# Oxygen Self-Diffusion in Fluorite High Entropy Oxides
## 1 Introduction  
There is a drive to maximize ionic diffusion for more efficient performance of the energy materials presently considered for electrochemical solid-state devices (fuel cells and battery applications) [1; 2]. For example, in solid oxide fuel cells (SOFC), the operation at intermediate temperatures requires materials with lower activation energies of diffusion to compensate [3; 4]. This is because, typically in SOFCs, the oxygen vacancy content and the mobility of the electrode material are linked to the surface exchange kinetics and impact the cathode performance [3; 4]. Accelerating ionic diffusion in ceramics can be achieved by doping, mechanical means (strain engineering), and designing the morphology (for example, grain boundary engineering and nanocomposites) [5; 6; 7; 8; 9]. In recent studies, [PERSON] et al. [10] introduced the High-Entropy Oxides (HEO) concept, in which mixed oxides are stabilized by entropy. HEO are in essence analogous to the High-Entropy Alloys (HEA), which are alloys where high configurational entropy (\(\Delta\)S\({}_{\text{config}}\)) that can stabilize metastable single phases (temperature is adequately high so as to account for the negative formation enthalpies of secondary phases) typically via high temperature sintering [11; 12; 13].  
Several studies have examined the oxygen ion self-diffusion in fluorite-structured oxides [14; 15; 16; 17]. Although the use of high-entropy compositions greatly expands the number of potential compounds, the values of oxygen diffusion have been shown to be relatively unnectrable and, in most cases, do not exceed the values reported for archetypal ternary oxides, such as ZrO\({}_{2}\) + 8% Y\({}_{2}\)O\({}_{3}\) (designated ZSY). Characteristically, in the study of fluorite HEO, [PERSON] et al. [15] synthesized numerous single-phase fluorite oxides. These solid solutions had equal molar fractions of HfO\({}_{2}\), ZrO\({}_{2}\), and CeO\({}_{2}\) as the base materials, whereas the oxides of yttrium (Y), gadolinium (Gd), ytterbium (Yb), titanium (Ti), magnesium (Mg), calcium (Ca), and lanthanum (La), in turn acted as fluorite structure stabilizers [15]. Irrespective of the composition, the determined activation energies of oxygen self-diffusion are unspectacular (1.14-1.29 eV) and the conductivity values are nearly an order of magnitude lower as compared to ZSY [15]. The low values of ionic diffusivity are likely due to trapping of oxygen ion vacancies by the high concentrations of aliovalent cation dopants used in the construction of near-equimolar HEOs. Therefore, one strategy to improve the ionic diffusivity is to reduce the level of aliovalent doping, replacing it with mixed but isovalent dopants.  
What is evident in the literature is that irrespective of a number of high-level experimental and theoretical studies (for example, Refs. [18; 19; 20; 21; 22; 23; 24; 25; 26; 27] and the references therein), there is no understanding on the impact of entropy stabilization on the oxygen self-diffusion in fluorite HEO. The key question is as follows: Does it improve the oxygen self-diffusion or not? In an attempt to address this in the present study, we use molecular dynamics (MD) simulations coupled with classical interatomic potentials to study the impact of composition and, in particular, the influence of the level of trivalent rare-earth dopants upon the oxygen ion diffusivity of a series of high-entropy alloy compositions. The focus will be on the HEOs of Zr\({}_{(1-x)/3}\)Hf\({}_{(1-x)/3}\)Ce\({}_{(1-x)/3}\)Gd\({}_{x/2}\)Y\({}_{x/2}\)O\({}_{2-x}\), Zr\({}_{(1-x)/3}\)Hf\({}_{(1-x)/3}\)Ce\({}_{(1-x)/3}\)Y\({}_{x/2}\)La\({}_{x/2}\)O\({}_{2-x}\) at \(x=0.14\) (8\(\%\)\(RE_{2}\)O\({}_{3}\)) compared with predicted oxygen ion diffusivity for Z8Y and C8Y.