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In this paper we use London Langevin molecular dynamics simulations toinvestigate the vortex matter melting transition in the highly anisotropichigh-temperature superconductor material Bi_2Sr_2CaCu_2O$_{8+\delta}$ in thepresence of low concentration of columnar defects (CDs). We reproduce withfurther details our previous results obtained by using Multilevel Monte Carlosimulations that showed that the melting of the nanocrystalline vortex matteroccurs in two stages: a first stage melting into nanoliquid vortex matter and asecond stage delocalization transition into a homogeneous liquid. Furthermore,we report on new dynamical measurements in the presence of a current thatidentifies clearly the irreversibility line and the second stage delocalizationtransition. In addition to CDs aligned along the c-axis we also simulate thecase of tilted CDs which are aligned at an angle with respect to the appliedmagnetic field. Results for CDs tilted by $45^{\circ}$ with respect to c-axisshow that the locations of the melting and delocalization transitions are notaffected by the tilt when the ratio of flux lines to CDs remains constant. Onthe other hand we argue that some dynamical properties and in particular theposition of the irreversibility line should be affected.

physical review. b, condensed matter and materials physics

Langevin Dynamics of the vortex matter two-stage melting transition in  Bi_2Sr_2CaCu_2O$_{8+\delta}$ in the presence of straight and of tilted  columnar defects