Based on the mass fraction model of multicomponent mixture, the interactions between weak shock wave and V shaped air/SF6 interface with different vertex angles are numerical simulated. The numerical scheme used in the simulation is the high-resolution finite volume method with minimized dispersion and controllable dissipation scheme, in which the dissipation can be adjusted without affecting the already optimized dispersion property of the scheme. The grid sensitivity study is performed to guarantee that the resolution is sufficient in the numerical simulation. After the shock wave interacts with the interface, the baroclinic vorticity is deposited near the interface due to the misalignment of the density and pressure gradient, which is the manifestation of the Richtmyer-Meshkov instability, leading to the vortical structures forming along the interface. The interface perturbations lead to the bubbles and spikes appearing. The predicted leftmost interface displacement and interface width growth rate in the early stage of interface evolution agree well with the experimental results. The process of transition to turbulence at the material interface is studied in detail. The numerical results indicate that with the evolution of the interfacial vortical structure due to Kelvin-Helmholtz instability, the array of vortices begins to merge. As a result, the vortices accumulate in several distinct regions. It is in these regions that the multi-scale structures are generated because of the interaction between vortices. It is shown clearly that in the regions where vortices are accumulated, the fluctuation energy spectrum has many large and smallscale elements, which indicates there may be turbulent structures in these regions. To further examine if there is mixing transition in these regions, the characteristic length scales of the flow fields are calculated. The separation between the Lipemann-Taylor scale and inner viscous scale is observed based on the circulation-based Reynolds number, leading to the appearance of an uncoupled inertial range. The classical Kolmogorov -5/3 power law is also shown in the fluctuation energy spectrum, which means that the inertial range is developed. The appearing of this inertial range confirms that the mixing transition does occur, and the flow field near the material interface will develop into turbulence.