Molecular simulations using the method of molecular dynamics have been carried out to examine heat transfer from the surface of a heated silicon nano-membrane/wafer. Our results and analysis of the interfacial resistance to heat transfer at the surface-fluid boundary has provided insight into the thermal Kapitza resistance RK. We have found that (1) increasing the fluid pressure, and (2) making the interface more hydrophilic lowers RK, which implies better acoustic matching. Overall, RK also decreases with increasing temperature.