Lever Rule

Ni-Cu phase diagram. Ni = A, while Cu = B in this blog. Phase 1 = liquid, Phase 2 =  α (solid phase).

One formula often used in phase diagram calculations is the so-called lever rule. It is a basic and interesting equation.

Consider an example of a binary alloy composed of atoms Ni and atoms Cu. (A ternary alloy has 3 elements.) It is possible for a NiCu alloy to have 2 phases, say phase 1 and phase 2.

A phase of a binary alloy is simply stated a distinct "character" of the alloy. If the alloy is heated to high temperature beyond its melting point, it will turn liquid. We say the alloy becomes a liquid phase when heated above its melting point. When it is cooled below the melting point, the alloy will return solid. A solid is another phase, different from the liquid phase. So, we could assign phase 1 = liquid phase, while phase 2 = solid phase.

A phase diagram then is a diagram that tells us what phase(s) the NiCu (binary) alloy would like to adopt at certain temperature, pressure, and amounts of Ni and Cu.

It is possible for an AB (binary) alloy to have more than one solid phase. Why? Because the diatomic A-A bond strength is different from the B-B bond strength, and the A-B bond strength can be different from both. The relative magnitudes of A-A, A-B, and B-B bond strengths motivate the binary alloy to have more than one solid phase. We can figure out the maximum number of phases using Gibbs' phase rule; for a binary alloy it is 4.

Lever rule is applicable for a region in a phase diagram, where 2 phases occur simultaneously. These 2 phases may be liquid phase coexisting with one solid phase, or two solid phases. Level rule tells us the total concentration of atom B given the concentration of atom B in both phases. It also tells us the total concentration of A if we know the concentration of A in both phases.

For phase 1, atoms A weigh m_A1 while atoms B weigh m_B1. The concentration of B in phase 1 is c_B1 = m_B1 ⁄ (m_A1 + m_B1). For phase 2, atoms A weigh m_A2 while atoms B weigh m_B2. The concentration of B in phase 2 is c_B2 = m_B2 ⁄ (m_A2 + m_B2).

Now, the total concentration of B is not equal to the sum of the individual concentrations of B in phase 1 and 2,

c_B ≠ c_B1 + c_B2,

since the correct expression is

c_B = (m_B1 + m_B2) ⁄ (m_A1 + m_B1 + m_A2 + m_B2).

Rarely though, each mass of A and B in both phases are all known. c_B is, however, often known beforehand. So, the correct expression above is usually solved for the variables on the right hand side of the expression. This is when we need the lever rule.

The lever rule relies on another fractional amount definition W₁ and W₂. It states that the total concentration of B must be equal to

c_B = W₁ c_B1 + W₂ c_B2,

if the sum of these fractional amounts is 1,

W₁ + W₂ = 1.

Solving for W₁ and W₂ using both equations we get the lever rule

W₂ = ( c_B − c_B1 )/( c_B2 − c_B1 )

and

W₁ = ( c_B2 − c_B )/( c_B2 − c_B1 ).

What do these W₂ and W₁ mean? When you substitute the values for these compositions, you'll find that W₂ corresponds to the mass of fraction of phase 2 with respect to the total mass (i.e., the mass sum of phase 1 and phase 2). W₁ is thus the mass fraction of phase 1.

What is interesting is that the lever rule does not solve the equation

c_B = (m_B1 + m_B2) ⁄ (m_A1 + m_B1 + m_A2 + m_B2)

directly. In practice, c_B is usually known, but some of the mass quantities on the right hand side are not. The lever rule makes a consistent connection to the c_B formula. More importantly, the lever rule is based on an assumption that the concentration c_B can be written as a linear combination of the respective concentrations of B atoms in the two coexisting phases. I am not aware of atomistic proof for this linear combination assumption.