Question 3 of 8: Be–Si Eutectic System — Solidification and Eutectic Fraction
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exam 04-BS-11, Properties of Materials — December 2013. 3 hours,
closed-book examination (approved Casio or Sharp calculator only). Any five questions constitute
a complete paper; only the first five questions as they appear in the answer book are marked. All
eight questions are solved below for completeness.
Given. $T_{m,Be}=1252^\circ$C, $T_{m,Si}=1414^\circ$C; eutectic at
$T_E=1090^\circ$C, $C_E=39\text{ wt\% Be}$; zero solid solubility of either component in the
other.
Find. The Be–Si phase diagram; the solidification path and final
microstructure for 90% Be and 30% Be alloys; the % eutectic in each cooled solid.
Fig. Q3 — Be–Si thermal equilibrium
(eutectic) diagram: two liquidus lines meeting at the eutectic point $E$ (1090°C, 39% Be); with
zero solid solubility, the terminal solid-solution fields collapse to the pure-component vertical
lines at 0% and 100% Be.
Approach
Because the two solids are completely insoluble, this is a simple binary eutectic: cooling any
off-eutectic composition first crosses a liquidus line, depositing primary (proeutectic) solid of
the nearer pure component while the remaining liquid composition slides along the
liquidus toward $C_E$; at $T_E$ all remaining liquid (exactly at $C_E$) transforms isothermally
into the fine eutectic mixture of both solids. The lever rule at a temperature just above $T_E$
(between the eutectic liquid at $C_E$ and the pure proeutectic solid) gives the fraction of liquid
that becomes eutectic.
(a) 90% Be alloy (Be-rich of the eutectic). On cooling, the liquid first
reaches the Be liquidus and deposits primary (proeutectic) Be; the remaining
liquid composition moves down the Be liquidus toward $C_E=39\%$ Be as temperature falls. Just
above $1090^\circ$C the system is (proeutectic Be) + (liquid at 39% Be). By the lever rule (using
the pure-Be solid at 100% Be as the other lever arm end, since Be holds no Si in solid solution):
$$\%\,\text{eutectic}=\frac{100-C_0}{100-C_E}\times100=\frac{100-90}{100-39}\times100=\boxed{16.39\%}.$$
The remaining $83.61\%$ is proeutectic Be. At $T_E$ the eutectic liquid solidifies isothermally
into alternating lamellae of Be and Si (the eutectic constituent), so the final room-temperature
microstructure is coarse primary Be grains ($83.6\%$) surrounded by fine eutectic (Be+Si)
($16.4\%$).
(b) 30% Be alloy (Si-rich of the eutectic). Here the liquid first reaches the
Si liquidus and deposits primary (proeutectic) Si; the remaining liquid slides
down the Si liquidus toward $C_E=39\%$ Be. Just above $1090^\circ$C the lever rule (now against
pure Si at 0% Be) gives
$$\%\,\text{eutectic}=\frac{C_0-0}{C_E-0}\times100=\frac{30-0}{39-0}\times100=\boxed{76.92\%}.$$
The remaining $23.08\%$ is proeutectic Si. Final microstructure: coarse primary Si grains
($23.1\%$) surrounded by a much larger fraction of fine eutectic (Be+Si) ($76.9\%$), since $30\%$
Be sits much closer to the eutectic composition than $90\%$ Be does.
