  We study the out-of-equilibrium time evolution after a local quench
connecting two anisotropic spin-1/2 XXZ Heisenberg open chains via an impurity
bond. The dynamics is obtained by means of the adaptive time-dependent
density-matrix renormalization group. We show that the entanglement entropies
(Von Neumann and R\'enyi), in the presence of a weakened bond depend on the
sign of the bulk interaction. For attractive interaction (\Delta< 0), the
defect turns out to be irrelevant and the evolution is asymptotically
equivalent to the one without defect obtained by conformal field theory. For
repulsive interaction (\Delta>0), the defect is relevant and the entanglement
saturates to a finite value. This out-of-equilibrium behavior generalizes the
well known results for the ground-state entanglement entropy of the model.
