Structural evolution and combustion energy of Al-B clusters

Structural evolution and combustion energy of Al-B clusters | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Structural evolution and combustion energy of Al-B clusters Elizaveta E. Vaneeva, Anastasiia A. Mikhailova, Sergey V. Lepeshkin, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9334205/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 14 You are reading this latest preprint version Abstract Systematic structure prediction of Al n B m clusters (0 ≤ n , m ≤ 16) was performed to explore their structural diversity, stability, and combustion-related properties. We identified structural motifs that evolve with boron content, ranging from isolated boron atoms embedded in aluminum frameworks to dense and perforated boron networks.We have analyzed the stability of Al n B m molecules using criteria adopted in nanocluster studies, where relative stability with respect to neighboring compositions serves as an indicator of “magic” clusters. Fragmentation analysis indicates that mixed Al n B m clusters typically dissociate through the loss of a single aluminum atom. At the nanoscale, pure boron and boron-rich Al n B m ​ clusters exhibit higher Gibbs free energies of combustion than aluminum clusters, in contrast to bulk behavior. Specific combustion energies for boron nanoclusters are higher too. Smaller clusters exhibit larger energy output because of their higher fraction of reactive surface atoms, although this size-dependent effect becomes less pronounced upon aluminum incorporation. Figures Figure 1 Figure 2 Figure 3 1. Introduction The development of high-energy-density fuels is a key challenge for next-generation propulsion and power systems. One promising approach is the incorporation of metal or metalloid nanoclusters into conventional liquid hydrocarbon fuels.[ 1 ] Due to their small size, nanoclusters possess a large fraction of surface atoms, which makes them highly reactive and allows for faster and more complete oxidation during combustion. This often results in increased heat release, shorter ignition delay, and faster flame propagation.[ 2 ] Among the various candidates, particularly aluminum and boron are the most widely explored. Their relatively low ignition temperatures, high heats of combustion, and compatibility with liquid-fuel formulations make them especially effective in improving the performance of traditional hydrocarbon fuels.[ 3 ] Both elements belong to group 13 and are isoelectronic, having three valence electrons, yet they exhibit different bonding characteristics and physical behavior. Aluminum has relatively diffuse valence orbitals, which give rise to delocalized, weakly directional metallic bonding, hence, it has a relatively low surface energy: breaking metallic bonds and forming surfaces is energetically inexpensive, and dangling bonds are not strongly penalized. Boron, in contrast, forms strongly directional, 2-center and multicenter covalent bonds and behaves as a frustrated semiconductor. In such systems, dangling bonds are energetically costly, leading to high surface energies and pronounced surface reconstruction. These electronic differences that govern the structural chemistry of aluminum and boron are directly reflected in their preferred cluster geometries. Small aluminum clusters favor compact, highly symmetric structures, most notably the icosahedral Al 13 “magic” cluster, and icosahedral motifs are also common in aluminum-based quasicrystals. In contrast, crystalline boron consists of hollow B 12 icosahedra arranged in complex networks, yet such units are absent in small free boron clusters, which instead adopt planar or quasi-planar structures over a wide size range. [ 28 ] Another important distinction lies in bond strengths. A single B–B bond (3.08 eV)[ 7 ] is significantly stronger than a single Al–Al bond (1.37 eV)[ 7 ] due to its covalent nature. From an energetic perspective, boron has long been considered as one of the most promising energetic additives for hydrocarbon fuels due to its exceptionally high gravimetric (≈ 58 MJ/kg) and volumetric (≈ 131 MJ/L) energy densities.[ 3 ] Its combustion releases large amounts of heat and can significantly increase the flame temperature, thereby improving the thermal efficiency and thrust of propulsion systems. However, the practical utilization of boron remains challenging. Although boron has a higher energy density than aluminum, its practical use is hindered by the rapid formation of a dense, B 2 O 3 layer that blocks oxygen diffusion to the boron core, leading to incomplete combustion and the generation of solid oxide particles that ultimately reduce performance and can cause nozzle clogging.[ 4 ][ 5 ] As a result, boron exhibits long ignition delays, incomplete oxidation, and reduced overall energy release.[ 6 ] These limitations hinder its direct use in propulsion systems despite its outstanding theoretical energy content. Aluminum, in contrast, ignites at lower temperature and exhibits faster oxidation kinetics under combustion conditions than boron, which together with its high heat of combustion (~ 31 MJ/kg) makes it an effective additive and ignition promoter in propellants and explosives.[ 7 ] When aluminum and boron are combined, the rapid ignition of aluminum can slow down the growth of the B 2 O 3 layer and partially disrupt the existing oxide film. As a result, oxygen gains easier access to the boron, enabling more complete oxidation and, consequently, more complete boron combustion.[ 5 ] Such mixed clusters or composite additives retain high theoretical energy density of boron while benefiting from superior ignition and oxidation characteristics of aluminum, making Al–B materials attractive candidates for next-generation high-energy fuels. However, clusters of mixed compositions (Al n B m ) have not been investigated properly until now. The experimental study of these clusters is a difficult task, therefore, it is advisable to start by identifying the most promising of them at the fundamental level. Only a few studies addressed the analysis of structure and properties of certain neutral Al n B m clusters of special composition: Al n B m ( n + m ≤ 7) [ 8 ] and AlB n ( n = 0 − 20).[ 9 ][ 10 ] Thus, most existing studies are limited to examining only a small number of fixed compositions, whereas multicomponent clusters, which are of greater interest for practical applications, may have unexpected stoichiometries not found at the macro scale. In this work, we conducted a systematic study of Al n B m nanoclusters in a wide compositional range (0 ≤ n ≤ 16, 0 ≤ m ≤ 16). We utilize global optimization methods in conjunction with density functional theory (DFT) calculations to determine ground-state structures. We also investigate the reactive (energetic) characteristics of Al–B clusters, including combustion energy that indicates their potential energy release upon oxidation, and identify the most stable (“magic”) compositions to elucidate the structure–stability relationships. 2. Computational Methods The structures of Al n B m clusters were determined using an evolutionary variable-composition global optimization algorithm implemented in the USPEX code. [ 9 , 10 ]. To ensure accurate and reliable structure prediction, the computational procedure consisted of several steps. First, the USPEX search was performed using an evolutionary algorithm in combination with the ab initio VASP code [ 11 ],[ 12 ], which employs the projector-augmented wave method[ 13 ] and the Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional.[ 14 ] Spin polarization was taken into account during these calculations. Subsequently, the 10 lowest-energy isomers from each composition were selected for further refinement using the GAUSSIAN software,[ 15 ] applying the B3LYP hybrid functional [ 16 , 17 ] and the 6-311G+(d,p) basis set, using spin multiplicity corresponding to the lower energetic state.[ 18 ] We calculated the vibrational frequencies in all obtained structures and verified that each possesses 3N-6 positive frequencies. For each ground state structure we calculated Gibbs free energy at T = 300K, applying partition functions both for vibrations, translations, rotations, and electronic configurations. The structural stability was assessed using the local stability, Δ 2 min ( n , m ), and fragmentation energy ( E frag ) as described in our previous studies.[ 19 – 26 ] 3. Results and Discussion 3.1. Structure and stability of nanoclusters We explored the ground-state structures of Al n B m clusters within a broad compositional range of 0 ≤ n , m ≤ 16. The xyz coordinates of all optimized ground-state structures are provided in ESI Section S2. Our results reproduced previous calculations for smaller clusters AlB m ( m = 0–20) [ 9 , 10 ], Al n B m ( n + m ≤ 7).[ 8 , 27 ]. The ground-state structures of Al n B m (up to 0 ≤ n , m ≤ 16) clusters were found for the first time. The ground-state structures of Al n B m nanoclusters can be classified into several distinct classes based on the structural characteristics of the boron fragments within each cluster (Fig. 1 ). Al-rich Al n B m nanoclusters ( m ≤ 4 and n + m ≥ 7) form 1st class (Fig. 1 a). The clusters have a non-planar structure: an aluminum framework with a small isolated boron fragment inside. As the boron content increases, boron replaces aluminum, starting from the inner sites and moving outward, while the atomic arrangement remains fixed. The second class of Al n B m (Fig. 1 b) nanoclusters is defined by dense boron fragments that form well-organized nets. Unlike Al-rich clusters, the boron atoms interact closely, forming continuous or semi-continuous networks that resemble fragments of larger boron sheets. Depending on the aluminum environment, nets could be planar, quasi-planar (exhibit a nearly planar geometry with slight deviations). There are the exceptions with non-planar boron motifs with 5 ≤ m ≤ 7. Notably, these boron networks correspond to the ground states of pure boron clusters.[ 28 ] Planar and quasi-planar nets are forming hexagonal or distorted hexagonal patterns, akin to borophene-like structures.[ 29 ] Structures along the diagonal n + m =22 exhibit pronounced layered motifs, reminiscent of those found in bulk aluminum boride AlB 2 . However, unlike the planar boron layers observed in the bulk phases, the boron-rich fragments in these clusters adopt curved, nonplanar geometries. As the boron concentration increases, aluminum atoms become embedded within the dense boron networks, displacing boron fragments and causing deviations from ideal compact structures (dense networks). Here gaps and irregularities appear, as if the once-unbroken sheet has been stretched and torn apart in places. This leads to the formation of a distinct class ( class 3 , Fig. 1 c) of Al n B m clusters known as perforated networks. It is worth noting that chain-like structures formed by boron in structures with n + m ≥ 24 are generally energetically unfavorable — previous studies consistently show boron's tendency to form tubular, cage-like and core-shell configurations. These perforated networks can be considered as transition states from dense nets to tubular or cage-like boron structures, such as B 19 (D 2d ) or the highly symmetric B 20 (D 10d ).[ 28 ] However, in this case, the incorporation of aluminum appears to stabilize such unconventional boron fragments by providing a metallic framework that supports the existence of perforated networks. Such structural modifications significantly impact the local stability of the clusters. Most of the small Al n B m clusters ( n + m ≤ 6 and m ≤ 4) are categorized as class 4 (Fig. 1 d) adopt a planar arrangement, with the exception of Al 2 B 2 and Al 3 B 4 , where boron atoms are repelled by aluminum, leading to a distorted geometry. The stability of binary nanoclusters, characterized by Δ 2 min and E frag , can be effectively visualized using 2D heatmaps (Fig. 2 ). In Fig. 2 a, red regions indicate highly stable magic clusters, while blue areas represent unstable compositions where Δ 2 min < 0. Light blue regions correspond to unstable molecules, where − 0.4 eV ≤ Δ 2 min ≤ 0. An initial examination of the stability maps reveals distinct diagonal stability trends at n + m = 6, 12, 16, 18, 20, and 22, particularly visible in the Δ 2 min map. Clusters along these diagonals exhibit closed-shell electronic structures and are predominantly associated with dense boron nets. For instance, clusters on the n + m = 6 diagonal transition from pure 3D-shaped Al 6 to planar B 6 , where boron atoms gradually replace aluminum, ultimately forming a separate boron sheet. Analogously, along the n + m = 12, 16, and 18 diagonals, clusters evolve from pure aluminum to Al-rich compositions, then to structures with dense boron nets, and finally to pure boron clusters. In contrast, clusters along the n + m = 20 and 22 diagonals are mainly composed of dense boron nets, which progressively transition into perforated boron structures. The upper right triangle of the Δ 2 min map ( where n + m ≥ 24), which primarily includes Al n B m clusters with perforated boron nets, displays a negative Δ 2 min value. This is due to the chain-like arrangement of boron in these structures, a configuration not typically associated with stable boron frameworks. However, when the aluminum content decreases to four or five atoms, some perforated nets transition to a planar geometry, thereby gaining local stabilit y. Notably, several clusters exhibit “magic” stability despite having an odd number of electrons, which is often associated with high structural symmetry, as in B 3 (D 3h ), B 5 (C 2v ), AlB 8 (C 7v ), Al 2 B 7 (D 7h ), Al 3 (D 3h ), Al 12 B (D 5d ), Al 13 (I h ) and Al 14 B 9 (C 2 ). However, magic behavior is not exclusively symmetry-driven, since several clusters without symmetry also display enhanced stability, including Al 3 B 14 (C 1 ), Al 12 B 11 (C 1 ), Al 13 B 10 (C 1 ), and Al 13 B 12 (C 1 ). The maps clearly highlight the magic Al 13 cluster, which adopts an I h geometry with one aluminum atom enclosed inside. This structure has been confirmed experimentally through mass‑spectrometric, which demonstrated the enhanced stability and characteristic electronic configuration of the 13-atom aluminum icosahedron.[ 30 ] This structural arrangement contributes to its exceptional stability, as confirmed by the fragmentation map (Fig. 2 b). Analysis of fragmentation behavior further demonstrates that these clusters are highly resistant to decomposition, with clusters most favorably losing a single aluminum atom with exception of pure boron clusters that tend to lose a boron atom instead, which is reflected in high values of fragmentation (Table S1 , Section 1 of ESI). We have used a third criterion, the HOMO−LUMO gap. (Fig. 2 c) shows interpolated heatmaps of gaps for Al n B m nanoclusters. This gap is a key factor in determining the electronic properties of a cluster, such as its electronic polarisability (and, hence, reactivity). Nanoclusters with a wide HOMO-LUMO gap are typically less reactive and thus more easily survive interactions with other molecules. The largest HOMO–LUMO gaps are observed for the B 8 ​ and AlB 8 ​ clusters. This enhanced gap can be attributed to their high structural symmetry, which promotes electronic shell closure and reduces the density of electronic states near the Fermi level. 3.2. Prospects of Al-B nanoparticles in application of high energy density materials We investigated the potential of Al n B m clusters as high-energy-density additives to organic fuels. We examined the combustion of any cluster into the most stable oxides of aluminum and boron. To better approximate real conditions, we consider the process at 300 K and ambient pressure, using Gibbs free energy. We calculated the energy output obtained from burning a given cluster in an oxygen atmosphere, leading to the formation of stable products — Al 2 O 3 and B 2 O 3 . We assume that each cluster reacts with O 2 , producing Al 2 O 3 and B 2 O 3 with the corresponding stoichiometric coefficients: Al n B m + ¾ ( n + m )O 2 -> ½ n Al 2 O 3 + ½ m B 2 O 3 The energetic output of the combustion reaction could be printed as: Δ c G (Al n B m ) =- [½n G (Al 2 O 3 ) +½m G (B 2 O 3 ) - G (Al n B m ) -¾ ( n + m ) G (O 2 )] = -Δ r G (Al n B m ), where G is Gibbs free energy of the corresponding compound. The Gibbs free energies of combustion for Al n B m nanoclusters are presented as an interpolated two-dimensional heatmap (Fig. 3 a). It shows the thermodynamic stability of the clusters. Here, Δ c G (Al n B m ) is taken with the opposite sign of the Gibbs free energy change, so that larger positive values correspond to more exothermic and thus more thermodynamically stable combustion. Figure 3 b illustrates the specific combustion energy: Δ c Q (Al n B m ) = Δ c G (Al n B m )/ M (Al n B m ), where M (Al n B m ) is the molar mass of the Al n B m ​ cluster. This quantity characterizes how intensely a given cluster “burns” per unit mass and allows direct comparison between clusters of different compositions. Pure boron clusters exhibit larger values of the Gibbs free energy of combustion (Δ c G ) than aluminum clusters, which is in contrast to their bulk phases, in which aluminum combustion releases more energy: Δ c G (Al 2 O 3 ) = 1690.96 kJ/mol, Δ c G (B 2 O 3 ) = 1288.12 kJ/mol. This reverse behaviour at the nanoscale occurs due to dangling B-B bonds, making the surface energy of boron clusters very high. Owing to the lower atomic mass of boron B-rich Al n B m nanoclusters exhibit the highest specific combustion energy, Δ c Q (Fig. 3 b). In addition, smaller clusters display enhanced values of Δ c Q , reflecting the increased fraction of under-coordinated surface atoms. 4. Conclusions In this work, we performed a systematic investigation of Al-B nanoclusters containing up to 16 atoms. Despite the isoelectronic nature of aluminum and boron, it is shown that their structural chemistry differs significantly both from each other and from that of mixed Al-B clusters. Pure aluminum clusters are prone to easy disruption of metallic bonding and the formation of weak delocalized interactions, whereas boron clusters tend to form multicenter covalent bonds. The structural motifs of Al-B clusters are determined by the interplay between directional B-B bonding and metallic interactions involving Al atoms. As a result, a structural evolution is observed from aluminum-rich frameworks to dense boron networks. Stability analysis shows that mixed Al-B clusters are locally stable, while fragmentation predominantly occurs through the loss of aluminum atoms, indicating stronger bonding within boron-rich frameworks. The highest stability is found for families of clusters satisfying the relation n + m = 2 k , k = 6, 12, 16, 18, 20, and 22, corresponding to closed electronic shells. However, additional stabilization is also observed for clusters with an odd number of electrons due to high structural symmetry, which leads to enhanced local stability. The combustion energetics reveal a pronounced nanoscale effect: boron-rich clusters exhibit higher Gibbs free energies of combustion and higher specific combustion energies than aluminum clusters, in contrast to the behavior observed in the bulk. This effect is associated with the low atomic mass of boron as well as the increased contribution of surface energy to the energetic characteristics of boron-containing clusters at the nanoscale. Accordingly, nanostructured covalently bonded materials, such as aluminum–boron clusters, may be of interest as high-energy additives to energetic mixtures due to the enhanced contribution of surface energy and their distinct structural chemistry compared to single-element clusters. Declarations Conflicts of interest The authors declare no competing financial interest. Author Contribution E.E.V. conceived the study, performed the structure prediction and quantum-chemical calculations, analyzed the data, and wrote the original draft of the manuscript. A.A.M. contributed to the methodology, data analysis, and manuscript editing. S.V.L. contributed to the conceptualization, methodology, analysis and interpretation of the results. A.P.M. contributed to the interpretation of the results and manuscript editing. A.R.O. supervised the project, contributed to the conceptualization and interpretation of the results, and critically revised the manuscript. All authors reviewed and approved the final manuscript. Acknowledgement This work was supported by the Russian Science Foundation (grant #25-43-20043). The calculations were performed on Oleg and Arkuda supercomputers at Skoltech and at the Joint Supercomputer Center of Russian Academy of Sciences and the Lobachevsky cluster at the University of Nizhny Novgorod. References I.A.S. Ferrão, M.A.A. Mendes, A.S.O.H. Moita, A.R.R. Silva, The addition of particles to an alternative jet fuel, Fuels 3 (2022) 184–206. https://doi.org/10.3390/fuels3020012 . M. Jones, C.H. Li, A. Afjeh, G. Peterson, Experimental study of combustion characteristics of nanoscale metal and metal oxide additives in biofuel (ethanol), Nanoscale Res Lett 6 (2011) 246. https://doi.org/10.1186/1556-276X-6-246 . D. Sundaram, V. Yang, R.A. Yetter, Metal-based nanoenergetic materials: Synthesis, properties, and applications, Prog. Energy Combust. Sci. 61 (2017) 293–365. https://doi.org/10.1016/j.pecs.2017.02.002 . A. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9334205","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":641575406,"identity":"04976ff1-2716-4cd8-870c-c0fed263bdbe","order_by":0,"name":"Elizaveta E. Vaneeva","email":"data:image/png;base64,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","orcid":"","institution":"Skolkovo Institute of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Elizaveta","middleName":"E.","lastName":"Vaneeva","suffix":""},{"id":641575407,"identity":"84b9c199-4b5b-47b8-897e-b85a08b85b83","order_by":1,"name":"Anastasiia A. Mikhailova","email":"","orcid":"","institution":"Skolkovo Institute of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Anastasiia","middleName":"A.","lastName":"Mikhailova","suffix":""},{"id":641575408,"identity":"28fdceb1-cab8-460d-8d72-5ad738961490","order_by":2,"name":"Sergey V. Lepeshkin","email":"","orcid":"","institution":"Skolkovo Institute of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Sergey","middleName":"V.","lastName":"Lepeshkin","suffix":""},{"id":641575409,"identity":"c3c98518-fb8f-4183-94d8-58f2ad026626","order_by":3,"name":"Alexey P. Maltsev","email":"","orcid":"","institution":"Skolkovo Institute of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Alexey","middleName":"P.","lastName":"Maltsev","suffix":""},{"id":641575410,"identity":"9f8a62db-ee85-4cc9-886c-ea13b039c940","order_by":4,"name":"Artem R. Oganov","email":"","orcid":"","institution":"Skolkovo Institute of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Artem","middleName":"R.","lastName":"Oganov","suffix":""}],"badges":[],"createdAt":"2026-04-06 13:09:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9334205/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9334205/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109759783,"identity":"c5957d0a-a7d9-457f-be8f-632d36538886","added_by":"auto","created_at":"2026-05-22 07:27:41","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":861461,"visible":true,"origin":"","legend":"\u003cp\u003eClassification of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e clusters into four categories based on their structural topology, illustrated using representative examples with Δ\u003csup\u003e2\u003c/sup\u003e\u003csub\u003emin\u003c/sub\u003e \u0026gt; 0: (a) Al-rich nanoclusters, (b) nanoclusters with dense networks, (c) nanoclusters with perforated networks, and (d) small Al–B nanoclusters. Blue and green spheres represent aluminum and boron atoms, respectively. (e) Schematic classification of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e nanoclusters, where the composition space is divided into four regions according to structural characteristics. Stars indicate “magic” stoichiometries.\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-9334205/v1/a3aedfda883e2bf6fe478e88.png"},{"id":109760415,"identity":"7f3abfbe-16ba-4211-830e-4e561feb9853","added_by":"auto","created_at":"2026-05-22 07:28:39","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":1427299,"visible":true,"origin":"","legend":"\u003cp\u003eInterpolated heatmaps of (a) Δ\u003csup\u003e2\u003c/sup\u003e\u003csub\u003emin\u003c/sub\u003e and (b) \u003cem\u003eE\u003c/em\u003e\u003csub\u003efrag\u003c/sub\u003e (c) HOMO-LUMO gap (in eV) for Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e nanoclusters as functions of n and m. Regions of structural instability are highlighted in blue. The dashed reference lines indicate “magic” nanoclusters located along the diagonals \u003cem\u003en\u003c/em\u003e + \u003cem\u003em\u003c/em\u003e = 6, 12, 16, 18, 20, and 22.\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-9334205/v1/f8234b3816b8607f0de0e12c.png"},{"id":109490155,"identity":"015d7517-9545-4e26-aa03-424a9d66a760","added_by":"auto","created_at":"2026-05-18 17:30:24","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":430254,"visible":true,"origin":"","legend":"\u003cp\u003eInterpolated heatmaps of (a) Gibbs free energies of combustion, Δ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eG\u003c/em\u003e and (b) specific combustion energy, Δ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eQ\u003c/em\u003e for A\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e nanoclusters. Stars highlight \"magic\" compositions of corresponding systems.\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-9334205/v1/b873cbb42f4f7b636222c481.png"},{"id":109765147,"identity":"57cfabce-c98e-4f4b-93f1-9f8c32b433b9","added_by":"auto","created_at":"2026-05-22 07:39:42","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2362204,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9334205/v1/13bdedae-b0c9-497c-a2b0-533c731abdc5.pdf"},{"id":109490153,"identity":"540bb7c2-8948-40d1-b9b9-89d56eb6e304","added_by":"auto","created_at":"2026-05-18 17:30:24","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":1173130,"visible":true,"origin":"","legend":"","description":"","filename":"AlBESIfinal.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9334205/v1/b01d54799480690e5bd1903c.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Structural evolution and combustion energy of Al-B clusters","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eThe development of high-energy-density fuels is a key challenge for next-generation propulsion and power systems. One promising approach is the incorporation of metal or metalloid nanoclusters into conventional liquid hydrocarbon fuels.[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] Due to their small size, nanoclusters possess a large fraction of surface atoms, which makes them highly reactive and allows for faster and more complete oxidation during combustion. This often results in increased heat release, shorter ignition delay, and faster flame propagation.[\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e] Among the various candidates, particularly aluminum and boron are the most widely explored. Their relatively low ignition temperatures, high heats of combustion, and compatibility with liquid-fuel formulations make them especially effective in improving the performance of traditional hydrocarbon fuels.[\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] Both elements belong to group 13 and are isoelectronic, having three valence electrons, yet they exhibit different bonding characteristics and physical behavior. Aluminum has relatively diffuse valence orbitals, which give rise to delocalized, weakly directional metallic bonding, hence, it has a relatively low surface energy: breaking metallic bonds and forming surfaces is energetically inexpensive, and dangling bonds are not strongly penalized. Boron, in contrast, forms strongly directional, 2-center and multicenter covalent bonds and behaves as a frustrated semiconductor. In such systems, dangling bonds are energetically costly, leading to high surface energies and pronounced surface reconstruction.\u003c/p\u003e \u003cp\u003eThese electronic differences that govern the structural chemistry of aluminum and boron are directly reflected in their preferred cluster geometries. Small aluminum clusters favor compact, highly symmetric structures, most notably the icosahedral Al\u003csub\u003e13\u003c/sub\u003e \u0026ldquo;magic\u0026rdquo; cluster, and icosahedral motifs are also common in aluminum-based quasicrystals. In contrast, crystalline boron consists of hollow B\u003csub\u003e12\u003c/sub\u003e icosahedra arranged in complex networks, yet such units are absent in small free boron clusters, which instead adopt planar or quasi-planar structures over a wide size range. [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e] Another important distinction lies in bond strengths. A single B\u0026ndash;B bond (3.08 eV)[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] is significantly stronger than a single Al\u0026ndash;Al bond (1.37 eV)[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] due to its covalent nature.\u003c/p\u003e \u003cp\u003eFrom an energetic perspective, boron has long been considered as one of the most promising energetic additives for hydrocarbon fuels due to its exceptionally high gravimetric (\u0026asymp;\u0026thinsp;58 MJ/kg) and volumetric (\u0026asymp;\u0026thinsp;131 MJ/L) energy densities.[\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] Its combustion releases large amounts of heat and can significantly increase the flame temperature, thereby improving the thermal efficiency and thrust of propulsion systems. However, the practical utilization of boron remains challenging. Although boron has a higher energy density than aluminum, its practical use is hindered by the rapid formation of a dense, B\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e layer that blocks oxygen diffusion to the boron core, leading to incomplete combustion and the generation of solid oxide particles that ultimately reduce performance and can cause nozzle clogging.[\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e][\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] As a result, boron exhibits long ignition delays, incomplete oxidation, and reduced overall energy release.[\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] These limitations hinder its direct use in propulsion systems despite its outstanding theoretical energy content.\u003c/p\u003e \u003cp\u003eAluminum, in contrast, ignites at lower temperature and exhibits faster oxidation kinetics under combustion conditions than boron, which together with its high heat of combustion (~\u0026thinsp;31 MJ/kg) makes it an effective additive and ignition promoter in propellants and explosives.[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] When aluminum and boron are combined, the rapid ignition of aluminum can slow down the growth of the B\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e layer and partially disrupt the existing oxide film. As a result, oxygen gains easier access to the boron, enabling more complete oxidation and, consequently, more complete boron combustion.[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] Such mixed clusters or composite additives retain high theoretical energy density of boron while benefiting from superior ignition and oxidation characteristics of aluminum, making Al\u0026ndash;B materials attractive candidates for next-generation high-energy fuels.\u003c/p\u003e \u003cp\u003eHowever, clusters of mixed compositions (Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e) have not been investigated properly until now. The experimental study of these clusters is a difficult task, therefore, it is advisable to start by identifying the most promising of them at the fundamental level. Only a few studies addressed the analysis of structure and properties of certain neutral Al\u003csub\u003en\u003c/sub\u003eB\u003csub\u003em\u003c/sub\u003e clusters of special composition: Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003em\u003c/em\u003e \u0026le; 7) [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] and AlB\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0\u0026thinsp;\u0026minus;\u0026thinsp;20).[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e][\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] Thus, most existing studies are limited to examining only a small number of fixed compositions, whereas multicomponent clusters, which are of greater interest for practical applications, may have unexpected stoichiometries not found at the macro scale.\u003c/p\u003e \u003cp\u003eIn this work, we conducted a systematic study of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e nanoclusters in a wide compositional range (0\u0026thinsp;\u0026le;\u0026thinsp;\u003cem\u003en\u003c/em\u003e\u0026thinsp;\u0026le;\u0026thinsp;16, 0\u0026thinsp;\u0026le;\u0026thinsp;\u003cem\u003em\u003c/em\u003e\u0026thinsp;\u0026le;\u0026thinsp;16). We utilize global optimization methods in conjunction with density functional theory (DFT) calculations to determine ground-state structures. We also investigate the reactive (energetic) characteristics of Al\u0026ndash;B clusters, including combustion energy that indicates their potential energy release upon oxidation, and identify the most stable (\u0026ldquo;magic\u0026rdquo;) compositions to elucidate the structure\u0026ndash;stability relationships.\u003c/p\u003e"},{"header":"2. Computational Methods","content":"\u003cp\u003eThe structures of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e clusters were determined using an evolutionary variable-composition global optimization algorithm implemented in the USPEX code. [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. To ensure accurate and reliable structure prediction, the computational procedure consisted of several steps. First, the USPEX search was performed using an evolutionary algorithm in combination with the ab initio VASP code [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e],[\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], which employs the projector-augmented wave method[\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] and the Perdew\u0026ndash;Burke\u0026ndash;Ernzerhof (PBE) exchange-correlation functional.[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] Spin polarization was taken into account during these calculations. Subsequently, the 10 lowest-energy isomers from each composition were selected for further refinement using the GAUSSIAN software,[\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] applying the B3LYP hybrid functional [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] and the 6-311G+(d,p) basis set, using spin multiplicity corresponding to the lower energetic state.[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] We calculated the vibrational frequencies in all obtained structures and verified that each possesses 3N-6 positive frequencies. For each ground state structure we calculated Gibbs free energy at T\u0026thinsp;=\u0026thinsp;300K, applying partition functions both for vibrations, translations, rotations, and electronic configurations. The structural stability was assessed using the local stability, Δ\u003csup\u003e2\u003c/sup\u003e\u003csub\u003emin\u003c/sub\u003e(\u003cem\u003en\u003c/em\u003e, \u003cem\u003em\u003c/em\u003e), and fragmentation energy (\u003cem\u003eE\u003c/em\u003e\u003csub\u003efrag\u003c/sub\u003e) as described in our previous studies.[\u003cspan additionalcitationids=\"CR20 CR21 CR22 CR23 CR24 CR25\" citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]\u003c/p\u003e"},{"header":"3. Results and Discussion","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Structure and stability of nanoclusters\u003c/h2\u003e \u003cp\u003eWe explored the ground-state structures of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e clusters within a broad compositional range of 0\u0026thinsp;\u0026le;\u0026thinsp;\u003cem\u003en\u003c/em\u003e, \u003cem\u003em\u003c/em\u003e\u0026thinsp;\u0026le;\u0026thinsp;16. The \u003cem\u003exyz\u003c/em\u003e coordinates of all optimized ground-state structures are provided in ESI Section S2.\u003c/p\u003e \u003cp\u003eOur results reproduced previous calculations for smaller clusters AlB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003em\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0\u0026ndash;20) [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e], Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003em\u003c/em\u003e \u0026le; 7).[\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. The ground-state structures of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e (up to 0\u0026thinsp;\u0026le;\u0026thinsp;\u003cem\u003en\u003c/em\u003e, \u003cem\u003em\u003c/em\u003e\u0026thinsp;\u0026le;\u0026thinsp;16) clusters were found for the first time. The ground-state structures of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e nanoclusters can be classified into several distinct classes based on the structural characteristics of the boron fragments within each cluster (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). Al-rich Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e nanoclusters (\u003cem\u003em\u003c/em\u003e\u0026thinsp;\u0026le;\u0026thinsp;4 and \u003cem\u003en\u0026thinsp;+\u0026thinsp;m\u003c/em\u003e \u0026ge; 7) form \u003cem\u003e1st class\u003c/em\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea). The clusters have a non-planar structure: an aluminum framework with a small isolated boron fragment inside. As the boron content increases, boron replaces aluminum, starting from the inner sites and moving outward, while the atomic arrangement remains fixed.\u003c/p\u003e \u003cp\u003eThe \u003cem\u003esecond class\u003c/em\u003e of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb) nanoclusters is defined by dense boron fragments that form well-organized nets. Unlike Al-rich clusters, the boron atoms interact closely, forming continuous or semi-continuous networks that resemble fragments of larger boron sheets. Depending on the aluminum environment, nets could be planar, quasi-planar (exhibit a nearly planar geometry with slight deviations). There are the exceptions with non-planar boron motifs with 5\u0026thinsp;\u0026le;\u0026thinsp;\u003cem\u003em\u003c/em\u003e\u0026thinsp;\u0026le;\u0026thinsp;7. Notably, these boron networks correspond to the ground states of pure boron clusters.[\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e] Planar and quasi-planar nets are forming hexagonal or distorted hexagonal patterns, akin to borophene-like structures.[\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e] Structures along the diagonal \u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003em\u003c/em\u003e=22 exhibit pronounced layered motifs, reminiscent of those found in bulk aluminum boride AlB\u003csub\u003e2\u003c/sub\u003e. However, unlike the planar boron layers observed in the bulk phases, the boron-rich fragments in these clusters adopt curved, nonplanar geometries.\u003c/p\u003e \u003cp\u003eAs the boron concentration increases, aluminum atoms become embedded within the dense boron networks, displacing boron fragments and causing deviations from ideal compact structures (dense networks). Here gaps and irregularities appear, as if the once-unbroken sheet has been stretched and torn apart in places. This leads to the formation of a distinct class (\u003cem\u003eclass 3\u003c/em\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ec) of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e clusters known as perforated networks. It is worth noting that chain-like structures formed by boron in structures with \u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003em\u003c/em\u003e \u0026ge; 24 are generally energetically unfavorable \u0026mdash; previous studies consistently show boron's tendency to form tubular, cage-like and core-shell configurations. These perforated networks can be considered as transition states from dense nets to tubular or cage-like boron structures, such as B\u003csub\u003e19\u003c/sub\u003e (D\u003csub\u003e2d\u003c/sub\u003e) or the highly symmetric B\u003csub\u003e20\u003c/sub\u003e (D\u003csub\u003e10d\u003c/sub\u003e).[\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e] However, in this case, the incorporation of aluminum appears to stabilize such unconventional boron fragments by providing a metallic framework that supports the existence of perforated networks. Such structural modifications significantly impact the local stability of the clusters.\u003c/p\u003e \u003cp\u003eMost of the small Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e clusters (\u003cem\u003en\u0026thinsp;+\u0026thinsp;m\u003c/em\u003e \u0026le; 6 and \u003cem\u003em\u003c/em\u003e\u0026thinsp;\u0026le;\u0026thinsp;4) are categorized as \u003cem\u003eclass 4\u003c/em\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ed) adopt a planar arrangement, with the exception of Al\u003csub\u003e2\u003c/sub\u003eB\u003csub\u003e2\u003c/sub\u003e and Al\u003csub\u003e3\u003c/sub\u003eB\u003csub\u003e4\u003c/sub\u003e, where boron atoms are repelled by aluminum, leading to a distorted geometry.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe stability of binary nanoclusters, characterized by Δ\u003csup\u003e2\u003c/sup\u003e\u003csub\u003emin\u003c/sub\u003e and \u003cem\u003eE\u003c/em\u003e\u003csub\u003efrag\u003c/sub\u003e, can be effectively visualized using 2D heatmaps (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). In Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea, red regions indicate highly stable magic clusters, while blue areas represent unstable compositions where Δ\u003csup\u003e2\u003c/sup\u003e\u003csub\u003emin\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0. Light blue regions correspond to unstable molecules, where \u0026minus;\u0026thinsp;0.4 eV\u0026thinsp;\u0026le;\u0026thinsp;Δ\u003csup\u003e2\u003c/sup\u003e\u003csub\u003emin\u003c/sub\u003e\u0026thinsp;\u0026le;\u0026thinsp;0.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAn initial examination of the stability maps reveals distinct diagonal stability trends at \u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003em\u003c/em\u003e = 6, 12, 16, 18, 20, and 22, particularly visible in the Δ\u003csup\u003e2\u003c/sup\u003e\u003csub\u003emin\u003c/sub\u003e map. Clusters along these diagonals exhibit closed-shell electronic structures and are predominantly associated with dense boron nets. For instance, clusters on the \u003cem\u003en\u0026thinsp;+\u0026thinsp;m\u003c/em\u003e = 6 diagonal transition from pure 3D-shaped Al\u003csub\u003e6\u003c/sub\u003e to planar B\u003csub\u003e6\u003c/sub\u003e, where boron atoms gradually replace aluminum, ultimately forming a separate boron sheet. Analogously, along the \u003cem\u003en\u0026thinsp;+\u0026thinsp;m\u003c/em\u003e = 12, 16, and 18 diagonals, clusters evolve from pure aluminum to Al-rich compositions, then to structures with dense boron nets, and finally to pure boron clusters. In contrast, clusters along the \u003cem\u003en\u0026thinsp;+\u0026thinsp;m\u003c/em\u003e = 20 and 22 diagonals are mainly composed of dense boron nets, which progressively transition into perforated boron structures. The upper right triangle of the Δ\u003csup\u003e2\u003c/sup\u003e\u003csub\u003emin\u003c/sub\u003e map ( where \u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003em\u003c/em\u003e \u0026ge; 24), which primarily includes Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e clusters with perforated boron nets, displays a negative Δ\u003csup\u003e2\u003c/sup\u003e\u003csub\u003emin\u003c/sub\u003e value. This is due to the chain-like arrangement of boron in these structures, a configuration not typically associated with stable boron frameworks. However, when the aluminum content decreases to four or five atoms, some perforated nets transition to a planar geometry, thereby gaining local stabilit\u003cem\u003ey.\u003c/em\u003e Notably, several clusters exhibit \u0026ldquo;magic\u0026rdquo; stability despite having an odd number of electrons, which is often associated with high structural symmetry, as in B\u003csub\u003e3\u003c/sub\u003e (D\u003csub\u003e3h\u003c/sub\u003e), B\u003csub\u003e5\u003c/sub\u003e (C\u003csub\u003e2v\u003c/sub\u003e), AlB\u003csub\u003e8\u003c/sub\u003e (C\u003csub\u003e7v\u003c/sub\u003e), Al\u003csub\u003e2\u003c/sub\u003eB\u003csub\u003e7\u003c/sub\u003e (D\u003csub\u003e7h\u003c/sub\u003e), Al\u003csub\u003e3\u003c/sub\u003e (D\u003csub\u003e3h\u003c/sub\u003e), Al\u003csub\u003e12\u003c/sub\u003eB (D\u003csub\u003e5d\u003c/sub\u003e), Al\u003csub\u003e13\u003c/sub\u003e (I\u003csub\u003eh\u003c/sub\u003e) and Al\u003csub\u003e14\u003c/sub\u003eB\u003csub\u003e9\u003c/sub\u003e (C\u003csub\u003e2\u003c/sub\u003e). However, magic behavior is not exclusively symmetry-driven, since several clusters without symmetry also display enhanced stability, including Al\u003csub\u003e3\u003c/sub\u003eB\u003csub\u003e14\u003c/sub\u003e (C\u003csub\u003e1\u003c/sub\u003e), Al\u003csub\u003e12\u003c/sub\u003eB\u003csub\u003e11\u003c/sub\u003e (C\u003csub\u003e1\u003c/sub\u003e), Al\u003csub\u003e13\u003c/sub\u003eB\u003csub\u003e10\u003c/sub\u003e (C\u003csub\u003e1\u003c/sub\u003e), and Al\u003csub\u003e13\u003c/sub\u003eB\u003csub\u003e12\u003c/sub\u003e (C\u003csub\u003e1\u003c/sub\u003e).\u003c/p\u003e \u003cp\u003eThe maps clearly highlight the magic Al\u003csub\u003e13\u003c/sub\u003e cluster, which adopts an \u003cem\u003eI\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e geometry with one aluminum atom enclosed inside. This structure has been confirmed experimentally through mass‑spectrometric, which demonstrated the enhanced stability and characteristic electronic configuration of the 13-atom aluminum icosahedron.[\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e] This structural arrangement contributes to its exceptional stability, as confirmed by the fragmentation map (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb). Analysis of fragmentation behavior further demonstrates that these clusters are highly resistant to decomposition, with clusters most favorably losing a single aluminum atom with exception of pure boron clusters that tend to lose a boron atom instead, which is reflected in high values of fragmentation (Table \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e, Section 1 of ESI).\u003c/p\u003e \u003cp\u003eWe have used a third criterion, the HOMO\u0026minus;LUMO gap. (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ec) shows interpolated heatmaps of gaps for Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e nanoclusters. This gap is a key factor in determining the electronic properties of a cluster, such as its electronic polarisability (and, hence, reactivity). Nanoclusters with a wide HOMO-LUMO gap are typically less reactive and thus more easily survive interactions with other molecules. The largest HOMO\u0026ndash;LUMO gaps are observed for the B\u003csub\u003e8\u003c/sub\u003e​ and AlB\u003csub\u003e8\u003c/sub\u003e​ clusters. This enhanced gap can be attributed to their high structural symmetry, which promotes electronic shell closure and reduces the density of electronic states near the Fermi level.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2. Prospects of Al-B nanoparticles in application of high energy density materials\u003c/h2\u003e \u003cp\u003eWe investigated the potential of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e clusters as high-energy-density additives to organic fuels. We examined the combustion of any cluster into the most stable oxides of aluminum and boron. To better approximate real conditions, we consider the process at 300 K and ambient pressure, using Gibbs free energy.\u003c/p\u003e \u003cp\u003eWe calculated the energy output obtained from burning a given cluster in an oxygen atmosphere, leading to the formation of stable products \u0026mdash; Al\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e and B\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e. We assume that each cluster reacts with O\u003csub\u003e2\u003c/sub\u003e, producing Al\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e and B\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e with the corresponding stoichiometric coefficients:\u003c/p\u003e \u003cp\u003eAl\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e + \u0026frac34; (\u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003em\u003c/em\u003e)O\u003csub\u003e2\u003c/sub\u003e -\u0026gt; \u0026frac12;\u003cem\u003en\u003c/em\u003eAl\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e + \u0026frac12;\u003cem\u003em\u003c/em\u003eB\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e \u003cp\u003eThe energetic output of the combustion reaction could be printed as:\u003c/p\u003e \u003cp\u003eΔ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eG\u003c/em\u003e(Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e) \u003cem\u003e=-\u003c/em\u003e[\u0026frac12;n\u003cem\u003eG\u003c/em\u003e (Al\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e) +\u0026frac12;m\u003cem\u003eG\u003c/em\u003e (B\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e) - \u003cem\u003eG\u003c/em\u003e (Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e) -\u0026frac34; (\u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003em\u003c/em\u003e)\u003cem\u003eG\u003c/em\u003e (O\u003csub\u003e2\u003c/sub\u003e)] \u003cem\u003e=\u003c/em\u003e-Δ\u003csub\u003er\u003c/sub\u003e\u003cem\u003eG\u003c/em\u003e (Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e),\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eG\u003c/em\u003e is Gibbs free energy of the corresponding compound.\u003c/p\u003e \u003cp\u003eThe Gibbs free energies of combustion for Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e nanoclusters are presented as an interpolated two-dimensional heatmap (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea). It shows the thermodynamic stability of the clusters. Here, Δ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eG\u003c/em\u003e(Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e) is taken with the opposite sign of the Gibbs free energy change, so that larger positive values correspond to more exothermic and thus more thermodynamically stable combustion. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb illustrates the specific combustion energy:\u003c/p\u003e \u003cp\u003eΔ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eQ\u003c/em\u003e(Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e) = Δ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eG\u003c/em\u003e(Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e)/\u003cem\u003eM\u003c/em\u003e(Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e),\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eM\u003c/em\u003e(Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e) is the molar mass of the Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e​ cluster. This quantity characterizes how intensely a given cluster \u0026ldquo;burns\u0026rdquo; per unit mass and allows direct comparison between clusters of different compositions.\u003c/p\u003e \u003cp\u003ePure boron clusters exhibit larger values of the Gibbs free energy of combustion (Δ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eG\u003c/em\u003e) than aluminum clusters, which is in contrast to their bulk phases, in which aluminum combustion releases more energy: Δ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eG\u003c/em\u003e(Al\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;1690.96 kJ/mol, Δ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eG\u003c/em\u003e(B\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;1288.12 kJ/mol. This reverse behaviour at the nanoscale occurs due to dangling B-B bonds, making the surface energy of boron clusters very high.\u003c/p\u003e \u003cp\u003eOwing to the lower atomic mass of boron B-rich Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e nanoclusters exhibit the highest specific combustion energy, Δ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eQ\u003c/em\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb). In addition, smaller clusters display enhanced values of Δ\u003csub\u003ec\u003c/sub\u003e\u003cem\u003eQ\u003c/em\u003e, reflecting the increased fraction of under-coordinated surface atoms.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4. Conclusions","content":"\u003cp\u003eIn this work, we performed a systematic investigation of Al-B nanoclusters containing up to 16 atoms. Despite the isoelectronic nature of aluminum and boron, it is shown that their structural chemistry differs significantly both from each other and from that of mixed Al-B clusters. Pure aluminum clusters are prone to easy disruption of metallic bonding and the formation of weak delocalized interactions, whereas boron clusters tend to form multicenter covalent bonds. The structural motifs of Al-B clusters are determined by the interplay between directional B-B bonding and metallic interactions involving Al atoms. As a result, a structural evolution is observed from aluminum-rich frameworks to dense boron networks.\u003c/p\u003e \u003cp\u003eStability analysis shows that mixed Al-B clusters are locally stable, while fragmentation predominantly occurs through the loss of aluminum atoms, indicating stronger bonding within boron-rich frameworks. The highest stability is found for families of clusters satisfying the relation \u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003em\u003c/em\u003e = 2\u003cem\u003ek\u003c/em\u003e, \u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;6, 12, 16, 18, 20, and 22, corresponding to closed electronic shells. However, additional stabilization is also observed for clusters with an odd number of electrons due to high structural symmetry, which leads to enhanced local stability.\u003c/p\u003e \u003cp\u003eThe combustion energetics reveal a pronounced nanoscale effect: boron-rich clusters exhibit higher Gibbs free energies of combustion and higher specific combustion energies than aluminum clusters, in contrast to the behavior observed in the bulk. This effect is associated with the low atomic mass of boron as well as the increased contribution of surface energy to the energetic characteristics of boron-containing clusters at the nanoscale. Accordingly, nanostructured covalently bonded materials, such as aluminum\u0026ndash;boron clusters, may be of interest as high-energy additives to energetic mixtures due to the enhanced contribution of surface energy and their distinct structural chemistry compared to single-element clusters.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eConflicts of interest\u003c/h2\u003e \u003cp\u003eThe authors declare no competing financial interest.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eE.E.V. conceived the study, performed the structure prediction and quantum-chemical calculations, analyzed the data, and wrote the original draft of the manuscript. A.A.M. contributed to the methodology, data analysis, and manuscript editing. S.V.L. contributed to the conceptualization, methodology, analysis and interpretation of the results. A.P.M. contributed to the interpretation of the results and manuscript editing. A.R.O. supervised the project, contributed to the conceptualization and interpretation of the results, and critically revised the manuscript. All authors reviewed and approved the final manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eThis work was supported by the Russian Science Foundation (grant #25-43-20043). The calculations were performed on Oleg and Arkuda supercomputers at Skoltech and at the Joint Supercomputer Center of Russian Academy of Sciences and the Lobachevsky cluster at the University of Nizhny Novgorod.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eI.A.S. Ferr\u0026atilde;o, M.A.A. Mendes, A.S.O.H. Moita, A.R.R. 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Luo, Pure metal clusters with atomic precision for nanomanufacturing, Nanomanufacturing Metrol. 5 (2022) 230\u0026ndash;239. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1007/s41871-022-00139-5\u003c/span\u003e\u003cspan address=\"10.1007/s41871-022-00139-5\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-cluster-science","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":" Learn more about [Journal of Cluster Science](https://www.springer.com/journal/10876) ","snPcode":"10876","submissionUrl":"https://mc.manuscriptcentral.com/jocl","title":"Journal of Cluster Science","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-9334205/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9334205/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSystematic structure prediction of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e clusters (0\u0026thinsp;\u0026le;\u0026thinsp;\u003cem\u003en\u003c/em\u003e, \u003cem\u003em\u003c/em\u003e\u0026thinsp;\u0026le;\u0026thinsp;16) was performed to explore their structural diversity, stability, and combustion-related properties. We identified structural motifs that evolve with boron content, ranging from isolated boron atoms embedded in aluminum frameworks to dense and perforated boron networks.We have analyzed the stability of Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e molecules using criteria adopted in nanocluster studies, where relative stability with respect to neighboring compositions serves as an indicator of \u0026ldquo;magic\u0026rdquo; clusters. Fragmentation analysis indicates that mixed Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e clusters typically dissociate through the loss of a single aluminum atom. At the nanoscale, pure boron and boron-rich Al\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eB\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e​ clusters exhibit higher Gibbs free energies of combustion than aluminum clusters, in contrast to bulk behavior. Specific combustion energies for boron nanoclusters are higher too. Smaller clusters exhibit larger energy output because of their higher fraction of reactive surface atoms, although this size-dependent effect becomes less pronounced upon aluminum incorporation.\u003c/p\u003e","manuscriptTitle":"Structural evolution and combustion energy of Al-B clusters","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-18 17:30:20","doi":"10.21203/rs.3.rs-9334205/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-05-22T11:56:10+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-21T18:13:38+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-21T16:05:32+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-15T18:00:30+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"309638802949474429681560598801962118439","date":"2026-05-13T06:38:48+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"87853954026894985383645495169977511339","date":"2026-05-12T14:21:37+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-11T20:08:07+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"243373904168954650598184786936291662378","date":"2026-05-11T08:49:25+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"269920894555170068341734823722236391880","date":"2026-05-08T17:38:08+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"35769920301450883905421707190018282944","date":"2026-05-07T20:40:07+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-05-07T15:52:06+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-04-29T11:33:36+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-04-28T14:15:37+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Cluster Science","date":"2026-04-06T13:02:50+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-cluster-science","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":" Learn more about [Journal of Cluster Science](https://www.springer.com/journal/10876) ","snPcode":"10876","submissionUrl":"https://mc.manuscriptcentral.com/jocl","title":"Journal of Cluster Science","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"22af5dbc-214d-4046-afc9-9c4c499e5f1a","owner":[],"postedDate":"May 18th, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Revision requested","date":"2026-05-22T11:56:10+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-21T18:13:38+00:00","index":26,"fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-21T16:05:32+00:00","index":25,"fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-15T18:00:30+00:00","index":24,"fulltext":""},{"type":"reviewerAgreed","content":"309638802949474429681560598801962118439","date":"2026-05-13T06:38:48+00:00","index":23,"fulltext":""},{"type":"reviewerAgreed","content":"87853954026894985383645495169977511339","date":"2026-05-12T14:21:37+00:00","index":22,"fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-11T20:08:07+00:00","index":21,"fulltext":""},{"type":"reviewerAgreed","content":"243373904168954650598184786936291662378","date":"2026-05-11T08:49:25+00:00","index":20,"fulltext":""},{"type":"reviewerAgreed","content":"269920894555170068341734823722236391880","date":"2026-05-08T17:38:08+00:00","index":17,"fulltext":""},{"type":"reviewerAgreed","content":"35769920301450883905421707190018282944","date":"2026-05-07T20:40:07+00:00","index":16,"fulltext":""},{"type":"reviewersInvited","content":"8","date":"2026-05-07T15:52:06+00:00","index":"","fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2026-05-22T12:09:16+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-18 17:30:20","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9334205","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9334205","identity":"rs-9334205","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}