Ice-ocean heat exchange in polar leads was examined using a large-eddy simulation model coupled to a slab ice model. Simulations were performed using an idealized square domain for a range of lead sizes, surface wind stress (0.05--0.1 N m-2), and lead temperature/salinity profiles. Particular emphasis was placed on understanding the role of fresh water in leads and how stratification controls the heat budget and ice edge melting rate. With uniform initial conditions we found that solar heating was not strong enough to develop lead freshening via ice edge melting; even weak winds (0.02 N m-2) generated circulations that maintained a well-mixed lead. In the weak wind case, adding a fresh water flux representative of surface melt runoff provided enough additional stratification so that the lead water became isolated from the rest of the simulated ocean boundary layer. However, stronger winds (0.1 N m-2) prevented the fresh water layer from forming. Experiments initialized with temperature/salinity profiles similar to observed cases (fresh water layer capping the lead) demonstrated that lateral melting rates increase with expanding lead size, agreeing with simple heat balance calculations for a square lead without vertical mixing. However, with stronger winds, lateral melting rates decreased because of greater turbulent mixing of cold water from beneath the fresh layer. Inspection of the lead circulation indicated that the strongest melting occurred where the ice edge currents were the largest. Overall, melting fluxes for a 24 m2 lead ranged from 200 to 400 W m-2, depending on the wind speed. Without the fresh layer, fluxes ranged from 50 to 60 W m-2, suggesting that fresh water stratification can have a dominate role in controlling ice edge melting.