21-Mat-B6 Ceramic Materials · December 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
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: Micrograph of a hypoeutectoid steel with two boxed/arrowed regions: a finely-hatched left region and a lighter right region. See the official exam paper or the cited reference text.]
The two arrowed boxes sit in visibly contrasting parts of the micrograph. The left-hand box encloses a finely, directionally hatched patch — this is pearlite, the fine lamellar eutectoid mixture of ferrite (α) and cementite (Fe3C) that forms when the last of the austenite is driven to the eutectoid composition (0.77 wt% C) at 727 °C. The apparent parallel-line texture is the alternating α/Fe3C lamellae; the lamellar spacing and, crucially, the line ORIENTATION differ from one pearlite colony to the next because each colony nucleates and grows independently — which is exactly what produces the several different hatch directions visible across the rest of the micrograph. The right-hand box encloses a lighter, largely unlined patch — this is proeutectoid (primary) ferrite, the single-phase α that nucleated and grew directly from austenite ABOVE the eutectoid temperature, before any pearlite had formed. Because it is a single phase holding very little dissolved carbon (≤0.022 wt% C at 727 °C) it etches lightly and shows no internal lamellar structure, appearing as the continuous, blocky, white-toned regions that trace out what were the prior austenite grain boundaries.
The specimen is a hypoeutectoid steel (overall carbon below the eutectoid composition of 0.77 wt% C) that was cooled slowly — e.g. furnace- or air-cooled — from the single-phase austenite field, allowing near-equilibrium transformation. As the temperature falls through the A3 line on the Fe–Fe3C diagram, proeutectoid ferrite nucleates heterogeneously at the highest-energy sites available in the parent austenite — principally the prior-austenite grain boundaries — and grows into the austenite grains. Ferrite can dissolve only a small amount of carbon, so as each increment of ferrite forms it rejects carbon into the shrinking pool of remaining austenite, progressively enriching that austenite as cooling continues. This partitioning continues until, at 727 °C, the remaining (untransformed) austenite has been driven to exactly the eutectoid composition; at that point it transforms isothermally (eutectoidally) into pearlite — cooperative, lamellar co-precipitation of ferrite and cementite. The micrograph is therefore a two-stage record of cooling: the white, blocky, grain-boundary-tracing ferrite formed FIRST (above 727 °C), and the finely lamellar pearlite formed LAST, filling in what had been the remaining austenite.
Given. Estimated weight fraction of white region (proeutectoid ferrite), $W_\alpha = 40\% = 0.40$. Standard Fe–Fe3C tie-line data just below 727 °C: ferrite carbon-solubility limit $C_\alpha = 0.022$ wt% C; eutectoid composition $C_{eutectoid} = 0.77$ wt% C.
Find. The overall (bulk) carbon content $C_0$ of the steel.
Approach. Apply the ordinary two-phase lever rule on the $\alpha$–(eutectoid) tie line at 727 °C: every part of the steel between $C_\alpha$ and $C_{eutectoid}$ that is not proeutectoid ferrite becomes pearlite, so the lever rule for $W_\alpha$ can be solved directly for $C_0$.
| Quantity | Value |
|---|---|
| Estimated proeutectoid-ferrite weight fraction (given) | 40% |
| Eutectoid composition, $C_{eutectoid}$ | 0.77 wt% C |
| Ferrite solubility limit at 727 °C, $C_\alpha$ | 0.022 wt% C |
| Implied overall carbon content, $C_0$ | ≈ 0.47 wt% C (a medium-carbon, hypoeutectoid steel, roughly the AISI 1045–1050 range) |
