Quantum mechanical calculations of the structure and properties of a range of BaxScyOz adsorbate layers on W (001), (110), and (112) surfaces reveal the importance of O chemical potential in controlling the surface structure, surface energy, equilibrium crystal shape, and work function of W particles in Sc-containing (scandate) cathodes. Using the experimentally observed shape of W particles present in a high-emission scandate cathodes as a constraint, screening of similar to 2000 possible combinations of surface configurations reveals that the observed W grains are terminated with Ba0.05O/W-(001), Ba0.25O/W-(110) and Ba0.50 O/W-(112) surfaces, and were equilibrated in an O-poor environment with mu(O) between similar to - 8.5 and - 8.0 eV/at. Examination of the surface structures studied reveals that competing Ba-O and O-W interactions control the net surface dipole and that this dipole directly correlates with computed work functions, implying that the surface dipole alone is sufficient to explain low work functions observed for scandate cathodes. Analysis of the present results suggests that the role of Sc in scandate cathodes is to tune mu(O) and that difficulties in manufacturing scandate cathodes likely arises from variability in the availability of O at the cathode surface during activation and operation.