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Recent experiments have reported intriguing trends for the molecular weight (MW) dependence of the conductivity of block copolymer lamellae that contrast with the behavior of homopolymer matrices. By using coarse-grained simulations of the sorption and transport of penetrant cations, we probe the possible mechanisms underlying such behavior. Our results indicate that the MW dependence of conductivity of homopolymeric and block copolymeric matrices arise from different mechanisms. On the one hand, the solvation energies of cations, and, in turn, the charge carrier concentrations, themselves, exhibit a MW dependence in block copolymer matrices. Such trends are shown to arise from variations in the thickness of the conducting phase relative to that of the interfacial zones. Moreover, distinct mechanisms are shown to be responsible for the diffusivities of ions in homopolymer and block copolymer matrices. In the former, diffusivity effects associated with the free ends of the polymers play an important role. In contrast, in block copolymer lamellae, the interfacial zone between the blocks presents a zone of hindered diffusivity for ions and manifests as a molecular weight dependence of the ionic diffusivity. Together, the preceding mechanisms are shown to provide a plausible explanation for the experimentally observed trends for the conductivity of block copolymer matrices.