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Composite electrolytes for lithium batteries typically combine materials with very different mechanical properties and ionic transport mechanisms and the degree to which these two phases affect each other is not well understood. In this work we used numerical simulations and experiments to investigate the transport in composite electrolytes consisting of polyethylene oxide (PEO) with lithium bis-trifluoromethanesulfonimide (LiTFSI) and Li1.3Al0.3Ti1.7(PO4)3 (LATP) lithium ion conducting glass-ceramic particles. In particular we are interested in how the introduction of a single ion conductor (SIC) changes the salt concentration gradients in the polymer electrolyte (PE) under applied potential. To study this, we performed numerical simulations and chronoamperometry experiments in electrolytes with different arrangements of the SIC and PE phases, i.e. layers and particulate composites. The results show that the particulate composites have the highest concentration gradients and take the longest time to reach steady state current. The best arrangement appears to have a layer of SIC impenetrable to anions in the polymer phase within the electrolyte membrane.