Sharp-interface limits of Cahn-Hilliard models and mechanics with moving contact lines

We construct gradient structures for free boundary problems with nonlinear elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phase-field models converge to certain sharp-interface limits when the interface thickness tends to zero $\varepsilon\to 0$. In particular, we study the scaling of the Cahn-Hilliard mobility $m(\varepsilon)=m_0\varepsilon^\alpha$ for $0\le \alpha \le \infty$. In the presence of interfaces, it is known that the intended sharp-interface limit is only valid for $\underline{\alpha}<\alpha<\overline{\alpha}$. However, in the presence of moving contact lines we show that $\alpha$ near $\underline{\alpha}$ produces significant errors.