The physical question

A planar solid–liquid interface can become unstable if the liquid immediately ahead of it is colder than the local equilibrium liquidus temperature. A small protrusion then enters liquid with a greater tendency to solidify, so it can grow faster than its surroundings.

A simple one-dimensional model

Take a dilute binary alloy with a planar interface advancing at speed V into liquid. Let z = 0 be the interface and z > 0 point into the liquid. Under steady growth, negligible solid diffusion and negligible liquid convection, the solute profile is

CL(z) = C0 [1 + ((1 − k)/k) exp(−Vz/DL)]

Here C₀ is nominal composition, k is the equilibrium partition coefficient, and Dₗ is the solute diffusivity in liquid. This ideal solution assumes a semi-infinite liquid and local equilibrium at the interface. For k < 1, solute rejected by the solid raises the interfacial concentration to C₀/k.

If the liquidus slope is m < 0, the local liquidus temperature is Tₗ(z) = Tₘ + mCₗ(z). Approximate the imposed temperature just ahead of the interface by T(z) = Tᵢ + Gz. At the interface these temperatures coincide.

The liquid becomes constitutionally supercooled close to the interface if its actual temperature rises with z more slowly than the local liquidus temperature. Comparing slopes at z = 0 gives the planar-growth condition

G / V ≥ |m| C0 (1 − k) / (k DL).

The right-hand side has the dimensions of G/V. Raising G, reducing V, increasing liquid mixing, or reducing solute rejection can move the system toward stability, but convection invalidates this particular steady diffusion profile.

What this criterion does not guarantee

This is a useful local criterion, not a complete prediction of cell or dendrite spacing. Capillarity, attachment kinetics, anisotropy, transient boundary layers, finite sample dimensions and convection affect the fastest growing perturbation. A full linear stability analysis adds perturbation wavelength and growth rate. In a real experiment, measure the local gradient and actual interface speed rather than substitute heater settings or translation speed without validation.