Understanding binary phase diagrams, lever rule, and phase transformations
A phase diagram is a graphical representation showing the equilibrium phases present in a material system as a function of temperature, pressure, and composition. For materials science, we typically work with temperature-composition diagrams at constant pressure (1 atm).
Phase diagrams answer critical questions:
Phase: A physically distinct and chemically homogeneous portion of a system with definite boundaries.
Component: Chemical species that make up the system (e.g., Cu and Ni in a Cu-Ni alloy).
Equilibrium: The state where no further changes occur with time at constant temperature and pressure.
Solubility Limit: Maximum concentration of a solute that can dissolve in a solvent while maintaining a single phase.
The Gibbs phase rule relates the number of phases, components, and degrees of freedom in a system at equilibrium:
Where: $F$ = degrees of freedom, $C$ = number of components, $P$ = number of phases
For systems at constant pressure (most materials applications):
Degrees of Freedom (F): Number of variables (temperature, pressure, composition) that can be changed independently without changing the number of phases in equilibrium.
For pure water (C = 1) at the triple point where ice, liquid water, and vapor coexist (P = 3):
$$F = 1 - 3 + 2 = 0$$
Zero degrees of freedom means the triple point occurs at exactly one temperature and pressure. You cannot change T or P without losing one of the phases.
The simplest type of binary phase diagram occurs when two metals are completely soluble in both liquid and solid states. The Cu-Ni system is a classic example.
At any point in the two-phase region:
The lever rule allows us to calculate the relative amounts (fractions) of phases present in a two-phase region. Think of it like a physical lever or seesaw.
Weight fraction of liquid phase:
$$W_L = \frac{C_S - C_0}{C_S - C_L}$$Weight fraction of solid phase:
$$W_S = \frac{C_0 - C_L}{C_S - C_L}$$Where: $C_0$ = overall composition, $C_L$ = liquid composition, $C_S$ = solid composition
A Cu-40wt%Ni alloy at 1250°C. From the phase diagram:
Fraction of liquid:
$$W_L = \frac{45 - 40}{45 - 32} = \frac{5}{13} = 0.385 = 38.5\%$$Fraction of solid:
$$W_S = \frac{40 - 32}{45 - 32} = \frac{8}{13} = 0.615 = 61.5\%$$Notice: $W_L + W_S = 1$ (as it must)
The lever rule works like a seesaw. The fraction of one phase equals the length of the lever arm on the opposite side divided by the total lever length. That's why the numerator for liquid fraction uses the solid composition ($C_S$).
A eutectic system contains a special point called the eutectic point, where a liquid transforms directly into two solid phases upon cooling. The Pb-Sn system is a common example used in soldering.
| Region | Phases Present | Description |
|---|---|---|
| L | Liquid | Single liquid phase |
| α | Alpha solid solution | Solute B dissolved in solvent A |
| β | Beta solid solution | Solute A dissolved in solvent B |
| α + L | Alpha + Liquid | Two-phase region |
| β + L | Beta + Liquid | Two-phase region |
| α + β | Alpha + Beta | Two solid phases |
Invariant reactions occur at specific temperatures and compositions where three phases coexist. At these points, F = 0 (no degrees of freedom).
Eutectic: One liquid becomes two solids upon cooling
$$L \rightarrow \alpha + \beta$$Example: Pb-Sn at 183°C
Eutectoid: One solid becomes two different solids
$$\gamma \rightarrow \alpha + \beta$$Example: Fe-C (austenite to ferrite + cementite) at 727°C
Peritectic: One solid plus liquid becomes a different solid
$$\alpha + L \rightarrow \beta$$Example: Pt-Ag system
The Fe-C phase diagram is one of the most important in materials science because it governs the heat treatment of steels. We focus on the Fe-Fe₃C (iron-cementite) metastable diagram.
The phases shown on a phase diagram translate directly into microstructures you can observe under a microscope. Understanding how to predict microstructure from a phase diagram is essential for materials design.
Cooling from austenite region to room temperature:
Using the lever rule at just above 727°C:
Fraction of proeutectoid ferrite = $(0.76 - 0.4)/(0.76 - 0.022) \approx 49\%$
Fraction of pearlite = $(0.4 - 0.022)/(0.76 - 0.022) \approx 51\%$
Proeutectoid: Phase that forms before the eutectoid reaction (proeutectoid ferrite or proeutectoid cementite).
Primary Phase: First solid phase to form from the liquid during solidification.
Pearlite: Lamellar structure of alternating ferrite and cementite layers, forms from eutectoid reaction.
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