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Solidification and cast structures

When a regular lamellar eutectic is directionally solidified, the lamellar spacing λ and growth rate V obey λ²V = constant. At a growth rate of 10 µm/s, λ = 2.0 µm. Which is closest to λ when the growth rate is raised to 160 µm/s?

  1. AAbout 0.13 µm
  2. BAbout 0.50 µm
  3. CAbout 1.0 µm
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Answer: B. About 0.50 µm

Since λ²V = constant, λ is inversely proportional to the square root of V. When the rate increases 16-fold, λ becomes 1/√16 = 1/4, giving 2.0/4 = 0.50 µm. About 0.13 µm assumes λ is inversely proportional to V and divides by 16, and about 1.0 µm assumes λ is inversely proportional to the fourth root of V. In the Jackson–Hunt theory this relation comes from coupled eutectic growth, in which two solid phases grow side by side while exchanging solute sideways by diffusion in the liquid ahead of the interface. A narrower spacing shortens the diffusion distance, but it adds phase boundaries and so increases the undercooling needed for interface energy; the spacing is set by the balance of the two. This is why eutectic alloys cooled faster have finer structures.

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