H$_2$ ortho-para spin conversion on inhomogeneous grain surfaces

We investigate the evolution of the ortho-to-para ratio of overall (gas + ice) H$_2$ via the nuclear spin conversion on grain surfaces coated with water ice under physical conditions that are relevant to star- and planet-forming regions. We utilize the rate equation model that considers adsorption of gaseous H$_2$ on grain surfaces which have a variety of binding sites with a different potential energy depth, thermal hopping, desorption, and the nuclear spin conversion of adsorbed H$_2$. It is found that the spin conversion efficiency depends on the H$_2$ gas density and the surface temperature. As a general trend, enhanced H$_2$ gas density reduces the efficiency, while the temperature dependence is not monotonic; there is a critical surface temperature at which the efficiency is the maximum. At low temperatures, the exchange of gaseous and icy H$_2$ is inefficient (i.e., adsorbed H$_2$ does not desorb and hinders another gaseous H$_2$ to be adsorbed), while at warm temperatures, the residence time of H$_2$ on surfaces is too short for the spin conversion. Additionally, the spin conversion becomes more efficient with lowering the activation barriers for thermal hopping. We discuss whether the spin conversion on surfaces can dominate over that in the gas-phase in star- and planet-forming regions. Finally, we establish a simple but accurate way to implement the H$_2$ spin conversion on grain surfaces in existing astrochemical models.