Question 8 of 8: Al-Si Casting Alloy Phase Diagram; Polymer Modulus vs. Temperature

Question 8: Al-Si Casting Alloy Phase Diagram; Polymer Modulus vs. Temperature (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) Al-Si system (Fig 3): pure-Al melting point $660^\circ$C, eutectic at $12.6\%$Si/$577^\circ$C, terminal $\beta$ (Si-rich) solid solubility $\approx99\%$Si at the eutectic temperature, liquidus rising to $1430^\circ$C at pure Si; alloy composition $C_0=20\%$Si.

Find. (a) Liquidus (start-of-solidification) and solidus/eutectic (end-of-solidification) temperatures; room-temperature microstructure; %eutectic microconstituent. (b) Schematic modulus-temperature curve for amorphous PE with $T_g$/$T_m$ marked, and how it changes with crystallinity and cross-linking.

[Figure not reproduced: Fig. Q8a — Al-Si phase diagram (redrawn from Fig 3) with the 20% Si alloy composition, its liquidus intercept, and the eutectic tie line marked. See the official exam paper.]

Approach

Because the alloy composition (20% Si) lies to the right of the eutectic composition (12.6% Si), this is a hypereutectic alloy: the primary phase that solidifies first on cooling is $\beta$ (nearly pure Si), not $\alpha$ — the same real alloy family (e.g. A390-type, ≈17% Si) used for wear-resistant automotive cylinder bores. The liquidus and eutectic temperatures are read directly off the diagram; the eutectic %microconstituent follows from the lever rule applied just above the eutectic temperature, with primary $\beta$ and liquid (which itself entirely transforms into the eutectic mixture at $577^\circ$C) as the two "phases."

1. Start of solidification (liquidus). Reading the printed silicon-side liquidus curve at 20% Si (it is strongly convex, so it must be followed, not chorded): $$\boxed{T_{liquidus}\approx685^\circ\text{C}}$$ — primary $\beta$ (essentially pure Si) crystals begin to form here.
2. End of solidification (eutectic isotherm). As cooling continues below the liquidus, primary $\beta$ grows and the remaining liquid is progressively enriched in Al, its composition sliding down the liquidus toward the eutectic point. Solidification cannot finish until the last liquid reaches the eutectic composition and undergoes the eutectic reaction, so $$\boxed{T_{solidus}=T_{eutectic}=577^\circ\text{C}}$$ (the same completion temperature as for a hypoeutectic alloy — only the path taken to reach it differs).
3. Room-temperature microstructure. Slow cooling gives: coarse primary $\beta$ (Si) particles/dendrites (formed above $577^\circ$C) embedded in a matrix of eutectic ($\alpha+\beta$) mixture (formed from the last liquid at $577^\circ$C) — essentially no further solid-state change occurs on cooling from $577^\circ$C to room temperature since the $\alpha$ solvus is very close to the Al-rich axis.
4. Fraction eutectic (lever rule, just above 577°C). Treating primary $\beta$ ($\approx99\%$Si) and liquid (eutectic composition, $12.6\%$Si, which becomes the eutectic mixture) as the two constituents at $C_0=20\%$: $$f_{\beta,primary}=\frac{C_0-C_{eut}}{C_\beta-C_{eut}}=\frac{20-12.6}{99-12.6} =\frac{7.4}{86.4}=0.086,$$ $$f_{eutectic}=1-f_{\beta,primary}=\boxed{\approx91\%}.$$ So the room-temperature structure is overwhelmingly ($\approx91\%$) the fine eutectic mixture, with only $\approx9\%$ coarse primary silicon.
5. (b) Modulus-temperature curve, fully amorphous PE. At low temperature the polymer is glassy (chain segments frozen, high modulus, weak temperature dependence). At the glass transition $T_g$, cooperative segmental motion becomes possible on the timescale of the test and the modulus drops sharply (typically 2–3 orders of magnitude) to a rubbery plateau (entangled chains resist flow only through physical entanglements, not a true network). At still higher temperature the modulus falls further into viscous flow as chains disentangle and slip past one another — a purely amorphous polymer has no true melting point $T_m$ (there is no crystal lattice to melt), so only $T_g$ is marked as a real transition on this curve.

6. (i) Effect of increasing crystallinity. Crystallites act like rigid, reinforcing physical cross-links embedded in the amorphous matrix, so the rubbery-plateau modulus rises (a semicrystalline polymer is stiffer above $T_g$ than the fully amorphous material) and the plateau extends further before a genuine, comparatively sharp $T_m$ drop now appears (the crystallites do melt, at a well-defined temperature, unlike the amorphous case) — $T_g$ itself is only weakly affected by crystallinity.
7. (ii) Effect of increasing cross-linking. Chemical cross-links tie the chains into a single covalently bonded network, which raises the rubbery-plateau modulus even further (and, at high cross-link density, essentially flattens it — the modulus becomes nearly temperature-independent above $T_g$) and, critically, eliminates the viscous-flow region and any $T_m$-like transition entirely: a covalently cross-linked network cannot flow or melt in the ordinary sense (it will char/decompose before it flows), unlike either the amorphous or the semicrystalline (physically entangled) cases above.