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We present a Monte Carlo model of carrier separation and recombination in nanostructured organic photovoltaic (OPV) devices which takes into account all electrostatic interactions, energetic disorder, and polaronic effects. This permits a detailed analysis of the strong morphology dependence of carrier collection efficiency. We find that performance is determined both by the orientation of the heterojunction relative to the external electric field as well as by carrier confinement due to polymer intermixing. The model predicts that an idealized interdigitated structure could achieve overall efficiencies twice as high as blends. The model also reproduces the weakly sublinear intensity dependence of short-circuit photocurrent (ISC) seen in experiment. We show that this is not the result of space-charge effects but of bimolecular recombination. Disconnected islands of polymer in coarser blends result in bimolecular recombination even at low intensities and should therefore be minimized. By including a microscopic description of dark injection, the model can describe the full current-voltage (J-V) characteristics of different OPV structures. We examine the effect of morphology, intensity, mobility, and recombination rate on key parameters such as short-circuit current, open-circuit voltage (VOC), and fill factor (FF). The model reproduces the intensity-dependent contribution to VOC in a bilayer above that of a blend observed in experiment. We find that performance in both bilayers and blends is very sensitive to the recombination rate across the heterojunction. The model also predicts a striking dependence of performance on mobility. Indeed it is shown that a tenfold increase in mobility dramatically improves ISC and FF and doubles the maximum power output in a bilayer device. As well as informing routes for improving device performance, the model also offers an improved microscopic understanding of OPV operation. [ABSTRACT FROM AUTHOR]