  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.
