Lever Rule Ternary Phase Diagram

Microstructural Changes

Refer to phase diagrams of preceding article. Liquid phase first changes to solid-liquid phase, and finally to solid phase during cooling. If variation in the rate of cooling is slow to fast, say 200°C per hour to 200°C per second, different microstructures will develop.

If we consider the slow rate of cooling which is a general procedure, the likely microstructural changes in overall composition of binary system will be shown by Figure 1(a – f).

Effect of Cooling on Microstructure Formation: Microstructure of single liquid phase above the liquidus is shown in Figure 1(b). On cooling we obtain two phase co-existing solid-liquid phase, Figure 1(d). The solid crystals float on the liquid phase.

On further cooling, the system crosses the solidus and crystallizes into a poly-crystalline solid, Figure 1(d). The above changes are true for the case when two components have almost similar melting points.

 However for eutectic solid solutions, Figure 1(c), the 13 crystals float in the liquid of eutectic composition below eutectic temperature.

Coring and dendritic structure: The eutectic composition of Pb-Sn alloy on solidification is shown by Figure (e). The α-crystals are embedded in the eutectic matrix. The compositional adjustments are brought by the atoms moving across the liquid-crystal boundary.

Such adjustments are a rare phenomenon in solids due to slow atomic movement. This induces inhomogeneity in composition that is called Coring.

In cubic crystals, the preferential grow occurs along <100> directions. During initial solidification, a tree like structure develops, Figure 1(f). This is called dendritic structure.

Lever Rule

Lever rule is employed to obtain relative amounts of co-existing phases of a binary system. For overall compositions Co lying on the tie-line in Figure 3 of preceding article, the compositions of the liquid and solid phases Cl and Cs remain the same. The tie-line is analogous to a lever arm of mass balance having its fulcrum at Co.

For arm to be horizontal i.e. the two components to be in phase equilibrium, the weight W to be hung at the ends of lever arm must be proportional to the length of the arm on two opposite sides of the fulcrum. The weight, here, corresponds to the amount of phase at that end.

lever rule ternary phase diagram

The relative amounts of solid and liquid phases at specified temperature and overall composition are determined by depiction of lever rule as shown in Figure. Weight fractions of liquid and solids are Wl and WS respectively on lever arm EFG whose fulcrum is at F. These are obtained as

Wl = FG/EG = (Cs – Co)/(Cs – Cl)

Ws = EF/EG = (Co – Cl)/(CsCl)

Non-applicability of Lever Rule: The lever rule is not applicable at eutectic or peritectic points as three phases exist in equilibrium at these temperatures. It can be applied just below or just above the invariant line. The lever rule may be used to calculate the following.

  1. The fraction of a proeutectic phase.
  2. The fraction of an eutectic mixture.
  3. The fraction of phase of an eutectic mixture.

In all the above cases, overall composition Co is always kept at the fulcrum. Refer to Figure 5 of preceding article, in which Co is the overall composition. The two ends of the lever arm correspond to phase boundary compositions of Cβe, and Ce where Ce is the average composition of eutectic mixture. We then obtain

Wproβ= (Co – Ce)/(Cβe – Ce)

and  Weut = (Cβe – Co)/(Cβe – Ce)

To obtain total β, which is sum of proeutectic β and the β eutectic mixture, we use

Wtotalβ= (Co – Cαe)/(Cβe – C αe)

Ternary Phase Diagram

These diagrams are drawn for three-component systems. According to Gibb’s phase rule, total number of phases can only be 5 when degree of freedom is zero. We obtain from the Equation 2 of preceding article that the total number of variables may be 12 in these diagrams.

A three-dimensional plot is required to show the ternary phase diagram. Use of three-components alloy system in engineering applications is very common. For example, German silver is an alloy of zinc, copper and nickel. It contains about 30% Zn, 55% Cu, and rest is nickel.

Another alloy is high speed steel that consists of mainly three components viz. about 20% tungsten, and 7% chromium in steel.

Pseudo Ternary Diagram

A two-dimensional depiction of a ternary phase diagram is shown in Figure 3. The sides of the equilateral triangle represents three binary alloy systems A and B, B and C; and C and A. The three components A, B and C are concentrated at apex of the triangle. There are total 4 variables associated with component A. These variables include two compositional variables between A and B; A and C; temperature and pressure.

Similarly components B and C each have 4 variables associated with them. In this way the number of total variables in the system is 12 as stated above. The side length α of the triangle is taken as 100%. Temperature (not shown in the diagram) axis T is plotted at right angles to the plane of triangle.

Designation of a ternary phase on a 2-dimensional diagram: Any point lying within the triangle specifies the composition of ternary alloy. To illustrate let 0 be a point within the triangle. Lines om, on and op are drawn parallel to the three sides. The geometrical rule that follows is

on + om + op = α = 100%

In Figure 3, on indicates 20% of component B, om specifies 40% of component A, and op depicts 40% of component C. Thus the above equation signifies the percentage composition of components in ternary alloy. The melting point of ternary eutectic is generally lower than the melting point of binary eutectic.

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