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Grain-Size Dependence of Thermal and Electrical Properties in (Cr,Mo,Ta,V,W)C1-{\delta} High-Entropy Carbide Ceramics

This study demonstrates that in near-fully dense (Cr,Mo,Ta,V,W)C $_{1-\delta}$ high-entropy carbide ceramics, varying grain sizes from approximately 9 to 29 $\mu$ m via spark plasma sintering has no systematic effect on thermal or electrical transport properties, suggesting that intrinsic disorder and defect chemistry dominate over grain-boundary scattering in determining these materials' behavior.

Original authors: Ali Sarikhani, Gregory E. Hilmas, David W. Lipke, Douglas E. Wolfe, Stefano Curtarolo, Shen J. Dillon, William G. Fahrenholtz

Published 2026-10-16

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CC BY 4.0

Original authors: Ali Sarikhani, Gregory E. Hilmas, David W. Lipke, Douglas E. Wolfe, Stefano Curtarolo, Shen J. Dillon, William G. Fahrenholtz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Materials science often focuses on the extreme: substances that can survive the searing heat of a jet engine or the crushing pressure of deep space. Among the most promising candidates for these harsh environments are high-entropy carbides. These are not simple compounds made of just two elements, but complex ceramics where five different metals—chromium, molybdenum, tantalum, vanadium, and tungsten—are mixed together in equal parts within a single crystal structure. This chaotic arrangement of atoms gives the material its unique strength and stability, making it a potential candidate for the next generation of heat shields and structural components. However, for these materials to be useful, engineers need to know how they handle energy. Specifically, they need to understand how heat and electricity move through the material, and whether the size of the tiny crystals that make up the material changes how well it performs.

In a recent study, researchers set out to solve a specific puzzle regarding these complex ceramics. When a material is made of many tiny crystals, or grains, the boundaries where these grains meet can act like roadblocks. In many materials, making the grains smaller increases the number of these roadblocks, which slows down the flow of heat and electricity. It is a well-known principle that if you want to control how a material conducts energy, you might try to shrink its grains. But for high-entropy carbides, it was unclear whether this rule still applied. The atoms inside these materials are already so disordered that they might scatter heat and electricity so effectively on their own that the size of the grains no longer matters. To find the answer, the team needed to create a series of these ceramics with different grain sizes but identical chemical makeup, and then measure how heat and electricity traveled through them.

The researchers began by mixing precise amounts of metal oxides and carbon powder. They subjected this mixture to intense heat and pressure in a specialized furnace, a process that fused the powders into solid, dense blocks. By carefully controlling the final temperature of this heating process, they were able to grow the grains to different sizes without changing the material's chemical composition or creating holes inside the block. The result was a set of five nearly perfect samples. One sample had an average grain size of about 9 micrometers, roughly the width of a human hair, while another had grains that grew to nearly 29 micrometers. Despite this threefold difference in grain size, every sample remained a single, solid phase with no cracks or voids, providing a clean test environment to see if the grain size made a difference.

With the samples prepared, the team measured how quickly heat moved through them and how easily electricity could pass. They found that the material conducted heat at a rate of roughly 6.8 to 7.1 watts per meter per degree Kelvin at room temperature. More importantly, this number did not change as the grain size increased. Whether the grains were small or large, the heat moved through the material at the same speed. Similarly, the electrical resistance, which measures how hard it is for electricity to flow, stayed within a narrow range of 151.9 to 177.0 microohm-centimeters, showing no pattern related to the size of the grains. The data suggested that the boundaries between the grains were not acting as significant barriers to the flow of energy.

To understand why this happened, the researchers looked deeper into the nature of the heat carriers. In these conductive ceramics, heat is carried by two things: vibrations in the crystal lattice, known as phonons, and moving electrons. The team calculated that electrons were responsible for about 58 to 68 percent of the total heat transport. They estimated that the distance an electron travels before hitting an obstacle, known as its mean free path, was incredibly short—only a few nanometers. In contrast, the grains in their samples were thousands of times larger, measuring in micrometers. Because the electrons were scattering off the disordered atoms inside the grains long before they ever reached a grain boundary, the size of the grains became irrelevant. The internal chaos of the material was doing all the work of slowing down the energy flow.

The study concludes that for this specific type of high-entropy carbide, once the material is fully dense and chemically uniform, the size of its grains does not significantly alter its thermal or electrical properties within the range they tested. The internal disorder of the five mixed metals is the dominant factor controlling how energy moves, effectively drowning out any influence the grain boundaries might have. This finding is crucial for engineers designing with these materials. It suggests that they do not need to obsess over minimizing grain size to improve thermal performance, as the material's complex chemistry already provides a high level of resistance to heat flow. Instead, the focus can remain on ensuring the material is fully dense and free of other defects, knowing that the grain size itself will not be the limiting factor in how the material handles extreme temperatures.

Technical Summary: Grain-Size Dependence of Thermal and Electrical Properties in (Cr,Mo,Ta,V,W)C1-δ High-Entropy Carbide Ceramics

Problem Statement
High-entropy carbides (HECs) are chemically complex refractory ceramics with potential for extreme-environment thermal and structural applications. While the phase formation and mechanical properties of HECs have been established, the influence of microstructural parameters—specifically grain size—on thermal and electrical transport remains uncertain. In polycrystalline ceramics, grain boundaries typically scatter phonons and charge carriers, potentially altering thermal conductivity and electrical resistivity. However, in chemically complex HECs, intrinsic disorder and defect chemistry may already limit carrier mean free paths significantly. The specific challenge lies in decoupling grain-size effects from concurrent changes in density, phase constitution, or defect states during processing. This study addresses the gap in understanding whether grain-boundary scattering measurably influences transport properties in near-fully dense, single-phase (Cr,Mo,Ta,V,W)C1-δ ceramics within a coarse-grained regime.

Methodology
The study utilized a model HEC composition, (Cr,Mo,Ta,V,W)C1-δ, synthesized via carbothermal reduction of mixed metal oxides and carbon black. To systematically vary grain size while maintaining phase purity and density, specimens were densified using Spark Plasma Sintering (SPS) at five distinct temperatures ranging from 1750 °C to 1950 °C. This approach generated a series of near-fully dense ceramics with average grain sizes ranging from 9.3 ± 4.9 µm to 28.8 ± 15.3 µm.

Characterization included:

Microstructure and Phase: Scanning electron microscopy (SEM) for grain size quantification (Feret diameter) and X-ray diffraction (XRD) to confirm single-phase rock-salt structure.
Density: Bulk density was measured via the Archimedes method, and theoretical density was calculated based on refined lattice parameters and EDS composition.
Thermal Properties: Heat capacity ( $C_p$ ) was estimated using the Neumann-Kopp rule and validated against Differential Scanning Calorimetry (DSC) and Laser Flash Analysis (LFA) data. Thermal diffusivity ( $D$ ) was measured via LFA from room temperature to ~200 °C. Total thermal conductivity ( $\kappa$ ) was calculated using the relation $\kappa = D \rho C_p$ .
Electrical Properties: Room-temperature electrical resistivity ( $\rho_e$ ) was measured using the Van der Pauw method.
Data Analysis: The electron contribution to thermal conductivity ( $\kappa_e$ ) was estimated using the Wiedemann-Franz law with a theoretical Lorenz number. Statistical analyses (linear regression and Spearman rank correlation) were performed to assess grain-size dependence.

Key Results

Microstructural Control: SPS successfully produced a series of single-phase, near-fully dense ceramics. Grain size increased systematically with sintering temperature, while phase constitution (FCC rock-salt) and relative density (>98%) remained essentially constant across the series.
Thermal Transport: Thermal conductivity values at room temperature ranged from 6.8 to 7.1 W m⁻¹ K⁻¹. Despite the systematic variation in grain size (9.3 to 28.8 µm), no clear, systematic dependence of thermal diffusivity or thermal conductivity on grain size was observed. Statistical analysis revealed no significant linear association between grain size and thermal diffusivity across the tested temperature range (room temperature to 200 °C), with the exception of an isolated data point at 100 °C that was not reproducible at adjacent temperatures.
Electrical Transport: Room-temperature electrical resistivity ranged from 151.9 to 177.0 µΩ cm. Similar to thermal properties, no systematic trend with grain size was detected.
Transport Mechanisms: The electron contribution to thermal conductivity was estimated to be approximately 58–68% of the total, indicating electron-dominated transport. The estimated electron mean free path ( $l_e$ ) was calculated to be 0.2–0.6 nm, which is several orders of magnitude smaller than the minimum grain size (9.3 µm).

Significance and Conclusions
The primary finding of this work is that within the investigated grain-size regime (9.3–28.8 µm) for near-fully dense (Cr,Mo,Ta,V,W)C1-δ, processing-induced grain coarsening does not produce clearly resolvable changes in thermal or electrical transport properties. The authors conclude that intrinsic disorder and defect chemistry (including carbon vacancies and chemical disorder) likely exert a stronger influence on transport properties than grain-boundary scattering in this specific system.

The study suggests that because the carrier mean free paths (both electrons and phonons) are limited to the nanometer scale by the complex chemistry and defect structure of the HEC, the micrometer-scale grain boundaries do not act as the dominant scattering mechanism. Consequently, for this class of materials, achieving full density and phase stability appears to be more critical for transport properties than further grain-size refinement within the coarse-grained regime. This work establishes a baseline for grain-size-property relationships in dense high-entropy carbides, indicating that property tuning via grain size alone may be limited unless grain sizes approach the nanometer scale comparable to carrier mean free paths.

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