Shock Hugoniot of an equiatomic high-entropy alloy NbMoTaW up to 143 GPa

High-entropy alloys (HEAs) are a class of alloys that contain multiple principal elements, in contrast to traditional alloys, which are based on one principal element with small amounts of additions to improve their mechanical or chemical properties. In HEAs, the elements form single-phase solid solutions other than intermetallic compounds owing to their high configurational entropy. HEAs typically contain four or more metallic elements; each has an atomic percentage between 5% and 35%. The high-entropy effect, lattice distortion effect, hysteresis diffusion effect, and “cocktail” effect endow HEAs with excellent mechanical and chemical properties, such as high hot hardness, high wear resistance, high corrosion resistance, low-temperature ductility, high thermal stability, and good oxidation-resistant at high temperature. Due to their unique characteristics, HEAs are poised to be used in a diverse range of applications, especially in extreme service conditions.^1–5 

The equations of state (EOSs) and phase stabilities of HEAs under high-pressure–temperature conditions are of paramount importance for engineering applications. Some studies have been published in this field including both static and dynamic high-pressure studies.^6–15 Most studies focused on Cantor's alloy (equiatomic CoCrFeMnNi), a prototype face-center cubic (fcc) structured HEA, and related materials. A fcc to hexagonal-close packing (hcp) structure transition was identified in Cantor's alloy by static compressions, the low- and high-pressure phases have extremely similar compressibility, and volume change is not observed at the phase transition. Transition pressures reported by different groups vary significantly (7–49 GPa), depending on the hydrostaticity of the pressure transmission media, sample grain size, and alloying elements.^6,7 Using the plane-impact method, Jiang et al. measured the Hugoniot of Cantor's alloy and a body-center cubic (bcc) structured equiatomic NiCoFeCrAl HEA up to 11 GPa.¹² Recently, the Hugoniots of fcc structured CoCrFeNiCu,¹¹ Al_0.1CoCrFeNi,⁹ and FeMnCrNi¹⁵ HEAs were measured up to 19.3, 19.5, and 9.5 GPa, respectively, and no shock-induced phase transitions were identified. These dynamic compression experiments were designed to investigate the strength properties and deformation mechanisms of HEAs under high-speed impact; therefore, the shock pressure generated in the experiments was relatively low.

The HfNbMoTaWV system is another class of HEAs that normally exhibit a bcc structure; they were completely composed of refractory metal elements. The first equiatomic refractory HEAs NbMoTaW and NbMoTaWV were synthesized by Senkov et al. They were found to have compressive strengths comparable to or higher than those of Ni-based superalloys, particularly at high temperatures.¹⁶ The compressive strength of NbMoTaW is 1211 MPa at room temperature and 405 MPa at 1600 °C. At temperatures exceeding 600 °C, the strengths of both materials decrease at a gradual rate with increasing temperature. Moreover, their single-phase BCC structure remains stable even at 1400 °C. These features render them potential candidates for structural applications in the aviation, aerospace, metallurgical, and nuclear energy industries. Recent studies have shown that incorporating a small amount of non-metallic elements, such as C or Si, into NbMoTaW HEA can further enhance its strength and plasticity.^3,17 Until now, few reports exist on the high-pressure–temperature properties of the HfNbMoTaWV system HEAs.

In this study, we synthesized the NbMoTaW HEA using the vacuum arc melting method and measured the elastic–plastic transition and Hugoniot data of the NbMoTaW HEA up to ∼143 GPa, using the planar-plate impact method. Subsequently, we studied the phase stability and EOS of the NbMoTaW HEA based on the measured data. It was found that the Hugoniots of CoCrFeNiCu and Al_0.1CoCrFeNi can be approximately reproduced by a simple interpolation model with the Hugoniot parameters of the compositional elements;^9,11 our experimental data enable us to verify the applicability of this model in a much wider pressure range. It should be noted that this is the first time that the Hugoniot of a HEA has been measured to pressures higher than 100 GPa.

The NbMoTaW HEA sample was synthesized by mixing equiatomic element powders in a planetary ball mill and melting them in a vacuum arc melting furnace. The sample was remelted four times to improve its uniformity, with each melting lasting approximately 15 min. Disks with a diameter of approximately 12 mm and thickness of 3 mm were cut from the sample button and prepared for shock experiments. The compositions of two randomly selected sample disks were analyzed using inductively coupled plasma-optical emission spectroscopy (ICP-OES), and the results, listed in Table I, indicate good uniformity. Figure 1 shows the x-ray diffraction (XRD) pattern of the sample; all the peaks could be assigned to the NbMoTaW solid solution with a single bcc phase, and no indication of ordering was observed. The longitudinal (C_l) and shear (C_s) sound velocities of the sample were measured using the pulse-echo method, yielding values of 4.077 (7) and 2.181 (6) km/s, respectively. The bulk sound velocity (C_b) and Poisson’s ratio (σ) calculated from C_l and C_s are 3.206 (7) km/s and 0.3, respectively. Scanning electron microscopy (SEM) backscatter images showed micropores with sizes ranging from several to tens of micrometers, indicating that the samples had some porosity. The average density of the samples (ρ₀₀) was measured using the immersion method and found to be 12.02 (1) g/cm³, which was smaller than the neutron diffraction density¹⁶ (ρ₀) of 13.77 g/cm³, suggesting a porosity of m = 1.146 (m = ρ₀/ρ₀₀).

In Eqs. (1) and (2), H′ = H/cosα and H the sample thickness; t₂ is the arrival time of the plastic wave on the sample surface; and U_fs1 is the free-surface velocity at the Hugoniot elastic limit (HEL). The latter two parameters can be obtained from the measured free-surface velocity profiles.²⁰ The in-material particle velocity of the elastic wave was approximated using the free-surface approximation (U_P1 = U_fs1/2), and the final particle velocity was calculated using the impedance-matching method.²¹ The wave interactions between elastic and plastic waves were neglected. For higher shock pressures (shots 3–6), where the elastic wave was overdriven, the shock velocities were measured using the electrical pin method described in detail in our previous study.¹⁸ Shock pressures and densities were calculated using the Rankine–Hugoniot conservation laws.²¹ The impact speeds were measured using an electromagnetic method with an accuracy greater than 0.5%. We used Ta, Cu, and Al as the flyer/driver plates; their Hugoniot parameters and HEA component materials are listed in Table II.

The calculated result is indicated by the blue dashed line in Fig. 6. Fitting the 300 K isothermal to the third-order Birch–Murnaghan equation⁴⁴ results in a bulk modulus K₀ = 238 GPa and its first-order pressure derivative K₀′ = 3.3. The bulk modulus obtained in this study aligns well with those obtained via first-principles calculations (247 or 236 GPa).⁴⁵ 

Different mixture models have been used to predict the Hugonoit of a mixture. In the studies of Zhang et al.⁹ and Li et al.,¹¹ it was found that the Hugoniot of CoCrFeNiCu and Al_0.1CoCrFeNi can be approximately reproduced by a simple interpolation model: $C o= ∑ i m i C o i$ and $S= ∑ i m i S i$⁠. Another commonly used method is the volume addition model, in which the volume of a mixture at a specific shock pressure P_H is calculated by $V( P H)= ∑ i m i V i ( P H )$⁠. We have calculated the Hugoniot of NbMoTaW HEA with both models using the parameters listed in Table II and the results are shown in Fig. 7. It is found that the calculated results by both models are in good agreement with the Hugoniot of the NbMoTaW HEA within the experimental uncertainty. This consistency indicates that these simple mixture models can be used to predict the EOS of HEAs over a wide pressure range, although high-entropy alloys are solid solutions rather than mixtures.

To summarize, we measured the Hugoniot of a NbMoTaW HEA up to 143 GPa and ∼6200 K. We observed a linear relationship between the shock and particle velocities of the HEA under high shock pressures. This observation suggests that the NbMoTaW HEA possibly remains stable within the pressure–temperature range examined in our study. This high stability makes HEA a potential important structural material that can be used under extreme conditions. Based on the measured data, we analyzed the EOS of the nonporous NbMoTaW HEA. We estimated the bulk modulus and pressure derivative to be K₀ = 238 GPa and K₀′ = 3.3, respectively. We found that the Hugoniot compression curve of NbMoTaW could be aptly described using the mixture rules, a principle that may also be applicable to other HEAs.

We acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 41974099).

The authors have no conflicts to disclose.

Yu Hu: Formal analysis (lead); Investigation (equal); Visualization (equal); Writing – original draft (equal). Yishi Wang: Formal analysis (equal); Visualization (equal); Writing – original draft (equal). Gang Yang: Data curation (equal); Formal analysis (equal). Xun Liu: Data curation (lead); Funding acquisition (lead); Investigation (equal); Methodology (lead); Project administration (lead); Writing – original draft (equal); Writing – review & editing (equal). Haijun Huang: Methodology (equal); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.