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The critical issue facing head-disk interface (HDI) design is determining the optimal combination of lubricant and overcoat materials. A high performing lubricant should have sufficient adhesion to the carbon overcoat (COC) layer to prevent spin-off, while the COC material protects the magnetic media from corrosion and tribological damage. In this paper, we examine at the atomistic level COC material performance by calculating the adhesive interactions between various types of model perfluoropolyether (PFPE) functional lubricants and COC. The COC materials we investigated include industry standard diamond-like carbon (DLC), composed of graphitic (sp2 bonded) and diamond (sp3 bonded) type carbon. In addition, we investigated the atomically thin graphene and pure diamond COCs to determine the effect of carbon structure on PFPE adhesion. These pure diamond and graphene results support the observation that the PFPE interacts more strongly with the graphitic content in DLC. The difference between the results for diamond and graphene are due to the sp2 bonding in graphene with delocalized electrons in the aromatic ring, whereas in the diamond structure, electrons are bound by the extra C-H bond. Our density functional theory study of PFPE interaction with these three COC materials shows that Ztetraol has the strongest adhesion on all of the COCs investigated. Based on our investigation, we would conclude that an ideal COC material should have high graphitic content to promote strong PFPE adhesion. We also evaluated the feasibility of pure graphene to promote PFPE adhesion. We extend our analysis to explore dimensional crossover to gauge the effect of COC thickness on adhesion.