  An idealized high entropy oxide is characterized by perfect chemical disorder
and perfect positional order. In this work, we investigate the extent to which
short-range order (SRO) and local structural distortions impede that idealized
scenario. Working in the entropy stabilized $\alpha$-PbO$_2$ structure, we
compare a two-component system, (Ti,Zr)O$_2$, with a four-component system,
(Ti,Zr,Hf,Sn)O$_2$, using a combination of experimental and computational
approaches. Special quasi-random structures are used in conjunction with
density functional theory calculations to investigate the local distortions
around specific elements revealing significant local distortions that are
relatively insensitive to the number of chemical constituents. Using finite
temperature Monte Carlo simulations, we are able to reproduce the previously
experimentally observed SRO and transition temperature for the two-component
system. However, ideal configurational entropy is never reached, so SRO is
expected even at synthesis temperatures. On the other hand, the order-disorder
transition temperature is dramatically lower and experimentally inaccessible
for the four-component system, while the configurational entropy is closer to
ideal and less sensitive to temperature. Total scattering measurements and pair
distribution function analysis of slow-cooled and quenched samples support this
view. In general, we demonstrate that SRO effects in high entropy materials are
less prevalent as more components are added in, provided the pairwise
interaction strengths remain comparable, while local distortions are less
affected by the number of components.
