A LOADING-DEPENDENT GEOMETRIC FRAMEWORK FOR CONFINED CO2 PACKING IN NaX (FAU-TYPE) ZEOLITE

Abstract

This work presents a loading-dependent free-volume geometric framework for CO₂ adsorption in NaX zeolite (FAU-type) based on the structural function interpreted as free pore volume within α-cavity. Unlike classical approaches, it is loading-dependent, enabling a continuous and purely analytical description of the geometric evolution of the molecular packing trajectory. Structural descriptors are derived considering the initial free pore volume and the average free volume consumed along the path and ψ is the cumulative molecular packing fraction. For two representative extreme conditions, η_Vmax = 3.57 mmol g^-1 (348 K, 1 bar) and η_Vmax = 7.26 mmol g^-1 (298 K, 10 bar), these descriptors show marked differences: low-loading condition leads to reduced initial free-pore volume (~ 157 ų molecule^-1), and ψ = 0.64 mmol g^-1, indicating weak molecular rearrangement within the adsorption pore. In contrast, the high-loading condition preserves higher F₀ and ψ values, reflecting a larger structural rearrangement of the confined adsorbed phase. These descriptors provide a robust measure of the geometric severity of molecular reorganization required to NaX-CO₂ reaches the saturation inside the confined α-cavity. This framework can be interpreted as a loading-dependent extension, adapted to confined adsorption systems, of the classical free-volume theory, establishing a bridge between macroscopic adsorption data and nanoscale structural interpretation.

Keywords:
CO₂ adsorption; NaX zeolite; free pore volume; molecular packing; structural function.

INTRODUCTION

Zeolites are crystalline aluminosilicate materials formed by SiO₄^4- and AlO₄^5- tetrahedra, in which the oxygen is shared between two tetrahedra to create a network of interconnected tunnels and cavities.¹ Faujasite NaX zeolite (FAU topology) is a widely used benchmark adsorbent, valued for its high crystallinity, large CO₂ uptake, and well-defined pore structure of α-cavities linked by 12-membered oxygen rings.² Classical models, such as the Langmuir isotherm³ and the Brunauer, Emmett, Teller (BET) multilayer theory,⁴ describe the macroscopic saturation capacity and the global pore volume of solids, but they do not directly yield nanometric descriptors such as molecular packing density. Figure 1 shows the arrangement of a FAU zeolite with its cavities. The basic FAU-family framework consists of sodalite octahedral small cavities (β) that are connected through double 6 rings (D6Rs) forming a large α-cavity with 12 MR window.² Barrer and Gibbons,⁵ in their investigation of CO₂ adsorption in NaX (a near-faujasite structure) with high loading, obtained an isotherm at 183.6 K and a pressure of 50 cmHg. The inferred free-saturation capacity was approximately 6.6 mmol g^-1, corresponding to roughly 14.8 CO₂ molecules adsorbed per α-cavity in the unit cell.

Figure 1
FAU structure with cavities depicted as silicon framework⁶

Many contemporary studies on carbon dioxide adsorption in zeolites, particularly FAU-type frameworks such as NaX, rely heavily on advanced structural and spectroscopic techniques to probe the adsorbed phase at the nanoscopic and atomic levels. For example, Thang et al.,⁷ in their investigation of adsorption of CO₂ in Liand Na-FAU zeolites, used Fourier-transform infrared (FTIR) spectroscopy and theoretical calculations based on density functional theory (DFT/CC) methods to determine that the molecules were adsorbed preferentially on sites II. These sites are more stable with a minor heterogeneity of adsorption sites present in the FAU samples, corresponding to a nanoscopic analysis of the system. Similarly, CO₂ adsorption in nanosized NaX and NaY zeolite samples of high crystallinity was investigated by Polisi et al.⁸ NaX and NaY zeolites exhibit the same FAU-type topology; however, the Si/Al ratio for NaX ranges from 1.0 to 1.5, while NaY presents a ratio above 1.5. They used X-ray powder diffraction (XRD) Rietveld refinement and FTIR spectroscopy to obtain structural information on nanometric scale, such as host-guest and guest-guest interactions, the number of CO₂ molecules adsorbed per unit cell, C-O bond lengths, and the identification of crystallographic independent Na⁺ sites in the NaX sample. Nonetheless, the exact number of strong active sites per α-cavity remained unresolved.

An investigation of CO₂ adsorption in NaY zeolite by Wong-Ng et al.⁹ showed that 45 of the apparent 76 molecules per unit cell were in only two crystallographically independent sites, bonding to the Na⁺ cations in α-cavity site II. Furthermore, Krachuamram et al.¹⁰ synthesized NaX zeolite where the surface area and pore volume were increased due to smaller crystal sizes, resulting in a significant enhancement of carbon dioxide uptake.

Stable single-unit-cell nanosheets of zeolite structure micropore (ZSM-5) were first reported by Choi et al.¹¹ and later by Na et al.¹² Micropore structures at the scale of several unit cells can form mesopore walls that reach the pore diameters required for heterogeneous catalysts in petroleum refining and the petrochemical industry.¹³,¹⁴ As reported by Polisi et al.,⁸ “The modifications of zeolite crystal size, from micron to nanoscale yielding biocompatible nanozeolites, become an important research field”. Indeed, Georgieva et al.,¹⁵ Komaty et al.,¹⁶ and Aufray et al.¹⁷ have shown that the administration of CO₂ in nanozeolite crystals can assist human health by controlling blood circulation and pH.

Consequently, the study of carbon dioxide adsorption in FAU-type zeolites, although highly valuable, often demands high-quality crystallographic data, sophisticated instrumentation, or computational modeling. This included the Rietveld method, DFT,¹⁸ molecular dynamics (MD),¹⁹ and FTIR spectroscopy.⁸ These complexities can limit their applicability for screening studies on large datasets, especially in an industrial research context.

In contrast, Araújo and Márquez²⁰ proposed an analytical macroto-nano theoretical framework to analyze relevant structural aspects of CO₂ adsorption on high-crystallinity NaX zeolites. Using data from Moura et al.,²¹ especially a loading of ~ 6.27 mmol g^-1 at 298 K and pressure of 1 bar, the framework required only the saturation capacity and BET pore volume as inputs. From these parameters, the molecular packing density of ~ 114.0 ± 8.1 ų per molecule was obtained, with approximately 85 CO₂ molecules hosted within the α-cavities of the unit cell, and approximately 6.44 strong-active sites per α-cavity. Building upon this foundation, the present work develops a complementary macro-to-nano framework for a more detailed structural interpretation of CO₂ adsorption phase in NaX under four representative temperature and pressure regimes.

Using the average pore volume available per molecule, V_{pore,molecule}^α (η_V; T, P), with unit ų molecule^-1; as defined by Araújo and Márquez,²⁰ we establish a macro-to-nano bridge centered on a F(η_V; T, P) descriptor in ų molecule^-1 that quantifies the average reduction of the free pore volume per molecule in the α-cavity of NaX. The independent loading variable η_V ranges for constant temperature and pressure. In the adsorption process, this descriptor is related to the packing density of the guest molecule.²² Furthermore, this descriptor serves as the basic kernel for the present framework. The present formulation of the structural function F(η_V; T, P) preserves a formal analogy with the classical free-volume theory of dense particle systems developed by Cohen and Turnbull²³ and Turnbull and Cohen.²⁴ In that theory, molecular mobility occurs when the available volume accessible to a particle exceeds its effective molecular volume, leading to the natural definition of the average free volume per molecule as V_free = V_available - V_molecule. In the present framework, an analogous relation arises within the confined adsorption α-cavity: the average pore volume available per molecule V_{pore,molecule}^α (η_V; T, P) plays the role of V_available, the effective molecular packing, Cut(T, P) corresponds to V_molecule, and the structural function F(η_V; T, P) representing V_free. Although the physical environments differ significantly, that is, a continuous dense medium in the classical theory versus geometrically structured confined pore space and discrete particles in the present framework, the strong correspondence between these quantities allows the adsorption process to be interpreted in terms of the structural function, that can be interpreted, with caution, as the extension of the free-volume concept to a confined adsorption domain, where the liquid-like environment is replaced by the α-cavity and the discrete particles are guest CO₂ molecules.

In this sense, the evolution of F(η_V; T, P) provides a geometric measure of how the accessible free pore volume per molecule decreases as adsorption proceeds, reflecting the dominant molecular rearrangements required to accommodate additional CO₂ molecules within the confined environment of NaX α-cavity. Unlike the classical free-volume description of dense particle systems, the accessible pore volume per molecule evolves continuously with adsorption loading giving to present framework a loading-dependence free-pore-volume description of molecular packing or equivalently characterizing F(η_V; T, P) as a dynamic model dependent of gravimetric saturation extending the classical free-volume-theory of dense particle systems interpretation to adsorption under nanoscale confinement, introducing a loadingdependent description of free-volume evolution driving the structural molecular rearrangement in microporous adsorption systems. From the structural function several descriptors are derived, including the initial free pore volume F₀ = F(η_V0; T, P), the average reduction of F₀ along de adsorption trajectory, ∆F_cons, and the practical robust dimensionless criterion of the severity of the molecular rearrangements, ψ = WF₀^-1. In particular, ψ can be interpreted as a cumulative molecular packing fraction, representing the fraction of the initial free pore volume that is cumulatively mobilized through molecular rearrangements along the adsorption packing trajectory.

Application of the proposed framework to CO₂ adsorption in NaX synthesized and analyzed by Moura et al.²¹ for four representative thermodynamic conditions of temperature and pressure was carried out in this work. For the lowest-capacity state, η_Vmax = 3.57 mmol g^-1 at T = 348 K and P = 1 bar, the analysis yields an initial free pore volume per molecule of F₀ ≈ 157 ų molecule^-1, an average reduction ∆F_cons = 93 ų molecule^-1, and a normalized indicator ψ ≈ 0.64 mmol g^-1. In contrast, for the highest-capacity, η_Vmax = 7.26 mmol g^-1 at T = 298 K and P = 10 bar, the analysis yields F₀ ≈ 258 ų molecule^-1, an average reduction of ∆F_cons = 182 ų molecule^-1, and a ψ >> 1 mmol g^-1 value indicating substantially stronger molecular rearrangement within the adsorption pore as the system NaX-CO₂ approaches the terminal packing state. Our framework deals with the molecular packing trajectory within the adsorption pore in the α-cavity, depending only on the physical domain of gravimetric saturation, and the BET pore volume, that is, it is a non-temporal model. Nevertheless, it exhibits properties analogous to those obtained with CO₂ diffusion trajectories in zeolites reported by Zhao et al.²⁵ Their diffusion study revealed strong interactions between CO₂ molecules and the pore walls, resulting in relatively straight and short migration pathways. These observations are consistent with the CO₂ loading trajectories predicted by the present framework, which likewise indicate nontortuous transport pathways. Our analytical framework does not attempt to explicitly describe microscopic mechanisms responsible for molecular rearrangements, such as local redistribution of molecules within α-cavity or cooperative packing effects among neighboring adsorbates.²⁶ Instead, the proposed loading-dependent geometric free-pore-volume framework should be interpreted as an effective geometrical representation of the collective molecular organization that occurs under confined adsorption conditions along the pore environment. Therefore, the framework is intended primarily as a geometrical-organizational description of confined molecular packing rather than as a predictive adsorption-capacity model. Because the formulation is based on general geometric relations, it may be naturally extended to other FAU-type zeolites of high crystallinity and potentially to a broader class of microporous adsorption systems.

METHODOLOGY

Our proposed framework depends on the pore volume per molecule in α-cavity of the NaX, which is a fundamental function because, as reported by Araújo and Márquez,²⁰ it captures the effective volume accessible to CO₂ inside the α-cavity being defined as:

(1) $${\mathrm{V}}_{\text{pore, molecule}}^{\alpha }({\eta }_{\mathrm{v}};\mathrm{T},\mathrm{P})=\frac{{10}^3{\mathrm{V}}_{\text{pore}}^{\mathrm{BET}}}{{\mathrm{N}}_{\mathrm{A}}{\eta }_{\mathrm{v}}(\mathrm{T},\mathrm{P})}\text{, in}{\text{Å}}^3{\text{molecule}}^{-1}$$

where V_pore^BET (cm³ g^-1) is the BET pore volume, used here as a geometric upper-scale reference, where the real accessibility of CO₂ is controlled by the experimental (T, P) conditions. N_A= 6.022 × 10²³ is the Avogadro constant, and η_V(T, P) is the gravimetric saturation (mmol g^-1) within the physical domain, represented as η_V0 → η_Vmax(T, P), where η_Vmax(T, P) is the saturation capacity at temperature T and pressure P. The V_pore^α (η_V = η_Vmax; T, P) is interpreted as the packing density for the guest molecule, according to Férey.²² When V_pore^BET is expressed in ų g^-1, Equation 1 is rewritten as:

(2) $${\mathrm{V}}_{\text{pore, molecule}}^{\alpha }({\eta }_{\mathrm{V}};\mathrm{T},\mathrm{P})=\frac{{\mathrm{K}}_{\text{geom}}}{{\eta }_{\mathrm{V}}(\mathrm{T},\mathrm{P})}$$

where ${\mathrm{K}}_{\text{geom}}=\frac{{10}^{27}{\mathrm{V}}_{\text{pore}}^{\mathrm{BET}}}{{\mathrm{N}}_{\mathrm{A}}}$ is a geometric constant with units of ų molecule^-1 (mmol g^-1).

The geometric constant K_geom encodes the structural constraints arising from zeolite architecture (e.g., pore size, shape, α-cavity geometry, the spatial distribution of adsorption sites under confinement and tortuosity).

From Equation 1 and 2, we obtain a macro-to-nano bridge centered in the key Equation 3 given by:

(3) $$\mathrm{F}({\eta }_{\mathrm{V}};\mathrm{T},\mathrm{P})={\mathrm{V}}_{\text{pore, molecule}}^{\alpha }({\eta }_{\mathrm{V}};\mathrm{T},\mathrm{P})-\mathrm{Cut}(\mathrm{T},\mathrm{P}),\text{in}{\text{Å}}^3\text{molecule}{}^{-1}$$

where Cut(T, P) = V_{pore,molecule}^α (η_Vmax; T, P).

Equation 3 quantifies the reduction of free-pore average volume per molecule within the α-cavity during the adsorption process, relative to the equilibrium saturated state Cut(T, P). For a fixed temperature T, the adsorption loading (η_V; T, P) is assumed to be continuous and strictly function of pressure over the interval P ∈ (0, P_max], as generally observed for adsorption isotherms and for the system under study. This monotonic relationship defines an invertible mapping between pressure and loading, allowing η_V to be used as an intrinsic state variable along the adsorption path. Consequently the structural function F(η_V; T, P) is well-defined over the physical domain η_V ∈ [η_V0, η_Vmax(T, P)], since the continuous variation of η_V is induced by the continum representation of pressure through the adsorption isotherm at fixed temperature. In this sense, the existence and admissibility of the structural function are ensured by the underlying thermodynamic consistency of the adsorption isotherm. Therefore, the geometric formulation does not require an explicit dependence on pressure, since the loading variable η_V already encapsulates the thermodynamic evolution of the system, providing a consistent and well-posed description of adsorption trajectories under confinement.

Some adsorption data for NaX zeolite reported by Moura et al.²¹ were used in this study: FAU-zeolite, high crystallinity, silicon aluminum ratio equal to unity; 96 Na⁺ extra-framework cations per unit cell; BET pore volume, V_pore^BET = 0.43 cm³ g^-1; four regimes (T = 298 K, P = 1 bar, η_Vmax = 6.27 mmol g^-1); (T = 348 K, P = 1 bar, η_Vmax = 3.57 mmol g^-1); (T = 298 K, P = 10 bar, η_Vmax = 7.26 mmol g^-1) and (T = 348 K, P = 10 bar, η_Vmax = 6.55 mmol g^-1). Based on the crystallographic composition reported in the literature,¹⁹ the framework was assumed to have the chemical formula Na₉₆Al₉₆Si₉₆O₃₈₄ and a dry molecular mass of M_dry = 13637 g mol^-1.

RESULTS AND DISCUSSION

The theoretical framework is based on Equation 3 that connects macroscopic BET-Langmuir data with the free-pore volume in α-cavity relative to the packing limit, enabling comparison of how confinement evolves as systems NaX-CO₂ approaches saturation. The fundamental saturation condition is F(η_Vmax; T, P) = 0, when η_V = η_Vmax. Therefore, F(η_V; T, P) quantifies how far the system is from the terminal packing density of guest molecule. From Equations 2 and 3, we have:

(4) $$\mathrm{F}({\eta }_{\mathrm{v}};\mathrm{T},\mathrm{P})={\mathrm{K}}_{\text{geom}}(\frac{1}{{\eta }_{\mathrm{v}}}-\frac{1}{{\eta }_{{\mathrm{V}}_{max}}(\mathrm{T},\mathrm{P})})\text{, in}{\text{Å}}^3{\text{molecule}}^{-1}$$

where K_geom ≅ 714 ų molecule^-1.

Figure 2 depicts the monotonic decay of the structural function F(η_V; T, P), which estimates the average free-pore volume per molecule available within α-cavity for adsorption before η_Vmax(T, P) is reached. Each curve tends toward zero at its own saturation point, where the cavity reaches its final saturation state. The behavior of these curves reflects the confinement effects imposed by the specific temperature and pressure regimes. While F(η_V; T, P) serves as a global geometric descriptor, it does not directely encode site-to-site heterogeneity; because local guest-guest interactions may become significant even at low global loadings. This occurs when adsorption concentrates around a small number of strong sites, promoting local co-adsorption (e.g., η_Vmax(348 K, 1 bar) = 3.57 mmol g^-1, as shown in Table 1).

Table 1
Structural metrics of the adsorbed CO₂ phase in NaX zeolite under variable thermodynamic conditions

T / K	P / bar	ηV / (mmol g-1)	Cut / (Å3 molecule-1)	NcellCO2	NαCO2	fmono	Nstrong-sitesα	α N CO2,site
298	1	6.27	113.9 ± 7.9	85.50	10.69	0.59	6.32	1.69
298	10	7.26	98.4 ± 6.8	99.00	12.38	0.69	8.00	1.54
348	10	6.55	109.0 ± 7.6	89.32	11.17	0.62	6.90	1.62
348	1	3.57	200.0 ± 14.0	48.68	6.09	0.34	2.05	2.97

Figure 2
Structural trajectories of average free-pore volume per adsorbed molecule in the α-cavity of NaX zeolite under various (T, P) conditions. The h_Vmax data were obtained from Moura et al.²¹

The low-capacity extreme, at T = 348 K, P = 1 bar, and η_Vmax(T, P) = 3.57 mmol g^-1, exhibits the steepest decay because only a limited volumetric reduction is required to reach its saturation capacity. Under these conditions, fewer CO₂ molecules are adsorbed per α-cavity; the terminal molecular volume Cut(T, P) remains high, approximately 200 ų molecule^-1. Consequently, the packing regime is comparatively loose, resulting in a low packing density. Conversely, high-capacity conditions, especially at T = 298 K, P = 10 bar, η_Vmax(T, P) = 7.26 mmol g^-1, show a much larger free volume at the beginning of the trajectory and a longer reduction pathway. This reflects the fact that reaching high saturation requires a pronounced and progressive volumetric reduction of the free pore space per molecule. This leads to smaller Cut(T, P) ≅ 98 ų molecule^-1 and, therefore, a high packing density. The progressive thickening of each trajectory as F → 0 reflects the increasing geometric constraints imposed on the system as saturation is approached. This visual encoding emphasizes that, although the average free pore volume continuously decreases along the adsorption path, the remaining residual free volume becomes progressively more restrictive as terminal packing is approached. Thus, even a small variation in F near saturation corresponds to increasingly confined and structurally constrained adsorption states. In this representation, the trajectories primarily describe the extent of adsorption pathway, indicating how far molecular packing progresses within the confined α-cavity under given (T, P) conditions.

The two intermediate cases lie between the extreme curves and preserve the same universal shape, indicating that the geometry of the packing pathway is dominated by the η_V^-1 dependence of the molecular pore-volume function. We evaluate F(η_V; T, P) over η_V0 → 7.26 mmol g^-1 with effective η_V0(T, P) = 2.0 mmol g^-1 as a practical lower-bound saturation. Lower loadings are of minor experimental relevance for high-capacity NaX samples and correspond to very weak structural crowding in the α-cavity. Using saturations below η_V0(T, P) value was to obtain better graphic resolution.

Figure 3 shows the monotonic decay of V_{pore,molecule}^α (η_V; T, P), within this physical domain. All four continuous curves start from an assumed common high-volume regime of approximately 350 ų molecule^-1 near η_V0 and progressively decrease until reaching their condition-specific terminal values at experimental saturation under various temperature and pressure conditions.

Figure 3
Evaluation of V_{pore molecule}^α _, (h_Vmax; T, P) with loading

The continuous segments are clearly separated by temperature and pressure severity. At T = 298 K and P = 10 bar, the system reaches its smallest packing density of CO₂ at approximately 99 ų molecule^-1 reflecting the most compact molecular arrangement. At 348 K and 1 bar, the system displays the most expanded configuration of 200 ų molecule^-1, consistent with weak confinement in the α-cavity. The intermediate cases lie between these limits.

Structural metrics of carbon dioxide packing obtained across the temperature and pressure conditions were based on the relations presented by Araújo and Márquez²⁰ and are not presented here. We compute the number of CO₂ adsorbed per α-cavity (N^α_CO2), per cell (N^cell_CO2), the fraction of the theoretical monolayer (f_mono), the resulting number of strong active sites per α-cavity (N_strong-sites^α ), the average number of guest molecules by site (N^α_{CO ,site2} ), and the packing density of CO₂ molecule (Cut). These quantities, which form the basis of the structural interpretation, are summarized in Table 1.

The values summarized in Table 1 illustrate how temperature and pressure conditions modulate the nanoscopic packing state of carbon dioxide inside the α-cavity. The accessibility factor f_mono represents the fraction of the theoretical monolayer available for adsorption. Although the extreme condition at T = 298 K and P = 10 bar yields the smallest packing density, 98.4 ± 6.8 ų per guest molecule, the largest total occupancy per α-cavity, N^α_CO2 ≅ 12.4CO₂ molecules, and the highest N^α_CO2,strong-site ≅ 8.5 capped at 8.0²⁷ with mean occupancy per site N^α_{CO ,site} ≅ 1.54 becomes slightly lower when compared with other cases. This effect reflects the progressive activation of a larger set of energetically relevant adsorption sites under strong confinement. As additional sites become occupied at high saturation, the molecular load tends to distribute more collectively through the α-cavity, reducing the average CO₂ occupancy per active site despite the overall crowding and strong co-adsorption conditions. Therefore, the molecular load is distributed more evenly among the available sites, reducing the average CO₂ occupancy per site, despite the overall crowding being maximal indicating strong co-adsorption process or guest-guest interactions on various sites. In Table 1, at conditions T = 348 K and P = 1 bar, each strong adsorption site can be accommodated up to three CO₂ molecules. Physically, this is consistent with the strong electrostatics field associated with high exposed Na⁺ sites, which stabilize the first CO₂ molecule and allows additional molecules to be co-adsorbed via quadrupolar orientation.²⁶ At high temperature and low pressure, the packing remains relatively loose, making such multiple occupancy of active sites feasible. Two distinct features emerge from Figure 2. (i) The magnitude of the average free pore volume at low loading. At initial η_V0 loading, the average free volume corresponds to the most compact final stage, T = 298 K and P = 10 bar. This is because its Cut(T, P) ≅ 98 ų molecule^-1, the system must be transverse a larger average free volume per molecule before reaching this tightly packed configuration. In contrast, the shorter final stage at T = 348 K and P = 1 bar, with Cut(T, P) ≅ 200 ų molecule^-1 shows the average free pore volumetric large at the same η_V0, the cavity is already relatively close to its loose saturation capacity even a modest loading. (ii) In terms of the average slope, this high temperature and low-pressure case display the steepest decay of F(η_V; T, P) trajectories consistent with shorter structural distance in the physical domain.

Summarizing, in F(η_V; T, P) representation, each curve can be interpreted as a molecular packing trajectory within α-cavity, describing the progressive exhaustion of the average free pore volume per molecule as loading increase with F → 0. The condition with the lowest terminal loading exhibits the steepest slope, indicating that the available geometric free-volume budget is depleted shorter per unit increase in η_V. Consequently, saturation F → 0 is achieved with a smaller geometric average free pore volume per molecule reduction required to reach the packing limit. In contrast, the highest saturation condition displays a more gradual slope, reflecting a larger geometric free pore volume reservoir that must be progressively exhausted before packing saturation is attained. These behaviors do not reflect time-dependent kinetics, but rather, a volumetric exhaustion process, that is, a kinetic geometry governed solely by volumetric constraints imposed by the thermodynamic state (T, P). The systematic analysis of the F(η_V; T, P) curves under different regimes reveals a common structural pattern: all trajectories originate at the positive free-volume state and converge monotonically toward the terminal packing condition F(η_Vmax; T, P) = 0 that constitutes an absorbing boundary in geometric volumetric space (η_V, F(η_V; T, P)) corresponding to complete exhaustion of the average free available pore volume per molecule inside α-cavity. This observation motivates the relevant geometric proposition that has been used formally throughout this discussion.

Although the present framework is formulated in purely geometric terms, the effects of intermolecular and host-guest and guest-guest interactions are implicitly embedded in this terminal packing condition through Cut(T, P). This parameter reflects the effective volumetric state reached at terminal saturation, incorporating the combined influence of CO₂-framework and CO₂-CO₂ interactions within the confined α-cavity. Consequently, interactions such as quadruple-quadruple effects are not explicitly represented in the formulation of F(η_V; T, P), but their impact is indirectly captured through the loading-dependent boundary conditions defined by η_Vmax(T, P) and Cut (T, P). In this sense, the molecular-based definition of F(η_V; T, P), remains geometrically consistent, while implicitly reflecting the interaction-driven organization of the adsorbed phase. Therefore, the model may be interpreted as geometrically explicit and energetically implicit, providing a reduced yet physically consistent description of confined molecular packing trajectories.

Proposition: geometric packing trajectory in FAU-type zeolite

For any FAU-type zeolite characterized by a well-defined BET pore volume and a fixed thermodynamic condition (T, P), the structural function F(η_V; T, P) defines a continuous geometric trajectory γ(η_V; T, P) in the state (η_V, F(η_V; T, P)), connecting the initial state (η_V0, F(η_V0; T, P)) to the terminal state (η_Vmax(T, P), 0), with η_V0 > 0 and η_V ranging in the physical domain between η_V0 and η_Vmax(T, P).

This proposition is not an abstract mathematical theorem, but rather a structural proposition derived from the analytical form of the function F(η_V; T, P). This result arises from the mathematical function V_{pore,molecule}^α (η_V; T, P) and is therefore architecture-intrinsic, independent of the specific chemical composition of the FAU framework.

Proof: for a fixed thermodynamic condition (T, P), the structural function is defined by Equation 3, in which V^α (η_V; T, P) α η ^-1.

As V _{V V}

η_V ∈ [η_V0, η_Vmax(T, P)], where η_V0 is a fixed positive value. Since $\frac{\mathrm{dF}}{\mathrm{d}{\eta }_{\mathrm{v}}}\propto (-\frac{1}{{\eta }_{\mathrm{v}}^2})<0$, then F(η_V; T, P) is strictly decreasing in the physical domain. This defines the decreasing continuous curve γ(η_V; T, P) = (η_V, F(η_V; T, P)) connecting the two boundary states (η_V0, F(η_V0; T, P)) and (η_Vmax(T, P), F(η_Vmax; T, P) = 0) ≤.

Consequently, the trajectory γ(η_V; T, P) can be interpreted as a continuous curve of the progressive depletion of the average free volume reservoir per molecule in two-dimensional geometric volumetric space (η_V, F(η_V; T, P)). Thus, each line shown in Figure 2 describes a true continuous path of the average free pore volume per molecule in NaX α-cavity in (η_V, F(η_V; T, P)) state space. In this framework, the adsorption trajectory is equivalently interpreted as a molecular packing trajectory within the confined α-cavity. The concept of trajectories introduced in the present framework finds a meaningful parallel in recent studies²⁵ of molecular CH₄ and CO₂ diffusion in zeolites, where the motion of adsorbates is described in terms of diffusion pathways governed by the atomic-level topology of the framework, the distribution of acid sites, host-guest interactions, and underlying free-energy landscape, that is, it is associated with a more energetically stable state and lower diffusion barriers. In such systems, molecular trajectories evolve in real space, reflecting the navigation of adsorbates across energetically favorable pathways in the pore network. In contrast, the present framework defines adsorbate trajectories in a geometric state space (η_V, F(η_V; T, P)), where the evolution is governed not by the time but the progressive depletion of the average free pore volume under increasing loading. Despite this fundamental difference, both descriptions share a common feature: the existence of constrained paths imposed by the zeolite architecture. While diffusion trajectories are shaped by energetic landscapes, adsorption trajectories in the present formulation are determined by geometric constraints associated with the confined packing. The present geometric framework exhibits characteristics analogous to those observed in diffusion of CO₂ in zeolites. In recent diffusion studies²⁵ with CH₄ in zeolites, weaker adsorption is associated with larger average nearest distances along the pore walls and more delocalized motion, whereas strong host-guest interactions lead to more localized and structured trajectories. In the present framework, an analogous behavior is also observed: low-saturation regimes correspond to weaker geometric confinement and reduced structural rearrangement, while high-saturation regimes require increasingly constrained molecular reorganization at the system approaches the terminal condition F → 0. Additionally, the recent diffusion studies have shown that CO₂ molecules tend to follow more direct and less tortuous pathways compared to other adsorbates such as methane, due to stronger interactions with the pore walls in AlPO₄^5- zeolite. The smooth and monotonic structural trajectories obtained from the structural function F(η_V; T, P) and shown in Figure 2 should not be interpreted as real-space diffusion pathways. Rather, the curves provide a qualitative geometric analogy for the constrained organization of CO₂ under confinement, distinct from kinetic diffusion trajectories reported in density functional theory (DFT) calculations or molecular simulations.²⁵,²⁸ Therefore, while the two approaches operate in different domains, real-space diffusion versus geometric state-space evolution, the present framework exhibits well-defined features which find relevant analogy with the observed in diffusion trajectories, providing a complementary perspective as how CO₂ molecular behavior is constrained by the zeolite architecture. In this sense, F(η_V; T, P) may be interpreted as a quantitative measure of the geometric constraint imposed on molecular packing under confinement. However, because the coordinates of γ(η_V; T, P) carry different physical units, Euclidean motions such as vector magnitude, arc length or geometric velocity are not physically meaningful. In this sense, the trajectory in the present formalism is physically interpretable as functional relation, not as a metric curve in a state space. Therefore, the structural function F(η_V; T, P) describes the exhaustion of the average free pore volume along the molecular packing trajectory, reflecting the severity of the progressive rearrangement of the confined adsorbed phase as the system NaX-CO₂ approaches the terminal packing state. The differential reduction of the average free pore volume per molecule associated with an incremental loading η_V is giving by the structural one-form,²⁹ ω = F(η_V; T, P)dη_V. Then, the cumulative geometric effect along this trajectory γ(η_V; T, P), is naturally obtained by the integrating this one-form along the path as

(5) $$\mathrm{W}(\mathrm{T},\mathrm{P})={\int }_{\gamma }\omega ={\int }_{{\eta }_{{\mathrm{v}}_0}}^{{\eta }_{{\mathrm{v}}_{max}}}\mathrm{F}({\eta }_{\mathrm{v}};\mathrm{T},\mathrm{P})\mathrm{d}{\eta }_{\mathrm{v}},\text{in}{\text{Å}}^3{\text{molecule}}^{-1}\left(\mathrm{mmol}{\mathrm{g}}^{-1}\right).$$

From Equations 2-4 we have

(6) $$\mathrm{W}(\mathrm{T},\mathrm{P})={\mathrm{K}}_{\text{geom}}[\ln \left(\frac{{\eta }_{{\mathrm{V}}_{max}}(\mathrm{T},\mathrm{P})}{{\eta }_{{\mathrm{V}}_0}}\right)-(1-\frac{{\eta }_{{\mathrm{V}}_0}}{{\eta }_{{\mathrm{V}}_{max}}(\mathrm{T},\mathrm{P})})]$$

A universal property of the packing trajectory is established from Equation 6 as

(7) $$\frac{\mathrm{W}(\mathrm{T},\mathrm{P})}{{\mathrm{K}}_{\text{geom}}}=\ln \left(\frac{{\eta }_{{\mathrm{v}}_{max}}(\mathrm{T},\mathrm{P})}{{\eta }_{{\mathrm{v}}_0}}\right)-(1-\frac{{\eta }_{{\mathrm{v}}_0}}{{\eta }_{{\mathrm{v}}_{max}}(\mathrm{T},\mathrm{P})})$$

showing that the normalized W(T, P) depends only on the ratio η_Vmax(T, P) (η_V0)^-1 or more precisely, it depends only on the η_Vmax, since η_V0 is fixed. The present model constitutes a confined nontemporal dynamic free pore framework in which the molecular packing trajectory is describe by a structural decreasing continuous curve γ(η_V; T, P) with the quantity W(T, P) emerging naturally as a non-mechanical geometric work, defined by the path integral of the structural one-form ω = F(η_V; T, P)dη_V. Therefore, W(T, P) should not be interpreted as a mechanical work associated with the pore compression. Instead, it represents a structural descriptor quantifying the cumulative reduction of the average free pore volume during adsorption trajectory. The molecular rearrangements responsible for this reduction arise from electrostatic adsorption interactions, guestguest interactions, redistribution of the confined adsorbed phase, and a possible residual framework deformation.

The natural packing parameter of the present framework ψ(η_Vmax(T, P)) is defined by

(8) $$\psi ({\eta }_{\mathrm{V}max}(\mathrm{T},\mathrm{P}))=\frac{\mathrm{W}(\mathrm{T},\mathrm{P})}{{\mathrm{F}}_0},\text{in}\mathrm{mmol}{\mathrm{g}}^{-1}$$

where F₀ = F(η_V0; T, P) is the initial free pore volume available per molecule in pore adsorption.

From the Equations 6 and 8 we have

(9) $$\psi =\psi ({\eta }_{max}(\mathrm{T},\mathrm{P}))=\frac{\ln \left(\frac{{\eta }_{max}(\mathrm{T},\mathrm{P})}{{\eta }_{{\mathrm{v}}_0}}\right)-(1-\frac{{\eta }_{{\mathrm{v}}_0}}{{\eta }_{{\mathrm{v}}_{max}}(\mathrm{T},\mathrm{P})})}{\frac{1}{{\eta }_{{\mathrm{v}}_0}}-\frac{1}{{\eta }_{{\mathrm{v}}_{max}}(\mathrm{T},\mathrm{P})}},\text{in mmol g}{\mathrm{g}}^{-1}$$

where ψ depends only on the ratio η_Vmax(T, P) (η_V0)^-1 or in other words of the extension of η_Vmax. The descriptor ψ = WF₀^-1, in mmol g^-1, Table 2. Intrinsic geometric work W(T, P) and packing parameter ψ(T, P) provides a loading-scaled measure of the cumulative structural consumption of the initial average free-pore volume available for confined adsorption packing. Since W(T, P) represents the cumulative geometric work associated with free-pore volume reduction along the adsorption trajectory, and F₀ corresponds to the initial average free-volume per molecule, the ratio ψ measures how much saturationequivalent molecular organization is required by the NaX-CO₂ system to structurally consume the initial pore-volume. Therefore, ψ provides a direct measure of the severity of molecular rearrangement associated with adsorption packing and can be interpreted as a universal scaling parameter of framework. Table 2 shows the measures of W(T, P) and ψ(T, P) for the four representative thermodynamic conditions of NaX zeolite.

Table 2
Intrinsic geometric work W(T, P) and packing parameter ψ(T, P)

T / K	P / bar	ηV / (mmol g-1)	F0 / (Å3 molecule-1)	W(T, P) / (Å3 mmol-1 (molecule g-1))	ψ(T, P) / (mmol g-1)
298	1	6.27	243.1	329.6	1.36
298	10	7.26	258.7	403.2	1.56
348	10	6.55	248.0	351.0	1.42
348	1	3.57	157.0	99.7	0.64

These results show that larger saturation capacities, η_V > 6.0 mmol g^-1 correspond to larger values of W(T, P) reflecting larger packing trajectories and greater cumulative molecular rearrangement within the confined adsorption pore. The smallest value (T = 348 K, P = 1 bar) corresponds to the shortest packing trajectory and the smallest cumulative reduction of the average free pore volume consistent with less significant molecular rearrangement, ψ ≅ 0.64 mmol g^-1. Thus, a value W ≅ 100 ų molecule^-1 (mmol g^-1) indicates that the adsorption pathway accumulates a geometric reduction equivalent 100 ų molecule^-1 of the average free pore volume per molecule within the physical loading interval. As seen in Equation 8, the packing parameter ψ emerges as dimensionless descriptor of the cumulative structural molecular reorganization required for adsorption packing within the α-cavity. Based on the four NaX-CO₂ thermodynamic conditions analyzed, the values of this relevant structural descriptor, shown in Table 2, provide a clear distinction between two extreme regimes of molecular pore reorganization. Therefore, for ψ << 1 mmol g^-1, the packing trajectory is characterized by limited molecular rearrangement, low packing efficiency, and relatively high values of Cut(T, P), indicating that only a small fraction of the initial average free pore volume per molecule is effectively mobilized for packing. For ψ ≅ 1 mmol g^-1, an intermediate regime is assumed, where moderate molecular rearrangement enables partial optimization of packing within the pore space. In contrast, for ψ >> 1 mmol g^-1, the system exhibits pronounced molecular mobility associated with structural rearrangement, high packing efficiency, and relative lower Cut(T, P), reflecting a more effective utilization of the available pore space, through successive structural reorganization. Within the present NaX system, this descriptor provides a robust geometric criterion for distinguishing between lowand highorganization regimes corresponding to different thermodynamic conditions (T, P). As reported previously, high ψ >> 1 mmol g^-1 values indicate strongly reorganized confined packing molecular states inside the α-cavity. Such highly organized configurations are qualitatively compatible with adsorption regimes where strong adsorbateframework interactions have also been associated in independent diffusion studies²⁵,²⁸ with more ordered and less dispersive CO₂ transport behavior. However, the present framework does not directly model diffusion trajectories or tortuosity effects. From dataset shown in Table 2, the states T = 298 K, P = 1 bar; T = 298 K, P = 10 bar; and T = 348 K, P = 10 bar belong to a high-rearrangement regime with 1.3 ≤ ψ ≤ 1.6, while the low state T = 348 K, P = 1 bar, belongs to a distinct low-rearrangement regime, ψ ≈ 0.64. From the structural function F(η_V; T, P), and by the mean theorem for integrals,³⁰ we have

(10) $$\mathrm{W}(\mathrm{T},\mathrm{P})={\int }_{{\eta }_{{\mathrm{V}}_0}}^{{\eta }_{\mathrm{Vax}}}\mathrm{F}({\eta }_{\mathrm{V}};\mathrm{T},\mathrm{P})\mathrm{d}{\eta }_{\mathrm{V}}=\mathrm{\Delta }{\eta }_{\mathrm{V}}\mathrm{F}({\eta }_{\mathrm{V}}^{\text{mean}};\mathrm{T},\mathrm{P})$$

where ∆η_V = η_Vmax - η_V0 and η_V^mean are in the physical domain ranging between η_V0 and η_Vmax.

Because the structural function F(η_V; T, P) decreases monotonically in this interval, the point is uniquely determined. Physically, the η_V^mean corresponds to the loading at which the instantaneous average free pore volume per molecule applied in η_V^mean, resulting in

(11) $${\mathrm{F}}_{\text{mean}}=\mathrm{F}({\eta }_{\mathrm{v}}^{\text{mean}};\mathrm{T},\mathrm{P})=\frac{\mathrm{W}(\mathrm{T},\mathrm{P})}{\mathrm{\Delta }{\eta }_{\mathrm{v}}},\text{in}{\text{Å}}^3{\text{molecule}}^{-1}.$$

Therefore, F_mean represents the average cumulative reduction in the mean free pore volume per reference molecule along the packing trajectory. From the structural form of Equation 11, F_mean reflects the average intensity of the cumulative reduction of the free pore volume per unit increase in loading. An interesting feature emerges for the three highest-capacity states considered, which correspond to terminal saturation η_Vmax > 6.0 mmol g^-1, the values η_V^mean(T, P) fall within a relatively narrow interval, between approximately 3.7-4.1 mmol g^-1, suggesting the existence of a characteristic structural regime of molecular rearrangement within adsorption states. In contrast, the low-capacity (T = 348 K, P = 1 bar), yields a significantly smaller value of η_V^mean(T, P) ≈ 2.71 mmol g^-1 reflecting the shorter packing trajectory and the reduced extent of molecular rearrangement inside adsorption pore. From Equations 4, 6 and 11, the point η_V^mean(T, P) can be effectively estimated by

(12) $${\eta }_{\mathrm{V}}^{\text{mean}}(\mathrm{T},\mathrm{P})={\{\frac{1}{{\eta }_{{\mathrm{V}}_{max}}(\mathrm{T},\mathrm{P})}+\frac{\ln \left(\frac{{\eta }_{{\mathrm{V}}_{max}}(\mathrm{T},\mathrm{P})}{{\eta }_{{\mathrm{V}}_0}}\right)-(1-\frac{{\eta }_{{\mathrm{V}}_0}}{{\eta }_{{\mathrm{V}}_{max}}(\mathrm{T},\mathrm{P})})}{\mathrm{\Delta }{\eta }_{\mathrm{V}}}\}}^{-1}$$

where ∆η_V = η_Vmax(T, P) - η_V0.

From F₀ and F_mean, we define the structural quantity

(13) $$\mathrm{\Delta }{\mathrm{F}}_{\text{cons}}(\mathrm{T},\mathrm{P})={\mathrm{F}}_0(\mathrm{T},\mathrm{P})-{\mathrm{F}}_{\text{mean}}(\mathrm{T},\mathrm{P}),\text{in}{\text{Å}}^3{\text{molecule}}^{-1},$$

representing the effective mean reduction of the average free pore volume per molecule along the packing pathway with respect to the initial state. From Equation 13, the dimensionless quantity Frac_mean(T, P) is given by

(14) $${\mathrm{Frac}}_{\text{mean}}(\mathrm{T},\mathrm{P})=\frac{\mathrm{\Delta }{\mathrm{F}}_{\text{cons}}(\mathrm{T},\mathrm{P})}{{\mathrm{F}}_0(\mathrm{T},\mathrm{P})}=1-\frac{{\mathrm{F}}_{\text{mean}}(\mathrm{T},\mathrm{P})}{{\mathrm{F}}_0(\mathrm{T},\mathrm{P})}$$

This variable should be interpreted as the mean fraction of the initial free pore volume per molecule consumed along the adsorption trajectory. These descriptors are referred to average descriptors. Table 3 shows the additional structural and average descriptors presented in this study.

Table 3
Structural and average free pore volume descriptors

T / K	P / bar	ηV / (mmol g-1)	F0 / (Å3 molecule-1)	Fmean / (Å3 molecule-1)	ηVmean / (Å3 molecule-1)	∆Fcons / (Å3 molecule-1)	Fracmean
298	1	6.27	243.1	77.2	3.74	165.9	0.688
298	10	7.26	258.7	76.7	4.08	182.0	0.704
348	10	6.55	248.0	77.1	3.84	170.8	0.689
348	1	3.57	157.0	63.5	2.71	93.5	0.596

Some important trends emerge from Table 3. For the three high-saturation regimes, F₀ remains in the narrow range from 243 to 259 ų molecule^-1 indicating that the initial volumetric space available for molecular rearrangement in the adsorption pore is remarkable similar under (T, P) conditions. F_mean and η_V^mean provide values that are nearly constant for these highest-capacity cases and only slightly larger than those obtained for the lowest saturation regime, indicating low sensitivity of these average descriptors. This fact is due to the averaging process, which reduces the sensitivity to structural variations. The pair (F₀, ∆F_cons) provides a particularly robust and direct measure of structural molecular rearrangement severity, clearly distinguishing between lowand high-packing regimes of CO₂ adsorption in NaX α-cavity. Moreover, ψ exhibits significant larger variations, as shown in Table 2, and can effectively distinguish the different packing regimes on its own. Differently from F_mean and η_mean average descriptors, F₀, ∆F_cons, η_v^mean, and ψ are structural descriptors, that is, are defined as quantities that preserve the geometric information encoded or embedded in the structural function, either in a quantitative sense, such as F₀, ∆F_cons, η_v^mean or in qualitative sense as ψ, which captures the cumulative normalized intensity of molecular rearrangement along the packing trajectory. In contrast, average descriptors collapse this information over the physical domain, reducing their sensitivity to distinguish clearly different molecular rearrangement regimes. Figure 4 consolidates the central interpretation of the proposed framework by combining molecularlevel representation and geometric analysis of the adsorption process. Using two extremes conditions (from η_vmax = 3.57 mmol g^-1 to η_vmax = 7.26 mmol g^-1), the results highlight how the same structural function F(η_V; T, P) can be interpreted not only in terms of the extent of adsorption trajectory as shown in Figure 2, but also as a quantitative measure of the geometric severity of molecular rearrangement under confinement. This dual interpretation provides a unified view linking molecular organization to the global packing trajectory, and establishes the framework as a purely analytical tool for representing the geometric evolution of CO₂ adsorption in NaX zeolites.

Figure 4
Geometric interpretation of CO₂ adsorption in NaX (FAU-type) based on structural function F(h_V; T, P). (³¹: sodium, : oxygen, : carbon. The colors follow the Corey-Pauling-Koltun (CPK) standard)

The surface schematically illustrates the molecular rearrangement within the confined adsorption site inside α-cavity for the two extremes conditions, where CO₂ molecules are progressively attracted to Na⁺ adsorption sites and reorganize to accommodate increasing loading determined by temperature and pressure. Figure 4 presents the corresponding structural trajectories γ(η_V; T, P) for these two extremes cases, interpreted as the average free pore volume available per molecule as a function of loading. While these extreme trajectories coincide with those introduced in Figure 2, they are here reinterpreted through a visual encoding based on thickness of lines and color intensity. The initially thicker and more intense segments from η_V ~ 1 to 2.5 mmol g^-1, indicate a higher degree of molecular rearrangement required within the pore, whereas the progressive thinning and fading of trajectories toward (F = 0) reflect the gradual reduction of the severity of rearrangement as free pore volume is depleted. Therefore, Figure 2 describes how far the adsorption trajectories extend, whereas Figure 4 reveals how severe the molecular CO₂ rearrangements are along these extreme pathways, together establishing a physical-geometric dualism of CO₂ adsorption within the α-cavity NaX (FAU-type) zeolite.

In summary, the structural function F(η_V; T, P) is not an empirical descriptor. Within the present framework, it represents the average free pore volume available per molecule within α-cavity, varying continuously along saturation and explicitly the loading-dependent nature of molecular occupation. Within this perspective, the packing trajectory is no longer a merely loading pathway but is instead described through a well-defined set of structural descriptors derived from the structural function. Namely, F₀ represents the initial free geometric volume available per reference adsorption molecule; ∆F_cons quantifies the average free volume consumed along the pathway; η_V^mean locates the point in the physical domain associated with the global descriptor F_mean; and ψ = WF₀^-1 corresponds to the cumulative structural consumption saturation-scaled of the mobilized initial free pore volume for adsorption of CO₂ molecule. These descriptors provide explicit and comparable measures of the geometric evolution of the system. The trajectory γ(η_V; T, P) can thus be reinterpreted as a measure of the geometric severity of molecular rearrangement imposed on confined adsorbed phase from the initial state to the final packing condition, with such severity being robustly captured by these individual or collective descriptors. Within the confined, loading-dependent dynamic free-volume framework developed in this work, CO₂ adsorption may therefore be interpreted as continuous geometric pathway of progressive exhaustion of the average free pore volume within α-cavity of NaX (FAU-type) zeolite. This establishes a direct link between the initial pore volume available for the NaX-CO₂ system to reach terminal saturation and the efficiency of molecular rearrangements within the confined adsorption environment.

CONCLUSIONS

A loading-dependent average free-pore-volume framework for CO₂ adsorption in highly crystalline NaX (FAU-type) zeolite was established based on a well-defined structural function F(η_V; T, P), grounded in the classical free-volume concept. The formulation provides a purely analytical (mathematical) description of the geometric evolution of molecular packing within the confined α-cavity. Relevant structural descriptors derived from F(η_V; T, P), including F₀, ∆F_cons, η_V^mean and ψ, preserve the geometric information associated with the progressive depletion of the average freepore volume per molecule, enabling the identification of distinct molecular rearrangement regimes under confinement. Application of the framework to the extreme conditions η_Vmax = 3.57 mmol g^-1 (T = 348 K, P = 1 bar) and η_Vmax = 7.26 mmol g^-1 (T = 298 K, P = 10 bar) showed that the higher-loading regime is associated with larger F₀ ~ 258 ų molecule^-1 and ψ ~ 1.56 mmol g^-1, consistent with a more extended and cumulative mobilized geometric packing trajectory within the α-cavity. In contrast, at low saturation, F₀ ~ 157 ų molecule^-1 and ψ ~ 0.64 mmol g^-1, these descriptors reflect a geometric relaxed packing state. In this sense, the descriptors quantitatively capture individually or collectively the severity of CO₂ molecular rearrangement within adsorption pore required for the system reaches the equilibrium. The geometric packing descriptor ψ, whose low and high values indicate, respectively, weakly organized and highly efficient packing states suggests that these regimes are consistent, but not necessarily equivalent with variations in CO₂ host-guest interactions observed in diffusion studies, where stronger interactions with the pore walls are associated with more structured and less tortuous pathways, whereas weaker interactions lead to more delocalized diffusion behavior. The framework establishes a direct bridge between macroscopic adsorption data and nanoscale structural information related to the CO₂ geometric packing trajectory within the adsorption pore. Because it is based on general geometric relations, the approach may be naturally extended to other highly crystalline FAU-type zeolites because the present framework operates at the level of experimentally observed macroscopic data and potentially to a broader class of microporous adsorption systems.

ACKNOWLEDGMENTS

This study was financed in part by the CAPES (finance code 001). The authors also thank FAPERJ for financial support (grants E-26/211.214/2021 and E-26/201.733/2025).

DATA AVAILABILITY STATEMENT

All data generated or analyzed during this study are included in this published paper.

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Edited by

• Executive Editor handled this article:
  Gustavo F. S. Andrade

Publication Dates

• Publication in this collection
  27 July 2026
• Date of issue
  2026

History

• Received
  15 Apr 2026
• Accepted
  28 May 2026
• Published
  16 June 2026