Sheet-shaped metallic glass is made by casting into a copper mold. Assume that the cooling rate at the center of the sheet is inversely proportional to the square of the thickness d, and that it is 1.0×10³ K/s for a thickness of 1.0 mm. Which is closest to the largest thickness at which an alloy with a critical cooling rate of 10 K/s can be made into glass?

Answer: C. About 10 mm

The cooling rate can be written R = 1.0×10³×(1.0/d)² K/s (d in mm), so the thickness at which R = 10 K/s is d = 1.0×√(1.0×10³/10) = 10 mm. The time for heat to escape to the mold is proportional to the square of the distance heat must travel, so the cooling rate is inversely proportional to the square of the thickness. About 100 mm assumes the cooling rate is inversely proportional to the thickness itself, and about 0.10 mm inverts the ratio of cooling rates. Early metallic glasses had large critical cooling rates of about 10⁵–10⁶ K/s, so only ribbons a few tens of µm thick made by melt spinning could be obtained, but multicomponent alloys lowered the critical cooling rate and made centimeter-scale bulk metallic glasses possible.