We present a computational study into the adsorption properties of CO$_2$ on amorphous and crystalline water surfaces under astrophysically relevant conditions. Water and carbon dioxide are two of the most dominant species in the icy mantles of interstellar dust grains and a thorough understanding of their solid phase interactions at low temperatures is crucial for understanding the structural evolution of the ices due to thermal segregation. In this paper, a new H$_2$O-CO$_2$ interaction potential is proposed and used to model the ballistic deposition of CO$_2$ layers on water ice surfaces, and to study the individual binding sites at low coverages. Contrary to recent experimental results, we do not observe CO$_2$ island formation on any type of water substrate. Additionally, density functional theory calculations are performed to assess the importance of induced electrostatic interactions.
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