The effect of a second phase on the sheath current carrying capacity of type II superconductors is investigated by measurements on suitable binary eutectic alloys. The results show that the presence of a normal second phase increases the sheath current. Furthermore, the orientation of the second phase with respect to the applied field is shown to affect the value of the sheath current. The results are explained by postulating surface sheath superconductivity occurring at the interfaces of the superconducting matrix with the second phase.