Compositionally Complex Ceramics

• Compositionally complex ceramics (CCCs) are ceramic solid solutions with at least three principal components that use compositional complexity and correlated disorder to tailor material properties.
• CCCs integrate low-, medium-, high-, and ultrahigh‑entropy regimes, accommodating both equimolar and non‑equimolar formulations with controlled short‑ and long‑range order.
• CCCs enable enhanced applications such as thermal barrier coatings and high‑temperature ceramics by optimizing properties like thermal conductivity, hardness, and phase stability through strategic compositional design.

Compositionally complex ceramics (CCCs) are ceramic solid solutions with at least three principal components. In this formulation, the category is not tied to a specific entropy threshold: it includes low‑, medium‑, high‑, and ultrahigh‑entropy compositions, and it explicitly accommodates non‑equimolar chemistry as well as short‑ and long‑range order. High‑entropy ceramics (HECs) and ultrahigh‑entropy ceramics (UECs) are therefore subsets of CCCs rather than exhaustive definitions of the field. The defining idea is that compositional complexity, correlated disorder, and access to vast non‑equimolar composition–structure spaces can be used as design variables to tailor and often enhance ceramic properties, rather than treating maximal configurational entropy as the sole objective (Luo, 7 Oct 2025).

1. Definitions, scope, and entropy nomenclature

The modern CCC framework emerged from the proposal to expand “high‑entropy ceramics” to “compositionally complex ceramics” so as to include medium‑entropy and non‑equimolar compositions and to further consider short‑ and long‑range orders, which reduce configurational entropies but offer additional dimensions and opportunities to tailor and improve various properties (Wright et al., 2020). In this broader usage, single‑phase equimolar HECs remain an important subset, but CCCs also encompass ordered, non‑equimolar, and even dual‑phase systems (Luo, 2022).

For a given cation sublattice, ideal configurational entropy is written as

$S_{\text{config}} = - R \sum_i x_i \ln x_i$

or, per atom or per cation,

$s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$

where $x_i$ is the mole fraction of species $i$, $R$ is the gas constant, and $k_B$ is Boltzmann’s constant (Luo, 7 Oct 2025). One widely used qualitative classification assigns low‑entropy ceramics (LECs) to $s_{\text{config}}<1\,k_B$, medium‑entropy ceramics (MECs) to $1\,k_B \le s_{\text{config}} \le 1.5\,k_B$, high‑entropy ceramics (HECs) to $s_{\text{config}}>1.5\,k_B$, and ultrahigh‑entropy ceramics (UECs) to $s_{\text{config}}>2\,k_B$ per cation, while other ultrahigh‑entropy studies use $s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$0 per cation as the corresponding threshold (Luo, 7 Oct 2025, Song et al., 6 Jul 2026). These cutoffs are explicitly described as subjective, somewhat arbitrary, and based on ideal random mixing; they ignore short‑range order, which lowers the actual entropy and is difficult to quantify (Luo, 7 Oct 2025).

A five‑component equimolar solid solution has

$s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$1

per cation, which only just satisfies the common “high‑entropy” criterion (Luo, 7 Oct 2025). That observation is central to the CCC viewpoint: entropy labels are useful descriptors, but CCCs are defined structurally and compositionally, not by a single entropy number. In this sense, “multicomponent ceramics” is a broader descriptive phrase that may or may not refer to solid solutions, whereas CCCs specifically emphasize multicomponent ceramic solid solutions and their associated defect, ordering, and microstructural degrees of freedom (Luo, 7 Oct 2025).

2. Entropy, enthalpy, and the critique of entropy maximization

A central thesis of the CCC literature is that maximizing entropy is neither necessary nor sufficient for optimal properties. For an equimolar five‑component HEC, the free‑energy contribution of configurational entropy is on the order of $s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$2 per atom, comparable to vibrational energy and modest relative to high‑enthalpy processes such as oxidation of non‑oxide ceramics (Luo, 7 Oct 2025). This implies that configurational entropy can stabilize a single phase in some systems, but it is only one term in the free‑energy balance and can be outweighed by enthalpy.

The practical consequence is that non‑equimolarity and ordering often improve performance even though they reduce ideal entropy. A canonical example is the YSZ‑like fluorite series

$s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$3

for which the non‑equimolar composition

$s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$4

exhibits a higher $s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$5 ratio than the equimolar five‑component HEC counterpart, giving superior potential as a thermal‑barrier coating (Luo, 7 Oct 2025). Earlier fluorite case studies reached the same conclusion in slightly different nominal compositions, emphasizing that medium‑entropy, non‑equimolar fluorites can achieve lower thermal conductivity while retaining high Young’s modulus and therefore higher $s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$6 than equimolar high‑entropy fluorites (Wright et al., 2019).

Pyrochlore CCCs reinforce the same point. In rare‑earth pyrochlore oxides, reduced thermal conductivity correlates more strongly with cation size disorder than with configurational entropy itself, and medium‑entropy compositions can have lower $s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$7 than higher‑entropy analogues (Luo, 7 Oct 2025). In a 20‑component fluorite–pyrochlore–weberite series,

$s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$8

the lowest thermal conductivities occur where weberite‑type short‑range order persists inside a pyrochlore long‑range ordered matrix, despite the entropy reduction associated with ordering (Luo, 7 Oct 2025). The resulting design principle is explicit: compositional complexity, not entropy maximization per se, is the primary lever.

3. Order, correlated disorder, and structural transitions

CCCs are distinguished not only by multicomponent chemistry but also by the coexistence of order and disorder across length scales. Long‑range order (LRO) refers to periodicity over many unit cells and defines the average crystal structure; short‑range order (SRO) denotes non‑random local arrangements over only a few nearest‑neighbor distances; and correlated disorder describes local chemical and structural order that deviates from the global symmetry, often producing local symmetries lower than the average crystal symmetry and manifesting as nanoscale domains (Luo, 7 Oct 2025).

Defect‑fluorite CCCs provide the most developed examples. A nominally disordered cubic fluorite lattice can contain weberite‑ordered nanoscale domains of order $s_{\text{config}} = - k_B \sum_i x_i \ln x_i,$9 nm, corresponding to a $x_i$0 or similar superstructure. Neutron total scattering and pair distribution function analyses have shown fluorite LRO with weberite‑type SRO, as well as weberite ordering persisting inside pyrochlore regimes of 20‑component systems (Luo, 7 Oct 2025). These locally ordered domains enhance phonon scattering, lower thermal conductivity, and illustrate that lowered symmetry and reduced entropy can be enthalpically favorable.

Ultrahigh‑entropy fluorite/pyrochlore systems further show that “more components and more entropy” does not simply mean “more disorder.” In 16–19‑component CCCs, ordered pyrochlores remain stable, fluorite–pyrochlore dual‑phase windows are suppressed relative to five‑component HECs, and the practical fluorite–pyrochlore transition correlates better with the fluorite‑based size‑disorder descriptor $x_i$1 than with a fixed cation‑radius ratio rule (Song et al., 6 Jul 2026). In the stoichiometric 19CCC‑P|Fy series,

$x_i$2

the pyrochlore–fluorite order–disorder transition occurs continuously at $x_i$3, whereas in the non‑stoichiometric 19CCC‑P|Fz series the transition is abrupt at $x_i$4, despite an order‑parameter extrapolation suggesting $x_i$5 (Song et al., 6 Jul 2026). Stoichiometry therefore controls not only where an order–disorder transition occurs, but also whether it is continuous or abrupt.

The same logic extends to redox and to even larger composition spaces. A reversible redox‑induced order–disorder transition was observed in the 10‑cation oxide

$x_i$6

which switches between single‑phase pyrochlore and single‑phase defect fluorite upon oxidizing versus reducing anneals at $x_i$7, with oxygen‑vacancy formation identified in situ by neutron diffraction (Zhang et al., 2021). At the extreme end of complexity, 21‑component systems yielded ultrahigh‑entropy pyrochlore, weberite, and fergusonite phases, together with an abrupt pyrochlore–weberite transition and thermal conductivities already near the amorphous limit (Qin et al., 2021). Ordered ultrahigh‑entropy phases are therefore not anomalies but a recurring feature of the CCC regime.

4. Composition space, microstructures, and processing strategies

Restricting design to equimolar HECs samples only isolated points within a much larger composition simplex. The fluorite example above already shows that varying $x_i$8 continuously in

$x_i$9

opens a high‑dimensional design space in which charge balance, phase stability, cation size disorder, valence disorder, and vacancy concentration can all be tuned (Luo, 7 Oct 2025). Analogous flexibility exists across borides, carbides, silicides, perovskites, pyrochlores, cuprates, and fluorite‑derived oxides.

The resulting microstructures need not remain single phase. Early HEC work favored single‑phase solid solutions, but the CCC framework explicitly admits dual‑phase and multiphase equilibria. In a carbide–diboride ultra‑high‑temperature ceramic, an initially equimolar Ti–Zr–Hf–Ta–Nb composition with B and C partitions at equilibrium into a five‑component diboride and a five‑component carbide; each individual phase becomes non‑equimolar, so the material is more accurately described as a dual‑phase CCC than a dual‑phase HEC. Its hardness exceeds both the weighted linear average of the single‑phase high‑entropy carbide and diboride counterparts and the constituent‑binary rule‑of‑mixtures expectations, demonstrating a microstructural strengthening contribution beyond compositional complexity alone (Luo, 7 Oct 2025).

Processing has become a decisive part of CCC design because many target systems are refractory, diffusion‑sluggish, and metastable. Reactive flash spark plasma sintering of

$i$0

from five binary borides produced a single‑phase, homogeneous microstructure with $i$1 theoretical density in about 2 minutes, including synthesis and densification (Luo, 7 Oct 2025). Such ultrafast routes enable high‑throughput exploration of non‑equimolar composition spaces and access to metastable single‑phase states that might decompose under slower processing.

Processing can also be used to tune metastability deliberately. In refractory high‑entropy nitrides, a non‑equimolar Al‑rich composition,

$i$2

was designed to be metastable enough to spinodally decompose at elevated temperature. The resulting coherent B1 nanodomains yielded a yield strength of $i$3 GPa at $i$4, about $i$5 higher than the best equimolar nitride studied under the same conditions (Pshyk et al., 2023). This establishes a metastability‑tuned CCC strategy in which non‑equimolarity is used to activate strengthening mechanisms that equimolar systems suppress.

Related structural effects appear in ordered multicomponent silicides. Both

$i$6

and

$i$7

form single‑phase homogeneous solid solutions, but the former stabilizes the hexagonal $i$8 ($i$9) phase even though all five constituent binary silicides are stable only in tetragonal $R$0 or $R$1 structures. X‑ray diffraction, Rietveld refinement, and aberration‑corrected STEM indicate cation ordering between the two metal sublattices, so configurational entropy is reduced while unexpected phase stability is gained (Shivakumar et al., 2021).

CCC processing has also moved into interface engineering. In polymer‑derived SiC, compositionally complex TiVCrMoC$R$2 MXene nanosheets introduced at the preceramic stage and then subjected to spark plasma sintering at $R$3 and 70 MPa partially transform into a single‑phase FCC $R$4 carbide. Reconstructed carbide/SiC interfaces locally disrupt stacking and promote $R$5-SiC, whereas coherent MXene/SiC interfaces preserve cubic stacking; at optimal loading, Young’s modulus increases by about $R$6 and fracture toughness by $R$7 (Gan et al., 26 Mar 2026). This extends CCC design from bulk solid solutions to interfacial phase selection and polytype control.

5. Properties and application domains

Thermal transport remains the best established property domain of CCCs. Fluorite and pyrochlore CCCs repeatedly break the conventional low‑$R$8/low‑$R$9 trade‑off by combining strong ionic–covalent bonding with intense phonon scattering from mass disorder, size disorder, aliovalent substitution, vacancy disorder, and correlated order (Luo, 7 Oct 2025). In non‑equimolar fluorite CCCs, medium‑entropy compositions can display amorphous‑like $k_B$0 while retaining $k_B$1–230 GPa and $k_B$2 GPa, making them attractive thermal‑barrier materials (Wright et al., 2019). In ultrahigh‑entropy fluorite–pyrochlore–weberite systems, some of the lowest thermal conductivities reported for bulk ceramics occur where weberite‑type correlated disorder is strongest (Luo, 7 Oct 2025).

Mechanical performance spans both structural ceramics and UHTCs. High‑entropy diborides and monoborides exceed simple rule‑of‑mixtures hardness expectations, and dissolving softer WB$k_B$3 or MoB$k_B$4 into a high‑entropy boride matrix can further increase hardness, plausibly through correlated disorder and complex local stress fields that impede dislocation motion (Luo, 7 Oct 2025). Single‑phase high‑entropy diborides maintain flexural strength at $k_B$5 greater than three times that of ZrB$k_B$6, while dual‑phase carbide–diboride CCCs achieve still higher hardness through phase‑distribution and interface effects (Luo, 7 Oct 2025). In nitrides, non‑equimolar metastability tuning provides a distinct, high‑temperature coherency‑strengthening route (Pshyk et al., 2023).

Phonon engineering has added a further complication: more cation disorder does not always imply lower thermal conductivity. Ab initio calculations on rock‑salt compositionally complex carbides showed that constituent selection and concentration can be used to tune phonon band structure, phonon bandgap, and phonon scattering; spatial‑domain thermoreflectance measurements on selected binaries, ternaries, and quinaries even showed that some five‑component carbides have higher thermal conductivity than certain ternary and binary alloys (Malakkal et al., 6 Aug 2025). This directly challenges monotonic “more disorder, lower $k_B$7” heuristics and reinforces the CCC emphasis on targeted compositional design.

Functional applications are similarly broad. Fluorite‑based CCCs with high oxygen‑vacancy concentrations and cation disorder are relevant to solid oxide fuel cells and protonic electrochemical cells (Luo, 7 Oct 2025). Compositionally complex ferroelectric relaxors with smaller and more numerous polar nanoregions exhibit ultrahigh dielectric energy storage densities (Luo, 7 Oct 2025). In solar thermochemical water splitting, the non‑equimolar compositionally complex perovskite

$k_B$8

achieves a maximum H$k_B$9 yield of $s_{\text{config}}<1\,k_B$0 in a 1‑hour redox duration, with $s_{\text{config}}<1\,k_B$1 cycles of stability under harsh interrupted conditions; the best composition is non‑equimolar rather than entropy‑maximizing, because it balances reduction thermodynamics and oxygen exchange kinetics (Zhang et al., 2022).

CCCs also extend into correlated‑electron oxides. Y‑site alloyed YBa$s_{\text{config}}<1\,k_B$2Cu$s_{\text{config}}<1\,k_B$3O$s_{\text{config}}<1\,k_B$4 with five equimolar rare‑earths on the Y site retains $s_{\text{config}}<1\,k_B$5 K in all reported series when near optimal oxygen doping, only about $s_{\text{config}}<1\,k_B$6 below pure YBCO. This shows that a compositionally complex cuprate can preserve high‑$s_{\text{config}}<1\,k_B$7 superconductivity while independently varying size and spin disorder on a non‑CuO$s_{\text{config}}<1\,k_B$8 sublattice (Raghavan et al., 2023).

Finally, CCCs have become central to extreme‑environment design. A recent perspective on high‑entropy ceramics emphasized radiation tolerance, corrosion resistance, high‑temperature robustness, self‑healing tendencies under irradiation, and continuously tunable magnetic, electronic, and optical responses as a combined consequence of configurational entropy, local disorder, and mixed bonding (Ward et al., 2024). In this sense, CCCs unify structural and functional ceramics under a common design language.

6. Open questions and emerging directions

The major unresolved problem is mechanistic. The field has established that compositional complexity and correlated disorder can produce ultralow thermal conductivity, superhardness, improved high‑temperature strength, and ultrahigh dielectric energy density, but quantitative links between local structure and macroscopic properties remain incomplete (Luo, 7 Oct 2025). The most immediate needs are advanced characterization of SRO, vacancy order, local strain, and nanoscale domains using neutron and x‑ray total scattering, pair distribution function analysis, high‑resolution TEM/STEM, spectroscopy, and in situ diffraction or spectroscopy under temperature and stress (Luo, 7 Oct 2025).

Thermodynamic and kinetic modeling are similarly underdeveloped. The CCC literature explicitly calls for CALPHAD‑like frameworks for multicomponent ceramic systems with multiple cation sublattices, charge balance, vacancy species, SRO, and correlated disorder, integrated with DFT and atomistic simulations including machine‑learned interatomic potentials (Luo, 7 Oct 2025). For order–disorder transitions in ultrahigh‑entropy fluorite/pyrochlore systems, defect energetics, vibrational entropy, and non‑stoichiometry still need to be incorporated quantitatively (Song et al., 6 Jul 2026).

Because the accessible composition spaces are combinatorially vast, data‑driven approaches are increasingly necessary. High‑throughput synthesis enabled by ultrafast reactive sintering is already viewed as a natural experimental platform for CCC exploration (Luo, 7 Oct 2025). More generally, neural‑network kinetics has been developed for path‑dependent vacancy migration barriers and diffusion‑induced chemical ordering in compositionally complex Nb–Mo–Ta, revealing a temperature of maximum diffusion multiplicity near the point of strongest B2 ordering; this suggests one computational route for studying diffusion heterogeneity and ordering in CCCs once suitable lattice descriptions are available (Xing et al., 2023).

Application‑driven optimization remains equally open. For TBCs, the unresolved targets include simultaneous optimization of $s_{\text{config}}<1\,k_B$9, $1\,k_B \le s_{\text{config}} \le 1.5\,k_B$0, phase stability, sintering resistance, and CMAS resistance. For UHTCs, oxidation resistance, fracture toughness, strength retention, and thermal shock resistance remain to be co‑optimized within CCC frameworks. For electrochemical devices, superconductors, dielectric capacitors, and catalytic oxides, the effect of compositional complexity on transport, redox behavior, and competing orders is still only partially mapped (Luo, 7 Oct 2025). The consistent implication across the literature is that CCCs should be treated not as a narrow extension of HECs, but as a general design framework in which entropy, non‑equimolarity, ordering, defects, interfaces, and processing kinetics are co‑engineered to access property combinations unavailable in simpler ceramic systems.