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Heat transport through a liquid-vapor interface is a complex and very
relevant phenomenon for cryogenics, heat-removal and energy-generation
applications. The physical coupling between  confining walls, a fluid and a
vapor is important to control and improve cooling rates or efficiency on
condensation processes. Surface-modification is a promising route to optimize
the heat transport in two-phase confined systems.  We used molecular-dynamics
simulations of  coarse-grained models, to study a liquid-vapor confined in a
nano-channel as a function of channel filling. We established an stationary
heat flow through the liquid-vapor  interface, with the liquid located on a
cold wall and the vapor on a hotter wall.  We performed simulations
progressively increasing the number of fluid particles (channel filling). We
calculated density, pressure and temperature profiles along the nano-channel
for different fillings and compared the thermal behavior of the system
when the hot wall are modified by grafting polymers on it. We calculated the flow
rate when bare walls or polymer-coated walls are used. We took polymers of
varying fluid  affinity (solvophylic or solvophobic) and stiffness. We analysed
extreme cases of polymer properties to elucidate the general behavior of the
polymers with the heat flow. We found that the walls covered by solvophylic
stiff polymers improve the heat flow by 6x, as compared to bare walls, when the
liquid phase is in contact with the polymers.  Once the liquid wets the hot
wall, the improvement on the heat transfer is much more subtle  and dominated by
the polymers' grafting density. For walls coated by highly rigid polymers, the
jump in the heat flux is found to happen at much lower channel fillings. We
found that the morphology of the polymer layer induces a "liquid bridge",
through which the heat is transported with a very high thermal conductivity, as
compared to that of a vapor phase.