The Dynamics of Three-Phase Triple Junction and Contact Points

We use the method of matched asymptotic expansions to study the sharp interface limit of the three-phase system modeled by the Cahn--Hilliard equations with the relaxation boundary condition. The dynamic laws for the interfaces, the triple junction, and the contact points are derived at different time scales. In particular, we show, at $O(t)$ time scale, the dynamic of the triple junction is determined by the balance of the chemical potential gradient along the three interfaces meeting at the triple junction. At faster time scale $O(\epsilon t)$, the motion of the triple junction is controlled by the contact point motions and geometric constraints.

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