams, 5(2):180–183, 1984.

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