Question 7 of 7: Phase Diagram

Question 7: Phase Diagram (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

[Figure not reproduced: Cu–Ag eutectic phase diagram (as printed on the exam), with the digitized 800°C tie line ($C_\alpha\approx7.5$, $C_L\approx67.0$ wt% Ag) and 700°C solvus readings ($4.6$, $94.4$ wt% Ag) overlaid in red. See the official exam paper.]

7.1 — Phases at 40wt%Cu–60wt%Ag, 800°C

Approach. Locate the 800°C isotherm; the alloy's overall composition (60 wt% Ag) is compared against the $\alpha$/($\alpha$+L) and ($\alpha$+L)/L boundary compositions at that temperature.

At 800°C the $\alpha$-phase (solidus) boundary reads $C_\alpha\approx7.5$ wt% Ag and the liquidus boundary reads $C_L\approx67.0$ wt% Ag. Since $C_\alpha \lt 60 \lt C_L$, the alloy sits inside the two-phase $\alpha+L$ field:

7.2 — Maximum Solubility at 700°C

Reading the two solvus branches at 700°C directly off the diagram:

• (a) Maximum solubility of Cu in Ag ($\beta$-phase solvus, G–H branch): the boundary reads $\approx94.4$ wt% Ag, i.e. $\boxed{\approx5.6\ \text{wt\% Cu dissolved in }\beta}$.
• (b) Maximum solubility of Ag in Cu ($\alpha$-phase solvus, B–C branch): the boundary reads $\boxed{\approx4.6\ \text{wt\% Ag dissolved in }\alpha}$.

Both solubilities are well below their eutectic-temperature maxima (8.0 and 8.8 wt%, respectively, at 779°C) — solid solubility falls as temperature drops, which is exactly the mechanism (retrograde solvus) that enables age/precipitation hardening in this and analogous alloy systems.

7.3 — The Eutectic Reaction

A eutectic reaction is the isothermal, invariant transformation of a liquid of one fixed composition directly into two distinct solid phases of two other fixed compositions, on cooling through the eutectic temperature: $L\rightleftharpoons\alpha+\beta$. For the Cu–Ag system: $$L(71.9\ \text{wt\%\,Ag})\ \xrightarrow[\text{cooling}]{779\,{}^{\circ}\text{C}}\ \alpha(8.0\ \text{wt\%\,Ag}) + \beta(91.2\ \text{wt\%\,Ag})$$

7.4 — Lever Rule: 55wt%Ag–45wt%Cu at 800°C

Given. $C_0=55$ wt% Ag; tie line at 800°C from 7.1: $C_\alpha=7.5$, $C_L=67.0$ wt% Ag.

Find. Mass fraction of each phase.

Approach. Lever rule: each phase's fraction is the ratio of the "opposite" tie-line segment to the total tie-line length.

1. Fraction of $\alpha$. $$f_\alpha=\frac{C_L-C_0}{C_L-C_\alpha}=\frac{67.0-55}{67.0-7.5}=\frac{12.0}{59.5}=\boxed{0.202\ (20.2\%)}$$
2. Fraction of liquid. $$f_L=\frac{C_0-C_\alpha}{C_L-C_\alpha}=\frac{55-7.5}{59.5}=\boxed{0.798\ (79.8\%)}$$ (Check: $f_\alpha+f_L=1.000$.)

7.5 — Microstructure Development, 20wt%Ag–80wt%Cu Cooled from 1100°C

Approach. Trace the vertical composition line at 20 wt% Ag down through the diagram: liquid $\to$ liquidus (primary-$\alpha$ nucleation) $\to$ eutectic isotherm (remaining liquid transforms) $\to$ solvus (secondary $\beta$ precipitation on further cooling).

At 1100°C the alloy is fully liquid ($1100\,{}^{\circ}\text{C}$ exceeds the liquidus temperature at 20 wt% Ag, which lies below Cu's own melting point of 1085°C). Cooling through the liquidus, primary (proeutectic) $\alpha$ dendrites nucleate and grow, rejecting Ag into the shrinking liquid, whose composition follows the liquidus toward the eutectic point. Because the bulk composition (20 wt% Ag) exceeds the maximum solid solubility at the eutectic temperature (8.0 wt% Ag), the liquid is not exhausted before reaching 779°C: just below the eutectic isotherm, the remaining liquid (now at 71.9 wt% Ag) transforms by the eutectic reaction into a fine lamellar mixture of $\alpha+\beta$, surrounding the pre-existing primary $\alpha$ grains. By the lever rule at $T_E^-$, the microconstituent fractions are $$f_{\text{primary }\alpha}=\frac{C_E-C_0}{C_E-C_{\alpha,E}}=\frac{71.9-20}{71.9-8.0}=\boxed{0.812\ (81.2\%)}\qquad f_{\text{eutectic}}=\boxed{0.188\ (18.8\%)}$$ On further cooling toward room temperature, the $\alpha$-solvus falls steeply (7.2 above), so both the primary $\alpha$ grains and the $\alpha$ lamellae within the eutectic mixture reject additional Ag by solid-state diffusion, precipitating fine secondary $\beta$ particles ($\beta_{II}$) throughout the primary $\alpha$. The room-temperature microstructure is therefore: primary $\alpha$ (with dispersed $\beta_{II}$ precipitates) + the eutectic $(\alpha+\beta)$ constituent, exactly the pattern sketched above.