Evolution of High-Temperature Oxygen Clusters and Radical Release: A Molecular Dynamics Study in Pure Oxygen and Titanium Tetrachloride Oxidation Environments

Highlights

What are the main findings?

• The “oxygen clustering–radical release” pathway is identified in high-temperature oxidation.
• Odd-numbered oxygen clusters (dominated by O₃) are reactive while even-numbered are inert.
• Reaching a critical size, the cluster releases radicals that persist in pure O₂ and TiCl₄-containing environments.

What are the implications of the main findings?

• Provides a novel mechanism for the controlled synthesis of functional oxide materials.
• Offers a new perspective for understanding the high-temperature oxidation corrosion mechanisms of alloy materials.
• Holds implications for the high-temperature combustion processes of fuel materials.

Abstract

Despite decades of research into high-temperature oxidation kinetics, the atomic-scale origin of oxygen radical formation—a critical driver of the synthesis of functional oxide materials, oxidation corrosion of alloys, and combustion of fuels—remains elusive. Here, we unveil a previously unrecognized “oxygen clustering–radical release” pathway at the molecular level, wherein transient van der Waals aggregates of oxygen molecules, rather than isolated O₂, serve as the primary radical source. Through an integrated computational strategy combining first-principles calculations and deep neural network potential function molecular dynamics, we demonstrate that oxygen clusters exhibit a pronounced odd–even oscillation in stability: even-numbered clusters (especially O₆/O₈ subunits) are chemically inert, whereas odd-numbered clusters (dominated by O₃) possess high reactivity. Kinetically, clusters evolve via reversible aggregation–dissociation, and upon reaching a critical size, undergo structural collapse accompanied by radical release—a process initiated predominantly by large odd-numbered clusters. This mechanism persists in both pure O₂ and TiCl₄-containing environments, with TiCl₄ modulating only the aggregation kinetics, not the fundamental pathway. Our work establishes oxygen clustering as a universal, spontaneous source of radicals in high-temperature oxidation, providing a new molecular-level mechanistic framework for understanding and controlling radical-driven processes in materials synthesis, corrosion and combustion.

1. Introduction

The controlled synthesis of functional oxide materials, alongside the mitigation of high-temperature corrosion and optimization of combustion, is closely related to a fundamental understanding of oxygen’s behavior under extreme conditions. In the high-temperature oxidation of titanium-based materials, oxygen (O₂) participates in complex physicochemical processes to form titanium dioxide (TiO₂), a key functional oxide with a broad spectrum of applications in coatings, photocatalysis, and solar cells. Although extensive studies have explored the kinetics and thermodynamics of oxygen in macroscopic high-temperature reactions [1,2,3], most models rest on a core assumption: oxygen consistently exists as isolated O₂ dimers [4,5,6]. This simplification may overlook the potential formation of intermolecular associations and transient polymers in oxygen under high-temperature, high-pressure conditions, along with their potential influence on the nucleation and growth of TiO₂ nanostructures. Despite some experimental hints of such clustered species, their role has been largely absent from mainstream kinetic process.

As early as the 1950s, Grundland [7] first observed an ion current with a mass number of 64 when oxygen passed through an ozone generator, suggesting the possible existence of O₄ molecules. With advancements in experimental techniques, particularly mass spectrometry and infrared spectroscopy, the existence of oxygen dimers (O₂-O₂) was gradually confirmed [8]. The assembly of oxygen clusters is governed by weak intermolecular forces, as the binding energy between O₂ molecules is significantly lower than that characteristic of conventional chemical bonds. O₄ is considered a “bound dimer” rather than a transient collision pair. Under cryogenic conditions, where oxygen exists in solid form, the geometric structures, electronic properties, and magnetic evolution of (O₂)_n clusters have been widely studied [9,10,11]. Research by Santoro [12] indicates that oxygen molecules within condensed phases tend to aggregate into clusters with long-range order. Notably, in solid oxygen, (O₂)₄ tetramers are formed through weak O₂-O₂ chemical interactions when subjected to pressures above 10 GPa. Based on high-accuracy quantum chemical calculations, Gadzhiev [13] computed the structures, energies, and vibrational frequencies of O_n (n = 1–6) to elucidate the properties of products formed during solid O₂ irradiation. Forte [14] predicted the free-space structures of O₆, O₈, and O₁₂. Besides neutral clusters, oxygen is capable of forming charged cluster ions (e.g., O₄⁺ and O₄⁻). The stability differences among cluster ions stem from charge distribution and polarization effects. O₄⁻ exhibits higher bond energy than O₄⁺ due to enhanced polarization from the extra electron. Conway [15,16,17,18] conducted an investigation of O₄⁺ to O₁₈⁺ ion clusters under conditions of low temperature and low pressure, determining their thermodynamic parameters. The study also identified the presence of “magic number” phenomena within the cluster size distribution at reduced temperatures.

Advancements in computational chemistry and characterization methodologies have enabled researchers to attain a more comprehensive understanding of the equilibrium properties and mechanistic pathways of oxygen radical reactions. Li [19] investigated the combustion characteristics of biogas under oxygen-enriched conditions, noting that O radicals are generated during combustion. Elevated oxygen concentrations facilitate the generation of H, O, and OH radicals, thereby accelerating the kinetics of exothermic reactions. Yamamoto [20] investigated the dynamic behavior of oxygen radicals within microwave plasma. In this context, high-energy electron collisions induce the dissociation of O₂ molecules, generating O radicals. These radicals subsequently react with hydrocarbon fuels or hydrogen radicals to produce hydroxyl (OH) species, thereby facilitating the acceleration of the oxidation process. Sun [21] demonstrated that, under conditions characterized by elevated temperature and energy, such as those present in plasma-assisted combustion, molecular oxygen (O₂) can undergo direct dissociation to produce oxygen radicals. Song [22] proposed that the interconversion processes among H, O and OH radicals is the principal origin of oxygen radicals in combustion systems, emphasizing that the surface activation mechanism of O radicals plays a critical role in catalytic combustion. Zhang [23], studying the combustion characteristics of the zero-carbon renewable fuel ammonia, found that oxygen radicals, specifically O and OH, are critical in facilitating NH₃ flame propagation by significantly enhancing reaction rates within the reaction zone. Chen et al. [24] proposed a radical cycle model for coal spontaneous combustion, demonstrating that initial reactions accumulate heat and promote the generation of active peroxyl radicals (R-O-O∙) and hydroxyl radicals (OH∙).

The studies referenced above demonstrate that, under cryogenic and condensed-phase conditions, oxygen molecules tend to aggregate into clusters predominantly composed of either oxygen molecules or ions. Conversely, in high-temperature combustion environments, oxygen primarily participates in reactions via dissociation into radicals. Bazarkina et al. [25] used in situ X-ray spectroscopy and electronic structure calculations to reveal the formation of uranium cubic octahedral clusters during UO₂ oxidation. This study confirms that cluster formation can occur in high-temperature oxidation processes. Nonetheless, there is a paucity of research investigating whether O₂ continues to form cluster structures similar to those observed in low-temperature environments when subjected to high-temperature conditions, as well as how the possible presence of such clusters may affect the production of oxygen radicals and the associated reaction kinetics. This gap arises partly from the substantial challenges associated with experimentally observing and theoretically modeling oxygen behavior under extreme high-temperature conditions, and partly from the inherent limitations of conventional research methodologies when applied to complex systems at elevated temperatures. In recent years, with the rapid development of artificial intelligence and high-performance computing methods, reaction mechanism research driven by first-principles calculation and deep learning is gradually maturing. Liu [26] employed Density Functional Theory (DFT) and AIMD to study the reaction pathways, thermodynamics, and kinetics of organic chlorine conversion to HCl during coal combustion. Ding [27] combined Deep Potential Molecular Dynamics (DPMD) and Ab Initio Molecular Dynamics (AIMD) simulations to study the interfacial structure and dynamics of amorphous titanium dioxide (a-TiO₂) in aqueous coating systems. Liu [28] utilized AIMD to theoretically investigate the thermal decomposition processes and primary reaction pathways of aqueous solutions of Hydroxylammonium Nitrate (HAN) as well as HAN mixtures with fuels such as methanol and ethyl ammonium nitrate. Qing [29] used LAMMPS simulations to study the combustion process of CH₄ and O₂ on a Pd catalyst surface within a spiral microchannel. The study involved calculating atomic-level behavior and thermodynamic parameters under varying initial temperatures and oxygen concentrations. Liu [30] integrated first-principles data to train neural network potential functions and employed the DPMD method to simulate the growth of SiC nanowire and track the aggregation behavior of CO/SiC molecules on a graphite substrate.

This study employs a methodology integrating first-principles calculations with deep neural network potential function molecular dynamics simulations to systematically investigate the dynamic evolution of oxygen clusters during high-temperature reaction. It reveals their microscopic formation mechanisms at the atomic scale, providing theoretical foundations and simulation frameworks for understanding oxygen behavior in high-temperature oxidation reactions.

2. Computational Methods

2.1. Cluster Configuration Calculations

The initial geometries of oxygen clusters O_n (n = 2–25) with diverse geometric features from the trajectories of DPMD (Deep Potential Molecular Dynamics) simulations at different temperatures were constructed using the Materials Studio (MS) 2017 [31]. To identify the most stable ground-state structures, geometry optimizations were performed using the Gaussian 16W (Revision A.03) software package at the M06L/6-311++G(d,p) level of theory [32], with convergence criteria of 1.0 × 10⁻⁶ Ha for energy, 0.00045 Ha/Å for force, and 0.0018 Å for displacement.

For each optimized structure, to accurately determine its ground spin state, we systematically calculated the energies for different multiplicities at the M06L/6-311++G(d,p) level. The ground state was identified as the spin configuration with the lowest total energy (E_min), based on the quantum mechanical variational principle. Subsequently, frequency calculations were performed on the optimized structures at the same level to confirm the absence of imaginary frequencies.

To systematically analyze the structures of oxygen clusters, we defined the following criteria. A distance threshold of 3.04 Å for O-O interatomic distances was set to identify atoms belonging to the same molecular subunit. This value is equal to the sum of the van der Waals radii of oxygen atoms [33], ensuring that we capture intermolecular interactions within a subunit. To determine whether two subunits form a larger cluster, we adopted the criterion that if the distance between any pair of oxygen atoms (O–O) from different subunits does not exceed 3.3 Å, the two subunits are considered connected and are classified as part of the same cluster.

The electrostatic potential (ESP) maps were generated and rendered using GaussView 6.0. Molecular orbital diagrams, including the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), were visualized using Visual Molecular Dynamics (VMD 1.9.1) software. The energy of the HOMO-LUMO gap was computed based on the orbital energies derived from the Gaussian single-point calculation.

2.2. Molecular Dynamics Simulations

Molecular dynamics (MD) simulations were performed using the LAMMPS (2 Aug 2023 version) package [34] with the deep neutral network potential model that trained by ourselves. The detailed training process and validation of this potential model can be found in our previous work, which is presented in the Supplementary Materials [35,36,37,38]. The initial simulation cells for both the pure oxygen system and the ternary Ti-O-Cl system were constructed using the Amorphous Cell module in Materials Studio, ensuring random molecular distribution without unrealistic atomic overlaps. The systems were defined as follows: The oxygen system consisted of 406 O₂ molecules and 12 O atoms, modeling a high-temperature oxygen-rich environment with a minor radical component. The Ti-O-Cl system contained 2624 O₂ molecules and 1914 TiCl₄ molecules, representing a stoichiometric mixture for the oxidation reaction.

All simulations were conducted in the isothermal-isobaric (NPT) ensemble. The temperature was maintained at the target reaction condition of 1573 K using a Nosé-Hoover thermostat [39]. The pressure was controlled independently at 3 atm and 6 atm using a Nosé-Hoover barostat to investigate pressure effects. The simulation was performed for 10 ns and 8 ns on the oxygen and Ti-O-Cl systems, respectively, with a time step of 0.2 fs.

3. Results and Discussion

3.1. The Ground-State Structures of O_n (n = 2–25) Clusters

Figure 1 illustrates the ground-state structures of O_n (n = 2–25) clusters, revealing their structural evolution patterns based on the assembly of fundamental subunits. Analytical results demonstrate that all clusters with even numbers are constituted by n/2 O₂ subunits, whereas clusters with odd numbers comprise a single O₃ subunit in conjunction with several O₂ subunits. In smaller clusters (n = 2–5), the structures preserve a degree of symmetry. Conversely, for clusters with n ≥ 6, the symmetry is diminished, resulting in all clusters belonging to the C1 point group. The O-O bond lengths in small clusters (O₂, O₃), measuring 1.21 Å and 1.30 Å respectively, characterize their stable covalent bonding characteristics. For n ≥ 4, the cluster structures transform into aggregates of O₂ and O₃ subunits through weak interactions. Specifically, O₄ and O₅ are constituted by (O₂)₂ and O₂·O₃ respectively. The inter-subunit distances increase from 2.2 Å to 3.04 Å, indicating a reduction in interaction strength. For larger clusters with n > 5, their structures are formed through the cooperative assembly of various smaller subunits (e.g., O₂, O₃, O₄, O₆, O₈). The O₆ cluster consists of three O₂ subunits, with a distance of 2.8 Å between two O₂ subunits (equivalent to an O₄ subunit), slightly longer than the corresponding bond length within an independent O₄ cluster. Analysis of the dihedral angles presented in Table S1 of Supplement S2 reveals that in isolated O₄ clusters, the characteristic dihedral angle approaches 180°, indicating a nearly ideal planar quadrilateral configuration with negligible torsional strain. However, when O₄ is embedded as a subunit in the O₆ cluster, its dihedral angle becomes 174.4°, showing a 5.6° distortion compared to the isolated O₄ cluster. Other critical dihedral angles within the O₆ cluster exhibit distortions approaching 10°, suggesting that the cluster exists in a relatively unstable high-energy state. The O₈ cluster is formed by the combination of O₂ and O₆ subunits, with a dihedral angle of 177.0° at the junction, indicating slight distortion. The configuration of the O₆ subunit exhibits notable differences compared to that of an isolated O₆ cluster. Specifically, in the isolated cluster, the O₂ subunits are positioned at greater intervals, whereas within the O₆ subunit, the separation between each pair of O₂ subunits is constrained to approximately 2.3–2.4 Å, accompanied by internal dihedral angles nearing 180°. The findings suggest that within the restricted environment of the O₈ cluster, the O₆ subunit displays intensified O-O interactions, resulting in markedly greater stability relative to the isolated O₆ cluster. Furthermore, in larger clusters ranging from O₁₀ to O₁₅, the presence of similarly robustly bound O₆ subunits is consistently evident. As the cluster size increases beyond n = 15, discernible cube-like or rhombus-like O₈ subunits begin to emerge. In O₁₆ and O₁₇ clusters, the O₂ subunit spacing within the O₈ subunits shortens to 2.1 Å, with critical internal dihedral angles highly approaching 180° (≥179.6°). This strongly suggests that both cube-like and rhombus-like O₈ subunits can form highly stable configurations with minimal torsional strain in large clusters. However, upon reaching a size of n ≥ 19, the O₂-O₂ bond lengths within the O₈ subunits predominantly elongate to a range of 2.6–2.8 Å, suggesting a decline in structural stability with the further increase in cluster size.

Notably, a newly emerging O₁₄ subunit was observed in larger clusters such as O₂₂ and O₂₅. The O₂₂ cluster consists of O₁₄ and O₈ subunits, while the O₂₅ cluster further incorporates an O₃ subunit, forming an O₁₄-O₈-O₃ combination. The O-O distance within the O₁₄ subunit of the O₂₅ cluster increases relative to that in O₂₂, which indicates that the incorporation of an O₃ unit reduces the structural compactness of adjacent even-atom subunits. Similarly, in the O₂₃ cluster (comprising O₁₂-O₈-O₃), the O-O bond on the O₃-adjacent side of the O₁₂ subunit elongates to 3.0 Å, significantly longer than typical values. This pronounced elongation suggests a region of marked instability that may function as an active site for cluster dissociation.

The structural analysis identifies distinct patterns in interatomic distances within the cluster. Specifically, the oxygen-oxygen bonds within O₂ or O₃ molecules exhibit short covalent bond lengths of approximately 1.2 to 1.3 Å, as shown by the solid line in Figure 1. The distances between O₂ molecules within the subunit vary from 2.1 to 3.04 Å as indicated by the dashed line in Figure 1, which are smaller than the combined van der Waals radii (3.04 Å). In contrast, the distances between subunits are observed to be greater than 3.04 Å (unlabeled in the figure for clarity). This clearly reflects the synergistic interaction between strong covalent bonds within subunits and weaker electrostatic or van der Waals forces between subunits. Furthermore, the emergence and distribution of characteristic subunit structures within oxygen clusters exhibit pronounced size dependence: O₄ subunits predominantly exist in small clusters with n < 10; O₆ subunits are common in medium-sized clusters with n = 8–20, while O₈ subunits consistently remain stable within large clusters with n ≥ 14. When O₆ or O₈ subunits are embedded within larger clusters as structural units, their stability significantly surpasses that of their independent cluster states. In contrast, other subunits unable to form such stable O₆ or O₈ configurations generally exhibit structural relaxation features such as increased bond lengths and distorted dihedral angles. The variation in structural stability observed at the subunit level constitutes a fundamental structural factor underlying the pronounced disparities in the overall stability of oxygen clusters of varying sizes.

3.2. Energetic Stability in the Clusters

The thermodynamic stability of atomic or molecular clusters can be quantitatively assessed through parameters such as the average binding energy (E_b) and the second-order differential energy (Δ₂E). Furthermore, their electronic structural stability is closely related to the HOMO-LUMO gap (ΔE_H-L) and the dipole moment.

For oxygen atom clusters, the average binding energy and second-order differential energy can be calculated using the following equations [40,41]:

E_b = (n∙E(O) − E(O_n))/n

Δ₂E(O_n) = E(O_n+1) + E(O_n−1) − ₂E(O_n)

where E(O): Energy of a single oxygen atom; E(O_n): Total energy of a cluster composed of n oxygen atoms; n: Number of atoms in the cluster; E(O_n+1) and E(O_n−1) are the energies of the adjacent clusters, respectively.

Figure 2a shows the variation in the average binding energy (E_b) of O_n (n = 2–25) clusters with increasing cluster size. The analysis reveals a pronounced odd–even oscillation in the binding energy E_b with respect to cluster size: The binding energy (E_b) values of clusters with even numbers of atoms are typically greater than those of their neighboring odd-atom clusters, suggesting that even-atom clusters exhibit increased thermodynamic stability. Further analysis reveals size-dependent variations in the binding energies of clusters with odd and even numbers of atoms. The binding energies of clusters with even numbers of atoms display minor fluctuations within a limited range, with variations not exceeding 0.15 eV per atom, indicating that their stability is largely independent of cluster size. Conversely, the binding energies of clusters containing an odd number of atoms demonstrate a marked positive correlation with size. This trend is especially evident in the size range of n = 3 to 9, where binding energies increase rapidly, subsequently followed by a more gradual reduction in the rate of growth. The maximum energy difference reaches 0.56 eV/atom, demonstrating that larger odd-atom clusters possess significantly enhanced thermodynamic stability compared to their smaller counterparts. This phenomenon not only confirms the inherent size dependence of odd-atom clusters but also validates the results from ground-state structural analysis: high-stability subunits such as O₆ and O₈ provide crucial structural frameworks for larger clusters (n > 8), which fundamentally contribute to their overall stability enhancement. Additionally, the binding energy of O₃ represents the lowest value among all clusters and is significantly lower than that of O₂. This computational result aligns remarkably well with established experimental observations: ozone (O₃) spontaneously undergoes disproportionation to form oxygen (O₂) under ambient conditions, whereas the conversion of O₂ to ozone requires external energy input (e.g., through electrical discharge), thereby energetically confirming the thermodynamic instability of O₃.

As shown in Figure 2b, the second-order energy difference (Δ₂E) serves as a crucial indicator for characterizing size-dependent cluster stability: when Δ₂E > 0, it indicates that the cluster of that particular size exhibits higher stability compared to its neighboring sizes, whereas Δ₂E < 0 suggests lower stability. The analysis demonstrates that Δ₂E exhibits significant odd–even oscillatory behavior with respect to cluster size n: all even-atom clusters display positive Δ₂E values, while odd-atom clusters consistently show negative Δ₂E values. This pattern clearly demonstrates that even-atom oxygen clusters possess superior thermodynamic stability compared to their odd-atom counterparts.

The HOMO-LUMO energy gap serves as a key parameter characterizing the stability of a system’s electronic structure and its chemical inertness. As shown in Figure 2c, O₂ exhibits the largest energy gap, indicating the highest kinetic stability and chemical inertness. This aligns with oxygen’s most stable natural existence as a diatomic molecule. The evolution of the energy gap with cluster size exhibits highly correlated even–odd oscillations with thermodynamic parameters (E_b, Δ²E): the energy gap of even-atom clusters show a systematic positive shift, averaging approximately 0.025 eV higher than odd-atom clusters. This further confirms, at the electronic structure level, that even-sized clusters possess higher chemical stability. Conversely, odd-atom clusters not only exhibit lower HOMO-LUMO gaps but also show a decreasing trend with increasing size (especially beyond n > 19), indicating that larger odd clusters possess less stable electronic structures and higher chemical reactivity.

Dipole moment results, as a key indicator of charge distribution symmetry, further corroborate the parity difference in cluster stability: all even clusters (especially O₂, O₄, O₆, O₈) exhibit dipole moments approaching zero, confirming their symmetric charge distribution—the very electronic foundation of even clusters’ high stability. Conversely, the significantly increased dipole moments of odd-atom clusters indicate asymmetric charge distribution and deviation from the positive and negative charge centers. At the electronic structure level, this is intrinsically linked to their lower thermodynamic stability.

Based on a comprehensive analysis of binding energy, second-order difference energy, HOMO-LUMO gap, and dipole moment, both O₆ and O₈ are identified as highly stable fundamental structural units. This conclusion is highly consistent with the distribution pattern observed in oxygen clusters: when n ≥ 10 and n ≥ 14, the structurally highly stable O₆ and O₈ subunits begin to exert their dominance, emerging as the defining features of the large-sized clusters. Furthermore, the overall stability of the O_n clusters does not vary monotonically with cluster size but is primarily governed by the parity (even/odd nature) of the number of oxygen atoms, demonstrating significant oscillatory behavior. Even-atom oxygen clusters exhibit superior thermodynamic stability and chemical inerticity from both thermodynamic and electronic structure perspectives. In contrast, while the thermodynamic stability of odd-atom clusters increases with size, their chemically activity, which is governed by electronic structure, decreases. This pattern indicates that as the size increases, the influences of geometric structure and electronic structure on the stability of odd-atom clusters engage in a competitive trade-off relationship, leading to a fundamental shift in the underlying stability mechanism.

3.3. Electrostatic Potential and Orbital Analysis

Based on the aforementioned analysis of binding energy, second-order difference energy, HOMO-LUMO gap, and dipole moment, it is collectively demonstrated that the stability of O_n clusters is closely related to their electronic structure and charge distribution. To intuitively reveal the underlying microscopic mechanisms, the Electrostatic Potential (ESP) and Frontier Molecular Orbitals (FMO) of the oxygen clusters will be subjected to visual analysis and discussion.

Figure 3a and Figure 4a illustrate the electrostatic potential distribution and frontier orbital diagrams for the smallest clusters in the series (n = 2–7). From an orbital perspective, the dense and continuous electron cloud bridges spanning oxygen atoms in O₂ and O₃ molecules indicate significant electron conjugation effects. The β-HOMO electron cloud of O₄ spans four oxygen atoms, demonstrating effective electron cloud overlap and charge transfer between two O₂ subunits. However, both α-LUMO and β-LUMO are localized within a single O₂ subunit, indicating that while O₄ exhibits electron delocalization and conjugation characteristics, its overall conjugation effect is weaker than that of O₂ and O₃. In the O₆ cluster, the HOMO and LUMO are located on different O₂ subunits without overlap, exhibiting certain chemical inertness. Analysis of the ESP diagrams of even-atom oxygen clusters reveals that both O₂ and O₄ display highly symmetric and uniform global electrostatic potential distributions, with no significant localized potential extrema on the surface. In the ground-state structure of the O₆ cluster, O₂ and O₄ subunits are combined through weak electrostatic interactions at distances of 3.3–3.4 Å, maintaining a uniform electrostatic potential characteristic without distinct positive or negative charge centers. These electronic structural features, combined with the aforementioned thermodynamic stability analysis, corroborate the higher intrinsic stability of small even-atom clusters.

In contrast to even-atom clusters, odd-atom clusters exhibit significant positive and negative charge centers (orange-red regions represent negative charge, deep blue represents positive charge), with both centers located within the O₃ subunit region. The V-shaped configuration of the O₃ molecule exhibits a degree of symmetry, characterized by a positively charged central oxygen atom and negatively charged terminal oxygen atoms. The negative charge center lies along the line connecting the terminal oxygen atoms, while the positive and negative charge centers do not coincide, resulting in a large permanent dipole. This constitutes key electronic structural evidence for the odd–even oscillation effect in cluster dipole moments (as shown in Figure 2d dipole moment trend). The diagram of frontier molecular orbitals reveals that the HOMO and LUMO of O₃ are degenerate, indicating its dual functionality as both a nucleophilic and electrophilic site, which accounts for its high reactivity. Coupled with its separated positive and negative charge centers and dipole, O₃ exerts significant electronic polarization and induction effects on other oxygen atoms in odd-atom oxygen clusters. The frontier molecular orbitals diagram of O₅ shows that the O₃ subunit remains the electrophilic and nucleophilic center of the entire cluster. Due to the inductive effect of O₃, strong electrostatic interactions occur between O₃ and O₂ subunits, resulting in uneven charge distribution on O₂ (+0.030, −0.027) which is indicated by the red arrows in Figure 3a for O₅. Similarly, O₇ also displays distinct positive and negative charge centers, with the induction effect of O₃ causing more uneven charges distribution on the O₂ subunit (+0.113, −0.033) shown by arrows in Figure 3a for O₇. The frontier molecular orbitals diagram demonstrates that the O₃ subunit polarizes the O₂ subunit, inducing degeneracy in its HOMO and LUMO and thereby reducing the energy gap, which significantly enhances the local reactivity. Thermodynamically, this active site facilitates cluster growth by incorporating new oxygen atoms, serving as an effective relaxation pathway to reduce the overall system energy and achieve stability.

Figure 3b and Figure 4b depict the electrostatic potential distributions and frontier orbitals for the intermediate-sized clusters (n = 8–17). Overall, the pattern of even–odd distribution differences aligns with that of small clusters (n = 2–7): even-atom clusters exhibit uniform electrostatic potential distributions, while odd-atom clusters still feature distinct positive and negative charge centers (deep blue and orange-red) located in the O₃ subunits. Electrostatic potential interactions exist between all subunits, with particularly pronounced effects between O₃ and other subunits. This further confirms the presence of weak interactions within the clusters, enabling oxygen atoms to form larger cluster structures.

Analysis of the frontier orbital distributions reveals partial overlap of the electron clouds on the O₆ and O₈ subunits in both odd and even clusters. This observation suggests a certain extent of electron delocalization within these subunits, implying that the O₆ and O₈ units inherently exhibit stability. Consequently, this intrinsic stability contributes to the enhanced overall stability of the larger cluster. In cluster O₉, both the β-HOMO and LUMO are localized on the O₃ subunit. A consistent localization pattern is observed for the α/β-LUMO in clusters O₁₁, O₁₃, O₁₅, and O₁₇. This persistent occupation of low-energy, unoccupied orbital space designates the O₃ subunit as a strong electron acceptor with pronounced electron-withdrawing character. Consequently, it exerts a strong inductive effect on the electrons of neighboring O atoms, consistent with the electrostatic potential results showing significant electrostatic interactions between O₃ and other subunits. For instance, cluster O₁₇ comprises three subunits: O₃, O₆, and O₈. O₈ exhibits pronounced internal conjugation effects and is less influenced by O₃. Conversely, O₆, being closer to O₃, experiences stronger induced dipole effects. Consequently, the bond distances between oxygen atoms of 1 and 3 (O1-O3) and atoms of 2 and 4 (O2-O4) within the O₆ subunit elongate, reducing its stability.

Figure 3c and Figure 4c display the electrostatic potential distribution and frontier orbital diagrams for the largest clusters (n = 18–25). As oxygen clusters grow in size, both the dimensions and number of their constituent subunits increase. While weak electrostatic interactions remain the dominant inter-subunit force, the internal bonding within odd-atom clusters becomes weakened due to reduced electron overlap. This leads to the formation of loosely bound internal configurations, such as O₆ + O₂, O₈ + O₂, and O₈ + O₂ + O₂. The large odd cluster O₁₉ is composed of two O₈ subunits and one O₃ subunit. Notably, one O₈ subunit is not a cuboid-like dense structure but instead forms an O₆ + O₂ structure. The HOMO and LUMO orbitals are localized precisely on this loose O₆ + O₂ structure and the O₃ subunit, indicating that the induced effect from O₃ reduces the local stability of the O₈ subunit, thereby increasing the atomic activity within that subunit. The electrostatic potential analysis of O₁₉ reveals that the O₃ subunit, owing to its pronounced electron-withdrawing capacity, engages in substantial electrostatic interactions with the adjacent O13 atom. This interaction consequently attenuates the O-O bonds within the O₈ subunit, thereby enhancing its reactivity. The O₂₅ cluster consists of O₁₄, O₈, and O₃ subunits. The LUMO orbitals are localized on the O₃ subunit and the oxygen atoms in the O₁₄ subunit adjacent to O₃, while the HOMO orbitals are localized on the O₁₄ subunit. The O₁₄ subunit forms a network-like structure, with local electron delocalization occurring among the central oxygen atoms. This causes the edge oxygen atoms near O₃, specifically O17 and O18, to be more susceptible to induction by the external O₃ subunit. Therefore, the two oxygen atoms near O₃ exhibit relatively higher activity and are prone to chemical reactions. Electrostatic potential analysis also shows that the O₃ subunit in the O₂₅ cluster, due to its strong electron-withdrawing ability, has significant electrostatic interaction with the nearby O18 atom.

A comprehensive analysis of the electrostatic potential distribution and frontier orbitals indicates that even-atom clusters exhibit stronger stability due to symmetrical structure and electron distribution. In contrast, odd-atom oxygen clusters experience internal electron density redistribution caused by the strong induction effect of the O₃ subunit. When the cluster is small, the induction effect of O₃ strongly polarizes the smaller subunits, making small clusters more inclined to capture external O₂ molecules through electrostatic interactions, thus promoting cluster growth. As the cluster size increases, the size and number of internal subunits also increase, dispersing the induction effect of a single O₃ subunit across multiple larger subunits. This leads to a reduction in the overall polarization strength on any single subunit. This “dilution” of the effect localizes the induction, thereby weakening a specific chemical bond and making dissociation (such as the dissociation of an oxygen radical) a more favorable reaction pathway than growth.

3.4. Molecular Dynamics Simulation of a Pure Oxygen System

Comprehensive analysis of the previous results indicates that the stability of oxygen clusters exhibits a pronounced odd–even oscillation trend as the number of oxygen atoms increases. The stability of odd-atom clusters is more closely related to their geometric and electronic structures, with the stability of larger odd-atom clusters diminishing. To further elucidate the growth kinetics mechanism of large-sized oxygen clusters, molecular dynamics simulations of a pure oxygen system were conducted at high temperature (1573 K) under different pressure conditions (3 atm and 6 atm). In our analysis, a cluster is counted if connectivity persists for ≥2 ps.

Figure 5 compares the oxygen cluster population results at different sizes simulated under varying pressures in a pure oxygen system, revealing a clear trend: As the pressure increases from 3 atm to 6 atm, the maximum cluster size observed during the simulation period expands from O₁₃ to O₃₁. Concurrently, the population of clusters in the O₄ and O₆–O₁₄ size ranges increases, while the number of O₂ clusters decreases relatively. This change reveals the promotional effect of increased pressure on cluster nucleation and growth kinetics: Higher pressure increases the number density of O₂ monomers and the effective collision frequency between molecules, significantly accelerating a series of elementary reaction rates—from initial nucleation (e.g., O₂ + O₂ → O₄) to subsequent monomer-addition growth (e.g., O_n + O₂ → O_n+2). This leads to a reduction in O₂ monomer numbers and an increase in large cluster numbers within the same time.

At pressures of 3 atm and 6 atm, the populations of O₂ and O₄ clusters are significantly greater than those of other sizes, indicating their superior kinetic stability. The average numbers of O₃, O₅, and O₆ clusters are comparatively lower, each remaining under 10. Conversely, clusters larger than O₁₀ exhibit a sharp decline, with average numbers below 0.1 during the simulation period, suggesting strong suppression of growth for large clusters. Within the interval spanning O₂ to O₆, even-atom oxygen clusters (O₂, O₄, O₆) generally exhibited higher populations than adjacent odd-atom clusters (O₃, O₅), indicating thermodynamic and kinetic advantages for even clusters. This finding strongly aligns with the pronounced “odd–even oscillation” effect observed in stability analysis. However, this effect gradually diminishes with increasing cluster size, diverging from earlier structural analysis and thermodynamic calculations. This suggests that over larger size ranges, kinetic factors exert a significant influence and predominantly govern the behavior of cluster distribution, as the formation of larger clusters necessitates surmounting greater free energy barriers. The formation of large clusters relies on greater energy and more complex structural rearrangements, the likelihood of which diminishes substantially as cluster size increases. Therefore, from a kinetic standpoint, the formation of large clusters is markedly disfavored.

Analysis of the size distribution of clusters under 6 atm revealed a non-monotonic variation within the O₁₀–O₂₈ range: the population gradually decreased in the O₁₀–O₁₅ range, consistent with conventional nucleation kinetics; however, within the O₁₅–O₂₅ range, the population rebounded, reaching a local maximum at O₂₅ before declining rapidly. This anomalous distribution is closely related to the structural evolution and stability of the clusters. Clusters comprising between 10 and 15 oxygen atoms predominantly include less stable O₄ subunits. In contrast, as the cluster size expands to the range of 15 to 25 oxygen atoms, the clusters are mainly constituted by more stable subunits, specifically O₆ and O₈. Furthermore, the inductive effect of O₃ is distributed among multiple subunits, which enhances the stability of the clusters. Consequently, the incidence of large clusters observed in the molecular dynamics simulations increases. The rapid decline after the O₂₅ peak suggests a structural transition (e.g., release of radicals), reflecting a strong correlation between the stability of the constituent subunits and the structural evolution of the clusters.

To elucidate the microscopic mechanisms underlying these size distribution characteristics, this study further focuses on the kinetic evolution pathways from small clusters to larger ones. Under 6 atm, the system not only forms larger clusters (e.g., O₃₁) but also exhibits structural transformation events triggered by the dissociation of large clusters, providing richer kinetic information for analyzing cluster formation and stability. Consequently, the following analysis will predominantly utilize the simulation trajectories obtained at 6 atm to perform a comprehensive investigation of the microscopic evolution mechanisms.

Figure 6a illustrates the evolutionary process, demonstrating that in a pure oxygen system, the formation of O₃ initiates from the combination reaction between O₂ molecules and initially introduced oxygen radicals (O∙). The generated O₃ tends to further interact with O₂ molecules during subsequent motion, forming chain-like O₅ clusters. However, due to their low stability, these chain-like O₅ clusters often rapidly dissociate, regenerating O₂ and O₃, thereby facilitating oxygen atom exchange. Kinetic simulation trajectories indicate that O₃ can also directly combine with O₄ to form O₇ clusters, whose structures exhibit dynamic structural characteristics. At 1536.8 ps in the kinetic simulation, a cyclic O₅ structure is observed to combine with O₂; as vibrations continue, their interaction gradually weakens, and they dissociate at 1537.3 ps, regenerating cyclic O₅ and O₂. Compared to chain-like O₅, cyclic O₅ exhibits higher kinetic stability due to electron delocalization effects, resulting in a longer residence time in the system (approximately 2 ns). After 2503.4 ps in the kinetic simulation, the cyclic O₅ undergoes ring-opening transformation, forming a chain-like configuration and combining with O₂ to generate O₇, which subsequently rapidly dissociates into O₄ and O₃.

During the evolution of oxygen clusters (Figure 6b), O₄ not only engages in reversible combination-dissociation behavior with O₃ but also interacts with O₂. The O₂-O₄-O₆ evolutionary pathway reveals a collision-driven reversible reaction mechanism. Initially, two O₂ molecules combine through collision to form an O₄ cluster. This O₄ cluster can further collide with free O₂ molecules in the system, generating O₆ clusters via the reversible reaction O₄ + O₂ ⇌ O₆. However, O₆ clusters are metastable with extremely short lifetimes, rapidly dissociating into O₂ and O₄. Through continuous reversible combination and dissociation with O₂ molecules, O₄ clusters not only achieve dynamic equilibrium between O₄ and O₆ but also promote oxygen atom exchange among different clusters, providing microscopic pathways for oxygen atom migration.

Molecular dynamics trajectory analysis of larger oxygen cluster evolutionary pathways (Figure 6c) indicates that their growth mechanism primarily relies on reversible adsorption and dissociation with smaller subunits (e.g., O₂, O₄, O₆). The initial O₉ cluster gradually increases in size through continuous reactions with such subunits. During the time interval from 5762.6 to 5764.5 ps, O₁₉ is observed to combine with O₆ to generate O₂₅, which rapidly dissociates, releasing two O₂ molecules and forming an O₂₁ cluster. This phenomenon suggests that, compared to larger subunits (e.g., O₆), the combination of smaller subunits (e.g., O₂) with clusters is kinetically more favorable.

As the size continues to increase, the cluster structure begins to undergo significant changes. At 6016.6 ps, O₂₇ combines with O₄ to form O₃₁, which dissociates after 6.2 ps, releasing one O₂ to become O₂₉. At this point, the large odd-atom cluster O₂₉ initiates O-O bond dissociation, releasing oxygen radicals and diminishing to O₂₂ within 45 ps (at 6114.2 ps). The detection of radical release from the sizable odd-atom cluster O₂₉ provides corroborative evidence supporting previous theoretical analyses: Structural analysis identifies relaxation features within the large cluster—including expanded interatomic distances both within and between subunits, as well as altered dihedral angles—providing a structural basis for its instability. The decrease in the HOMO-LUMO gap and the increased local atomic activity within large subunits (e.g., O₈, O₁₄), as revealed by electronic structure analysis, are the electronic structural origins of the enhanced reactivity. The release of free radicals is a necessary outcome of the kinetic instability caused by the aforementioned structural relaxation and increased local atomic activity. At 6172.2 ps, the cluster subsequently re-enters a phase of aggregated growth with small subunits such as O₂. This entire process reveals a dynamic growth pattern for oxygen clusters: their size does not increase in a strictly monotonic manner; rather, it undergoes dynamic modulation through reversible interactions among subunits and internal induced dipole effect, which initiate bond cleavage and the release of oxygen radicals, thereby promoting atomic exchange. This “growth-dissociation” dynamic cycle demonstrates that oxygen cluster evolution is jointly governed by thermodynamic stability and kinetic activity: elevated kinetic activity enables atomic exchange and structural rearrangement, while thermodynamic stability ultimately guides the system toward forming more structurally stable cluster morphologies, establishing a dynamic equilibrium between these two processes.

3.5. MD Simulation of the Titanium Tetrachloride-Oxygen System

To investigate the formation of oxygen clusters and radical release mechanisms under realistic combustion conditions, molecular dynamics simulations were further conducted in the titanium tetrachloride oxidation system (temperature: 1573 K, pressure: 6 atm).

Figure 7a indicates the variation in the number of oxygen clusters with cluster size, demonstrating that O₂ and O₄ are the most abundant species, consistent with results from pure oxygen systems. This indicates that O₂ and O₄ possess optimal kinetic stability in both oxygen and oxygen-rich oxidation environments. As the cluster size n increases further, the population of large clusters rapidly decreases; however, after n > 13, the population rises again, forming a peak in the O₁₅–O₁₇ range, with O₁₆ exhibiting the highest population and greatest stability. When n > 16, the number of clusters decreases again. Although the maximum cluster size extends to O₂₈, clusters in the O₂₂–O₂₈ range are very scarce. Larger clusters quickly enter metastable states and rapidly release free radicals. This evolutionary pattern is basically consistent with that in pure oxygen environments. The stability of oxygen clusters does not change monotonically with size but shows a clear nonlinear relationship. Stable O₆ and O₈ subunits also serve as characteristic structural units, enhancing the stability of large clusters larger than O₁₃. This further confirms that O₆ and O₈ subunits play a key stabilizing role in large clusters under complex reaction conditions.

Similar to the pure oxygen system, Figure 7b shows the process of molecular dynamics simulations of the titanium tetrachloride oxidation system at 6 atm, revealing that oxygen clusters continue to grow through the dynamic association and dissociation of smaller O_n (n = 2–6) clusters with existing clusters. Upon reaching O₂₁, the oxygen clusters commence a rapid and continuous liberation of free radicals. As the clusters reduce to O₁₆, they subsequently merge with smaller clusters, including O₂ and O₄, to facilitate further growth, entering a cycle of releasing free radicals (O∙) and merging with O_n clusters. As demonstrated in Section 3.3, high pressure (6 atm) in the pure oxygen system drives rapid cluster growth up to O₂₉ before any radical release occurs. However, in the TiCl₄ oxidation system at the same pressure, radical release occurs at a smaller cluster size of O₂₁, indicating that TiCl₄ addition has a dual effect on oxygen cluster evolution. Firstly, the steric hindrance effect of TiCl₄ molecules in physical space impedes effective contact and collision between oxygen molecules (O₂) and existing oxygen clusters, thereby suppressing the formation of large clusters and shifting the maximum cluster size distribution towards smaller dimensions. More significantly, the chemical catalysis and energy supply role of TiCl₄ profoundly participates in the oxidation reaction, releasing substantial energy during the process. This high-density energy is preferentially absorbed by adjacent, unstable oxygen clusters, particularly those containing strongly inductive O₃ subunits in odd-atom clusters, significantly accelerating the desorption rate of free radicals (O·). This suggests that TiCl₄ may alter the pathway of free radical formation, shifting the system from a physical transformation pathway dependent on large clusters to a chemical pathway reliant on rapid free radical generation from smaller clusters, thereby enhancing the overall reactivity of the system. This observation underscores the critical role of kinetic factors in influencing free radical release.

In both the pure oxygen system and the high-temperature TiCl₄ oxidation system, the evolution pathways of oxygen clusters and the process of radical release exhibit a high degree of consistency. This finding indicates that oxygen radicals are primarily generated through the intrinsic cluster evolution mechanism of oxygen molecules themselves, rather than necessitating direct participation of TiCl₄ molecules. The role of TiCl₄ appears to be more inclined towards modulating the thermodynamic equilibrium and kinetic rates of oxygen cluster formation, rather than constructing an entirely new reaction pathway. This oxygen cluster evolution mechanism, serving as a universal channel for radical generation, not only deepens the understanding of the titanium tetrachloride oxidation system but also provides a novel microscopic perspective and theoretical model for interpreting a broader range of catalytic oxidation and combustion phenomena.

4. Conclusions

In summary, through multi-scale computational simulations, we have revealed a previously unrecognized yet fundamental pathway for radical generation in high-temperature oxidation: the formation, evolution, and collapse of transient oxygen clusters. Contrary to the conventional view of oxygen as discrete O₂ molecules, we demonstrate that oxygen at high temperatures exists as dynamic van der Waals aggregates whose stability follows a distinct odd–even oscillation pattern. This oscillation is governed by the interplay between the geometric stability of even-numbered subunits (O₂, O₄, O₆, O₈) and the electronic activity of O₃ units, with the latter serving as the key driver of radical release.

Mechanistically, cluster evolution proceeds via a reversible growth–dissociation pathway, wherein radical release is not determined by a specific cluster size but is regulated by both system pressure and chemical environment. The persistence of this “oxygen clustering–radical release” mechanism in both pure O₂ and TiCl₄-containing systems underscores its universality, with TiCl₄ primarily modulating aggregation kinetics rather than altering the fundamental pathway.

These findings shift the paradigm for understanding high-temperature oxidation from a molecular to a supramolecular perspective, establishing oxygen clustering as a spontaneous, intrinsic source of radicals. The proposed model provides a new theoretical framework for explaining non-equilibrium radical generation and offers mechanistic insights relevant to materials synthesis, corrosion, combustion chemistry, and other high-temperature processes where radical-mediated reactions play a decisive role.