Past exam of the materials science course of the University of Cambridge / 2018 / Paper 1 / 4 / b

Use the line fraction occupied by the tall narrow ridges as an estimate of cementite volume fraction, assuming the line is representative and its etched height classification identifies the phases correctly. The displayed profile gives roughly $17vol\%$ cementite and $83vol\%$ ferrite.

An operational measurement at a height threshold near $75\mathrm{nm}$ gives four ridge widths totaling about $0.88\mu \mathrm{m}$ over a profile about $5.3\mu \mathrm{m}$ long:Changing the threshold over a reasonable part of the high ridges gives roughly $16$–$18\%$, so this is an image-based estimate, not an exact phase analysis. The occasional low protrusions should not all be counted as full cementite lamellae merely because they are above a noisy baseline. The reference line and threshold convention must be stated.

For eutectoid pearlite, apply the lever rule between ferrite and cementite at the transformation temperature. Using standard approximate Fe–C values $C_e=0.76$, $C_{\alpha }=0.022$ and $C_{\mathrm{cem}}=6.67$ in weight percent carbon,These are mass fractions. To compare with an unbiased geometric line fraction, convert using the phase densities:With representative densities ${\rho }_{\mathrm{cem}}\simeq 7.69$ and ${\rho }_{\alpha }\simeq 7.87\mathrm{Mg}{\mathrm{m}}^{-3}$, this gives about $11.3vol\%$ cementite and $88.7vol\%$ ferrite. Because the densities are similar, the mass and volume fractions differ only modestly. The phase-diagram values here are conventional approximations, not a claim that the exact 2018 Data Book has been recovered.

The profile estimate and equilibrium lever-rule estimate need not agree exactly. The height trace contains only four major ridges, making it a small sample of the microstructure. Pixel size, ridge-threshold choice, tip geometry, contamination, polishing damage and unequal etching change apparent widths. A finite-radius AFM tip can broaden the narrow raised cementite regions, biasing their inferred fraction upward.

The equilibrium calculation assumes eutectoid pearlite with the chosen carbon composition and tie-line values. Off-eutectoid steel can include proeutectoid ferrite or cementite outside the pearlite; alloying, carbon segregation and non-equilibrium cooling alter the simple Fe–C estimate. Carbon solubility also varies with temperature. The mass-to-volume conversion requires phase densities.

The observed difference between roughly $17\%$ ridge fraction and $11\%$ equilibrium phase volume is therefore a reason to examine sampling and the etched-surface measurement, not to force the image widths to match the lever rule. Multiple colonies, repeated calibrated traces and a phase-sensitive measurement would improve the comparison.

Ferrite etches faster in Nital, so it recedes and leaves cementite standing as the higher ridges. The identification is based on the surface-relief preparation, not on assuming that every bright feature in every microscopy mode is cementite. In the stated AFM image, brightness is a displayed height contrast.