Chapter 1
High-Entropy Alloys and Compounds: Fundamentals

Haojie Liu, Yonggang Yao*

School of Materials Science and Engineering, State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, P.R. China

*Corresponding author: yaoyg@hust.edu.cn

1.1 Introduction (Historical Background and Development)

High-entropy alloys (HEAs) were first conceptualized and introduced in the early 2000s (Figure 1.1), pioneered by J.W. Yeh and his colleagues in 2004 [1]. Traditional alloy design typically revolves around a single principal element, supplemented with smaller amounts of secondary elements as dopants to achieve desired property enhancements. However, HEAs challenge this paradigm by combining five or more elements in nearly equal proportions, creating a fundamentally distinct multicomponent system characterized by high configurational entropy and complex microstructures or atomic coordination. The elevated entropy arising from the random distribution of multiple principal elements can stabilize simple solid-solution phases, such as face-centered cubic (FCC) or body-centered cubic (BCC) structures rather than the formation of more complex intermetallic compounds. This stabilization is a direct consequence of entropy-driven thermodynamics, where the high configurational entropy counteracts enthalpic contributions that typically drive phase separation.

Figure 1.1 Rising trend of alloy chemical complexity versus time (IMs: intermetallics, HEA: high-entropy alloy) [2].

Source: [2] / with permission of Springer Nature.

The initial HEAs, often consisting of transition metals like Fe, Ni, Co, Cr, and Mn, exhibited outstanding properties [3-5], including high hardness, excellent wear resistance, and remarkable thermal stability. These properties stem from the synergistic combination of elements, which collectively enhance the material's performance under extreme conditions. The high configurational entropy in these alloys plays a key role in suppressing the formation of brittle intermetallic phases but instead promoting ductile and tough solid solutions [6].

Building on the foundational work of Yeh and others, the field of HEAs rapidly expanded in scope and application. Researchers began exploring diverse compositional systems, incorporating elements from various groups of the periodic table, including refractory metals (e.g., Mo, W, Ta), lightweight elements (e.g., Al, Mg, Li), and even nonmetallic elements (e.g., Si, B, C) [7]. This diversity allowed for the tailoring of HEA properties to meet the specific demands of various applications. For instance, refractory HEAs demonstrated exceptional high-temperature strength and oxidation resistance, while lightweight HEAs became promising candidates for aerospace applications requiring materials with low density and high strength. HEAs have also shown remarkable potential in harsh environments, such as those encountered in aerospace, nuclear reactors, energy generation, and deep-sea exploration. These environments demand materials that can withstand extreme temperatures, radiation, corrosion, and oxidation. For example, corrosion-resistant HEAs have been developed for marine and chemical processing industries, while high-strength refractory HEAs are being evaluated for use in hypersonic vehicles and fusion reactors [8, 9].

The continued exploration of HEAs is fueled by advancements in computational modeling, materials processing techniques, and experimental characterization. Computational tools such as high-throughput simulations and machine learning algorithms are enabling the design and optimization of novel HEA compositions with tailored properties [10, 11]. These tools allow researchers to predict phase stability, mechanical performance, and environmental resistance with unprecedented accuracy, significantly reducing the time and cost associated with traditional trial-and-error approaches. In parallel, innovations in materials processing techniques, including additive manufacturing (AM), powder metallurgy, and advanced casting methods, are facilitating the scalable production of HEAs with complex geometries and microstructures [12-14]. Moreover, state-of-the-art characterization tools, such as atom probe tomography (APT), transmission electron microscopy (TEM), and synchrotron-based techniques, provide deep insights into the atomic-scale structure and properties of HEAs, driving a better understanding of the underlying mechanisms that govern their exceptional performance [15, 16].

HEAs hold great promise for addressing critical challenges in materials science and engineering. Their unique combination of properties-including high strength, excellent thermal stability, superior corrosion resistance, and exceptional wear resistance-makes them highly attractive for applications in next-generation technologies. Furthermore, HEAs are poised to play a pivotal role in sustainable development by enabling materials with enhanced durability, reduced resource consumption, and improved energy efficiency. As research continues to advance, the integration of interdisciplinary approaches, combining theoretical modeling, experimental validation, and industrial-scale implementation, will be essential to unlocking the full potential of HEAs. The development of HEAs tailored to specific applications will not only expand the boundaries of materials science but also contribute to the creation of environmentally resilient and economically viable technologies. These advancements position HEAs as key enablers of innovation and sustainability across a broad range of industries (Figure 1.2).

Figure 1.2 Schematic diagram of the development of high-entropy nanoparticles with multielemental composition and enhanced functionality [17].

Source: [17] / with permission of American Association for the Advancement of Science.

1.2 Definition of HEAs and Compounds

The concept of high-entropy materials (HEMs), which are composed of at least five elements or components, represents a shift in how materials are designed and understood, leveraging entropy as a key factor in their creation. Typical examples are HEAs and high-entropy compounds (HECs), offering a wide array of unique properties derived from their multicomponent systems. While the effects of entropy have long been recognized in traditional materials and could be explained through thermodynamic principles, the impact of entropy in systems with multiple principal elements was historically overlooked or considered insignificant. This changed with the development of HEAs, which prompted a new focus on multicomponent systems and introduced a paradigm shift in materials science.

Composition Definition. Alloys containing five or more elements with an atomic percentage of each element greater than 5% and less than 35% are defined as HEAs. The idea behind HEAs is to take advantage of multielement mixing and the large configurational entropy (discussed later) that results from the mixing of these elements into a single-phase alloy. HEMs are a broader category of materials that, like HEAs, are composed of multiple elements in near-equal or equimolar proportions to replace the used one/few components position. The term "HEMs" extends beyond just metallic alloys, which also include nonmetallic systems, such as ceramics and polymers, that exhibit the same principle of high entropy. For example, high-entropy ceramics are ceramic materials wherein multiple metal or nonmetal elements are mixed to create a stable phase with desirable properties such as high hardness, thermal stability, and oxidation resistance, as in complex oxide ceramics of (Ti, Zr, Hf, Nb, Ta)O². In the later text, we will use HEA as the most important example in HEMs to illustrate those basic concepts.

Entropy Definition. The key defining characteristic of HEAs is their "high-entropy" composition, where the term "entropy" refers to the number of ways atoms can be arranged in a system. By mixing several elements in nearly equal amounts, the number of possible atomic arrangements increases, leading to a higher-entropy state, which can stabilize the solid-solution phase at room temperature, resulting in improved properties. This definition specifies the kinds and ratios of components and introduces the concept of high entropy, which aims to overcome the formation of intermetallic phases through high mixing entropy, thereby obtaining a single-phase solid-solution structure. According to Boltzmann formula, the entropy of a system is related to a given state as the following relationship:

where is the Boltzmann constant and is the probability that a given state exists. The entropy of the system includes four parts: configurational entropy , vibrational entropy , magnetic entropy , and electronic randomness entropy . The total mixing entropy of HEAs is depicted by the equation:

But compared with vibrational entropy, magnetic entropy, and electronic entropy, configurational entropy plays a dominant role, in general. Therefore, to avoid the difficulties caused by the calculation of vibrational entropy, magnetic entropy, and electron entropy, the configuration entropy of the system is generally regarded as the mixing entropy of the ideal solid solution, which can be expressed as:

where is the molar gas constant , is the mole fraction of each constituent element. The mixing...