A free-boundary problem for concrete carbonation: Front nucleation and rigorous justification of the $\sqrt{t}$-law of propagation

Abstract

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 $\sqrt{t}$ asymptotic behavior. Previously we obtained the upper bound $s(t)\le C^{\prime }\sqrt{t}$ for some constant $C^{\prime }$; now we show the optimality of the rate by proving the right nontrivial lower estimate, i.e. there exists $C^{\prime \prime }>0$ such that $s(t)\ge C^{\prime \prime }\sqrt{t}$.

(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 nucleation of the moving carbonation front – a scenario that until now was open from the mathematical analysis point of view.