We study the diffusion of gas molecules through a two-dimensional network of polymers with the help of Monte Carlo simulations. The polymers are modelled as non-interacting random walks on the bonds of a two-dimensional square lattice, while the gas particles occupy the lattice cells. When a particle attempts to jump to a nearest-neighbour empty cell, it has to overcome an energy barrier which is determined by the number of polymer segments on the bond separating the two cells. We investigate the gas current J as a function of the mean segment density ρ, the polymer length and the probability q m for hopping across m segments. Whereas J decreases monotonically with ρ for fixed , its behaviour for fixed ρ and increasing depends strongly on q. For small, non-zero q, J appears to increase slowly with . In contrast, for q = 0, it is dominated by the underlying percolation problem and can be non-monotonic. We provide heuristic arguments to put these interesting phenomena into context. © 2005 IOP Publishing Ltd.