  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.
