  Due to the vast compositional space of multi-principal element alloys
(MPEAs), the rational design of MPEAs for optimized microstructures is
difficult. Therefore, a high-throughput first-principles study of
Mo-V-Nb-Ti-Zr, a refractory MPEA, was conducted to gain insights into the
underlying microstructures. Using Monte-Carlo simulations powered by cluster
expansion, we uncover the principles governing the MPEA's microstructures
across a large compositional space that includes non-equiatomic compositions
and encompasses the constituent binaries, ternaries, and quaternaries. In the
spirit of Hume-Rothery rules for complete solid solubility, we present a
quantitative expression for predicting solid solution formation from the
composition. Within a consistent framework, our results reproduce the
microstructural observations (solid solution vs. segregation) from numerous
experiments and provide microstructural predictions for unexplored regions in
the compositional space. Our work illuminates the MPEA's microstructures in
terms of the separation and clustering tendencies of the elements, presenting a
set of simple but powerful design principles for future experiments to
rationally design MPEAs with the desired microstructures for superior
mechanical properties.
