Sub-additive ionic transport across arrays of solid-state nanopores

Nanopores, either biological, solid-state, or ultrathin pierced graphene, are powerful tools which are central to many applications, from sensing of biological molecules to desalination and fabrication of ion selective membranes. However, the interpretation of transport through low aspect-ratio nanopores becomes particularly complex as 3D access effects outside the pores are expected to play a dominant role. Here, we report both experiments and theory showing that, in contrast to naïve expectations, long-range mutual interaction across an array of nanopores leads to a non-extensive, sub-linear scaling of the global conductance on the number of pores N. A scaling analysis demonstrates that the N-dependence of the conductance depends on the topology of the network. It scales like G ∼ N/log N for a 1D line of pores, and like $G\sim \sqrt{N}$G∼N for a 2D array, in agreement with experimental measurements. Our results can be extended to alternative transport phenomena obeying Laplace equations, such as diffusive, thermal, or hydrodynamic transport. Consequences of this counter-intuitive behavior are discussed in the context of transport across thin membranes, with applications in energy harvesting.