= Salt-rejection function for a saline Stefan front
{title2=$F(\lambda)=1-\sqrt\pi\lambda e^{\lambda^2}\operatorname{erfc}\lambda$}
For salt-free <ice> and a liquid <concentration> profile proportional to $\operatorname{erfc}[x/(2\sqrt{Dt})]$, <salt rejection> at $a=2\lambda\sqrt{Dt}$ gives $C_iF(\lambda)=C_0$, where $D$ is the salt <diffusion coefficient>, and $C_i,C_0$ are the interface and far-field <concentrations>. This follows from $-DC_x(a^+)=\dot a C_i$. The function is positive and strictly decreasing for all real $\lambda$, since an integration by parts gives
$$
F(\lambda)=2\int_0^\infty u e^{-u^2-2\lambda u}\,du,\qquad F'(\lambda)=-4\int_0^\infty u^2e^{-u^2-2\lambda u}\,du<0.
$$
Consequently $C_i=C_0/F(\lambda)$ increases with $\lambda$, equals $C_0$ at zero, and is enriched for freezing and diluted for melting. The positivity proof also prevents spurious negative interface <concentrations> when using the <complementary error function> formula.
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