THE SYMMETRIC HIERARCHY A GROUP-THEORETIC FRAMEWORK FOR PREDICTIVE DESIGN OF REFRACTORY MATERIALS (First-Principles Insights into Carbides, Spinels, and Amorphous Ceramics for Extreme Environments)

The design of materials for extreme environments requires predictive frameworks that transcend empirical trial-and-error. Here we establish a unified computational methodology that integrates abstract algebraic principles—crystallographic symmetry analysis, group-subgroup relationships, and representation theory—with first-principles density functional theory simulations to enable rational design of refractory materials. We demonstrate that the symmetric hierarchy provides a quantitative foundation for predicting solid-state reaction pathways, assessing phase stability, and engineering thermal transport properties. In transition metal carbide systems M-W-C (M = Ti, Zr, Hf, V, Nb, Ta), we show that the most probable reaction pathways minimise the symmetric gap between reactants and products, achieving 94% predictive accuracy against experimental observations. Formation energies follow the hierarchy M6W6C < M3W3C < M2W2C, with energy differences correlating directly with symmetric gap magnitudes: ΔEf = –2.12 + 0.43·Δsym (R2 = 0.94). Percolation analysis of interstitial carbon networks reveals that carbon diffusion becomes rate-limiting below the site percolation threshold pc = 0.311, explaining the transient appearance of M2W2C phases in group V systems. For entropy-stabilised spinel refractories (MgxZn(1–x)/2Cu(1–x)/2)(Al0.9Fe0.1)2O4, configurational entropy increases monotonically with x from 0.90R to 1.03R, producing a linear reduction in thermal conductivity at 1000°C from 3.20 to 2.78 W·m–1·K–1. The correlation κ = 3.85 – 1.04·Sconfig (R2 = 0.98) provides a quantitative design rule. Phonon mode analysis demonstrates that increased configurational entropy decreases phonon participation ratios from 0.88 to 0.70 for high-frequency modes, directly causing reduced thermal transport. In amorphous systems a-Si3N4 and a-SiBN, we establish that local coordination symmetry—96% tetrahedral Si sites (Td symmetry) and 92% planar N sites (D3h symmetry)—is remarkably preserved despite the absence of long-range order. BN domain segregation in a-SiBN creates symmetry-breaking interfaces that scatter phonons, reducing thermal conductivity beyond predictions from simple mixing rules: Δκ = –1.2·finterface (R2 = 0.94). For HfO2–SiO2 glasses, thermal conductivity decreases with HfO2 content despite increasing density, due to low-frequency vibrational modes from heavy Hf atoms (mass contrast mechanism). Based on these validated principles, we predict 23 new refractory compositions with projected use temperatures exceeding 2000°C, including 12 high-entropy carbides and 11 entropy-stabilised spinel oxides. The most promising candidates—(TiZrHfNbTa)C and (MgCoNiZnCu)(AlCrFeGa)2O4—have configurational entropies of 1.61R and projected thermal conductivities below 15 W·m–1·K–1 and 2 W·m–1·K–1, respectively. This work establishes symmetry analysis as a fundamental predictive tool for materials design, bridging abstract algebra and engineering applications, and provides a roadmap for the next generation of extreme-environment ceramics.