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arxiv: 2604.08961 · v1 · submitted 2026-04-10 · ❄️ cond-mat.mtrl-sci

Grain Growth Kinetics in (Cr,Mo,Ta,V,W)C1-{δ} High-Entropy Carbide Ceramics

Ali Sarikhani , Gregory E. Hilmas , David W. Lipke , Douglas E. Wolfe , Stefano Curtarolo , Shen J. Dillon , Ahmad Mirzaei , William G. Fahrenholtz This is my paper

Pith reviewed 2026-05-10 17:30 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords grain growth kineticshigh-entropy carbidesspark plasma sinteringactivation energydiffusion-controlled growthmicrostructure evolutiondensification
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The pith

Grain growth in high-entropy (Cr,Mo,Ta,V,W)C carbide ceramics is diffusion-controlled with an activation energy of 620 kJ/mol.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This work studies grain growth and densification in single-phase (Cr,Mo,Ta,V,W)C1-δ high-entropy carbide ceramics processed by spark plasma sintering at temperatures from 1750 to 1950 °C for 10 minutes. The authors apply a normal grain growth model assuming a growth exponent of n=3 to extract temperature-dependent growth rates and perform Arrhenius analysis. They report an apparent activation energy of about 620 kJ/mol, similar to diffusion in other refractory carbides, along with rapid initial densification followed by temperature-driven grain coarsening and improved chemical homogeneity. These quantitative kinetics provide a basis for predicting and controlling microstructure in ultra-high-temperature carbide materials.

Core claim

The paper establishes that grain growth kinetics in (Cr,Mo,Ta,V,W)C1-δ high-entropy carbide follow the normal grain growth equation with n=3, leading to an apparent activation energy of approximately 620 kJ mol^{-1} consistent with diffusion-controlled processes. Elemental mapping indicates reduced Ta segregation at higher temperatures, and densification occurs mostly before the peak temperature is reached.

What carries the argument

Normal grain growth model with assumed exponent n=3 applied to temperature series of grain sizes, followed by Arrhenius plot of the growth factor.

If this is right

  • Grain coarsening is dominated by temperature after initial rapid densification.
  • Chemical homogenization improves with increasing sintering temperature.
  • Microstructural stability can be related to diffusion kinetics in high-entropy carbides.
  • Quantitative links between sintering temperature and grain size enable process control.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the activation energy reflects bulk diffusion, then grain growth rates in similar high-entropy systems could be predicted from known diffusion data of constituent elements.
  • Testing other growth exponents on the same dataset might reveal whether boundary pinning or other mechanisms dominate.
  • The observed lattice parameter increase suggests possible vacancy annihilation or compositional adjustments during sintering that warrant further property measurements.

Load-bearing premise

The grain growth data conform to the normal grain growth model specifically with exponent n=3, rather than other values or mechanisms like pore drag or abnormal growth.

What would settle it

Plotting log(grain size) versus log(time) at fixed temperature to verify if the slope is 1/3 as assumed, or obtaining independent diffusion activation energies for comparison to 620 kJ/mol.

Figures

Figures reproduced from arXiv: 2604.08961 by Ahmad Mirzaei, Ali Sarikhani, David W. Lipke, Douglas E. Wolfe, Gregory E. Hilmas, Shen J. Dillon, Stefano Curtarolo, William G. Fahrenholtz.

Figure 3
Figure 3. Figure 3: Elemental homogeneity and average EDS quantification for the S1950 specimen. Representative EDS elemental maps for the specimen sintered at 1950 °C (S1950), showing spatially uniform distributions of Cr, Mo, Ta, V, and W across the analyzed field of view. The accompanying plot reports the average elemental composition measured by EDS (all-atom basis) for all specimens. Error bars represent the standard dev… view at source ↗
read the original abstract

Understanding grain-boundary mobility during spark plasma sintering can enable microstructure control in high-entropy carbides, yet quantitative grain-growth kinetics remain scarce. In this work, grain growth kinetics and densification behavior were investigated for single-phase fully dense (Cr,Mo,Ta,V,W)C1-{\delta} high-entropy carbide ceramics. Specimens were densified by spark plasma sintering for a constant dwell time of 10 min at temperatures between 1750 {\deg}C and 1950 {\deg}C to isolate the role of temperature on microstructural evolution. Increasing sintering temperature produced grain growth and increased lattice parameter, while maintaining a single-phase rock salt structure. Elemental mapping showed a progressive reduction of Ta segregation with increasing sintering temperature, suggesting enhanced chemical homogenization at elevated temperatures. Grain growth kinetics were analyzed using a normal grain growth model with an assumed growth exponent of n=3, physically reasonable for grain-boundary-controlled growth influenced by solute and vacancy pinning. Arrhenius analysis of the growth factor yielded an apparent activation energy of approximately 620 kJ mol-1, comparable to diffusion-controlled processes in refractory transition-metal carbides. Densification curves revealed rapid consolidation prior to reaching the peak temperature followed by temperature-dominated grain coarsening. These results establish quantitative relationships between densification temperature, grain growth, and diffusion kinetics in a carbide system, providing insight into the microstructural stability of high-entropy, ultra-high-temperature carbide ceramics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates grain growth and densification in single-phase (Cr,Mo,Ta,V,W)C_{1-δ} high-entropy carbide ceramics densified by spark plasma sintering. Samples were held for a fixed 10 min dwell at temperatures from 1750 °C to 1950 °C. Grain growth is analyzed via the normal grain growth equation with the growth exponent fixed at n=3; an Arrhenius plot of the resulting growth factor yields an apparent activation energy of ~620 kJ mol^{-1}. The work also reports temperature-dependent lattice expansion, reduction in Ta segregation, and rapid early densification followed by coarsening.

Significance. If the extracted activation energy is robust, the study supplies one of the few quantitative kinetic benchmarks for grain-boundary mobility in high-entropy carbides, directly comparable to diffusion data in binary refractory carbides. Such numbers are useful for process modeling and microstructure design in ultra-high-temperature ceramics.

major comments (2)
  1. [Grain growth kinetics analysis (abstract and corresponding results section)] The central activation-energy result rests on the assumption that n=3 in the normal grain growth model (G^n - G_0^n = Kt). With grain-size data reported only after a single fixed 10 min dwell at each temperature, n cannot be fitted from the measurements and must be imposed a priori. Because the growth factor scales with n, an incorrect choice alters the temperature dependence of the plotted quantity and therefore the slope that yields Q ≈ 620 kJ mol^{-1}. No residual comparison for n=2 or n=4, no multi-time isothermal series, and no sensitivity analysis are described.
  2. [Grain growth kinetics analysis] The manuscript states that n=3 is 'physically reasonable for grain-boundary-controlled growth influenced by solute and vacancy pinning,' yet provides no supporting evidence from the present data set (e.g., goodness-of-fit metrics or literature values specific to this five-cation carbide). Because the activation energy is the primary quantitative claim, the untested exponent choice is load-bearing.
minor comments (2)
  1. [Abstract] The abstract reports an activation energy 'of approximately 620 kJ mol^{-1}' but supplies neither the raw grain-size values, the number of grains measured per condition, nor error bars on the growth factor; these should appear in the main text or a supplementary table.
  2. [Densification behavior] The densification curves are described qualitatively ('rapid consolidation prior to reaching the peak temperature'); quantitative plots of relative density versus time or temperature would strengthen the separation of densification and grain-growth regimes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review of our manuscript on grain growth kinetics in (Cr,Mo,Ta,V,W)C_{1-δ} high-entropy carbide ceramics. The comments focus on the grain growth analysis, which we address point by point below. We will revise the manuscript to improve the justification and include additional analysis as outlined.

read point-by-point responses

  1. Referee: The central activation-energy result rests on the assumption that n=3 in the normal grain growth model (G^n - G_0^n = Kt). With grain-size data reported only after a single fixed 10 min dwell at each temperature, n cannot be fitted from the measurements and must be imposed a priori. Because the growth factor scales with n, an incorrect choice alters the temperature dependence of the plotted quantity and therefore the slope that yields Q ≈ 620 kJ mol^{-1}. No residual comparison for n=2 or n=4, no multi-time isothermal series, and no sensitivity analysis are described.

    Authors: We agree that n cannot be fitted directly from our dataset, which uses a single fixed 10 min dwell time at each temperature to isolate the effect of sintering temperature. The choice of n=3 follows standard practice for grain-boundary diffusion-controlled growth with pinning in refractory carbides. In the revised manuscript, we will add a sensitivity analysis recalculating the growth factor and apparent activation energy for n=2, n=3, and n=4, including a discussion of how Q varies with the assumed exponent. We cannot add new multi-time isothermal series without additional experiments, but the sensitivity analysis will quantify the robustness of the reported Q value. revision: partial

  2. Referee: The manuscript states that n=3 is 'physically reasonable for grain-boundary-controlled growth influenced by solute and vacancy pinning,' yet provides no supporting evidence from the present data set (e.g., goodness-of-fit metrics or literature values specific to this five-cation carbide). Because the activation energy is the primary quantitative claim, the untested exponent choice is load-bearing.

    Authors: We acknowledge that the original text lacked explicit supporting references or quantitative assessment. Goodness-of-fit metrics are not feasible without time-series data at constant temperature. In revision, we will cite relevant literature on grain growth exponents in binary and multicomponent refractory carbides (including cases with solute pinning) where n=3 has been applied to similar diffusion-controlled mechanisms. Combined with the sensitivity analysis, this will provide stronger justification for the choice of n=3 and its impact on the activation energy. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper measures grain sizes after fixed 10 min dwells at multiple temperatures, assumes n=3 a priori in the normal grain growth equation to compute a growth factor K at each temperature, then extracts activation energy Q from the slope of an Arrhenius plot of those K values. This chain relies on experimental data plus an explicitly stated conventional assumption rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. No equations reduce to their own inputs by construction, and the abstract provides no evidence of uniqueness theorems or ansatzes imported from prior author work. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The kinetic analysis rests on the domain assumption that normal grain growth with n=3 applies and that Arrhenius behavior can be extracted from the limited temperature series; no free parameters beyond the assumed exponent are introduced.

free parameters (1)
  • grain growth exponent n
    Fixed at 3 because it is described as physically reasonable for grain-boundary-controlled growth with solute and vacancy pinning; no fit to other values is shown.
axioms (1)
  • domain assumption Normal grain growth model with n=3 describes the observed coarsening
    Invoked to convert grain-size data into a growth factor for Arrhenius analysis.

pith-pipeline@v0.9.0 · 5601 in / 1330 out tokens · 81550 ms · 2026-05-10T17:30:23.400743+00:00 · methodology

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    High-entropy high-hardness metal carbides discovered by entropy descriptors,

    P. Sarker, T.J. Harrington, C. Toher, C. Oses, M. Samiee, J.-P. Maria, D.W. Brenner, K.S. Vecchio, and S. Curtarolo, “High-entropy high-hardness metal carbides discovered by entropy descriptors,” Nature Communications 9, 4980 (2018)

    work page 2018

  2. [2]

    Hardness of single phase high entropy carbide ceramics with different compositions,

    P.M. Brune, G.E. Hilmas, W.G. Fahrenholtz, J.L. Watts, C.J. Ryan, C.M. DeSalle, D.E. Wolfe, and S. Curtarolo, “Hardness of single phase high entropy carbide ceramics with different compositions,” Journal of Applied Physics 135(16), 165106 (2024)

    work page 2024

  3. [3]

    High-entropy carbide: A novel class of multicomponent ceramics,

    J. Zhou, J. Zhang, F. Zhang, B. Niu, L. Lei, and W. Wang, “High-entropy carbide: A novel class of multicomponent ceramics,” Ceramics International 44(17), 22014-22018 (2018). 23

    work page 2018

  4. [4]

    Processing and Properties of High-Entropy Ultra-High Temperature Carbides,

    E. Castle, T. Csanádi, S. Grasso, et al., “Processing and Properties of High-Entropy Ultra-High Temperature Carbides,” Scientific Reports 8, 8609 (2018)

    work page 2018

  5. [5]

    Microstructural development in equiatomic multicomponent alloys,

    B. Cantor, I.T.H. Chang, P. Knight, and A.J.B. Vincent, “Microstructural development in equiatomic multicomponent alloys,” Materials Science and Engineering A 375-377, 213- 218 (2004)

    work page 2004

  6. [6]

    Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes,

    J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, C.H. Tsau, and S.Y. Chang, “Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes,” Advanced Engineering Materials 6(5), 299-303 (2004)

    work page 2004

  7. [7]

    A universal configurational entropy metric for high- entropy materials,

    O.F. Dippo and K.S. Vecchio, “A universal configurational entropy metric for high- entropy materials,” Scripta Materialia 201, 113974 (2021)

    work page 2021

  8. [8]

    High-entropy ceramics,

    C. Oses, C. Toher, and S. Curtarolo, “High-entropy ceramics,” Nature Reviews Materials 5, 295-309 (2020)

    work page 2020

  9. [9]

    High-entropy ceramics: propelling applications through disorder,

    C. Toher, C. Oses, M. Esters, D. Hicks, G. Kotsonis, C.M. Rost, D.W. Brenner, J.-P. Maria, and S. Curtarolo, “High-entropy ceramics: propelling applications through disorder,” MRS Bulletin 47, 194-202 (2022)

    work page 2022

  10. [10]

    Enhanced Hardness in High-Entropy Carbides through Atomic Randomness,

    Y. Wang, T. Csanádi, H. Zhang, J. Dusza, M.J. Reece, and R.-Z. Zhang, “Enhanced Hardness in High-Entropy Carbides through Atomic Randomness,” Advanced Theory and Simulations 3, 2000111 (2020)

    work page 2020

  11. [11]

    Low-temperature densification of high- entropy (Ti,Zr,Nb,Ta,Mo)C-Co composites with high hardness and high toughness,

    S.C. Luo, W.M. Guo, K. Plucknett, et al., “Low-temperature densification of high- entropy (Ti,Zr,Nb,Ta,Mo)C-Co composites with high hardness and high toughness,” Journal of Advanced Ceramics 11, 805-813 (2022). 24

    work page 2022

  12. [12]

    Phase stability, mechanical properties and melting points of high-entropy quaternary metal carbides from first principles,

    S.-Y. Liu, S. Zhang, S. Liu, D.-J. Li, Y. Li, and S. Wang, “Phase stability, mechanical properties and melting points of high-entropy quaternary metal carbides from first principles,” Journal of the European Ceramic Society 41(13), 6267-6274 (2021)

    work page 2021

  13. [13]

    The role of entropy and enthalpy in high entropy carbides,

    X. Tang, G.B. Thompson, K. Ma, and C.R. Weinberger, “The role of entropy and enthalpy in high entropy carbides,” Computational Materials Science 210, 111474 (2022)

    work page 2022

  14. [14]

    Optimized process and superior toughness of a (HfNbTaTiW)C high entropy carbide ceramic,

    J. Li, L. He, F. Peng, B. Liu, Q. Li, S. Zhao, and Z. Wu, “Optimized process and superior toughness of a (HfNbTaTiW)C high entropy carbide ceramic,” Ceramics International 50(18A), 32129-32137 (2024)

    work page 2024

  15. [15]

    Phase stability and mechanical properties of novel high entropy transition metal carbides,

    T.J. Harrington, J. Gild, P. Sarker, C. Toher, C.M. Rost, O.F. Dippo, C. McElfresh, K. Kaufmann, E. Marin, L. Borowski, P.E. Hopkins, J. Luo, S. Curtarolo, D.W. Brenner, and K.S. Vecchio, “Phase stability and mechanical properties of novel high entropy transition metal carbides,” Acta Materialia 166, 271-280 (2019)

    work page 2019

  16. [16]

    Preparation of High-Entropy (Ti, Zr, Hf, Ta, Nb) Carbide Powder via Solution Chemistry,

    P. Šolcová, M. Nižňanský, J. Schulz, P. Brázda, P. Ecorchard, M. Vilémová, and V. Tyrpekl, “Preparation of High-Entropy (Ti, Zr, Hf, Ta, Nb) Carbide Powder via Solution Chemistry,” Inorganic Chemistry 60(11), 7617-7621 (2021)

    work page 2021

  17. [17]

    Low-temperature sintering of single- phase, high-entropy carbide ceramics,

    L. Feng, W.G. Fahrenholtz, and G.E. Hilmas, “Low-temperature sintering of single- phase, high-entropy carbide ceramics,” Journal of the American Ceramic Society 102, 7217-7224 (2019)

    work page 2019

  18. [18]

    Preparation of high-entropy carbides by different sintering techniques,

    J. Pötschke, M. Dahal, M. Herrmann, et al., “Preparation of high-entropy carbides by different sintering techniques,” Journal of Materials Science 56, 11237-11247 (2021). 25

    work page 2021

  19. [19]

    Thermal and electrical properties of a high entropy carbide (Ta, Hf, Nb, Zr) at elevated temperatures,

    E. Schwind, M.J. Reece, E. Castle, W.G. Fahrenholtz, and G.E. Hilmas, “Thermal and electrical properties of a high entropy carbide (Ta, Hf, Nb, Zr) at elevated temperatures,” Journal of the American Ceramic Society 105, 4426-4434 (2022)

    work page 2022

  20. [20]

    Thermal and electrical properties of single-phase high entropy carbide ceramics,

    P.M. Brune, G.E. Hilmas, W.G. Fahrenholtz, J.L. Watts, and S. Curtarolo, “Thermal and electrical properties of single-phase high entropy carbide ceramics,” Journal of the American Ceramic Society 107, 5893-5902 (2024)

    work page 2024

  21. [21]

    Thermal and Electrical Properties of (Cr,Mo,Ta,V,W)C High-Entropy Carbide Ceramics,

    A. Sarikhani, S.M. Smith, S. Filipovic, W.G. Fahrenholtz, and G.E. Hilmas, “Thermal and Electrical Properties of (Cr,Mo,Ta,V,W)C High-Entropy Carbide Ceramics,” Journal of the European Ceramic Society, 118347 (2026)

    work page 2026

  22. [22]

    Disordered enthalpy-entropy descriptor for high-entropy ceramics discovery,

    S. Divilov, H. Eckert, D. Hicks, C. Oses, C. Toher, R. Friedrich, M. Esters, M.J. Mehl, A.C. Zettel, Y. Lederer, E. Zurek, J.-P. Maria, D.W. Brenner, X. Campilongo, S. Filipović, W.G. Fahrenholtz, C.J. Ryan, C.M. DeSalle, R.J. Crealese, D.E. Wolfe, A. Calzolari, and S. Curtarolo, “Disordered enthalpy-entropy descriptor for high-entropy ceramics discovery,...

    work page 2024

  23. [23]

    The Development of Hard Materials and Fibrous Monolith Composites for Friction Stir Welding,

    P.M. Brune, “The Development of Hard Materials and Fibrous Monolith Composites for Friction Stir Welding,” Doctoral dissertation, Missouri University of Science and Technology, Dissertation 3340 (2024)

    work page 2024

  24. [24]

    Grain growth kinetics and densification mechanism of (TiZrHfVNbTa)C high-entropy ceramic under pressureless sintering,

    W. Zhang, L. Chen, C. Xu, X. Lv, Y. Wang, J. Ouyang, and Y. Zhou, “Grain growth kinetics and densification mechanism of (TiZrHfVNbTa)C high-entropy ceramic under pressureless sintering,” Journal of Materials Science & Technology 110, 57-64 (2022)

    work page 2022

  25. [25]

    Zirconium Carbide Produced by Spark Plasma Sintering and Hot Pressing: Densification Kinetics, Grain Growth, and Thermal Properties,

    X. Wei, C. Back, O. Izhvanov, C.D. Haines, and E.A. Olevsky, “Zirconium Carbide Produced by Spark Plasma Sintering and Hot Pressing: Densification Kinetics, Grain Growth, and Thermal Properties,” Materials 9(7), 577 (2016). 26

    work page 2016

  26. [26]

    Densification kinetics and mechanical properties of tantalum carbide,

    A. Nisar, S. Ariharan, and K. Balani, “Densification kinetics and mechanical properties of tantalum carbide,” International Journal of Refractory Metals and Hard Materials 73, 221-230 (2018)

    work page 2018

  27. [27]

    A study of the densification mechanisms during spark plasma sintering of zirconium (oxy-)carbide powders,

    M. Gendre, A. Maître, and G. Trolliard, “A study of the densification mechanisms during spark plasma sintering of zirconium (oxy-)carbide powders,” Acta Materialia 58, 2598- 2609 (2010)

    work page 2010

  28. [28]

    Synthesis of zirconium oxycarbide powders: Influence of stoichiometry on densification kinetics during spark plasma sintering and on mechanical properties,

    M. Gendre, A. Maître, and G. Trolliard, “Synthesis of zirconium oxycarbide powders: Influence of stoichiometry on densification kinetics during spark plasma sintering and on mechanical properties,” Journal of the European Ceramic Society 31, 2377-2385 (2011)

    work page 2011

  29. [29]

    Constitutive modeling of spark-plasma sintering of conductive materials,

    E.A. Olevsky and L. Froyen, “Constitutive modeling of spark-plasma sintering of conductive materials,” Scripta Materialia 55(12), 1175-1178 (2006)

    work page 2006

  30. [30]

    Impact of Thermal Diffusion on Densification During SPS,

    E.A. Olevsky and L. Froyen, “Impact of Thermal Diffusion on Densification During SPS,” Journal of the American Ceramic Society 92, S122-S132 (2009)

    work page 2009

  31. [31]

    On description of grain growth kinetics,

    V.Yu. Novikov, “On description of grain growth kinetics,” Scripta Materialia 39(7), 945- 949 (1998)

    work page 1998