Disordered enthalpy-entropy descriptor for high-entropy ceramics discovery

High-entropy ceramics offer exceptional performance in extreme environments (thermal barrier protection, wear and corrosion resistance, thermoelectrics, batteries, catalysts) but their discovery has been predominantly experimental. Prior computational success was largely limited to high-entropy carbides using the entropy-forming-ability (EFA) descriptor. The authors identify the need for a general, theory-guided approach to predict when multicomponent ceramics will form single-phase solid solutions under a given synthesis route. They introduce the concept of functional synthesizability—synthesizability defined relative to a specific process (here, hot-pressed sintering of transition-metal disordered ceramics)—and propose a new descriptor, DEED, that balances entropy gains with enthalpy costs across chemistries and structures to predict solid-solution formation versus multiphase outcomes.

The EFA descriptor (2018) effectively ranked solid-solution formation in high-entropy carbides but struggled for systems with non-homogeneous enthalpy landscapes. Historically, limited availability of ab initio convex-hull data for high-entropy ceramics hindered rigorous assessment of enthalpy costs. The authors address this by expanding the aflow.org repository with appropriate thermodynamic data and computational modules for recursive phase diagram searches, enabling a descriptor that simultaneously accounts for the thermodynamic density of states spread (entropy gain) and distances to the convex hull (enthalpy cost). Alternative rigorous techniques (cluster expansion; Lederer-Toher-Vecchio-Curtarolo) are computationally prohibitive for the very large configuration spaces of multicomponent systems, motivating the use of AFLOW POCC and the new convolutional cPOCC acceleration.

Descriptor formulation and computation:

• The thermodynamic density of states Ω(E)δE measures the number of configurationally excited states in an energy interval. Using AFLOW POCC, a random alloy is represented by an ensemble of ordered representative states (POCC tiles) computed by DFT.
• Statistical moments of the POCC energy spectrum relative to the convex hull are computed: the average distance to the hull <ΔH_hull> as enthalpy cost, and the variance (or standard deviation σ) of the energy spectrum as a measure of configurational spread (inverse spread ~ entropy gain). DEED is defined as the ratio between entropy gain and enthalpy loss and is physically connected to the inverse of an order–disorder temperature (miscibility gap critical temperature surrogate).
• Compensation temperature Θ is introduced as Θ = [k × DEED]^{-1}, expected to correlate monotonically with the miscibility gap critical temperature T_mg.
• To make calculations tractable for large composition spaces, the authors develop convolutional POCC (cPOCC), which partitions the problem into subsystems (e.g., M, C2N, and M_xC_yN_z for rock-salt carbonitrides), each with a much smaller set of unique tiles. Energies are then convolved to approximate the full spectrum, drastically reducing the number of DFT evaluations while retaining accuracy.
• Global stoichiometries were chosen equiatomic on the transition-metal sublattice, and crystal structures were fixed to those relevant for the target classes at those compositions (rock salt for carbides and carbonitrides; AlB2 for borides).

Computational campaign:

• Characterized 952 disordered ceramics within AFLOW++: 548 metal carbides, 70 metal carbonitrides, and 334 metal borides. DEED, EFA (σ^+), Θ, and convex-hull distances were computed and compared.

Experimental validation – carbonitrides:

• Selected 9 new carbonitrides for synthesis: 5 with large DEED (targeting single-phase) and 4 with small DEED and structural incompatibility among precursors (targeting multiphase/microstructure formation).
• Precursors: TiC, VC, NbC, TaC, WC (hP2), Mo2C (oP12), TiN, ZrN, HfN, NbN (hP4), TaN (hP8); +5 at% C to aid oxide reduction without destabilizing phases.
• Processing: dry-mixed 24 h at 200 rpm (ball-to-powder 9:1) with 2 mm YSZ media in 125 ml HDPE bottles; sieved 60-mesh; hot pressing via FAST at various soaks (e.g., 2200 °C for 30 min; 2300 °C for 60 min). XRD and SEM used for phase and microstructure analysis. AFLOW-Spinodal used to quantify lamellar wavelengths in pearlite-like microstructures.

Experimental validation – borides:

• Selected 8 new borides: four with highest DEED (expected single-phase): (HfMoNbTaZr)B2, (HfNbTaTiV)B2, (CrMoTiVW)B2, (CrHfNbTiZr)B2; and four from lower DEED (expected multiphase): (CrHfMoTaW)B2, (HfMnTiVZr)B2, (CrHfTiYZr)B2, (CrHfNbVY)B2.
• Precursors: MnO2, WO3, HfO2, Nb2O5, Ta2O5, TiO2, ZrO2, V2O3, Cr2O3, MoO3, Y2O3; reductants: carbon black and B, C.
• Processing: ball-milled mixtures pressed into discs at 2 MPa; boro-/carbothermal reduction at 1660 °C for 2.5 h under mild vacuum; two-step SPS: 1650 °C under 15 MPa for 5 min, then 1900 °C under 50 MPa for 10 min; XRD and SEM characterization.

Analysis and benchmarking:

• Compared DEED’s predictions with EFA and VEC against literature and new experiments. Examined misclassification regions and correlation of Θ with observed single-phase versus multiphase formation and microstructures.

• DEED descriptor: A thermodynamically grounded ratio balancing entropy gain (inverse spectral spread) against enthalpy cost (average distance to convex hull) provides an observable linked to the order–disorder transition; its inverse correlates with the miscibility gap critical temperature. Larger DEED predicts easier solid-solution formation under hot-pressed sintering.
• Scope: Computed DEED for 952 ceramics (548 carbides, 70 carbonitrides, 334 borides).
• Performance vs. baselines: DEED consistently outperforms EFA and VEC in ranking functional synthesizability across carbides, carbonitrides, and borides, with notably smaller misclassification regions; VEC shows large misclassification, especially for borides.
• Carbides: DEED improves ordering by penalizing Group VIB components (Cr, Mo, W) that require structural transformations, which appear near the bottom of rankings; Θ trends align with experimental transition behavior (e.g., Θ[(HfNbTaTiZr)C] < Θ[(HfScTaTiZr)C] < Θ[(HfNbTaTiV)C] < Θ[(MoNbTaVW)C]).
• Carbonitrides (9 synthesized): All five high-DEED systems formed single-phase rock-salt FCC matrices: (HfNbTiVZr)CN, (HfNbTaTiV)CN, (NbTaTiVZr)CN, (HfTaTiVZr)CN, (MoNbTaTiZr)CN. Four low-DEED systems formed multiphase microstructures with FCC and hexagonal WC phases: (HfTaTiWZr)CN, (MoNbTiWZr)CN, (HfTiVWZr)CN, (HfMoNbTaW)CN. W-rich regions appeared as irregular grains; pearlite-like lamellae observed with principal wavelengths ≥0.5 μm, consistent with synthesis within the miscibility gap.
• Borides (8 synthesized): High-DEED predictions validated: (HfMoNbTaZr)B2, (HfNbTaTiV)B2, (CrMoTiVW)B2, (CrHfNbTiZr)B2 showed clean AlB2 hexagonal phases. Low-DEED systems displayed multiple phases or mixed AlB2 variants: (CrHfMoTaW)B2, (HfMnTiVZr)B2, (CrHfTiYZr)B2, (CrHfNbVY)B2, with corresponding microstructural signatures (e.g., grain size reduction due to pinning, core-shelling, microcracking).
• New single-phase discoveries guided by DEED: five carbonitrides and four borides listed above.
• Practical utility: DEED can also map effects of changing chemical potentials (e.g., added carbon) by recomputing stability landscapes and moments, predicting if single-phase stability persists or if lower-solubility phases are favored.

By defining functional synthesizability from the process perspective and introducing DEED as a balance of entropy and enthalpy contributions, the study provides a predictive, physically meaningful metric closely tied to the order–disorder transition and miscibility gap. DEED’s superior classification across distinct chemistries and structures (carbides, carbonitrides, borides) demonstrates its generality. Its correlation with an effective transition temperature (Θ) explains why larger DEED values correspond to single-phase formation under given synthesis conditions, while smaller DEED values predict multiphase products and even microstructural motifs (lamellar pearlite-like structures) consistent with synthesis inside a miscibility gap. The successful synthesis and characterization of 17 previously unexplored systems validate the predictive power of DEED and highlight its advantage over EFA and VEC. Integration in AFLOW++ and the cPOCC acceleration enable rapid prescreening and guide experimental efforts, reducing trial-and-error and enabling structure-property predictions for new high-entropy ceramics.

The work introduces DEED, a thermodynamic descriptor that captures the entropy–enthalpy balance governing solid-solution formation and frames synthesizability as a process-dependent property. Using AFLOW POCC and the convolutional cPOCC algorithm, the authors computed DEED for 952 candidate ceramics and experimentally validated predictions by synthesizing nine carbonitrides and eight borides. DEED accurately classified functional synthesizability, outperforming EFA and VEC, and guided the experimental discovery of single-phase compositions: (HfNbTiVZr)CN, (HfTaTiVZr)CN, (HfNbTaTiV)CN, (NbTaTiVZr)CN, (MoNbTaTiZr)CN, (HfMoNbTaZr)B2, (HfNbTaTiV)B2, (CrMoTiVW)B2, and (CrHfNbTiZr)B2. The approach provides many new predictions for future exploration, and its extension to other disordered ceramics (e.g., oxides) is straightforward given coordination-corrected enthalpy formalisms and DEED/cPOCC integration in AFLOW++.

• Θ is an estimate of the miscibility gap critical temperature and inherits approximations from DEED: finite configurational sampling, neglect of precursor-specific effects, and finite-size effects in calculations.
• For increasing numbers of species, exact spectrum calculations become impractical; cPOCC provides an efficient approximation but remains an approximation of the full configurational landscape.
• Synthesizability is process-dependent; mixed reports in the literature for some borides indicate sensitivity to specific processing routes and conditions, which may shift thresholds for single- vs multiphase formation.