Introduction to Phase Diagrams
Phase diagrams are crucial in understanding how materials behave during phase transitions. They provide a visual representation of the equilibrium states of materials at different temperatures and compositions. One of the simplest and most common binary phase diagrams is the copper-nickel (Cu-Ni) system, which exhibits complete solid solubility between the two components.
An essential tool for analyzing phase diagrams is the lever rule, which helps determine the proportion of phases present in a system at equilibrium. In this article, we’ll explore the use of the lever rule in the Cu-Ni binary alloy system.
The Cu-Ni Alloy System
The Cu-Ni alloy system is characterized by a single-phase field (α, which is a solid solution) and a two-phase region (L + α, a mixture of liquid and solid). Due to complete solubility between Cu and Ni, the Cu-Ni phase diagram has no intermediate compounds or eutectic points, making it ideal for studying phase transformation concepts such as the lever rule.
Cu-Ni Phase Diagram
Lever Rule Overview
The lever rule is a graphical method used in phase diagrams to determine the relative quantities of phases in a two-phase region. It is based on the mass balance between the two phases at equilibrium and uses the analogy of a simple lever (like a seesaw) to explain phase proportions.
Mathematically, the lever rule is expressed as:
Lever Rule –> WL*R = Wα*S
The lever rule is particularly useful when analyzing the two-phase region (L + α) of the Cu-Ni phase diagram, where both solid and liquid phases coexist.
Case Study: Applying the Lever Rule with R and S in the Cu-Ni System
Let’s apply this analogy to the Cu-Ni system with an alloy composition of 35 wt%Ni and 65 wt% Cu at a temperature of 1250°C.
Step 1: Locate the Alloy Composition and Temperature
From the Cu-Ni phase diagram, locate the point where the alloy composition (C0), 35 wt% Ni intersects with the 1250 °C temperature line. This point falls within the two-phase region (L + α), meaning that both liquid and solid phases are present.
Step 2: Identify the Phase Compositions
Draw a horizontal line (isotherm) at 1250 °C to the liquidus and solidus lines:
- The liquid phase composition at 1250°C (CL) is about 31 wt% of Ni.
- The solid phase composition at 1300°C (Cα) is about 42 wt% of Cu.
Step 3: Calculate the Phase Fractions Using Lever Rule
Now, use the lever rule to find the fraction of liquid and solid phases:
- R = C0-CL = 35-31 = 3
- S = Cα-C0 = 42-31 = 11
Now, calculate the phase fractions:
Wα = R/(R+S) = 3/(3+11) = 0.214
WL = S/(R+S) = 11/(3+11) = 0.785
Thus, at 1250°C, the alloy consists of 78.5% liquid and 21.4% solid phases.
Implications of the Lever Rule in Cu-Ni Composition
From the calculation above, using a tie line, we can also determine the composition of each phase at point B represented in the following microstructure schematic representation:
