the eutectoid reaction
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The eutectoid is the eutectic's solid-state twin; the ending -oid means eutectic-like. Instead of a liquid, here a solid phase transforms on cooling into two different solid phases at once. This is the single reaction that makes steel behave like steel, turning austenite into the layered mixture called pearlite.
More precisely, the eutectoid reaction is solid gamma to solid alpha plus solid beta on cooling, at an invariant temperature (F = 0). In the iron-carbon system this is austenite (0.76 wt% C) to ferrite (0.022% C) plus cementite (6.7% C) at 727 degrees C. Because every phase involved is solid, diffusion is slower than in a eutectic, so the cooling rate strongly controls how fine the product comes out. At equilibrium it forms pearlite, alternating lamellae of soft ferrite and hard cementite.
This one reaction underlies most heat treatment of steel. A steel right at 0.76% C (eutectoid) becomes 100 percent pearlite; a leaner one (hypoeutectoid) first grows proeutectoid ferrite, a richer one (hypereutectoid) first grows proeutectoid cementite, and the rest then goes eutectoid. Cool faster than equilibrium and the eutectoid reaction is bypassed entirely for bainite or martensite, which is precisely why those non-equilibrium structures never appear on the equilibrium diagram.
Fe-C: austenite (0.76% C) to ferrite + cementite at 727 degrees C, forming pearlite.
A solid splits into two solids, the mechanism behind pearlite.
Eutectoid means solid to two solids (versus eutectic's liquid to two solids). Pearlite is the equilibrium product; martensite and bainite come from non-equilibrium cooling and are not on the phase diagram.