Layered Li(Ni,Mn,Co,)O$_2$ (NMC) presents an intriguing ternary alloy design space for optimization as a cathode material in Li-ion batteries. Recently, the high cost and resource limitations of Co have added a new design constraint and high Ni-containing NMC alloys have gained enormous attention despite possible performance trade-offs. It is not fully understood if this material space is a disordered solid solution at room temperature and any arbitrary combination can be used or if there exist distinct transition metal orderings to which meta-stable solid solutions will decay during cycling and affect performance. Here, we present a high fidelity computational search of the ternary phase diagram with an emphasis on high-Ni, and thus low Co, containing compositional phases to understand the room temperature stability of the ordered and disordered solid solution phases. This is done through the use of density functional theory training data fed into a reduced order model Hamiltonian that accounts for effective electronic and spin interactions of neighboring transition metal atoms at various lengths in a background of fixed composition and position lithium and oxygen atoms. This model can then be solved to include finite temperature thermodynamics into a convex hull analysis to understand the regions of ordered and disordered solid solution as well the transition metal orderings within the ordered region of the phase diagram. We find that for the majority of transition metal compositions of the layered material, specifically medium to high-Ni content, prefer transition metal ordering and predict the collection of preferred compositions in the ordered region.
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