= Small-solid-diffusivity melting concentration
{title2=$C_a(-r)=C_s H(r)/(1+H(r))$}
For $r=-\lambda>0$ and vanishing solid-to-liquid diffusivity ratio, $H(r)=\sqrt\pi r e^{r^2}\operatorname{erfc}(-r)$ gives the interfacial liquid <concentration>. It increases from zero to $C_s$ as $r$ increases. Its <derivative> is $C_sH'/(1+H)^2$, which decreases from $\sqrt\pi C_s$ to zero. For a direct proof, write $A=H/r=\sqrt\pi e^{r^2}\operatorname{erfc}(-r)$. Then $A'=2rA+2$, so $A\geq\sqrt\pi+2r$. Differentiation gives $2H'^2-(1+H)H''=2[(1+r^2+2r^4)A^2+r(4r^2-1)A+2r^2-2]>0$: the bracket increases with $A$, and substitution of its lower bound gives only positive <polynomial> coefficients, including constant $\pi-2$. Thus the <concentration> is strictly concave as a function of melting speed parameter $r$. Combining this with a linearized thermal balance produces one fold on the melting branch and, with the freezing branch, an interval of three possible interface speeds.
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