Cluster-model-embedded first-principles study of thermodynamic stability and elastic properties in (Zr, Ti)C_x carbides

Abstract

Multi-component carbide ceramics have garnered significant attention as ultra-high-temperature structural materials due to their exceptionally high melting points and excellent mechanical properties. In this work, we systematically investigate the synergistic effects of C vacancies and Ti alloying on the thermodynamic stability and elastic behavior of (Zr, Ti)C_x carbides using first-principles calculations. Specific cluster structural models of [C-M₆](C,□)₅ (M = Zr/Ti, □ = vacancy) were constructed by considering the local chemical short-range orders of elemental distribution and the ordering of vacancies on C sublattice, which were then employed as inputs for first-principles calculations. The results reveal that the introduction of C vacancies decreases the free energy at high temperatures and enhances the thermodynamic stability, whereas Ti substitution for Zr tends to reduce stability. Notably, the ternary carbide Zr₅Ti₁C₅ ([C-Zr₅Ti₁](C,□)₅) with an equimolar ratio of Ti-to-vacancy exhibits superior high-temperature thermodynamic stability. Analysis of entropy contributions indicates that both vacancies and Ti addition primarily alter the free energy by modifying the lattice vibration modes, an effect dominated by the vibrational entropy. These two types of defects weaken the M-C bond strength, resulting in reduced binding energy and Young’s modulus. Furthermore, this synergistic effect considerably lowers the critical temperature required to stabilize the single-phase solid solution structure in multi-component carbides, which is attributed to a decrease in mixing enthalpy and an increase in configurational entropy caused by vacancies. The cluster-model-embedded first-principles approach offers valuable insight for designing high-performance carbides in complex ceramic systems.

Keywords

First-principles calculations / MC-type carbides / cluster model / free energy / young’s modulus

References

[1] Wyatt BC,Hilmas GE,Anasori B.Ultra-high temperature ceramics for extreme environments.Nat Rev Mater2023;9:773-89

[2] Xiang H,Dai F.High-entropy ceramics: present status, challenges, and a look forward.J Adv Ceram2021;10:385-441

[3] Shojaie-Bahaabad M,Najafizadeh M.Ultra high temperature ceramic coatings in thermal protection systems (TPS).Ceram Int2024;50:9937-51

[4] Lun H,Xiong X.Oxidation behavior of non-stoichiometric (Zr,Hf,Ti)C_x carbide solid solution powders in air.J Adv Ceram2021;10:741-57

[5] Harrington TJ,Sarker P.Phase stability and mechanical properties of novel high entropy transition metal carbides.Acta Mater2019;166:271-80

[6] Wang XG,Kan YM,Wang PL.Densification behavior and properties of hot-pressed ZrC ceramics with Zr and graphite additives.J Eur Ceram Soc2011;31:1103-11

[7] Yang Q,Bao W.Influence of equiatomic Zr/(Ti,Nb) substitution on microstructure and ultra-high strength of (Ti,Zr,Nb)C medium-entropy ceramics at 1900 °C.J Adv Ceram2022;11:1457-65

[8] Huang S,Xiang X.Strengthening or softening: on the impact of off-stoichiometry on the mechanical properties of ZrC.Acta Mater2025;289:120892

[9] Dai F,Ren Y,Zhou Y.Segregation of solute atoms in ZrC grain boundaries and their effects on grain boundary strengths.J Mater Sci Technol2022;101:234-41

[10] Hossain MD,Oses C.Entropy landscaping of high-entropy carbides.Adv Mater2021;33:e2102904

[11] He L,Peng F.Host lattice and solid solution formation in an octal-cation (NbTaZrTiHfVWMo)C high entropy carbide ceramic.J Eur Ceram Soc2023;43:5792-801

[12] Deadmore DL.Vaporization of tantalum carbide-hafnium carbide solid solutions.J Am Ceram Soc2006;48:357-9

[13] Han X,Sedlak R.Improved creep resistance of high entropy transition metal carbides.J Eur Ceram Soc2020;40:2709-15

[14] Ye B,Huang K,Chu Y.First‐principles study, fabrication, and characterization of (Hf_0.2Zr_0.2Ta_0.2Nb_0.2Ti_0.2)C high‐entropy ceramic.J Am Ceram Soc2019;102:4344-52

[15] Zeng Y,Xiong X.Ablation-resistant carbide Zr_0.8Ti_0.2C_0.74B_0.26 for oxidizing environments up to 3,000 °C.Nat Commun2017;8:15836 PMCID:PMC5474735

[16] Sheindlin M,Petukhov S,Vasin A.Recent advances in the study of high-temperature behaviour of non-stoichiometric TaC_x, HfC_x and ZrC_x carbides in the domain of their congruent melting point.Adv Appl Ceram2018;117:s48-55

[17] Liu D,Meng J,Han J.The significant influence of carbon content on mechanical and thermal properties of (VNbTaMoW)_0.5C_x high entropy carbides.J Eur Ceram Soc2022;42:5262-72

[18] Schmid B,Ntemou E.Mechanical properties of VC/ZrC and VC/HfC superlattices.Acta Mater2024;270:119852

[19] Wang Z,Zhang L.Tailoring structural, mechanical, and electronic properties of (Ti_0.25Ta_0.25Zr_0.25Nb_0.25-La )C high-entropy carbide ceramics via rare-earth La incorporation: a comprehensive first-principles study.Ceram Int2025;51:33721-30

[20] Chen L,Liu Q.Covalent bonds enhancing and microstructure evolution induced by carbon content in multi-component (TiZrNbMo)C ceramics.J Materiomics2025;11:101048

[21] Zhang Y,Wang J.Theoretical investigations of the effects of ordered carbon vacancies in ZrC_1-x on phase stability and thermo-mechanical properties.Acta Mater2016;111:232-41

[22] Zunger A,Ferreira LG.Special quasirandom structures.Phys Rev Lett1990;65:353-6

[23] Kim J.First-principles investigation of the elastic properties and phase stability of (Ti_1-xNi_x)C_1-y ternary metastable carbides.J Alloys Compd2021;853:157349

[24] Liu S,Zhang H.Design of superhard high-entropy diborides via high-throughput DFT and thermodynamics calculations.Ceram Int2024;50:17977-87

[25] Liu S,Liu S.Stability and mechanical properties of single-phase quinary high-entropy metal carbides: first-principles theory and thermodynamics.J Eur Ceram Soc2022;42:3089-98

[26] Liu Y,Wang WY.Effects of solutes on thermodynamic properties of (TMZrU)C (TM = Ta, Y) medium-entropy carbides: a first-principles study.J Mater Inf2023;3:17

[27] Meng H.Surface energies in high‐entropy carbides with variable carbon stoichiometry.J Am Ceram Soc2022;105:5835-42

[28] Ma Y,Jiang B.Controlled formation of coherent cuboidal nanoprecipitates in body-centered cubic high-entropy alloys based on Al₂(Ni,Co,Fe,Cr)₁₄ compositions.Acta Mater2018;147:213-25

[29] Yuan J,Yang Y.Applications of machine learning method in high-performance materials design: a review.J Mater Inf2024;4:14

[30] Wang WY,Wang Y.Atomic and electronic basis for the serrations of refractory high-entropy alloys.NPJ Comput Mater2017;3:23

[31] Bie X,Zhou X.Sub-stoichiometry and vacancy structures in V/Nb carbide precipitates by cluster expansion and first-principles calculations.Acta Mater2024;269:119806

[32] Hu W,Zhang Y.Superstructural nanodomains of ordered carbon vacancies in nonstoichiometric ZrC_0.61.J Mater Res2012;27:1230-6

[33] Gusev AI.Phase equilibria, phases and compounds in the Ti-C system.Russ Chem Rev2002;71:439-63

[34] Gusev AI.Order-disorder phase transition channel in niobium carbide.Phys Stat Sol1986;93:71-80

[35] Xiang J,Liu S.Spark plasma sintering of the nonstoichiometric ultrafine-grained titanium carbides with nano superstructural domains of the ordered carbon vacancies.Mater Chem Phys2011;130:352-60

[36] Hong Q.Prediction of the material with highest known melting point fromab initiomolecular dynamics calculations.Phys Rev B2015;92:020104

[37] Gusev AI.Superstructures of non-stoichiometric interstitial compounds and the distribution functions of interstitial atoms.Phys Stat Sol1993;135:15-58

[38] Zeng Y,Li G,Sun W.Microstructure and ablation behavior of carbon/carbon composites infiltrated with Zr-Ti.Carbon2013;54:300-9

[39] Kresse G.Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set.Comput Mater Sci1996;6:15-50

[40] Kresse G.Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set.Phys Rev B Condens Matter1996;54:11169-86

[41] Kresse G.From ultrasoft pseudopotentials to the projector augmented-wave method.Phys Rev B1999;59:1758-75

[42] Perdew JP,Ernzerhof M.Generalized gradient approximation made simple.Phys Rev Lett1996;77:3865-8

[43] Momma K.VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data.J Appl Crystallogr2011;44:1272-6

[44] Wang V,Liu J,Geng W.VASPKIT: a user-friendly interface facilitating high-throughput computing and analysis using VASP code.Comput Phys Commun2021;267:108033

[45] Grimme S,Goerigk L.Effect of the damping function in dispersion corrected density functional theory.J Comput Chem2011;32:1456-65

[46] Ma D,Körmann F,Raabe D.Ab initio thermodynamics of the CoCrFeMnNi high entropy alloy: importance of entropy contributions beyond the configurational one.Acta Mater2015;100:90-7

[47] Togo A.First principles phonon calculations in materials science.Scr Mater2015;108:1-5

[48] Togo A,Tanaka I.First-principles phonon calculations of thermal expansion in Ti₃SiC₂,Ti₃AlC₂, and Ti₃GeC₂.Phys Rev B2010;81:174301

[49] Olsson P,Bjerkén C.Ab initio thermodynamics investigation of titanium hydrides.Comput Mater Sci2015;97:263-75

[50] Ye B,Nguyen MC,Wang C.First-principles study, fabrication and characterization of (Zr_0.25Nb_0.25Ti_0.25V_0.25)C high-entropy ceramics.Acta Mater2019;170:15-23

[51] Page Y, Saxe P. Symmetry-general least-squares extraction of elastic data for strained materials fromab initio calculations of stress.Phys Rev B2002;65:104104

[52] Wu X,Hamann DR.Systematic treatment of displacements, strains, and electric fields in density-functional perturbation theory.Phys Rev B2005;72:035105

[53] Zakarian D,Firstov S.Universal temperature dependence of Young’s modulus.Metal Powder Rep2019;74:204-6

[54] Réjasse F,Léchelle J.Study of the TiC_1-x - TiO₂ reactive interface.Acta Mater2018;146:225-36

[55] Luo Z,Mao H,Peng Y.Phase field simulation of the lamellar precipitation in the TiC-ZrC system.Ceram Int2018;44:22041-4

[56] Lei J,Brittain JO.The electrical resistance of the group IV transition metal monocarbides and mononitrides in the temperature range 20-1000 °C.Mater Sci Eng A1990;123:129-40

[57] Mahday AA,Ahmed H.Mechanically induced solid state carburization for fabrication of nanocrystalline ZrC refractory material powders.J Alloys Compd2000;299:244-53

[58] Sara RV.The system zirconium - carbon.J Am Ceram Soc2006;48:243-7

[59] Zhou Y,Li C,Hilmas GE.Vacancy ordering in zirconium carbide with different carbon contents.J Eur Ceram Soc2023;43:5814-21

[60] Lun H,Zeng Y,Wang Q.Mechanisms responsible for enhancing low‐temperature oxidation resistance of nonstoichiometric (Zr,Ti)C.J Am Ceram Soc2022;105:5309-24

[61] Lu W,Huang S.The coupling of carbon non-stoichiometry and short-range order in governing mechanical properties of high-entropy ceramics.NPJ Comput Mater2025;11:64

[62] Kim H,Kim J.Enhancing the hydrogen storage properties of (Ti, M)C_1-x materials (M = W, Mg, Ni, and Al) depends on the carbon vacancies: potential for the recycling of scraps.Int J Refract Met Hard Mater2024;118:106475

[63] Baranov VM,Korostin OS.The temperature dependence of the elastic constants of nonstoichiometric zirconium carbides.Strength Mater1973;5:1074-7

[64] Sangiovanni D,Harrington T,Vecchio K.Temperature-dependent elastic properties of binary and multicomponent high-entropy refractory carbides.Mater Des2021;204:109634

[65] Evitts L,Kardoulaki E,Rushton M.Influence of boron isotope ratio on the thermal conductivity of uranium diboride (UB₂) and zirconium diboride (ZrB₂).J Nucl Mater2020;528:151892

[66] Esters M,Hicks D.Settling the matter of the role of vibrations in the stability of high-entropy carbides.Nat Commun2021;12:5747 PMCID:PMC8484556

[67] Markström A,Frisk K.Combined ab-initio and experimental assessment of A_1-xB_xC mixed carbides.Calphad2008;32:615-23

[68] Razumovskiy V,Odqvist J,Korzhavyi P.Effect of carbon vacancies on thermodynamic properties of TiC-ZrC mixed carbides.Calphad2014;46:87-91