  We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for
solutions to the curve diffusion flow if the maximal time of existence is
finite. In our setting, we consider an evolving family of curves driven by
curve diffusion flow, which has free boundary points supported on a line. The
evolving curve has fixed contact angle $\alpha \in (0, \pi)$ with that line and
satisfies a no-flux condition. The proof is led by contradiction: A compactness
argument combined with the short time existence result enables us to extend the
flow, which contradicts the maximality of the solution.
