= Binary alloy Stefan similarity solution
{title2=$a(t)=2\lambda\sqrt{D_lt}$}
Consider a planar <binary alloy> with solid at $x<a(t)$ and liquid at $x>a(t)$, equal constant phase <mass densities>, no bulk flow, constant thermal properties and constant solute <diffusion coefficients> $D_s,D_l>0$. A <similarity solution> places the interface at $a(t)=2\lambda\sqrt{D_lt}$. In the liquid, far-field <concentration> $C_\infty$ and constant interfacial <concentration> $C_a$ give
$$
C_l(x,t)=C_\infty+(C_a-C_\infty)
\frac{\operatorname{erfc}[x/(2\sqrt{D_lt})]}{\operatorname{erfc}\lambda}.
$$
This solves the <diffusion equation>, has $C_l(a,t)=C_a$ and tends to $C_\infty$ far ahead. The solid has its own diffusion profile, with coordinate $x/(2\sqrt{D_st})$. Similarly, thermal profiles use each phase's <thermal diffusivity>, not automatically $D_l$.
Local <liquidus> equilibrium gives $T_a=T_m-mC_a$, with slope magnitude $m>0$, and the <solid-liquid segregation coefficient> $k_D$ gives $C_s(a,t)=k_DC_a$. For the normal directed toward the liquid, the heat and solute jump conditions are
$$
\rho L\dot a=k_s\partial_xT_s(a^-)-k_l\partial_xT_l(a^+),\qquad
(1-k_D)C_a\dot a=D_s\partial_xC_s(a^-)-D_l\partial_xC_l(a^+).
$$
The first is the <Stefan condition>; the second conserves solute rejected by the moving boundary. Together with the far-field thermal data, these determine $\lambda$ and $C_a$. For $k_D=0$ and negligible solid solute flux, the second reduces to <salt rejection>. Neglecting solid diffusion is an additional model limit: finite diffusion on both sides can matter near a nearly stationary interface $\lambda=0$.
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