Question 3 of 7: Pb–Sn Lever Rule; Precipitate Coherency; Age-Hardening Candidates

Question 3: Pb–Sn Lever Rule; Precipitate Coherency; Age-Hardening Candidates (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.

Given. (a) At 200°C: solid $\alpha$ solubility $=18\%$ Sn; liquid solubility $=43\%$ Pb ($=57\%$ Sn); alloy is 60% liquid + 40% solid $\alpha$ by mass. (c) A generic Y–Z eutectic phase diagram with four labelled alloy compositions (1–4); $\beta$ forms a coherent precipitate in $\alpha$.

Find. (a) Overall alloy composition. (b) Coherent vs. incoherent precipitates. (c) Which alloys are age-hardenable, and the age-hardening procedure.

Approach

Part (a) is a direct lever-rule mass balance between the two phase compositions. Part (c) hinges on recognizing that age hardening requires an alloy composition that is single-phase $\alpha$ at some elevated temperature (so it can be solution-treated) but becomes two-phase $\alpha+\beta$ on cooling to room temperature (so $\beta$ can precipitate) — this is exactly the composition window between the room-temperature (terminal) solid solubility limit and the maximum solid solubility at the eutectic temperature.

1. (a) Lever rule. Taking composition in %Sn: the solid $\alpha$ is $C_\alpha=18\%$ Sn (18% Sn dissolved in solid Pb) and the liquid is $C_L=100-43=57\%$ Sn (57% Sn / 43% Pb). With mass fractions $f_L=0.60$, $f_\alpha=0.40$, $$C_0=f_LC_L+f_\alpha C_\alpha=0.60(57)+0.40(18)=34.2+7.2=\boxed{41.4\%\ \text{Sn},\ 58.6\%\ \text{Pb}}.$$
2. (b) Coherent vs. incoherent precipitates. A coherent precipitate has a crystal lattice that matches up atom-for-atom with the surrounding matrix lattice across the interface; a small lattice mismatch is accommodated elastically (coherency strain) rather than by a dislocation array, and it is exactly this coherency strain field that most effectively obstructs dislocation motion — coherent zones (e.g. GP zones, $\theta''$ in Al–Cu) give the strongest precipitation-hardening effect. An incoherent precipitate has its own distinct crystal structure and orientation, separated from the matrix by a sharp interphase boundary containing misfit dislocations; it is typically the coarser, equilibrium-stage particle (e.g. overaged $\theta$ in Al–Cu) and strengthens mainly by Orowan dislocation bypass, a weaker mechanism than coherency strengthening.
3. (c) Identifying the hardenable alloys. Age hardening needs both conditions: the alloy must be two-phase ($\alpha+\beta$) at low temperature, so there is a $\beta$ precipitate to form, and it must be single-phase $\alpha$ at some higher temperature, so it can be solution treated. On the Y–Z diagram the solvus (the right-hand boundary of the $\alpha$ field) rises from the terminal solubility at the lowest temperature shown — which falls between Alloy 1 and Alloy 2 — up to the maximum solubility at the eutectic isotherm (the "knee" where solidus meets solvus), which falls between Alloy 3 and Alloy 4. Reading each alloy's vertical line against those two limits:
   • Alloy 1 lies to the left of the terminal solubility limit, so its vertical line never crosses the solvus: it is single-phase $\alpha$ at every temperature down to the bottom of the diagram, never becomes supersaturated, and has no $\beta$ to precipitate — not age-hardenable.
   • Alloys 2 and 3 fall between the terminal and maximum solubility limits: each crosses the solvus on cooling (two-phase $\alpha+\beta$ at low temperature) yet lies inside the single-phase $\alpha$ field just below the eutectic isotherm, so each can be solution treated, quenched to trap a supersaturated solid solution, and aged to precipitate coherent $\beta$ — both are age-hardenable.
   • Alloy 4 lies to the right of the knee (between the maximum solubility composition and the eutectic composition): even at the highest sub-solidus temperature available its overall composition exceeds the maximum $\alpha$ solubility, so it solidifies directly into an $\alpha+\beta$ (proeutectic $\alpha$ plus eutectic) microstructure with no temperature at which it exists as single-phase $\alpha$ — there is nothing to "solution treat," so it is not age-hardenable by this mechanism.
4. Age-hardening procedure (for Alloys 2 and 3).
   1. Solution heat treatment — heat above the alloy's solvus (but below the eutectic/solidus) to dissolve all $\beta$ into a homogeneous single-phase $\alpha$ solid solution; hold long enough for complete dissolution.
   2. Quench — cool rapidly (e.g. water quench) to room temperature. This suppresses diffusion so the $\beta$ phase cannot precipitate on cooling, trapping a metastable, supersaturated single-phase $\alpha$.
   3. Age — hold at room temperature (natural aging) or reheat to a moderate intermediate temperature (artificial aging) to allow controlled, fine, coherent $\beta$ (or GP-zone) precipitation. This is the step that produces the actual strengthening; aging past the peak-hardness point (overaging) coarsens the precipitate into the incoherent form and softens the alloy again.