Direct contact membrane distillation (DCMD) is a thermal desalination process that is capable of treating high salinity waters using low-grade heat. As a water treatment process, DCMD has several advantages, including the utilization of waste heat (below 100 degrees C), perfect rejection of nonvolatile solutes, low areal footprint, and high scalability. However, the energy efficiency of DCMD is relatively low compared to other work-based and thermal desalination processes. In this study, we aim to quantify how membrane properties and process conditions affect the exergy or second-law efficiency (eta(II)) of a DCMD desalination system with external heat recovery. In particular, we analyze how the membrane permeability coefficient (B) and thermal conduction coefficient ((K) over bar) impact MD performance. We show that increasing the B value of a membrane by reducing its thickness, initially leads to an increase in eta(II) before conductive heat loss through the membrane causes eta(II) to fall. For a typical MD membrane with a porosity of 0.90, material thermal conductivity of 0.20 W m(-1) K-1, and a nominal pore diameter of 0.6 mu m, we find that the optimal permeability coefficient is 1.59x10(-6)kg m(-2)s(-1)Pa(-1) (572 kg m(-2)h(-1)bar(-1)). This value corresponds to an optimal membrane thickness of around 95 mu m. Our analysis stresses the importance of effective heat recovery in DCMD. We show that an external heat exchanger with a minimum approach temperature of 5 degrees C reduces energy consumption by 72%. Finally, we demonstrate that increasing the ratio B/(K) over bar, rather than just the B value, is key to increasing the exergy efficiency of DCMD desalination. For example, increasing membrane porosity from 0.70 to 0.90, which yields a 160% increase in B/(K) over bar, leads to a 42% increase in eta(II) from 5.3% to 7.6%. The advantages of reducing the bulk pressure (P) in the membrane pores are also explored. For a typical membrane, halving P from 1.0 bar to 0.5 bar, results in a 21% increase in eta(II) from 7.0% to 9.2%. We conclude by identifying that the maximum exergy efficiency achievable as membrane porosity tends to unity is 10% for a bulk membrane pressure of 1.0bar and 12% for a bulk membrane pressure of 0.5 bar, given perfect heat recovery.