A free-boundary problem for concrete carbonation : rigorous justification of the $\sqrt t$-law of propagation

We study a one-dimensional free-boundary problem describing the penetration of carbonation fronts (free reaction-triggered interfaces) in concrete. Using suitable integral estimates for the free boundary and involved concentrations, we reach a twofold aim: (1) We fill a fundamental gap by justifying rigorously the experimentally guessed tv asymptotic behavior. Previously we obtained the upper bound s(t)=C'tv for some constant C'; now we show the optimality of the rate by proving the right nontrivial lower estimate, i.e. there exists C''>0 such that s(t)=C''tv. (2) We obtain weak solutions to the free-boundary problem for the case when the measure of the initial domain vanishes. In this way, we allow for the {\em nucleation of the moving carbonation front} – a scenario that until now was open from the mathematical analysis point of view. Keywords: Large-time behavior; free-boundary problem; concrete carbonation; integral estimates