In a binary eutectic phase diagram (e.g., Pb-Sn), a liquid of eutectic composition is cooled slowly through the eutectic temperature. What microstructure forms, and why is it different from cooling a liquid of off-eutectic composition?
Think about your answer, then reveal below.
Model answer: At the eutectic composition, the liquid transforms simultaneously into two solid phases (alpha + beta) at a single temperature, producing a fine-grained lamellar or rod-like microstructure where the two phases alternate on a micrometer scale. This occurs because both solids must nucleate and grow cooperatively from the liquid at the invariant eutectic temperature. An off-eutectic (hypoeutectic or hypereutectic) composition first precipitates primary crystals of one phase as it cools through the liquidus, forming large primary grains. The remaining liquid enriches toward the eutectic composition, and when it reaches the eutectic temperature, the residual liquid undergoes the eutectic transformation, producing a fine lamellar mixture surrounding the primary grains. The result is a two-scale microstructure: coarse primary phase plus fine eutectic colonies.
The eutectic microstructure is technologically important because the fine interphase spacing provides good mechanical properties (Hall-Petch strengthening from closely spaced phase boundaries) and the eutectic composition has the lowest melting point in the system, making it useful for solders, brazing alloys, and casting. The Pb-Sn eutectic (63Sn-37Pb, mp 183C) was the basis of electronics soldering for decades before lead-free regulations.
Question 2 True / False
The lever rule states that in a two-phase region of a binary phase diagram, the fraction of phase alpha equals (C_beta - C_0) / (C_beta - C_alpha), where C_0 is the overall composition and C_alpha and C_beta are the compositions of the two phases. This rule is derived from conservation of mass.
TTrue
FFalse
Answer: True
The lever rule is simply a mass balance. If a system of overall composition C_0 splits into two phases with compositions C_alpha and C_beta, then f_alpha * C_alpha + f_beta * C_beta = C_0, where f_alpha + f_beta = 1. Solving gives the lever rule. The name comes from the analogy to a mechanical lever: the fulcrum is at C_0, and the 'arms' are the distances to C_alpha and C_beta. The fraction of each phase is inversely proportional to its distance from C_0, just as weights on a balanced lever are inversely proportional to their arm lengths.
Question 3 Multiple Choice
Spinodal decomposition and nucleation-and-growth are both mechanisms for phase separation in a miscibility gap. What is the fundamental thermodynamic difference between them?
ASpinodal decomposition requires higher temperatures than nucleation-and-growth
BInside the spinodal (where d2G/dC2 < 0), the system is unstable to infinitesimal composition fluctuations — no nucleation barrier exists, and the system spontaneously unmixes by uphill diffusion. Between the spinodal and the binodal, the system is metastable — small fluctuations increase free energy, so decomposition requires nucleation over an energy barrier
CSpinodal decomposition produces large precipitates while nucleation produces fine-scale structures
DNucleation-and-growth only occurs in metallic systems while spinodal decomposition only occurs in ceramics
The distinction is thermodynamic stability vs. instability. In the metastable region (between binodal and spinodal curves), the free energy curve is concave up (d2G/dC2 > 0), so small composition fluctuations increase the free energy. Phase separation requires forming a nucleus large enough that the volume free energy gain exceeds the interfacial energy cost. Inside the spinodal (d2G/dC2 < 0), ANY fluctuation lowers the free energy, so the system decomposes spontaneously without nucleation. This produces a characteristic interconnected, wavelike microstructure rather than discrete precipitates. Spinodal decomposition is exploited in some glass-ceramics (e.g., Vycor) and in spinodal-hardened Cu-Ni-Sn alloys.