Gibbs Phase Rule & Solubility
A Phase is a homogeneous, physically distinct, and mechanically separable portion of a material. In metallurgy, pressure is usually constant (1 atm), so we use the condensed Gibbs Phase Rule to determine the degrees of freedom ($F$) for a system at equilibrium.
The Condensed Phase Rule
$$F = C - P + 1$$
- $C =$ Number of Components (e.g., Fe and C $\rightarrow 2$)
- $P =$ Number of Phases present (e.g., Liquid + Solid $\rightarrow 2$)
- $F =$ Degrees of Freedom (variables like Temp/Comp you can change independently)
Key Boundaries
- Liquidus Line: The temperature above which the material is completely liquid.
- Solidus Line: The temperature below which the material is completely solid.
- Solvus Line: The limit of solid solubility (where a single solid phase separates into two solid phases).
The Binary Eutectic System
Some alloys (like Lead-Tin solders or Cast Irons) feature an invariant point where three phases coexist in equilibrium. This is the Eutectic Point.
Eutectic Reaction (Cooling)
$$L \xrightarrow{\text{Cooling}} \alpha + \beta$$
- ➤ At the eutectic composition (lowest melting point), the liquid transforms instantly into a two-phase solid.
- ➤ Because atoms must partition into A-rich ($\alpha$) and B-rich ($\beta$) phases rapidly, it typically forms a lamellar (layered/striped) microstructure.
- ➤ Applying the Phase Rule at the eutectic point ($P=3$, $C=2$): $F = 2 - 3 + 1 = 0$. The temperature and composition are strictly fixed!
The Lever Rule (Mass Balances)
If we are inside a two-phase region (e.g., $L + \alpha$), we need to know how much of the material is Liquid and how much is Solid $\alpha$. We draw a horizontal isotherm (tie line) and use the inverse lever rule.
Given alloy composition $C_0$, Liquidus boundary $C_L$, and Solidus boundary $C_S$:
$$W_{Liquid} = \frac{C_S - C_0}{C_S - C_L}$$
$$W_{Solid (\alpha)} = \frac{C_0 - C_L}{C_S - C_L}$$
Rule of thumb: To find the fraction of a phase, take the length of the tie-line on the opposite side of the fulcrum ($C_0$), divided by the total tie-line length.
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