The condition for solidification with a planar interface and no constitutional supercooling is G/V ≥ ΔT₀/D_L (G is the temperature gradient in the liquid at the interface, V the growth rate, D_L the solute diffusion coefficient in the liquid, and ΔT₀ = mC₀(1 − k)/k the freezing range). When an Al-2.0 wt%Cu alloy (magnitude of the liquidus slope m = 3.4 K/wt%, k = 0.17, D_L = 3.0×10⁻⁹ m²/s) is grown at V = 1.0×10⁻⁵ m/s, which is closest to the minimum temperature gradient needed to keep the interface planar?

Answer: B. About 1.1×10⁵ K/m

The freezing range is ΔT₀ = 3.4 × 2.0 × (1 − 0.17)/0.17 ≈ 33.2 K. The required gradient is G = ΔT₀V/D_L = 33.2 × 1.0×10⁻⁵/(3.0×10⁻⁹) ≈ 1.1×10⁵ K/m, or about 110 K/mm. About 1.9×10⁴ K/m comes from taking ΔT₀ as mC₀(1 − k) and forgetting to divide by k, and about 1.1×10² K/m mistakes the number for 110 K/mm as K/m. Temperature gradients in ordinary casting are of the order of 10³ to 10⁴ K/m, so an alloy can rarely keep a planar interface and solidifies with cells or dendrites. Directional solidification with a planar interface is used in experiments and crystal growth with a large temperature gradient and a very small growth rate.