  Microscopic structures for fcc-based quaternary high-entropy alloys (HEA) in
thermodynamically equilibrium state is examined based on first-principles (FP)
calculation combined with our recently-developed theoretical approach. We find
that (i) hierarchical ordering tendency for whole quaternary, and ternary and
binary subsystems for the present five HEAs cannot be reasonably characterized
by conventional Goldschmidt atomic radius or by those obtained from unary
system for FP calculation, but can be systematically characterized by atomic
radius from specially fluctuated configuration. (ii) ordering tendency for
whole- and sub-systems can be simultaneously treated with individual definition
of atomic radius ratio, and that linear correlation between ordering tendency
and atomic radius ratio naturally decreases with decrease of number of
constituents for whole (or sub-) system due to the counterbalance breaking of
like- and unlike-atom neighboring pair. We also find that by introducing
appropriate normalization, ordering tendency for systems with different number
of constituents (here, quaternary and ternary) can also be simultaneously
treated by the atomic radius ratio.
