Simulation of gas adsorption on single-walled carbon nanotubes

Abstract

This study combined Grand Canonical Monte Carlo molecular simulations with density functional theory calculations to systematically examine the adsorption of N₂, O₂, H₂, CO₂, and CH₄ on nineteen single-walled carbon-nanotube (SWCNT) architectures. The effects of temperature, pressure, nanotube diameter, chirality, and vacancy defects on adsorption energies and isosteric heats are quantified. Binary N₂/O₂ separation within a (22,18) SWCNT is modelled by analyzing energy-distribution functions and spatial adsorption fields. Intermolecular interactions are represented with the Universal Force Field and Lennard–Jones potentials. Lower temperatures and higher pressures enhanced adsorption capacity, while adsorption energies and isosteric heats decreased accordingly. Furthermore, smaller-diameter SWCNTs exhibited superior selectivity for air separation. Neglecting electrostatic and hydrogen-bonding terms for non-polar gases is demonstrated to reduce computational cost without sacrificing accuracy. These findings establish a robust framework for rationalizing SWCNT-based adsorbents for gas-separation applications.

Introduction

Over the past few decades, interest in adsorption on solid surfaces has grown steadily¹. Adsorption is notably important across several fields within the natural sciences and serves as the basis for many technological processes². The practical applications of adsorption are vast and encompass a diverse range of areas, including gas storage^3,4, heterogeneous catalysis⁵, gas sensing⁶, air and water purification^7,8, electrochemistry^9,10, chromatography¹¹, and numerous others. Utilizing solid adsorbents for gas separation is compelling due to simple regeneration, minimal energy consumption, prevention of equipment corrosion, and environmentally sustainable properties¹². Solid adsorbents that exhibit promise for use in gas adsorption include activated carbons¹³, metal–organic frameworks (MOFs)¹⁴, porous organic polymers (POPs)^15,16, zeolites¹⁷, and carbon nanotubes (CNTs)^18,19. CNTs have been used for numerous separation processes, such as membrane and adsorbent²⁰. Some experimental and simulation studies have indicated that CNTs have an excellent permeability for various gas and liquid systems. Hinds et al. used an array of CNTs incorporated across a polymer film to form a well-ordered nanoporous membrane structure for N₂ adsorption. They authenticated that CNTs have incredible potential for chemical separations and sensing²¹. Beyond their use as general carbon nanotube adsorbents, single-walled carbon nanotubes (SWCNTs) constitute the simplest CNT architecture, formed by rolling a single graphene sheet into a seamless cylinder with diameters typically between 0.7 and 2 nm and lengths up to several micrometers²². This one-dimensional geometry endows SWCNTs with an exceptionally high surface-area-to-volume ratio and uniform pore size, facilitating rapid gas diffusion and creating well-defined adsorption sites. Moreover, the electronic structure of SWCNTs is tunable via their chirality indices (n,m), which enable selective interactions with gas molecules, enhancing separation performance²³. SWCNTs are exceptionally well-suited for physisorption applications owing to their very high specific surface area (~ 1000 m² g⁻¹). For example, Tian et al. employed (6,5)-enriched semiconducting SWCNTs as NO₂ gas sensors and demonstrated an exponential correlation between the Raman 2D-band shift and NO₂ concentration, enabling quantitative evaluation of charge-transfer interactions and adsorption behavior²⁴. Tsuruta et al. synthesized semiconducting SWCNT networks via floating-catalyst vapor deposition. They demonstrated their application as hydrogen sensors, achieving rapid response and recovery kinetics at ambient temperature with a detection threshold as low as 0.1 vol % H₂²⁵. Majumder et al.²⁵ demonstrated that liquid transport through membranes composed of aligned carbon nanotube arrays exhibits volumetric fluxes four to five orders of magnitude greater than those predicted by classical continuum fluid‐flow models. Molecular simulation provides predictive insight into a system’s behavior at the atomic scale, enabling detailed exploration of intermolecular forces and adsorption mechanisms before any experimental work is undertaken²⁶. As such, molecular simulation has become indispensable for guiding adsorbent design and optimizing process conditions across chemical engineering and related disciplines²⁷. Density Functional Theory (DFT) is a type of first-principles method that can forecast the properties of materials for unexplored systems without requiring any experimental data, and it has become a widely accepted approach due to its relatively low computational demands²⁸. Skoulidas et al. performed molecular simulations to quantify light-gas diffusivities within carbon nanotubes and zeolites of comparable pore size. They found that diffusion rates in CNTs exceed those in zeolitic materials by one to two orders of magnitude. From these elevated diffusivities, they inferred that CNT-based membranes could achieve permeation fluxes far greater than conventional zeolite membranes²⁹. In another research, Skoulidas et al. used molecular simulation to investigate the adsorption and transport diffusion of CO₂ and N₂ in SWCNTs at room temperature. The results demonstrated that the transport diffusivities of molecules inside carbon nanotubes are incredibly faster than in other porous materials³⁰. Over the past decade, numerous molecular simulation studies have explored the separation of light-gas binary mixtures. Notably, Kowalczyk et al. employed Grand Canonical Monte Carlo (GCMC) simulations to examine the adsorption and separation of flue-gas components (CO₂, CO, N₂, H₂, O₂, and CH₄) on double-walled carbon nanotubes (DWNTs). Their computed equilibrium selectivities for equimolar CO₂–X pairs exhibited significantly enhanced separation performance under low-pressure conditions³¹. Huang et al. applied GCMC simulations to assess equimolar CO₂/CH₄ adsorption in five CNTs (0.678–1.356 nm diameter) over 283–343 K and 1–30 MPa. They demonstrated that lower temperatures, higher pressures, and optimal pore size markedly increase CO₂uptake, and that CNTs outperform activated carbons, 13X zeolites, and various MOFs in CO₂/CH₄ selectivity under the same conditions³². Razavi et al. employed Canonical Monte Carlo (CMC) simulations to investigate CO₂/N₂ separation by physisorption on SWCNTs. This study evaluated five CNTs with 0.807–1.35 nm diameters across 300–343 K, 0.15–10 MPa, and CO₂ mole fractions of 0.3–0.7³³. Yusfi et al. performed DFT calculations to elucidate the adsorption behavior of acetylene (C₂H₂) and ethylene (C₂H₄) on Ni-doped single-walled carbon nanotubes (Ni-SWCNTs, (10,0)), demonstrating that Ni incorporation markedly narrows the electronic band gap and that C₂H₄ exhibits a higher adsorption energy than C₂H₂, indicative of preferential ethylene binding on the modified nanotube surface³⁴. Jonuarti performed density functional theory calculations on eight boron-doped (4,0) SWCNT structures and showed that charge redistribution from boron toward adjacent carbon atoms gives rise to electrostatic attractions that stabilize the doped framework³⁵. Functionalization of SWCNTs with platinum atoms creates discrete, catalytically active sites that perturb the local electronic structure and enhance molecular binding affinity. Demir et al. employed density functional theory (B3LYP/6-31G(d,p) for C, O, H; LANL2DZ for Pt) to assess Pt-doped (4,0) SWCNTs as CO-sensing elements at 298 K, demonstrating that CO preferentially coordinates at the Pt center forming a Pt–C bond with significant charge transfer and elevated adsorption energy³⁶. Hong et al. employed GCMC simulations to quantify the adsorption of CH₄, N₂, H₂O, CO₂, and CO within open-ended SWCNTs (10.85 Å diameter) at 273.15 K and 298.15 K over 1–101.325 kPa. The effects of temperature, pressure, tube diameter, and chirality on adsorption capacity, isosteric heat, and energy distribution were quantified. Open ends enhanced adsorption kinetics and yielded uniform adsorption profiles. Binary-gas simulations produced selectivity heat maps, while vacancy defects reduced binding strength. All interactions were described by UFF and Lennard–Jones potentials, enabling precise mapping of CO₂ binding sites and energy–distance curves³⁷.

Although adsorption on SWCNTs has been extensively studied through equilibrium isotherms, operating-condition analyses, and selectivity measurements, significant knowledge gaps persist. In this work, we systematically characterize the thermodynamic behavior of N₂, O₂, H₂, CO₂, and CH₄ gases spanning a wide range of molecular sizes, polarizabilities, and quadrupole moments across nineteen SWCNT chiralities. Then we integrate GCMC simulations of pure gases with targeted O₂/N₂ mixture studies to generate high-resolution energy-distribution maps and detailed adsorption-energy profiles. Finally, by incorporating vacancy defect models, we assess adsorption stability and defect-mediated selectivity under air-separation conditions. This comprehensive approach deepens mechanistic insight into gas–SWCNT interactions and provides a robust framework for rationalizing nanotube-based separation materials.

Theory and modeling

Thermodynamics of adsorption

The isosteric heat of adsorption, q_st, represents the molar enthalpy change when a small increment of adsorbate transfers from the bulk gas to the surface at constant coverage. Operationally, q_st is extracted from adsorption isotherms measured at two or more temperatures using the Clausius–Clapeyron relation (Eq. 1):

$$q_{st} = - R\left( {\frac{{\partial \left( {Ln P} \right)}}{{\partial \left( \frac{1}{T} \right)}}} \right)_{q}$$

(1)

P represents the equilibrium pressure at a fixed loading q, T represents the temperature, and R denotes the ideal gas constant. Similarly, q_st signifies the difference between the partial molar enthalpies in the adsorbed and bulk phases for component i. While the partial molar enthalpies of the components in the bulk phase can be measured or calculated using an equation of state like Peng-Robinson, the partial molar enthalpies of the adsorbed phase cannot be directly measured. For each component in the bulk, the partial molar enthalpy can be calculated using the Gibbs − Helmholtz relation (Eq. 2):

$$\Delta \overline{h}_{i} = \overline{h}_{i}^{adsorbed} - \overline{h}_{i}^{bulk}$$

(2)

$$\overline{h}_{i} = \left( {\frac{{\partial \left( {\frac{{\mu_{i} }}{T}} \right)}}{{\partial T^{ - 1} }}} \right)_{n}$$

(3)

Physical equilibrium is necessary for the equality of temperature and chemical potential for each component in the adsorbed and bulk phases.

$$\begin{array}{*{20}c} {T_{i}^{bulk} = T_{i}^{absorbed} } \\ {\mu_{i}^{bulk} = \mu_{i}^{absorbed} } \\ \end{array}$$

(4)

The partial molar enthalpy of each component in the adsorbed phase can be determined through Eq. (5):

$$\overline{h}_{i}^{absorbed} = \left( {\frac{{\partial \left( {\frac{{\mu_{i}^{absorbed} }}{T}} \right)}}{{\partial T^{ - 1} }}} \right)_{{q_{i}^{absorbed} }} = \left( {\frac{{\partial \left( {\frac{{\mu_{i}^{bulk} }}{T}} \right)}}{{\partial T^{ - 1} }}} \right)_{{q_{i}^{absorbed} }} \ne \overline{h}_{i}^{bulk} = \left( {\frac{{\partial \left( {\frac{{\mu_{i}^{bulk} }}{T}} \right)}}{{\partial T^{ - 1} }}} \right)_{{n_{i}^{bulk} }}$$

(5)

Equation (6) defines the isosteric heat of adsorption, q_st, as the enthalpy change accompanying the transfer of an infinitesimal amount of adsorbate from the bulk gas to the adsorbed phase at constant coverage. This thermodynamic parameter directly reflects the strength of adsorbate–adsorbent interactions and its variation with surface loading, thereby revealing the energetic heterogeneity of adsorption sites. A higher or more rapidly changing q_st indicates the presence of high‐energy adsorption sites and pronounced energetic heterogeneity across the surface³⁸.

$$\begin{aligned} & q_{st} = k_{B} T - \frac{{f\left( {U,N} \right)}}{{f\left( {N,N} \right)}} \\ & f\left( {X,Y} \right) = XY - XY \\ \end{aligned}$$

(6)

Grand-Canonical Monte Carlo

The Monte Carlo method is a standard tool for molecular‐level adsorption studies. In particular, GCMC simulations employ the grand–canonical ensemble, holding chemical potential (μ), temperature (T), and volume (V) constant while allowing particle numbers to fluctuate between the bulk and adsorbed phases³⁹. Equilibrium requires equal μ and T in both phases, and the fixed-volume constraint makes the grand–canonical ensemble ideal for physisorption modeling. The Lennard–Jones potential (Eq. 7) commonly describes intermolecular forces, a pairwise function introduced by Jones in 1924 that captures attractive van der Waals and short‐range repulsive interactions in gases, liquids, and solids⁴⁰.

$$\Phi \left( {r_{ij} } \right) = k\varepsilon \left( {\left( {\frac{\sigma }{{r_{ij} }}} \right)^{n} - \left( {\frac{\sigma }{{r_{ij} }}} \right)^{m} } \right)\quad r_{ij} < r_{c}$$

(7)

In this equation, $\varepsilon$ and $\sigma$ are Lennard–Jones (L–J) constants, and $k$ and $r_{ij}$ are calculated through:

$$k = \frac{n}{n - m}\left( \frac{n}{m} \right)^{{\frac{n}{n - m}}} ,\quad r_{ij} = \left| {r_{i} - r_{j} } \right|$$

(8)

In the Lennard–Jones formulation, the repulsive and attractive exponents are conventionally fixed at 12 and 6, respectively, defining the steepness of the short-range repulsion and long-range attraction. The cutoff distance r_c specifies the maximum separation for which pairwise interactions are computed, beyond which forces are neglected. The depth parameter ε quantifies the interaction strength, while the size parameter σ represents the effective molecular diameter. For consistency in Grand Canonical Monte Carlo simulations, the same potential parameters used for fluid–fluid interactions should be applied to solid–liquid interfaces, ensuring a unified description of intermolecular forces at the adsorbent–adsorbate boundary⁴¹.

$$\Phi_{wall} \left( r \right) = 4\alpha \left( {\left( {\frac{{\sigma_{sf} }}{{r_{ij} }}} \right)^{12} - \left( {\frac{{\sigma_{sf} }}{{r_{ij} }}} \right)^{6} } \right)$$

(9)

In Eq. (9), the mixed interaction parameters ε_ij⁴² and σ_ij are obtained via combining rules. The most common of these is the Lorentz–Berthelot scheme, which defines^43,44.

$$\sigma_{ij} = \frac{{\sigma_{i} + \sigma_{j} }}{2}$$

(10)

$$\varepsilon_{ij} = \sqrt {\varepsilon_{i} \varepsilon_{j} }$$

(11)

GCMC simulations are widely used to predict the adsorption of single and mixed gas species in porous solids, including activated carbons, zeolites, and metal–organic frameworks, by explicitly sampling the grand–canonical ensemble. In GCMC, a fixed‐volume simulation cell serves as a “focal sphere” in equilibrium with an ideal gas reservoir at constant temperature, volume, and chemical potential μ, allowing the number of adsorbate molecules to fluctuate naturally⁴⁵. In experimental gas‐adsorption systems, adsorption equilibria occur within microscopic interfacial elements where adsorbate molecules simultaneously interact with the solid surface and the external gas phase. Within each such element, temperature and chemical potential remain uniform, and its volume may be treated as constant for modeling purposes. System configurations evolve via stochastic Monte Carlo moves, each accepted or rejected according to the Metropolis criterion to ensure correct grand‐canonical sampling⁴⁶.

Computational details

In two stages, gas–solid adsorption was modeled using GCMC simulations. First, a C⁺⁺ code generated energy–distance profiles under specified thermodynamic conditions⁴⁷. Second, the Adsorption Locator module of Materials Studio produced quantitative data on adsorption energies, isosteric heats, spatial adsorption fields, preferred binding sites, and energy‐distribution functions (Table 1). The Universal Force Field (UFF) was selected for this study due to its broad applicability to diverse material classes, including carbon-based structures such as SWCNTs. UFF has been extensively validated for inorganic and carbonaceous materials, making it a reliable choice for simulating gas adsorption in SWCNTs. Simulation parameters are given in Table 1. Electrostatic terms were omitted, consistent with the nonpolar symmetry of N₂, O₂, H₂, CO₂, and CH₄ and the neutral character of SWCNTs⁴⁸. This study fully incorporated van der Waals (dispersion) interactions via the Lennard–Jones potential. The neglect mentioned in the manuscript pertains only to electrostatic interactions, justified by the nonpolar nature of the studied gas molecules and neutral carbon nanotubes, thus not affecting the accuracy or reliability of our results. Table 2 compiles the Lennard–Jones parameters employed in our GCMC simulations. Their suitability for accurately representing gas–solid interactions has been confirmed through B3LYP (Becke’s three-parameter exchange with Lee–Yang–Parr correlation) hybrid-DFT benchmarks reported in Reference⁴⁹.

Table 1 Parameters of simulations.

Table 2 Lennard–Jones parameter used in GCMC simulation.

A representative set of SWCNTs namely, chiral tubes (6,5), (7,6), (9,6), (10,5), (12,4), (20,5), (21,7) and (22,18), together with armchair configurations (8,8), (14,14) and (20,20) were employed in GCMC simulations (Fig. 1). Before adsorption calculations, all nanotube and gas-molecule geometries were fully optimized at the DFT level using the DMol³ package; the resulting structural parameters are listed in Table 3.

Fig. 1

Structure of SWCNT (n, m).

Table 3 Open-ended SWCNT dimensions and unit cell volume in 3D crystal model.

Although SWCNTs are polyatomic assemblies, our simulations explicitly account for the individual atomic parameters of each carbon site to ensure accurate modeling of gas–surface interactions.

Results and discussion

Initially, individual O₂, H₂, CO₂, CH₄, and N₂ molecules were placed at the geometric center of an armchair SWCNT (14,14) and translated incrementally along its axial (x-) direction to compute Lennard–Jones interaction energies, with electrostatic contributions neglected (Fig. 2). The resulting energy–distance profiles (Fig. 3) directly yield the equilibrium adsorption separations.

Fig. 2

Structure of SWCNT (14,14) and simulation box.

Fig. 3

Energy–distance curves for SWCNT (14,14) interacting with (a) O₂, (b) N₂, (c) CO₂, (d) H₂, and (e) CH₄.

Figure 3 presents the Lennard–Jones potential energy curves for O₂, N₂, CO₂, H₂, and CH₄ as a function of radial separation from the wall of an armchair SWCNT (14,14). The position of each energy minimum identifies the equilibrium adsorption distance. At the same time, its depth reflects the relative binding strength: CO₂ exhibits the deepest well, indicative of the strongest physisorption, whereas H₂ shows the shallowest interaction consistent with its low polarizability. These gas-specific minima pinpoint preferred adsorption sites and explain the selectivity trends observed in GCMC simulations. This analysis delivers critical mechanistic insight for the rational design and optimization of SWCNT-based separation materials by quantifying equilibrium separations and adsorption energies.

The simplified energy–distance profiles (Fig. 3) compared with full-scale GCMC adsorption data in an extended simulation cell (Fig. 4) reveal excellent concordance, confirming that the pairwise Lennard–Jones approach reliably reproduces adsorption equilibria in larger nanotube environments.

Fig. 4

CO₂ adsorption snapshots inside SWCNTs of different diameters: (a) (8,8), (b) (14,14), and (c) (20,20).

The intrinsic electronic properties of CO₂ can rationalize the adsorption trends observed in Figs. 3 and 4. To quantify these characteristics, we performed density functional theory optimizations and frequency analyses for O₂, N₂, H₂, CO₂, and CH₄ using the B3LYP functional with a 6–31 + G (d,p) basis set in Gaussian. From these calculations, we obtained fully relaxed molecular geometries, frontier orbital energies, and vibrational modes. Figure 5 displays the optimized structures of O₂, N₂, CO₂, H₂, and CH₄.

Fig. 5

Optimized geometries of (a) O₂, (b) N₂, (c) CO₂, (d) H₂, and (e) CH₄ adsorbed on the SWCNT surface.

By using the energy levels of the HOMO and LUMO levels obtained from DFT, indices such as electron affinity (EA) and ionization energy (IP) can be estimated, and the newer and sharper index of electrophilicity calculated based on the following formulas⁵⁰:

$$EA = - LUMO.$$

(12)

$$IP = - HOMO.$$

(13)

$${\upomega } = \frac{{\left( {HOMO + LUMO} \right)^{2} }}{{4\left( {HOMO - LUMO} \right)}}.$$

(14)

Table 4 summarizes key DFT‐derived electronic descriptors for O₂, N₂, CO₂, H₂ and CH₄, including HOMO and LUMO energies, IP, EA, ω, and permanent dipole moments. These values directly link each molecule’s electronic structure and physisorption behavior on SWCNT surfaces.

Table 4 DFT-derived frontier-orbital energies, electronic descriptors, and dipole moments for the five-adsorbate molecules.

In the energy–distance profiles shown in Fig. 3, three distinct regimes emerge. At large separations, the curve forms a near-constant plateau that indicates negligible interaction between the CNT wall and the adsorbate. As the molecule moves closer, the energy falls to a pronounced minimum at the equilibrium distance, reflecting attractive van der Waals forces. When the separation becomes shorter than this equilibrium position, the energy rises sharply because of Pauli repulsion from overlapping electron clouds. This transition from valley to wall illustrates the interplay of long-range attraction and short-range repulsion that governs physisorption on SWCNT surfaces. Considering the data in Table 4 and the graphs in Fig. 3, the deepest potential is related to molecules having higher electron affinity and electrophilicity. Because the orbital hybridization in the CNT structure is of the sp² type, it can be concluded that the carbon tube has resonant electrons in the structure, which causes these molecules to be attracted to the tube to acquire unpaired electrons⁵¹. We calculated the electron density distribution for the optimized SWCNT–methane complex to elucidate nanotube-adsorbate interactions at the electronic level. Methane being chosen for its minimal attractive and maximal repulsive behavior in the energy distance profiles (Fig. 3). The resulting electron density map (Fig. 6) highlights charge accumulation and depletion regions around the contact interface, clearly visualizing repulsion-dominated interactions at sub-equilibrium separations.

Fig. 6

Electron-density map of the optimized SWCNT–CH₄ adsorption complex (Drawn by Gauss View 6).

DFT analysis of the SWCNT–CH₄ complex reveals a contiguous electron density bridging the nanotube wall and the methane molecule. The computed equilibrium separation of 3–5 Å agrees well with the energy-distance minima identified in Fig. 3. Binding-energy calculations performed at the B3LYP level indicate that the physisorption interaction approaches half the strength of a typical covalent bond (≈ 0.5 eV), confirming that van der Waals forces dominate the adsorption mechanism.

Single-component gas adsorption

Adsorption isotherms for O₂, N₂, CO₂, H₂, and CH₄ on the (22,18) SWCNT were obtained via Grand Canonical Monte Carlo simulations. For each gas, twenty-five independent simulations were carried out at five discrete temperatures, with corresponding pressures (1–20 bar) calculated from the Peng–Robinson equation of state for loadings of 2–10 molecules per cell. As depicted in Fig. 7, adsorption capacity increases with pressure and decreases with temperature, consistent with the exothermic nature of physisorption. Isosteric heats of adsorption were then determined for all eleven SWCNT chiralities to quantify the binding enthalpies of each adsorbate.

Fig. 7

Adsorption isotherms of H₂, O₂, N₂, CO₂, and CH₄ on SWCNT (22,18).

v

Figure 8 presents the mean, minimum, and maximum isosteric heats of adsorption for each gas–SWCNT pairing, thereby capturing both the strength and heterogeneity of adsorbate–surface interactions across distinct binding sites. A fitted mathematical model further relates these values to SWCNT structural parameters with high predictive accuracy. As shown in Fig. 8, the isosteric heat decreases systematically as SWCNT diameter increases, indicating that larger nanotubes exhibit weaker adsorption enthalpies due to reduced van der Waals overlap at their surfaces.

Fig. 8

Isosteric Heat for adsorption for gases on SWCNT (22, 18), (a) Oxygen, (b) Nitrogen, (c) Carbon Dioxide, (d) Hydrogen, and (e) Methane.

To validate our adsorption energy calculations and rule out edge-induced artifacts, we constructed a finite SWCNT “nanobelt” model and optimized its geometry via DFT [75]. The resulting electrostatic potential surface (Fig. 9) exhibits a uniform potential distribution with negligible end-cap perturbations, confirming that terminal effects do not materially influence the adsorption energetics.

Fig. 9

Electrostatic-potential surface of the SWCNT nanobelt: (a) overall view and (b) colour map, where warmer tones denote regions of negative potential (Drawn by Gauss View 6).

Electrostatic potential mapping of the DFT-optimized nanobelt reveals localized regions of partial negative charge on specific carbon atoms. To elucidate how these charge distributions influence electronic structure, Fig. 10 presents the spatial contours of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), highlighting their localization and orientation within the nanobelt framework.

Fig. 10

Spatial distribution of the HOMO and LUMO orbitals on the CNT nanobelt, shown from top and side perspective.

Analysis of the HOMO orbital reveals its delocalized π-electron density along the nanotube framework, which gives rise to Pauli-repulsion when an adsorbate approaches closer than the HOMO interaction distance. In contrast, the LUMO orbital distribution identifies regions of low-lying vacant states that favor electron transfer from the gas, with the depth of the LUMO potential well directly correlating to the adsorbate’s binding strength. Notably, as SWCNT diameter decreases, the spatial overlap between the adsorbate’s frontier orbitals and the tube’s LUMO increases, resulting in higher isosteric heats of adsorption. This orbital-overlap mechanism thus underpins the inverse relationship between nanotube diameter and adsorption enthalpy observed in our simulations. Figure 11 presents density-of-states (DOS) spectra for two DFT-optimized SWCNT nanobelts of identical width but differing radii, illustrating how the LUMO energy level shifts to lower values as the nanotube radius decreases.

Fig. 11

Density of states (DOS) diagrams around the Fermi level for two nanobelts with different radii and the same width to compare the depth of the LUMO position.

DOS analysis shows that the nanobelt with a smaller radius has its LUMO level at a lower energy, corresponding to a deeper potential well and greater electron affinity, which enhances its adsorption strength compared with the nanobelt of larger radius. Curvature also influences charge localization; Fig. 12 compares Mulliken charge distributions for two nanobelts of identical width but different radii, revealing more pronounced charge polarization in the smaller radius structure.

Fig. 12

The electric charge distribution in two nanobelts with different radii and the same width for comparison at the Mulliken scale.

Air separation

This section compares oxygen and nitrogen adsorption on single-walled carbon nanotubes. We begin by evaluating the mean isosteric heat of adsorption. As shown in Fig. 13, the average isosteric heat for pure oxygen, pure nitrogen, and their N₂/O₂ mixture decrease with increasing nanotube diameter. Under identical nanotube conditions, oxygen exhibits a higher mean isosteric heat than nitrogen, indicating stronger physisorption interactions between oxygen molecules and the carbon framework. A review of the energy–distance profiles in Fig. 3 confirms that oxygen’s minimum adsorption energy is lower than nitrogen’s. These differences reflect the distinct electronic structures of the two diatomic gases and are corroborated by quantum mechanical simulations, such as density functional theory.

Fig. 13

Average of the Isosteric heat for pure components, the ideal and real solution.

Beyond adsorption energies and isosteric heats, the energy‐distribution profiles and equilibrium separations of adsorbates provide critical insight into competitive adsorption. Figure 14 compares the probability density of adsorption energies for O₂ and N₂ within the simulation cell, revealing distinct, non-overlapping peak positions and intensities. The higher and sharper distribution peak for O₂ indicates that, at thermodynamic equilibrium in a 20% O₂/80% N₂ mixture, oxygen molecules experience deeper potential wells and thus higher adsorption probabilities than nitrogen. This preferential binding is visualized in the spatial adsorption field of Fig. 15, which confirms the stronger affinity of O₂ for the SWCNT surface under the same conditions.

Fig. 14

Energy distribution for adsorption of binary mixture 20% O₂ and 80% N₂ on: (a) SWCNT (6, 5), (b) SWCNT (7, 6), (c) SWCNT (9, 6), (d) SWCNT (10, 5), (e) SWCNT (12, 4), (f) SWCNT (20, 5), (g) SWCNT (21, 7), and (h) SWCNT (22, 18). Red is for O₂, and green is for N₂.

Fig. 15

Adsorption field for physisorption of binary mixture O₂/N₂ with mole ration 1:4 on SWCNT (22, 18). Green dots for O₂ and red dots for N₂.

Hence, carbon nanotubes appear well suited for O₂/N₂ separation, as illustrated in Fig. 16. Energy‐distribution and isosteric‐heat analyses confirm that O₂ occupies deeper potential wells and binds more strongly to the CNT surface than N₂. Nonetheless, Fig. 16 shows a higher absolute uptake of N₂, which reflects the 1:4 O₂:N₂ feed ratio used in our simulations. Thus, although oxygen exhibits superior per‐molecule affinity, the overall loading is governed by the gas‐phase composition. By adjusting feed ratios and process conditions, CNTs can exploit this disparity in binding strength to achieve highly selective O₂/N₂ separation.

Fig. 16

Adsorbed binary mixture O₂/N₂ with a mole ratio 1:4 on SWCNT (22, 18). Red molecules are, N_2, and blue molecules are O₂.

Effect of temperature and pressure

Temperature and pressure strongly govern adsorption performance. Lower temperatures reduce molecular motion while higher pressures increase gas density at the surface, both favoring CO₂ uptake. As shown in Fig. 17, the average isosteric heat of CO₂ on (22,18) SWCNTs falls with rising temperature and pressure. At low pressure, adsorption occurs mainly in the first layer, producing a high heat of adsorption; as pressure increases, additional layers form and lateral interactions weaken, lowering the average enthalpy. Higher temperature gives molecules more kinetic energy, shortening their surface residence time and reducing adsorption heat. These results demonstrate that CO₂ capture on SWCNTs is optimized under low-temperature, high-pressure conditions.

Fig. 17

Average of isosteric heat for CO₂ adsorption on SWCNT (22, 18) for various temperatures and pressures.

Defect on SWCNT

Carbon nanotube synthesis often produces vacancy defects that perturb the local adsorption landscape. To account for this, we introduced a single-atom vacancy into the (22,18) SWCNT framework and recalculated CO₂ physisorption at 300 K and 10 bar. As shown in Fig. 18, the defective SWCNT exhibits a reduced average isosteric heat of adsorption, indicating weakened adsorbate–surface affinity. Figure 19 displays a more dispersed CO₂ distribution around the vacancy site, confirming that structural defects diminish binding sites’ uniformity and strength.

Fig. 18

Average of Isosteric heat for SWCNT (22, 18) and defective SWCNT (DSWCNT) (22, 18) in various temperatures.

Fig. 19

Adsorption field for physisorption of CO₂ on DSWCNT (22, 18) at T = 300 K and P = 10 bar.

Conclusion

This investigation delivers a comprehensive, multiscale view of gas physisorption on single-walled carbon nanotubes (SWCNTs) by coupling Grand Canonical Monte Carlo (GCMC) simulations with density functional theory (DFT) calculations. Nineteen chiralities covering the full practical diameter range of 7.5–27 Å were examined, which are routinely encountered in synthesized samples. Five prototypical gases, O₂, N₂, CO₂, H₂, and CH₄, were selected to span a broad spectrum of molecular size, polarizability, and quadrupole moment. Systematic variation of temperature (250–350 K), pressure (1–30 bar), and structural integrity (perfect versus single-vacancy defects) enabled quantitative determination of uptake capacity, adsorption energy landscapes, and isosteric heats. The GCMC calculations produced full adsorption isotherms and three-dimensional energy-distribution maps, while complementary DFT optimizations furnished binding geometries, equilibrium separations, and frontier-orbital descriptors for each gas–tube combination. Integration of these datasets revealed that adsorption is strongly diameter-dependent: smaller tubes (< 12 Å) generate deeper Lennard–Jones potential wells and correspondingly larger isosteric heats, whereas curvature effects diminish as the radius increases. Vacancy defects universally attenuate binding enthalpies by up to 25% for CO₂ because local bond rehybridization reduces van der Waals overlap and disrupts charge-induced polarisation. To explore competitive adsorption, a realistic feed comprising 20% O₂ and 80% N₂ was simulated on eleven representative SWCNTs. Oxygen consistently occupied deeper potential wells, confirmed by DFT-derived electron affinities and electrostatic-potential surfaces, yet the bulk composition governed overall loading. This dichotomy underscores that separation performance hinges on intrinsic affinity and process conditions. Spatial adsorption fields showed that O₂ preferentially populates groove and interstitial sites, whereas N₂ is restricted to less energetically favourable axial regions. Together, these results establish clear structure–property relationships: high curvature, defect-free SWCNTs maximise adsorption enthalpy and selectivity, whereas larger or defective tubes sacrifice binding strength and increase energetic heterogeneity. The study, therefore, identifies small-diameter, SWCNTs as optimal candidates for energy-efficient O₂/N₂ separations, and the same design principles can be extrapolated to other gas pairs with contrasting electronic signatures. Beyond its specific findings, the work demonstrates the power of a unified GCMC–DFT workflow. The approach reconciles quantum–mechanical insight with macroscopic observables by aligning atomistic energetics with statistical sampling, providing a transferable protocol for screening and optimizing nanostructured adsorbents across various environmental and industrial gas-separation challenges.