Hydrogen evolution reaction in high-entropy MXenes: Insights into atomic configurations

Summary

High-entropy (HE) MXenes have recently gained attention for enhancing catalytic activity and improving stability through diverse active sites and compositional tuning. However, solute segregation during synthesis causes preferential atomic distributions that affect catalytic performance. In this study, HE variants of M₂C(-T)₂ MXenes with different surface terminations were evaluated for hydrogen evolution reaction using density functional theory and crystal orbital Hamilton population analysis. Substituting Ti with other transition metals improved H∗ adsorption and diffusion. Mo atoms showed short-range ordering, with strong Mo–Mo interactions forming preferred H∗ diffusion pathways. The Volmer-Heyrovsky mechanism displayed a 0.40 eV energy barrier, while hydrogen spillover reduced the Volmer-Tafel barrier from 0.94 to 0.48 eV. OH-terminated HE MXenes demonstrated both thermodynamic and mechanical stability, with Young’s and shear moduli of approximately 170 and 68 N/m, respectively. Engineering the composition can favorably induce short-range ordering of atoms, facilitating HER by enabling multiple H∗ diffusion routes.

Graphical abstract

Highlights

• •HE MXenes enhance H∗ adsorption and diffusion, boosting HER performance
• •Short-range Mo ordering optimizes H∗ diffusion, lowering energy barriers for HER
• •Hydrogen spillover reduces the Tafel step barrier by 50%, improving reaction efficiency
• •Stability analyses confirm HE MXenes as durable catalysts for sustainable energy

Catalysis; Applied sciences

Introduction

The rise in energy consumption, primarily fueled by fossil fuels, has led to critical challenges such as resource depletion and significant environmental issues.¹ Consequently, there is an urgent demand for sustainable and renewable energy sources. Hydrogen with its energy content of 142 MJ per kilogram (MJ/kg) is a promising renewable energy source.¹ To harvest hydrogen, the electrocatalytic processes such as water hydrolysis in which water can be split into H₂ and O₂ through hydrogen evolution and oxygen evolution reactions (HER and OER) are utilized.^2,3,4 However, the current most efficient and enduring anode materials in electrocatalysts rely on precious metals like platinum, ruthenium, and iridium that are rare and expensive.^1,5,6,7,8

To overcome this issue, graphene and transition metals dichalcogenides were recommended to use though each of them had its benefits and drawbacks. For instance, graphene with several advantages, including a high specific surface area, good conductivity, and electrochemical stability has attracted huge attention over the past decades; however, it has inadequate active sites and exhibits limited electrocatalytic activity.^5,9 The concept of high entropy (HE) which suggests that maximizing configurational entropy by composing at least four elements promotes the creation of a stable solid solution, can potentially trigger a significant interest in electrocatalysts where each element can contribute to catalytic activity.^{10,11,12,13,14} Introducing HE in two dimensional (2D) materials can foster the catalytic performance by combination of the high-entropy effect, the lattice distortion effect, and geometrical features, such as variety in adsorption energy, and high density of active sites.^5,15 While the high-entropy effect satisfies the thermodynamic stability and mechanical durability of 2D electrocatalysts, the high density of various adsorption sites offers concurrent adsorption/desorption tendencies that expedites the adsorbate diffusion kinetics.

Transition metal carbides (MXenes) with high configurational entropy, also called HE MXenes, can be potential 2D materials with improved stability and electrocatalytic performance as compared to primary (pristine) MXenes.^16,17,18 While primary MXenes hold remarkable performance as functional materials due to their promising electrical conductivity, extensive surface areas, structure adjustability and versatility, their catalytic performance is hindered by their tendency to stack together easily and their limited thermodynamic stability in oxidative conditions.^19,20,21,22 HE MXenes maintain a similar chemical formula as primary MXenes where the term “M” in HE MXenes is a multicomponent transition-metal compound that usually contains 3 to 5 elements.^23,24,25 The lattice distortion effect in HE MXenes that induces a gradient in strain localized through the lattice plays a significant impact on the enhancement of electrocatalytic efficiency in strain-rich electrocatalysts, as it can consistently and effectively modify the adsorption characteristics and catalytic performance of a surface reaction entity.^26,27,28 Aside from that, inspired by the principle of entropy-driven stabilization, transition metal elements with varying atomic radii can form stable solid solutions, resulting in a variety of characteristics, including tailorable physical and chemical properties, multiple bonding sites, and high HER catalytic activity.^29,30

While primary MXenes have been extensively studied for HER applications, research on HE MXenes remains largely unexplored.^31,32,33 In particular, the kinetics of different HER steps, i.e., Volmer-Heyrovsky-Tafel, and the influence of atomic configuration on the HER performance have yet to be investigated. These aspects become even more critical given that achieving a homogeneous atomic distribution in HE MXenes is inherently challenging, as atomic ordering and solute segregation are nearly inevitable during the exfoliation process.^11,16 In this study, an HE MXene composed by different transition metal elements is introduced as a promising candidate for HER for the first time. The electrocatalytic performance of HE M₂C(-T)₂ MXenes, featuring various termination groups and atomic configurations, is evaluated using density functional theory (DFT). Multiple compositions are investigated to identify the most suitable configuration for HER, wherein the H∗ adsorbate can diffuse easily at low energy barrier. Crystal orbital Hamilton population (COHP) analysis is performed to determine the bond strength between different atom pairs, shedding light on the role of short-range ordering in bonding/anti-bonding affinities and the adsorbate diffusion pathway. Finally, thermodynamic and mechanical stability of the selected HE MXene are assessed, offering insights into its synthesizability.

Results

Density of states and bond strength

HE variants of M₂C(-T)₂ with both -O and -OH terminals were modeled using ATAT software by randomly substituting surface Ti atoms on both sides of Ti₂C(-T)₂ with Mo, Cr, and V. Various stoichiometries and configurations are shown in Figures 1 and S1, including (Ti_0.25V_0.25Mo_0.25Cr_0.25)₂C(-T)₂, (Ti_0.25V_0.125Mo_0.50Cr_0.125)₂C(-T)₂, (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-T)₂, (Ti_0.375V_0.375Mo_0.125Cr_0.125)₂C(-T)₂, (Ti_0.50V_0.125Mo_0.25Cr_0.125)₂C(-T)₂, (Ti_0.125V_0.25Mo_0.50Cr_0.125)₂C(-T)₂, (Ti_0.125V_0.375Mo_0.375Cr_0.125)₂C(-T)₂, and (Ti_0.125V_0.50 Mo_0.25Cr_0.125)₂C(-T)₂. The M₂C(-T)₂ MXene is composed of five layers in the sequence T-M(1)-C-M(2)-T. Figures 1 and S1 present the density of states (DOS) for primary MXenes and HE MXenes with both termination groups. Except for the primary Ti₂C(-O)₂ MXene, the HE MXenes do not exhibit a bandgap. With higher configurational entropy, the peaks become smoother, resulting in a wider band spread.

Figure 1.

Density of states in HE MXenes

DOS plots in O-, and OH-terminated HE MXenes (other MXenes have been shown in Figure S1).

As illustrated in Figure 2, the projected crystal orbital Hamilton population (pCOHP) diagrams reveal bonding and anti-bonding contributions to the band-structure energy. Here, the bands crossing the Fermi level predominantly reflect bonding interactions among various pairs in the HE MXene, resulting in a notable DOS, as previously shown in Figure 1. Comparable to molecular orbital theory, the high-energy anti-bonding states (positive pCOHP) indicate electronic instability.³⁴

Figure 2.

COHP plots of HE MXenes

Projected crystal orbital Hamilton population (pCOHP) analysis for various transition metals bonding in (A–D) primary MXenes, and (E and F) (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ MXene.

In the primary MXenes (Figures 2A–2D), the transition metal atoms in (Mo or Ti)₂C(-T)₂ exhibit stronger bonding interactions, resulting in higher pCOHP values compared to (V or Cr)₂C(-T)₂. The variation in bonding-antibonding magnitudes among different transition metal atoms suggests that the coexistence of these elements in HE MXenes introduces a diverse range of adsorption sites, with varying levels of activity.

The energy integral of the pCOHP, called as the integrated (ICOHP), indicates the contribution of a specific bond to the band energy. ICOHP quantifies bond strength, where a more negative ICOHP value generally suggests a stronger bonding interaction between atoms, as additional bonding states contribute to bond stabilization. As given in Table 1, Mo-based primary MXenes exhibit the strongest bond strength between transition metal atoms, whereas Cr-based primary MXenes display the weakest. Also, the magnitudes of the bond strengths between the stronger pairs (Ti-Ti and Mo-Mo) and their interactions with other transition metals (Cr and V) in an HE MXene are also reported in this table.

Table 1.

Bond strength representation in various MXene denoted by ICOHP

Lattice	Bond	ICOHP	Bond	ICOHP
Primary MXenes (First row: (TM∗)2C(O)2) (second row: (TM)2C(OH)2) ∗ TM: Transition metal	Cr-Cr	−0.05	Mo-Mo	−0.16
	Cr-Cr	−0.06	Mo-Mo	−0.18
	V-V	−0.07	Ti-Ti	−0.09
	V-V	−0.09	Ti-Ti	−0.13
(Ti0.375V0.125Mo0.375Cr0.125)2C(-OH)2 HE MXene	Ti-Ti	−0.16	Mo-Mo	−0.45
	Ti-Mo	−0.16	Mo-V	−0.15
	Ti-V	−0.15	Mo-Cr	−0.07
	Ti-Cr	−0.06	V-Cr	−0.07

Adsorption energies and HER mechanisms

Following the analysis of DOS and pCOHP in HE MXenes, the adsorption energy of H∗ on different adsorption sites is calculated for each HE MXene, as shown in Figure 3; Tables S1 and S2. As shown in Figure 3, except two cases of (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ and (Ti_0.375V_0.375Mo_0.125Cr_0.125)₂C(-OH)₂ HE MXenes, all others exhibit positive adsorption energy for H∗ on certain sites.

Figure 3.

H∗ adsorption energy in HE MXenes

Adsorption energy of H adsorbate on transition metal sites in OH-terminated HE MXenes.

After identifying the weakest (most positive $\Delta E_{H\ast }$) and strongest (most negative $\Delta E_{H\ast }$) adsorption sites for each HE MXenes, the potential for adsorbate diffusion along the most favorable path is examined to ensure that the energy barrier for the rate-limiting step remained below 1.0 eV, enabling HER at ambient temperature.³⁵ The energy barrier at each transition state is monitored using the CI-NEB technique, revealing that a longer diffusion path for the adsorbate correlated with better HER performance. Considering the stochastic nature of HE lattices using special quasi-random structures (SQS), and after applying CI-NEB approach to each HE MXene, the one terminated with OH groups and exhibiting a short-range ordering of Mo atoms in its lattice, (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂, has been identified as the most suitable candidate.

Figure 4A illustrates the $\Delta E_{H\ast }$ values for 16 top sites in (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene. The purple area represents the diffusion window, where the most favorable sites for the Heyrovsky and Tafel reactions have $\Delta E_{H\ast }$ values ranging from −0.14 to −0.07 eV. Regardless of the type of transition metal atom, the adsorption energy of the hydrogen absorbate ranges from −0.86 to −0.07 eV, indicating a relatively uniform potential across the HE MXenes’ transition metal surface for facilitating the Volmer reaction. The arrows connecting the data points between adsorption sites in Figure 4A represent the change in hydrogen adsorption energy ($\Delta E_{H\ast }$) as a function of position on the surface—essentially illustrating how H∗ may diffuse across different adsorption sites.

Figure 4.

Energy barriers for Volmer-Heyrovsky-Tafel steps in HER

(A) Distribution of $\Delta E_{H\ast }$ on the surface of (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene, (B) Volmer–Heyrovsky mechanism of HER, (C) Volmer–Tafel mechanism of HER on the HE MXene (111), (D–G) comparison of reaction free energies for the Volmer–Heyrovsky mechanism between the HE MXene and primary MXenes, and (H and I) diffusion pathways of H∗ over the surface of the HE MXene (the jump steps of H∗ were enumerated from 1 to 5).

Both the Volmer-Heyrovsky and Volmer-Tafel mechanisms for the HER were analyzed on the (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene (111), as shown in Figures 4B and 4C. In these subfigures, H^A and H^B stand for the initial, most stable adsorption site for H following the Volmer step, and a neighboring site, possibly slightly less or similarly stable, accessible via surface diffusion, respectively. In Figure 4H, the pathway of H∗ while diffusing from the strongest site to the weakest site is elucidated. Although -OH termination groups are not considered the primary sites for H adsorbate landing, they are crucial in facilitating H∗ diffusion along the diffusion path.

Figures 4D–4G depict a comparison of the adsorption energy levels of Volmer-Heyrovsky mechanism between the (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene and primary MXenes composed of a single transition metal with both -O and -OH termination groups. In all cases of Ti₂C(-T)₂, V₂C(-T)₂, Mo₂C(-T)₂, and Cr₂C(-T)₂, the -O termination group results in a lower energy barrier at the transition state compared to the -OH groups. However, this energy level remains significantly higher than the adsorption energy range observed within the diffusion window of HE MXenes.

Figures 5A and 5B illustrate the spillover mechanism in (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene. In this Figure, H^C represents another nearby transition metal site that serves as an active site for H∗ spillover. This process involves the diffusion of the first hydrogen adsorbate along the most favorable diffusion path, from the strongest to the weakest adsorption sites, followed by the diffusion of a second hydrogen adsorbate via a different diffusion path.³⁶

Figure 5.

H∗ spillover in HE MXenes

(A and B) H∗ spillover within other diffusion path, reducing the energy barrier for Volmer-Tafel mechanism, and (C and D) H∗ adsorption energy across short-range-ordered pathways.

Considering the same chemical composition of (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene, but with a slight variation in lattice design illustrated in Table S4—where all transition metals are ordered within short ranges—Figure 5C presents the adsorption energy of H adsorbates on various sites. As shown in Figures 5C and 5D, although both Cr and V, as minor constituents, exhibit short diffusion paths over two neighboring atoms, one site shows a high positive adsorption energy, making the diffusion barrier dominant. Additionally, due to their high positive magnitudes, these sites are likely thermodynamically unstable at ambient temperature. In contrast, Mo and Ti short-range ordering introduces a narrower adsorption energy window between adsorption sites. Ti still exhibits positive adsorption energy values at sites 4 and 6; however, since these magnitudes along the Ti diffusion pathway are close to zero, they can potentially contribute to HER through the Heyrovsky step and H∗ diffusion, as they indicate weak H∗ binding.

Thermodynamic and mechanical stabilities

After detailing the HER performance in (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene, it is crucial to verify that this structure is both thermodynamically and mechanically stable. To assess thermodynamic stability, the average cohesive energy of the system can be considered as a representative factor, calculated as³⁷:

$$E_{coh}=\frac{E_{Tot}-\left(\sum _{i=1}^n{n}_i{E}_i\right)}{N}$$ (Equation 1)

in which $E_{tot}$ represents the total energy of the (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene, while $N$ is the total number of atoms in the supercell. The subscript $i$ denotes the element present in the composition, and $n$ refers to the number of atoms of that element in the system, such as 16 carbon atoms. $E_i$ is the total energy of a single isolated atom of element $i$, with no interaction with other atoms. The magnitudes of each term in Equation 1 have been reported in Table S3. The average cohesive value for the HE MXene has been reported in Table 2.

Table 2.

Stability and properties of (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ MXene

Material	Avg. cohesive energy (eV)	Young’s Modulus (N/m)	Shear Modulus (N/m)	Poisson’s ratio	Thermodynamic stability	Mechanical stability
(Ti0.375V0.125Mo0.375Cr0.125)2C(-OH)2	−4.99	169.27	77.15	0.09	Stable	Stable

To confirm mechanical stability of the (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene, Voigt notation for 2D materials was applied. In this context, the full matrix of stiffness tensors (a 3 × 3×3 × 3 tensor) is compacted into a 3 × 3 matrix for an isotropic symmetrical 2D material, as given in Equation 2^38,39:

$$C_{ij}=\left[\begin{array}{ccc}{C}_{11}& C_{12}& 0\\ C_{12}& C_{11}& 0\\ 0& 0& C_{66}\end{array}\right]$$ (Equation 2)

where $C_{11}$ represents the stiffness tensor along the x-direction, and $C_{12}$ is the coupling stiffness tensor between the x and y axes, explaining how stress applied in the y-direction affects the strain in the x-direction, and vice versa. $C_{66}$ represents the material’s response to in-plane shear. An isotropic symmetrical 2D material is mechanically stable only if the strain stability criteria listed in Equation 3 are met.³⁸

$$C_{11}>0;C_{11}>\left|C_{12}\right|,\text{or}\left(C_{11}+C_{12}\right)>0;C_{66}=\left(C_{11}-C_{12}\right)>0$$ (Equation 3)

Based on DFT calculations, the elastic constants are assessed by evaluating the energy variation of the lattice strained at various strains within the elastic deformation (0.000, 0.007, and 0.015). The stiffness tensor in (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene is calculated as follows:

$$C_{ij-HEMXene}=\left[\begin{array}{ccc}170.9& 16.6& 0\\ 16.6& 170.9& 0\\ 0& 0& 77.1\end{array}\right]$$

Discussion

As shown in Figures 1 and S1, at Fermi energy level, the high DOS is largely due to contributions from the d orbitals of various transition metals, depending on the composition. For instance, in (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂, the d orbitals of Ti and Mo significantly contribute to the DOS. These delocalized orbitals often overlap with neighboring atoms, leading to metallic behavior, and enhanced electrical conductivity. Conversely, in HE MXenes with -O terminations, where the Fermi energy exhibits more negative values, oxygen is the main contributor to the DOS near the Fermi level, introducing localized p orbitals.

Transition metals are essential for catalytic activity, as they introduce more conducting states and multiple adsorption sites, enhancing catalytic performance for HER. When DOS at the Fermi level is dominated by transition metal d orbitals, MXenes tend to exhibit improved catalytic performance, particularly in redox reactions such as HER or ORR.³⁶ On the other hand, oxygen atoms in MXenes with -O termination, due to their high electronegativity, increase the material’s chemical reactivity and adsorption properties, often resulting in sluggish adsorbate diffusion. Therefore, tuning the atomic composition and configuration in HE MXenes is critical for optimizing their electrocatalytic properties. Partitioning the band-structure energy into orbital-pair interactions, i.e., a bond-weighted DOS between a pair of adjacent atoms is conducted using the pCOHP analysis.

In Figure 2, stronger bonding interactions between specific atom pairs reduce the likelihood of bonding with other first-order neighboring atoms, while weaker interactions increase this likelihood. In other words, strong Mo-Mo bonding suggests that the Mo atoms are highly coordinated and may exhibit reduced availability of electronic states for bonding with H adsorbates in HER reaction. This aligns with the idea that Mo sites might not bind H∗ strongly, as their d orbitals are already involved in stabilizing the Mo-Mo bonds. Weak adsorption on Mo sites would favor hydrogen diffusion rather than trapping, allowing H∗ to move efficiently across the surface, as will be discussed. This property is crucial for HER since rapid diffusion and desorption of H∗ are required to facilitate the reaction’s kinetics.

Additionally, functionalizing MXenes with OH groups, rather than O groups, narrows the pCOHP amplitude, thereby better regulating the adsorption/desorption tendencies of adsorbates. A narrower bonding-antibonding amplitude can fine-tune the diffusion kinetics of adsorbates on the surface, providing a trade-off between different HER mechanisms in HE MXenes. Based on this analysis, pCOHP calculations for HE MXenes reveal that Mo-Mo bonding, and to a lesser extent Ti-Ti bonding, contribute more significantly to the DOS and bonding strength than other interactions in OH-terminated HE MXenes illustrated in Figures 2E and 2F.

As given in Table 1, in HE MXenes, employing multicomponent transition-metal compounds enhances the bond strength between similar atoms (e.g., Ti-Ti and Mo-Mo). This demonstrates that composition engineering or alloy design can effectively optimize the overlap of d orbitals between different constituents, leading to increased thermodynamic stability and reduced likelihood of lattice dissociation under various environmental conditions. The findings indicate that the short-range ordering of Mo atoms, where Mo-Mo interactions exhibit maximum bond strength, likely represents the preferred diffusion pathway for the H∗ adsorbate. Further investigation is conducted in the following to validate this.

Focusing on Figure 3 in which the adsorption energy of H∗ on different adsorption sites is calculated for each HE MXene, the results indicate that the thermodynamic stability of six HE MXene catalysts could be challenged under various conditions, such as elevated temperatures or within an electrochemical environment. This is due to the positive adsorption energy for H∗ on various transition metal sites in these MXenes.

Exploring the HER mechanisms of HE MXenes, the Volmer-Heyrovsky mechanism in Figure 4A shows the Heyrovsky step with ${\Delta E}_{Hey}$ = −0.40 eV, indicating an exothermic reaction. The adsorbed H∗ transitions through the diffusion path shown in Figure 4H elucidates the energy difference of 0.94 eV (range of adsorption energy over the diffusion window, cf. Figure 4B). For the Volmer-Tafel mechanism, the initial two Volmer steps are exothermic reactions on HE MXene (111), with ${\Delta E}_{Vol-1}$ = −0.53 eV and ${\Delta E}_{Vol-2}$ = −0.40 eV. The subsequent Tafel step on HE MXene (111) encounters a significant energy barrier of 0.94 eV that is considered as the rate-limiting step, cf. Figure 4C. As will be discussed further, the only mechanism that can make the Tafel reaction competitive with the Heyrovsky reaction is the spillover of H∗.

Considering Figure 4H, the H atoms in the -OH termination groups exhibit a strong affinity for the H∗ atom, helping it navigate past significant energy barriers typically encountered at bridge adsorption sites. Nonetheless, HE MXenes with -O termination groups do not benefit from this facilitation; they either lack a diffusion window or have only a very short diffusion path between neighboring transition metal atoms as the energy barrier between their sites is large.

Apart from the role of OH termination groups in navigating the H∗ during the HER reaction, the role of short-range ordering of Mo atoms should not be neglected. As previously discussed, the strong bonding between Mo-Mo atoms within the lattice leads to highly overlapping d orbitals, which reduces the availability of electronic states for bonding with H∗ during the HER reaction, resulting in weak adsorption sites. Short-range ordering or localized solute segregation is a common phenomenon in HE MXenes, as experimentally confirmed by Leong et al.¹¹ Similarly, in the HER reaction, the localized segregation of transition metal atoms plays a crucial role. However, the influence of first- and second-order neighboring atoms is also significant, as they define the local energy landscape of the lattice.

Referring to Figures 4D–4G, while the HE MXenes exhibit exothermic Volmer and Heyrovsky steps with a rate-limiting step occurring within a moderate diffusion barrier ($\Delta E<1.0$ eV), the primary MXenes undergo an endothermic Volmer step as the rate-limiting step, characterized by a significantly higher energy barrier ($1.0<\Delta E<3.2$ eV), which results in sluggish H∗ diffusion across the lattice sites. Moreover, the diffusion pathway on the lattice sites of HE MXenes allows the adsorbate to overcome energy barriers through multiple transitional steps, rather than encountering a single dominant barrier as observed in primary MXenes. Additionally, as illustrated in Figure S2, the sole driving force for the diffusion of H adsorbates in primary MXenes is the variation in adsorption energy between adjacent sites with different spatial arrangements, such as hollow and bridge sites. This occurs because, when all transition metal atoms are identical, no preferential top site exists. Therefore, if the diffusion of H∗ occurs via thermal activation in this case, the diffusion path in primary MXenes is significantly shorter compared to that in (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene, where the driving force for diffusion can be tuned by varying atom configurations through a long-range diffusion path. These figures clearly elucidate the superior performance of (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene in Volmer-Heyrovsky HER mechanism.

According to Figures 5A and 5B, the energy barrier for the Tafel step, in the presence of H∗ spillover, decreases significantly by nearly 50%, from 0.94 eV (cf. Figure 4) to 0.48 eV (cf. Figure 5). This reduction enhances the likelihood of the Volmer-Tafel mechanism occurring in (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene, as less deriving force to overcome the rate-limiting step is needed. The second H∗ spillover occurs along the diffusion path depicted in Figure 5B, with a difference of 0.46 eV in reaction adsorption energy shown through the diffusion window II. With the consideration of H∗ spillover, all the electrochemical steps including Volmer and Tafel can become spontaneous. It is important to note that the final adsorption site toward which the H∗ spillover occurs is not a top site of transition metal, but rather a hollow adsorption site, as illustrated in Figure 5B.

In Figures 5C and 5D, Mo short-range ordering features a long diffusion path with short energy barriers between its active sites, leading to thermodynamically stable Mo–H∗ interactions within a range of −0.75 to −0.05 eV. This stability facilitates H∗ adsorption and diffusion through both the Volmer and Heyrovsky mechanisms. Referring to Table 1, it can be concluded that the stronger the bond between two identical atoms, the lower the availability of electronic states for bonding with H∗, resulting in tunable adsorption and diffusion of the adsorbates.

Although much progress has been made in developing cost-effective electrocatalysts, challenges remain regarding their robustness under varied conditions.^7,40,41 For instance, Uwadiunor et al.⁷ synthesized primary Ti₃CN and Ru-doped Ti₃CN MXenes to develop sustainable and cost-effective catalysts for HER. The primary Ti₃CN catalyst demonstrates better performance in alkaline conditions compared to acidic conditions, attributed to its strong water adsorption energy but weak hydrogen adsorption energy. In alkaline media, water adsorption accelerates the Volmer step, but weak hydrogen bonding limits further reactions. In acidic media, the Volmer step is significantly restricted by low hydrogen adsorption energy. Upon doping with Ru, active sites enhance hydrogen bonding, shifting the rate-determining step to the Heyrovsky step and improving catalytic efficiency. Ru-doped Ti₃CN performs better in alkaline than acidic conditions. In alkaline media, Ti₃CN facilitates water reduction, and weakly bound hydrogen atoms transfer to Ru sites, where hydrogen desorption occurs efficiently. In acidic media, Ru primarily handles hydrogen adsorption and electrochemical desorption, consistent with the observed activity trends.

The concept of HE in MXenes enhances their catalytic activity across both alkaline and acidic media by engineering the chemical composition, which contrasts with the performance of primary or nanoparticle-doped MXenes.⁴² Both the composition and atomic configuration are crucial in tailoring electrocatalytic properties. Various transition metals behave differently in acidic and alkaline media due to their distinct hydrogen adsorption properties. Metals with weak hydrogen adsorption potentially accelerate the Volmer step in alkaline media but limit subsequent reactions, while in acidic media, their low hydrogen adsorption restricts the Volmer step. Hence, a combination of weak and strong adsorption sites, achieved by introducing different transition metal atoms in HE MXenes, is beneficial for HER.

The influence of lattice size on HER performance can be discussed along both the thickness (c direction) and lateral dimensions (a and b directions) of the simulated system. In the c direction, variations in thickness come from different MXene phases such as M₂C, M₃C₂, and M₄C₃, with thicker structures like M₄C₃ exhibiting more layers of transition metals and carbon. This increased thickness tends to promote greater degrees of transition metal segregation, both in-plane and out-of-plane, which can alter the distribution and nature of active sites, thereby influencing HER activity.^11,43 In contrast, changes in the lateral dimensions (x and y directions) primarily affect the physical size of the supercell but not the periodic electronic environment due to the use of periodic boundary conditions in DFT. The structural and electronic properties remain consistent when extending the lattice laterally, and the catalytic trends observed are preserved. Nevertheless, DFT-based simulations are inherently constrained by computational cost, typically limiting supercell sizes to a few hundred atoms. In this work, a 112-atom supercell was employed, which offers a representative model of the HE MXene system.

In the end, to verify mechanical stability of the presented HE MXene as the HER catalyst, the values obtained using strain-energy method to calculate the elastic constant of the most suitable HE MXene candidate and the strain stability criteria presented in Equation 3 indicate that the material is mechanically stable. This theoretically assures that the material can tolerate stresses below the plastic deformation zone without failure. The elastic properties of the HE MXene is presented in Table 2.

To sum up, the (Ti_0.375V_0.125Mo_0.375Cr_0.125)₂C(-OH)₂ HE MXene stands out as an excellent candidate for the HER, combining efficient catalytic performance, robust thermodynamic stability, and mechanical durability. The result underscores that even minor compositional variation in high-entropy MXenes can substantially impact HER performance due to the intrinsic tendency of transition metals to segregate during synthesis elucidated in other study.^11,43 Specifically, the strong short-range ordering and d-orbital overlap of Mo atoms reduce the availability of active sites for hydrogen adsorption, highlighting the critical role of atomic-scale arrangements in tuning catalytic behavior. These insights emphasize that theoretical models must account for realistic synthesis-induced inhomogeneities, and that compositional fine-tuning—though challenging to reproduce experimentally—can offer valuable pathways for designing more robust and efficient HER catalysts. These findings not only validate the potential of HE MXenes in sustainable energy applications but also highlight the importance of composition engineering in optimizing materials for energy conversion reactions.

Limitations of the study

While our study provides valuable insights into the influence of composition and atomic configurations on the HER performance of the HE MXenes, several limitations should be acknowledged. First, experimental validation through the synthesis of the computationally designed chemistries is essential to assess the extent to which compositional control can be achieved during the fabrication of MXenes. Second, the potential for solute segregation or clustering within the transition metal (TM) layers of these M₂C(-T)₂ HE MXenes should be further investigated using advanced characterization techniques, such as transmission electron microscopy and atom probe tomography. In this context, Leong et al.¹¹ reported preferential segregation of TMs in HE MXenes; however, their observations were limited to the M₄C₃ phases. Lastly, the role of the energy landscape in H∗ adsorption and diffusion requires further exploration at larger scales. This could be achieved by developing interatomic potentials (force fields) tailored to the studied chemistries, enabling atomic-scale simulations via molecular dynamics. Such efforts would enhance our understanding of the reaction kinetics, particularly the influence of time and temperature on HER activity.

Resource availability

Lead contact

For further information or resource requests, please contact the lead author, Chandra Veer Singh (chandraveer.singh@utoronto.ca); he will respond to and fulfill such inquiries.

Materials availability

Lattice structures and their corresponding data are available upon request from the lead contact.

Data and code availability

All data and original code reported in this study will be made available by the lead contact upon reasonable request.

Acknowledgments

The authors acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), the New Frontiers Research Fund-Transformation (NFRFT-2022-00197), and the University of Toronto. We also acknowledge Digital Research Alliance of Canada for providing computing resources at the SciNet, CalculQuebec.

Author contributions

Conceptualization, investigation, analysis, data curation, writing – original draft: M.H.G. conceptualization, writing, reviewing, and editing: Z.W.C. data curation, writing, reviewing, and editing: P.G.D. conceptualization, funding acquisition, supervision, writing, reviewing, and editing: C.V.S.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Software and algorithm
Vienna ab initio simulation package	Kresse et al.50	https://www.vasp.at/
Crystal Orbital Hamilton Populations	Dronskowski et al.58	http://www.cohp.de/
Vaspkit	Wang et al.57	https://vaspkit.com/
Alloy Theoretic Automated Toolkit	van de Walle et al.44	https://axelvandewalle.github.io/www-avdw/atat/

Method details

HE configurations in the MXene M₂C(-T)₂ with both -O and -OH termination groups were designed. The structure, characterized by a layered hexagonal arrangement, belongs to the space group P63/mmc.^19,21 In this structure, a layer of carbon atoms is sandwiched between two layers of titanium atoms, with the titanium atoms being further passivated by oxygen atoms on both sides. Special quasi-random structures (SQS) of these HE MXenes were generated using the Alloy Theoretic Automated Toolkit (ATAT) software.⁴⁴ Eight different compositions were created by varying the ratios of the transition metal atoms (Mo, Cr, V) to partially replace Ti atoms on both sides while maintaining charge neutrality. For all calculations, a 4×4×1 HE MXene configuration supercell, containing 80 or 112 atoms depending on the termination group (O or OH), was used. This supercell was deemed sufficient to investigate the adsorption properties and diffusion of hydrogen on the surfaces of the HE MXene monolayers.

To fully harness the exceptional properties enabled by the high-entropy concept, increasing configurational entropy alone is not sufficient. A more critical factor is the way solute atoms are distributed throughout the lattice to form a thermodynamically stable solid solution, rather than metastable or ordered phases such as intermetallics.^45,46 To satisfy these criteria, material selection should be based on prior studies that have demonstrated both the synthesizability of the material and the homogeneous distribution of solute atoms. Among high-configurational entropy MXenes, numerous studies have shown that the Ti-V-Mo-Cr alloying system not only remains stable under various functionalization groups, but also forms a randomly distributed solid solution, as confirmed by microscopy techniques.^{16,26,27,47,48,49} Therefore, the material system chosen for this study is based on this composition to ensure its practical applicability.

First-principle DFT was employed using the Vienna Ab-initio Simulation Package (VASP).⁵⁰ The ion-electron interaction was treated with the projected augmented wave (PAW) method.⁵¹ Exchange and correlation interactions were addressed using the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional.⁵² An energy cut-off of 550 eV was utilized. For sampling the Brillouin zone, a k-mesh of 3×3×1 Gamma-centered Monkhorst-Pack was employed for geometry optimization, and 9×9×1 for density of states (DOS) calculations.⁵³ Convergence criteria of 10^-5 eV for total energy were applied. Van der Waals interactions were included using the DFT-D3 method with the Grimme scheme.^54,55 The dissociation barrier of the adsorbed H∗ was calculated using the climbing image-nudged elastic band (CI-NEB) method.⁵⁶ Theoretical elastic constants and stiffness tensors, involving the assessment of energy variation when small strains were applied to the equilibrium lattice configuration, were measured using VASPKIT package.⁵⁷ The Crystal Orbital Hamilton Population (COHP) analysis was performed using the LOBSTER software to evaluate the bonding characteristics in HE MXenes. This approach leverages plane-wave DFT calculations, utilizing the PAW method for accurate electronic structure modeling.^34,58

The total adsorption energy of hydrogen ($\Delta E_{H\ast }$) is calculated using Equation 4 as follows³⁶:

$$\Delta E_{H\ast }=E_{H\ast }-E_{\left(HE-MXene\right)}-E_H$$ (Equation 4)

where $E_{H\ast }$, $E_{\left(HE-MXene\right)}$, and $E_H$ represent the total adsorption energy of the HE MXene with the H adsorbate, isolated HE MXene, and the isolated H atom, respectively. Also, the Volmer, Heyrovsky, and Tafel reaction steps are presented in Equation 5, and their corresponding energy barriers are calculated using the CI-NEB method.⁵⁹

$$H^{+\left(aq\right)}+e^{-}\to H_{ads}\left(\text{Volmer}\text{reaction}\right)$$ (Equation 5)

$$H^{+\left(aq\right)}+e^{-}+H_{ads}\to H_2\left(g\right)\left(\text{Heyrovsky}\text{reaction}\right)$$ (Equation 6)

$$H_{ads}+H_{ads}\to H_2\left(g\right)\left(\text{Tafel}\text{reaction}\right)$$ (Equation 7)

Quantification and statistical analysis

In all Figures except Figure 2, the calculations for adsorption energies, NEB analysis, and DOS were conducted through various transition metal atoms as adsorption sites using VASP. In Figure 2, COHP calculations were repeated three times, and the data was represented as the mean.

Published: August 25, 2025