A simple model for the dynamics of contact lines freezing on a cold substrate
Résumé
Starting from de Gennes model of contact line dynamics, we develop a simple, analytical, description of a four phase contact line, at which a liquid advances on a cold substrate (of infinite therml conductivity) and at the same time freezes on the solid. The key idea consists in a subtle balance between thermical and hydrodynamical divergences at contact line, heat produced by dissipation in the liquid phase limitating the solid growth. We calculate the apparent liquid contact angle and its equivalent for the solid phase as a function of the contact line velocity U and the other parameters involved, in the limit of small angles. We show that the arrest condition (U=0) implies a divergence of these angles, that are presumably reaching 90°, while the liquid film thickness vanishes, which is reminiscent of previous works of Sonin et al in the 90's. Remarkably, at low contact line velocity, we show that a stick-slip behaviour is to be expected, which has been recently observed experimentally. The temperature dependance of the critical velocity differs from what is found in recent experiments, but this model is to our opinion a necessary first step to understand these complex behaviours.
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