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 Incoming links: Morphological instability of a kinetically limited solidification front

In the frame of a planar interface moving at speed

V

, the steady liquid temperature obeys

- V T_{z} = κ T_{zz}

and approaches

T_{\infty}

. Thus

T = T_{\infty} + (T_{i} - T_{\infty}) e^{- V z / κ} .

The Stefan condition, with negligible solid-side heat flux, gives

L V = c_{p} (T_{i} - T_{\infty}) V,

T_{i} - T_{\infty} = L / c_{p} = S Δ T

V = G (T_{m} - T_{i}) = G Δ T (1 - S),

which requires

S < 1

for advancing solidification.

Scale length with

κ / V

, time with

κ / V^{2}

, and temperature with

Δ T

Θ_{0} = S e^{- z}

. For an interface displacement

ζ e^{i αx + σ t}

, write the thermal perturbation as

θ = A e^{- λ z + i αx + σ t} .

λ^{2} - λ - α^{2} - σ = 0, λ = \frac{1}{2} [1 + 1 + 4 (α^{2} + σ)] .

Linearizing the Stefan condition gives

λ A = S (1 + σ) ζ,

while the kinetic law with stabilizing curvature undercooling gives

A = [S - (1 - S) (σ + Γ α^{2})] ζ, Γ = \frac{γ G}{κ} .

Eliminating

A

gives the general implicit dispersion relation

\frac{S ( 1 + σ )}{λ} = S - (1 - S) (σ + Γ α^{2}), λ^{2} - λ = α^{2} + σ .

σ = 0

λ = \frac{1}{2} (1 + 1 + 4 α^{2}),

\frac{S}{1 - S} = Γ \frac{λ}{λ - 1} α^{2} .

Since

α^{2} = λ (λ - 1)

on this curve, the right-hand side is simply

Γ λ^{2}

. A positive cutoff wavenumber therefore exists exactly when

\frac{S}{Γ ( 1 - S )} > 1, that is, S > \frac{Γ}{1 + Γ} .

The unstable band is

0 < α < α_{c}

, where

λ_{c} = [\frac{S}{Γ ( 1 - S )}]^{1/2}, α_{c}^{2} = λ_{c} (λ_{c} - 1) .

A sketch of

Γ λ^{2}

against the constant

S / (1 - S)

shows no crossing above

α = 0

below threshold and one cutoff above it. As

S ↑ 1

λ_{c}

and

α_{c}

diverge like

(1 - S)^{- 1/2}