We study scale effects on the conductivity of crystalline contacting interfaces. The approach focuses on the role played by lattice vibrations in the thermal conductivity using finite elements and molecular dynamics models. A special effort is made at calibrating the continuum model directly from molecular dynamics simulations. An innovative method that uses the temperature evolution issued from an impulse boundary condition is employed to compute heat conductivity, which is notoriously known as difficult to measure. Using this approach, a parametric study is conducted on a set of contacting surfaces on which we specify asperities wavelengths. It is shown that the usual power laws that relate contact area ratio with thermal conductivity do not apply at the nanoscale.