  We study the phase behaviour of the Al$_x$CrFeCoNi high-entropy alloy. Our
approach is based on a perturbative analysis of the internal energy of the
paramagnetic solid solution as evaluated within the Korringa-Kohn-Rostoker
formulation of density functional theory, using the coherent potential
approximation to average over disorder. Via application of a Landau-type linear
response theory, we infer preferential chemical orderings directly. In
addition, we recover a pairwise form of the alloy internal energy suitable for
study via atomistic simulations, which in this work are performed using the
nested sampling algorithm, which is well-suited for studying complex potential
energy surfaces. When the underlying lattice is fcc, at low concentrations of
Al, depending on the value of $x$, we predict either an $\mathrm{L}1_2$ or
$\mathrm{D}0_{22}$ ordering emerging below approximately 1000 K. On the other
hand, when the underlying lattice is bcc, consistent with experimental
observations, we predict $\mathrm{B}2$ ordering temperatures higher than the
melting temperature of the alloy, confirming that this ordered phase forms
directly from the melt. For both fcc and bcc systems, chemical orderings are
dominated by Al moving to one sublattice, Ni and Co the other, while Cr and Fe
remain comparatively disordered. On the bcc lattice, our atomistic modelling
suggests eventual decomposition into $\mathrm{B}2$ NiAl and Cr-rich phases.
These results shed light on the fundamental physical origins of atomic ordering
tendencies in these intriguing materials.
