Refractory high-entropy alloys (RHEAs) with body-centered-cubic (BCC) structure are a new class of alloy materials and have great potential for high temperature applications. The present work investigated the BCC structural stabilities of Ti-Zr-Nb-Hf RHEAs with both first-principles calculations and experimental characterization. The cluster-plus-glue-atom model (cluster model) for the presentation of chemical short-range ordering (CSRO) in solid solutions was applied to construct the model input for the first-principles method, in which the density functional theory was used. Three cluster structural units were considered, [Ti-Hf₁₄]Nb₃, [Ti-Zr₈Hf₆]Nb₃, and [Ti-Zr₈Hf₄Ti₂]Nb₃. The corresponding alloys were fabricated, and microstructural characterization was performed. The calculated results of the formation energies and free energies of these three alloys indicate that the [Ti-Zr₈Hf₄Ti₂]Nb₃ has the highest BCC structural stability due to its lowest formation energy and free energy. It is well consistent with the experimental observation, where the [Ti-Zr₈Hf₄Ti₂]Nb₃ alloy exhibits a single BCC structure without any precipitation, while a small amount of α and/or ω phases would precipitate from the BCC matrix in the other alloys. The cluster-model-embedded first-principles method could provide a new approach for accurate calculations.

具有体心立方（BCC）结构的难熔高熵合金（RHEA）是一类新型合金材料，在高温应用方面具有巨大潜力。目前的工作通过第一性原理计算和实验表征研究了 Ti-Zr-Nb-Hf RHEA 的 BCC 结构稳定性。用于表示固溶体中化学短程排序（CSRO）的簇加胶原子模型（簇模型）被应用于构造第一性原理方法的模型输入，其中使用密度泛函理论. 考虑了三个簇结构单元，[Ti-Hf ₁₄ ]Nb ₃、[Ti-Zr ₈ Hf ₆ ]Nb ₃和[Ti-Zr ₈ Hf _{4 ]}Ti ₂ ]Nb ₃。制造了相应的合金，并进行了微观结构表征。这三种合金的形成能和自由能的计算结果表明[Ti-Zr ₈ Hf ₄ Ti ₂ ]Nb ₃由于其最低的形成能和自由能而具有最高的BCC结构稳定性。这与实验观察非常吻合，其中 [Ti-Zr ₈ Hf ₄ Ti ₂ ]Nb ₃合金表现出单一的 BCC 结构，没有任何沉淀，而少量 α 和/或 ω 相会从其他合金的 BCC 基体中沉淀出来。聚类模型嵌入的第一性原理方法可以为精确计算提供一种新方法。