Alloying (continued)

Tie and Lever Rules
Simply by looking at a phase diagram it is possible to tell what phase or phases an alloy will have at a given temperature. But, it is also possible to get quantitative information from the diagram. Consider the alloy at the temperature shown on the phase diagram. It is easy to see that at this temperature, it is a mixture of alpha and liquid phases. Using a tie line it is also possible to determine the composition of the phases at this temperature. A tie line is an isothermal (constant temperature) line drawn through the alloy's position on the phase diagram when it is in a two phase field. The points where the ends of the tie line intersect the two adjacent solubility curves indicate the compositions of the two phases that exist in equilibrium at this temperature. In this example, the tie line shows that the alpha phase is 5.2%B and the liquid phase is 34.5%B at this temperature. It is important to keep in mind that the tie rule addresses the determination of the compositions of the constituent phases within the sample and it does not address the overall chemical composition of the sample, which remains unchanged.

It is also possible to determine how much of each phase exists at the given temperature using the lever rule. It is important to know the amounts of each phase present because the properties of the alloy depend on the amount of each phase present. The lever rule uses the tie line and the basic scientific principle of the conservation of mass to determine the ratio of the two phases present. The tie-line gives the chemical compositions of each of the two phases, and the combined amounts of these two compositions must add up to the alloy's overall composition (C_o), which is known. In other words, C_o must be composed of the appropriate amount of α at composition C_α and of liquid at C_liq. So basically, the proportions of the phases present are given by the relative lengths of the two sections of the tie line.

The fraction of alpha phase present is the given by the ratio of the C_o to C_liq portion of the tie line and the total length of the tie line (C_liq to C_α). Mathematically the relationships can be written as f_α << (C_liq – C_o)/(C_liq - C_α). The fraction of liquid phase present is given by the ratio of the C_o to C_α portion of the tie line and the total length of the tie line (C_liq to C_α). Mathematically this relationships can be written as f_liq << (C_o - C_α)/(C_liq - C_α). Of course, the two values must total to equal one.

Note that the right side of the tie line gives the proportion of the phase on the left (α phase in this example) and left side of the tie line gives the proportion of the phase to the right (liquid phase in this example). It is easy to keep this relationship straight by simply considering what the ratio would be near one of the tie line intersect points. For example, if C_o were near the liquidus line the ratio of the liquid section of the line to the total length of the line will be nearly one.