Adsorption mechanism in crystalline micropores: multimodal fluctuations, phase coexistence and phase transformations in nanoconfinement

Adsorption mechanism in crystalline micropores: multimodal fluctuations, phase coexistence and phase transformations in nanoconfinement | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Adsorption mechanism in crystalline micropores: multimodal fluctuations, phase coexistence and phase transformations in nanoconfinement Bogdan Kuchta, Małgorzata Stankiewicz, Anthony Dorhauer, Carlos Wexler, and 5 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7148852/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Understanding molecular adsorption in microporous materials is key to advancing gas separation, storage, and catalysis. Here, we study CO₂ and CH₄ adsorption in crystalline metal-organic frameworks (IRMOF-1, 8, 10, and 14), emphasizing the emergence of metastable states. Molecular simulations reveal that adsorption is governed by a fine balance between fluid–fluid and fluid–framework interactions, leading to transitions between low- and high-density pore-filling states. These metastable features are highly sensitive to pore geometry and thermodynamic conditions, especially near the adsorbate’s triple point. In contrast, water adsorption displays more complex behavior: strong hydrogen bonding induces stable clusters, multiple free energy minima, and exceptionally slow equilibration. These features often escape conventional simulations. Our results provide fundamental insight into adsorption mechanisms and underscore the importance of metastability in accurately modeling and designing advanced nanoporous materials for practical applications. Physical sciences/Materials science/Theory and computation/Atomistic models Physical sciences/Nanoscience and technology/Nanoscale materials/Structural properties metastability adsorption mechanism nano-pores phase coexistence MOFs water methane carbon dioxide Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Introduction Adsorption of gases on the surfaces is often perceived as a conceptually simple phenomenon. Its fundamental principles were formulated by Irving Langmuir in 1918, who wrote: “ when gas molecules impinge against any solid or liquid surface, they do not in general rebound elastically, but condense on the surface, being held by the field of force of the surface atoms. These molecules may subsequently evaporate from the surface. The length of time that elapses between the condensation of a molecule and its subsequent evaporation depends on the intensity of the surface forces. Adsorption is the direct result of this time lag ” [ 1 ]. This description intuitively captures the dynamic nature of the adsorbed phase, whose composition continuously fluctuates and is consistent with both thermodynamic and kinetic stability. While this definition remains valid across a wide range of environments, adsorption under nanoconfinement reveals additional, intriguing behaviors. One of them is the existence of metastability. The analysis of sorption isotherms remains a fundamental and widely applied method for characterizing adsorption/desorption phenomena in a broad range of materials. This approach relies on the analysis of the isotherms’ shapes and their classification according to established schemes, such as those proposed by IUPAC [ 2 , 3 ]. Although the IUPAC classification provides a valuable phenomenological framework, it offers only limited insight into the microscopic mechanisms driving sorption processes. These mechanisms are often complex and system-specific, emerging from the interplay of multiple factors, including structural heterogeneity of the adsorbent surface, a broad distribution of pore sizes and geometries, and spatial variations in the adsorption energy landscape. This complexity is particularly pronounced in crystalline microporous materials. This aspect has also been emphasized in the recent paper discussing adsorption hysteresis in nanoporous MOFs (Metal − Organic Frameworks) [ 4 ], revealing two different mechanisms of rapid changes in the adsorption uptake: one resembling capillary condensation (gas-to-liquid transition) and the other more akin to liquid-to-solid transition. According to IUPAC classification, microporous systems – defined by pore sizes below 2 nm - are associated with Type I adsorption isotherms, characterized by a steep, near-vertical uptake at very low relative pressures, indicative of rapid pore filling. However, both experimentally measured and computationally derived adsorption isotherms frequently show significant deviations from this idealized Type I behavior in such systems [ 4 – 10 ]. These deviations are particularly pronounced in materials with crystallographic symmetry and/or well-ordered pore-wall structures. They primarily result from heterogeneities in the adsorption energy landscape and from the complex interplay between different interaction components, modulated by the thermal energy of the surrounding environment. In particular, strong fluid-fluid interactions can compete with fluid-framework interactions, potentially giving rise to metastable states [ 8 ] and specific pore filling mechanisms, including structural transitions [ 9 ]. Metastability plays a critical role in molecular simulations. Systems that become trapped within local - rather than global-thermodynamic minima are described as kinetically stable or persistent (long-lived). Although such metastable states may be unobservable in a macroscopic experimental viewpoint, where observation timescales typically range from minutes to hours, they are often essential for interpreting microscopic behavior. In molecular simulations, which are generally limited to nanosecond timescales, metastability can significantly hinder reaching thermodynamic equilibrium, increase statistical uncertainty, and substantially increase the computational cost required to sample representative equilibrium ensembles [ 11 ]. In this study, we investigate the emergence of metastable states during gas adsorption in microporous materials. We show that this phenomenon is common across ordered or crystalline microporous systems, including metal-organic frameworks (MOFs) and zeolites. The formation of metastable states is attributed to a confluence of factors: the spatial distribution and crystallographic symmetry of adsorption sites, the restricted pore dimensions (typically 1–2 nm), and the competitive interplay between gas–gas and gas–framework interactions. Together, these factors create a complex free energy landscape characterized by multiple local minima, leading to non-trivial adsorption pathways and mechanisms [ 4 , 8 ]. The formation of metastable states during gas adsorption in microporous systems is analyzed using methane and carbon dioxide adsorption in selected model IRMOF structures. Our results demonstrate how the structural features and crystallographic symmetry of pore walls influence the pore-filling process, frequently leading to metastable configurations that can temporarily trap the system. The lifetimes of these states are found to be temperature-dependent, reflecting a delicate balance between thermodynamic driving forces and kinetic barriers. These findings underscore the intricate and dynamic nature of adsorption in nanoporous crystalline materials and provide new insights into the mechanisms governing molecular confinement and transport in such systems. Metastability: fundamental element of multimodal adsorption mechanism In mesoporous materials (pore dimensions > 2 nm), a well-documented example of a metastability-driven phenomenon is the hysteresis observed during capillary condensation, consisting in rapid pore filling. In contrast, microporous materials exhibit different filling mechanisms [ 8 – 10 ]. While capillary condensation is primarily governed by fluid-fluid interaction and often displays hysteresis, micropore filling is controlled by a more nuanced balance between fluid-fluid and fluid-framework interactions and is typically reversible. Moreover, micropore filling can induce structural transformations in the adsorbed phase [ 4 , 9 ], potentially leading to the formation of new stable phases or metastable intermediate states. This stands in contrast to capillary condensation, where pores filling proceeds rapidly without altering the structure of the already adsorbed fluid layers. Previous experimental and computational studies have reported atypical features in the adsorption behavior of methane and carbon dioxide in microporous MOF-5 (IRMOF-1) framework [ 8 , 9 , 12 , 13 ]. Similarly, water adsorption isotherms in microporous materials have been shown to adopt a variety of shapes [ 12 , 14 – 19 ] and are well-known to involve long living metastable states. In many crystalline microporous systems, adsorption isotherms deviate from the classical IUPAC Type I profile, characterized by rapid pore filling at very low pressures. Specifically, at low temperatures and pressures, the isotherms often exhibit an initial monolayer-like adsorption, followed by an S-shaped profile, and ultimately a sharp, step-like uptake [ 13 ]. In our previous work [ 9 ], this abrupt pore filling was interpreted as the nano-scale coexistence of low- and high-density adsorbate states, driven by large-amplitude, correlated fluctuations between these configurations. Such a mechanism requires the presence of at least one metastable state [ 8 ], with transitions between phases governed by a relatively low (and depending on temperature) free energy barrier separating the macroscopic low- and high-density phases (Fig. 1). As a result, adsorption proceeds not through gradual, incremental accumulation of adsorbate molecules, but rather via a collective, synchronized, and cooperative structural transformation of the adsorbed phase. At higher temperatures, the free energy barrier between states of different density diminishes, leading to more frequent fluctuations and a gradual transition to a continuous adsorption profile. Metastability plays a critical role in shaping the dynamics and kinetics of the adsorption process. In our previous paper [ 8 ], we demonstrated how metastability between low-density and high-density methane phase structures within IRMOF-1 pores governs the adsorption mechanism of methane. Building on these results, the present study explores how metastability evolves as a function of pore size and temperature in a series of IRMOF-type frameworks (IRMOF-X, where X = 8, 10, 14), for both methane (CH₄) and carbon dioxide (CO₂). Results CO 2 adsorption in IRMOF-X materials (X = 1, 8, 10, 14) We begin our analysis of adsorption mechanisms by examining CO 2 adsorption in IRMOF-type structures. Figure 2 presents the experimental isotherms of CO 2 and CH 4 adsorption in IRMOF-1. These results serve as a reference for understanding the distinct adsorption behaviors observed in this class of materials and guide the interpretation of corresponding simulation results. At temperatures near the bulk triple point of CO₂ (216.58 K, NIST database), the adsorption isotherms display a fully reversible, step-like character (Fig. 2, left panel). A similar behavior is observed for CH 4 adsorption at temperatures approaching its bulk triple point (90.67 K), as shown in Fig. 2 (right panel) [ 8 ]. In both cases, increasing the temperature leads to a progressive transformation of the sharp, quasi-discrete transition into a smoother, S-shaped adsorption profile, indicating a more continuous adsorption process. The comparison of CO₂ and CH 4 adsorption isotherms suggests a shared underlying mechanism governing transitions between low- and high-density nanophases. This resemblance persists despite differences in molecular symmetry of guest molecules, adsorption conditions (temperature and pressure), and the absence of electrostatic interactions in CH 4 adsorption. These findings point to a degree of universality in the adsorption mechanism that transcends specific structural and chemical properties of the adsorbate. This observation raises a fundamental question: what physicochemical parameters dictate the onset and nature of transitions between nanophases during pore filling in microporous materials? TM-GCMC calculations of the free energy landscapes (Fig. 3 , right column) offer critical insight into this question [ 8 ]. The calculated free energy reveals distinct minima over a specific pressure range, corresponding to low- and high-density adsorption states, separated by a finite energy barrier. When this barrier is relatively low (here, below 10 kJ/mol for CO₂), thermal fluctuations enable spontaneous transitions between the two states, at pressures corresponding to rapid pore filling (Fig. 3 , middle columns). Within the pressure range defining the step on the adsorption isotherm, the free energy consistently shows at least one metastable state. However, this state remains macroscopically undetectable, as the experimentally measured uptake represents a time-average value, weighted by the residence time in each of the coexisting states [ 8 ]. As the temperature increases, the energy barrier progressively diminishes, and the two minima ultimately coalesce into a single, broader minimum. This marks the disappearance of distinct nanophases, and the onset of a continuous adsorption regime. We define the corresponding temperature as the pore critical temperature T pc . As an initial step toward generalization, we extended the analysis to CO₂ adsorption in larger-pore IRMOF structures: IRMOF-8, IRMOF-10, and IRMOF-14. Structural parameters for all investigated frameworks are provided in the Supplementary Information (Table S1 in the Supplementary Information). CO 2 adsorption isotherms for all studied IRMOF-X materials at T = 220 K and 230 K are presented in Fig. 4. Qualitatively, the isotherms exhibit similar features across the series. However, the pressure range associated with the sharp, step-like adsorption in IRMOF-X (X > 1) is systematically shifted to higher values relative to IRMOF-1. This shift primarily reflects the variation of the pore volume resulting from the incorporation of longer organic linkers. Interestingly, the shift between IRMOF-8 and IRMOF-14 isotherms is relatively small, despite the larger pore volume of IRMOF-14 (see Fig. 2S in the Supplementary Information). This behavior can be attributed to a compensating effect: the longer and more massive linker in IRMOF-14 introduces additional strong adsorption sites, which facilitate adsorption at lower pressures. For the same reason, the pressure shift between IRMOF-8 and IRMOF-10 is larger than that between IRMOF-8 and IRMOF-14, even though the unit cell volumes of IRMOF-10 and IRMOF-14 are comparable. In the case of IRMOF-10, the linker structure results in weaker adsorption forces, allowing the low-density phase to persist over a broader pressure range. Figure 3 S- 5 S in the Supplementary Information show the results of free energies and the corresponding bimodal fluctuations of the CO 2 uptake in IRMOF-8, IRMOF-10 and IRMOF-14 Bimodal behavior: the limit of reversible pore filling. The calculated free energy profiles (Fig. 3 ) indicate that the bimodal behavior observed at pressures corresponding to step-like adsorption can be captured in Monte Carlo simulations when the energy barrier E B separating the low- and high-density states remains sufficiently low (typically E B < 10 kJ/mol for CO 2 ). In this regime, the transition probability is high enough to allow fluctuations between the two states within the simulation timescale. To quantify this effect, we explicitly calculated the probabilities of crossing the barrier, assuming the transition probability between the low-density state (with energy E L ) and the high-density state (with energy E H ) as a two-step process. The total transition probability was expressed as the product of two subsequent transition probabilities, with the intermediate state defined by the barrier height E B : $$\:p\left({E}_{L}\to\:{E}_{H}\right)=p\left({E}_{L}\to\:{E}_{B}\:\right)p\left({E}_{B}\to\:{E}_{H}\right)$$ , $$\:p\left({E}_{H}\to\:{E}_{L}\right)=p\left({E}_{H}\to\:{E}_{B}\:\right)p\left({E}_{B}\to\:{E}_{L}\right)$$ , proportional to the Boltzmann factors when the final state has higher energy, and equal to 1 when the final state has lower energy: $$\:p\left({E}_{L}\to\:{E}_{B}\:\right)=\text{e}\text{x}\text{p}(-\frac{{E}_{BL}}{{k}_{B}T})\le\:1\:\text{w}\text{h}\text{e}\text{r}\text{e}\:{\Delta\:}EBL\:=\:{E}_{B}-{E}_{L}$$ $$\:p\left({E}_{H}\to\:{E}_{B}\:\right)=\text{exp}\left(-\frac{{E}_{BH}}{{k}_{B}T}\right)\le\:1\:\:\:\text{w}\text{h}\text{e}\text{r}\text{e}\:{\Delta\:}EBH\:={\:E}_{B}-{E}_{H}$$ $$\:p\left({E}_{B}\to\:{E}_{H}\right)=\:p\left({E}_{B}\to\:{E}_{L}\right)=1$$ These transition probabilities satisfy the detailed balance (microscopic reversibility condition), which leads to the following relation: $$\:p\left({E}_{L}\right)p\left({E}_{L}\to\:{E}_{H}\right)=p\left({E}_{H}\right)p\left({E}_{H}\to\:{E}_{L}\right)$$ Therefore, we can estimate the transition probabilities \(\:p\left({E}_{L}\to\:{E}_{H}\right)\:\) and \(\:p\left({E}_{H}\to\:{E}_{L}\right)\:\) : $$\:p\left({E}_{L}\to\:{E}_{H}\right)=p\left({E}_{L}\to\:{E}_{B}\:\right)p\left({E}_{B}\to\:{E}_{H}\right)\:=\:p\left({E}_{L}\to\:{E}_{B}\:\right)$$ $$\:p\left({E}_{H}\to\:{E}_{L}\right)=p\left({E}_{H}\to\:{E}_{B}\:\right)p\left({E}_{B}\to\:{E}_{L}\right)=p\left({E}_{H}\to\:{E}_{B}\:\right)$$ Figure 5 shows the estimated transition probabilities for IRMOF-X (X = 1, 8, 10, 14) at equilibrium pressures, i.e. when the free energies of the low- and high-density states are equal E L = E H , plotted as a function of temperature. As the transition probability approaches p = 0.6, the distinction between the two well defined nanophases disappears. We define the corresponding temperature at which this occurs as the pore critical temperature T pc . The pore critical temperature T pc defines the upper temperature limit at which bimodal adsorption behavior can be observed. This condition corresponds to the point where the free energy barrier separating low- and high-density states is comparable to, or smaller than, the thermal energy k B T . Our results indicate that T pc is governed primarily by two factors: pore size and linker topology. We observe an increase in the T cp from IRMOF-1 to both IRMOF-8 and IRMOF-10, consistent with their larger pore volume. However, the T cp of IRMOF-14 is approximately equal to that of IRMOF-8, despite IRMOF-14 requiring higher pressure for pore filling (Table 1 ). This suggests that, beyond pore size, the linker structure also plays a critical role in stabilizing specific adsorbed states. For CO 2 adsorption, all studied IRMOFs exhibit qualitatively similar free energy profiles, each characterized by two metastable states. In contrast, CH 4 adsorption reveals more complex behavior in some frameworks, with free energies featuring multiple distinct minima indicative of additional metastable configurations. Table 1 Triple point ( T tr ) and critical ( T c ) temperatures for bulk (CH 4 and CO 2 ), and pore critical ( T pc ) temperatures for CO 2 and CH 4 adsorbed in IRMOF-X (X = 1, 8, 10, 14). T cp (IRMOF-X) is the lowest temperature where continuous transformation from the low density to high density adsorption state is observed (estimated accuracy of ± 4 K). T tr (bulk) T cp (IRMOF-1) T cp (IRMOF-8) T cp (IRMOF-10) T cp (IRMOF-14) T c (bulk) pore volume --- 9.38 nm 3 27.25 nm 3 40.29 nm 3 40.64 nm 3 --- CH 4 90.67 K ∼ 110 K ∼ 125 K ∼ 120 K ∼ 125 K 190.6 K CO 2 216.58 K ∼ 230 K ∼ 240 K ∼ 255 K ∼ 240 K 304.13 K CH 4 adsorption in IRMOF-X materials (X = 1, 8, 10, 14) The mechanism of CH 4 adsorption in IRMOF-1 has been recently investigated and reported in detail [ 8 ]. Its free energy profile is qualitatively like that discussed above for CO 2, exhibiting two well-defined metastable states. However, in larger pore frameworks, specifically IRMOF-8, IRMOF-10, and IRMOF-14, the adsorption behavior becomes more complex, with free energy landscape changes significantly displaying multiple minima (Fig. 1). This suggests that CH 4 adsorption in these systems involves a series of structural transformations as a function of adsorption uptake. As the temperature increases, the adsorption landscape progressively smooths: the discrete minima begin to merge, and the energy barriers separating them decrease, ultimately leading to a more continuous adsorption process. Figure 6 illustrates the key features of CH4 adsorption in IRMOF-8. At T = 85 K and 92 K, the free energy profile shows a multi-minima structure within a very narrow pressure range, revealing the presence of intermediate configurations at the uptake of ∼ 150 molecules per unit cell. These intermediate states are not thermodynamically stable and can only be inferred from the evolution of uptake fluctuations observed during the equilibration phase of simulations. Around pressures 53–54 Pa, the system becomes temporarily trapped in a low-density configuration (analogous to monolayer adsorption) before reaching a stable, high-density state. The pressure range associated with these metastable states is extremely narrow and is not resolved in the corresponding adsorption isotherms. As temperature increases, the isotherms’ shape evolves. Metastable low-density states disappear, and uptake increases continuously with pressure. At higher pressure, where the pores are partially filled pores and the accessible volume is reduced, only equilibrium fluctuations are observed (Fig. 6 , bottom row: P = 1365 Pa, T = 110 K). In IRMOF-14, CH₄ uptake increases smoothly at low pressures until approximately half of the pore volume is filled (Fig. 7 ). Within the temperature range of 85–125 K, two high-density metastable adsorption states are observed, accompanied by large-amplitude uptake fluctuations in the upper part of the adsorption step. The adsorption mechanism resembles that of capillary condensation, where adsorption of initial layers is followed by more abrupt pore filling. However, the nature of the filling process varies with temperature. At lower temperatures (e.g., T = 115), pore filling proceeds via more scattered events, while at higher temperatures (e.g., T = 125 K), the process becomes more continuous (Fig. 7 ). Influence of the size of Monte Carlo simulation box on barriers between metastable states Two primary input parameters govern the outcome of the adsorption simulation. The first is the intermolecular potential model, which reflects the underlying physical interactions. The critical role of the model is well established and widely recognized (Fig. 1S). The second parameter concerns the simulation methodology: the size of the simulation box. To evaluate the impact of box dimensions and geometry on the system’s tendency to become trapped in metastable, yet physically unrealistic states, we performed CH₄ adsorption simulations in IRMOF-8 using Monte Carlo boxes of varying sizes. Three configurations were considered, corresponding to the following three-dimensional ( x , y , z ) unit cell arrangements: (1) 1×1×1, (2) 2×2×1, and (3) 3×3×1. These simulations were performed at a temperature of 110 K and a pressure of 1385 Pa. The free energy profile for CH 4 in the smallest simulation box (1×1×1) is shown in Fig. 7 (right column), while those for larger boxes (2×2×1, and 3×3×1) are presented in Fig. 8 . In the smallest box, the equilibrium free energy landscape exhibits a barrier of approximately 3 kJ/mol and the system displays fluctuations between a medium density state ( N ∼ 150) and a high-density state ( N ∼ 300) during the simulation. As the box size increases, the free energy barrier also rises, reaching approximately 15 kJ/mol for the biggest box. According to the relation p ≈ exp(- \(\:{E}_{B}\) / k B T ), the probability of observing fluctuations over such a barrier at T = 110K is of the order of p ≈ 10 − 8 . This vanishingly small probability effectively precludes the observation of fluctuations within standard MC simulations comprising ∼ 10 6 steps. For comparison, in molecular dynamics simulations where the simulation timestep is usually of the order of 0.5 fs, 10 6 steps correspond to simulated times shorter than 1 ns. This limitation is purely technical and could be mitigated in longer simulations or with more powerful computer resources. In contrast, real experiments are not constrained by such restrictions and can probe systems over macroscopic timescales (minutes to hours), which are many orders of magnitude longer than those accessible in molecular simulations (nanoseconds to microseconds). Interestingly, increasing the size of the simulation box alone does not lead to the emergence of additional intermediate metastable states (i.e., additional local minima) in the free energy landscape of CH 4 adsorption. This observation supports the interpretation of the transformations as a discontinuous, first-order process, in which intermediate macrostates represent weighted (statistical) averages over the two coexisting metastable microscopic states, corresponding to either fully filled or fully empty pores. A similar conclusion was reached in our previous study on CH 4 adsorption in IRMOFs [ 8 ]. These results reinforce the hypothesis that pore filling proceeds via a first-order transformation, independent of the Monte Carlo simulation box size. At the same time, they show that the metastability observed in this system originates from a mechanism fundamentally different than that reported for water adsorption [ 14 – 19 ], where the metastable states originate from the formation and rearrangement of molecular clusters. In the case of water, the number of metastable states may depend on the size of the simulated sample (MC box), and their presence is associated with extremely slow equilibration dynamics [ 16 , 18 , 19 ]. Origin of the metastability A fundamental question in the study of adsorption is the origin of metastability. Simulations of CO₂ adsorption reveal that at low temperatures, certain macrostates are effectively inaccessible—neither thermodynamically stable nor kinetically reachable. Rather than stabilizing in a single well-defined macrostate (which would correspond to lower entropy), the system establishes a dynamic equilibrium between two metastable states, each statistically populated over time. These states are separated by a finite free energy barrier, making the probability of transitions between the states dependent on temperature. Pore symmetry plays a crucial role in governing adsorption behavior. Adsorbate configurations that preserve the symmetry of the framework are generally lower in energy and thus thermodynamically more favored. In contrast, intermediate macrostates that break this symmetry tend to be both higher in energy and lower in entropy, which renders them thermodynamically inaccessible. As a result, the system avoids these asymmetric configurations, and metastability emerges because of the combined enthalpic and entropic penalties associated with symmetry-breaking intermediate states. At higher temperatures, the behavior of the system changes. Increased thermal energy introduces greater disorder into the adsorbed phase, enhancing the entropy and allowing the system to explore a broader region of configuration space. As a result, the free energy barriers separating macrostates become easier to overcome (or disappear), and the adsorption process becomes more continuous in nature. A complementary perspective emerges from the analysis of CH₄ adsorption, where additional metastable states are observed, particularly at low adsorbate densities. In this case, the spherical shape of the methane molecule and the symmetric distribution of adsorption sites within the pore play a critical role. In our previous work [ 9 ], we identified a distinct metastable configuration that disappeared at elevated temperatures. This structure was stabilized by the geometric symmetry of the pore walls and corresponded to a low-density gas molecules arrangement aligned with shallow adsorption sites (Fig. 9 ). At very low loading, methane molecules preferentially populate these shallow, yet energetically favorable, sites. As the adsorbate density increased, the distribution of CH₄ molecules became more localized, giving rise to a configuration that was also symmetric, but distinct from the low-density state. This newly formed structure—though metastable—was stabilized by the interplay of molecule–pore interactions and packing constraints. It occupied a region of the free energy landscape near the barrier separating macrostates, illustrating how geometric symmetry and molecular shape contribute to the emergence and persistence of metastable states in adsorption processes. In summary, at low temperatures and within highly symmetric pores, the emergence of two metastable states corresponding to low- and high-density states appears to be a natural consequence of the system’s energetic and geometric constraints. At low adsorbate densities, the ordered pore environment energetically favors only symmetric molecular arrangements. As the density increases, the balance between the gas-gas and gas-pore interactions begins to influence the stability and structure of the final configuration. With increasing temperature, thermal energy introduces disorder and raises the system’s entropy, enabling stabilization of a broader ensemble of macrostates. Additional metastable states may arise when specific distributions of adsorption sites provide sufficient energetic stabilization (see also Fig. 6 S). Finally, we compare the metastability observed in CH₄ and CO₂ adsorption with water adsorption in micropores of Al(OH)(1,4-ndc) MOF [ 20 ], where strong water-water interactions play a central role. These interactions promote the formation of multiple free energy minima [ 14 – 16 ], corresponding to clusters of water molecules that assemble within the pores into energetically stable networks resembling the structure of bulk liquid water [ 16 ]. These macrostates represent metastable configurations of the system (Fig. 10 ). The number of free energy minima increases with the size of the Monte Carlo simulation box, suggesting that in real (macroscopic) systems, the transition between the low- and high-density states of adsorbed water may be continuous. In the thermodynamic limit, where system size approaches infinity, the number of metastable minima likewise becomes infinite [ 14 ]. In this regime, water adsorption proceeds via sequential filling of pores, with each occupancy level corresponding to a distinct metastable state. This behavior is consistent with the previously reported simulation studies [ 14 – 16 ]. The nature of metastability in water adsorption differs from that observed for CO 2 and CH 4 in IRMOF structures. Water clusters are stabilized by strong intermolecular hydrogen bonding, resulting in relatively deep energy minima. This makes the addition or removal of individual molecules energetically unfavorable, leading to discrete cooperative adsorption events. Such a collective behavior is absent in the gas-like adsorption of CO₂ and CH₄. Consequently, water fills the pores sequentially, each step corresponding to the formation of a stable cluster. The number of resulting metastable states scales with the system size. In contrast, for CO 2 and CH 4 in IRMOFs, intermediate metastable states emerge only under specific symmetry conditions and are stabilized by the delicate competition between fluid–fluid and fluid–framework interactions. These states are typically observed at low temperatures and in highly ordered pore environments. In the absence of such symmetry constraints or when thermal energy introduces a structural disorder, these intermediate, partially filled configuration states become thermodynamically unstable, and no metastable states persist. The systems then transition smoothly between low- and high-density phases, resulting in continuous adsorption behavior. In conclusion, water represents a unique case among adsorbates due to its pronounced hydrogen bonding, which results in strong, directional electrostatic interactions. These interactions substantially slow down adsorption kinetics and promote the formation of stable molecular clusters within the pores. In narrow or hydrophilic pores, water often undergoes phase-like transitions (such as pore filling or emptying) that result in metastable states with long residence times, posing significant challenges for numerical simulations. For the same reasons, water adsorption is often cooperative: the initial adsorption of a few water molecules reduces the energetic barriers for subsequent uptake. Furthermore, confinement imposes substantial constraints on both rotational and translational degrees of freedom, leading to notable entropy losses. These constraints, in combination with directional bonding, contribute to the complexity and distinct behavior of water under nanoconfinement. Metastability and the mechanism of water adsorption need more fundamental studies. Table 2 below provides a comparative summary of the main features associated with the adsorption and metastability of water and CO 2 /CH 4 in crystalline microporous materials. Table 2 Key differences between the mechanisms of adsorption: water versus CO₂ / CH₄ Feature Water CO₂ / CH₄ Intermolecular interactions strong, directional hydrogen bonds, which drive cluster formation inside pores. weak, non-directional van der Waals or quadrupolar interactions. Adsorption behavior cooperative adsorption: once a few molecules adsorb, it becomes energetically favorable for more to follow rapidly. gradual, smoother isotherms with sharp transitions only at temperature close to the triple point temperature Free energy surface rugged, with deep multi-minima smooth and continuous, with a limited number of minima GCMC acceptance rate often very low: inserting new molecules into a dense, hydrogen-bonded cluster has a very low acceptance rate. moderate to high: insertion energies fluctuate mildly, and moves are more often accepted. Metastability pronounced (e.g., pore filling): it leads to metastable states with long residence times. only at very low temperature (below the triple point) Sampling bottlenecks severe only at very low temperature (below the triple point) Discussion Metastability plays a fundamental role in molecular simulations of adsorption, particularly in systems characterized by complex energy landscapes or undergoing rare events such as phase transitions. In this study, we examined the adsorption mechanisms of CO₂, CH₄, and H₂O in crystalline microporous materials, with a focus on how metastable states influence dynamic equilibrium and sorption behavior. We show that the presence of free energy barriers, separating the metastable states, can significantly hinder transitions within simulation-accessible timescales. These barriers—and the corresponding rare fluctuations—are key to understanding adsorption in highly structured porous materials, where conventional interpretations based on equilibrium thermodynamics may be insufficient. While many microporous materials display IUPAC Type I isotherms, this classification oversimplifies the behavior observed in highly ordered frameworks [ 2 ]. Our simulations reveal that, particularly near the adsorbate’s triple point, adsorption proceeds through a low-density phase followed by a sharp pore-filling transition. As the temperature increases, this behavior shifts toward smoother, Type V-like isotherms. The geometry of micropores, in combination with framework symmetry and fluid–fluid interactions, gives rise to complex dynamics and unexpected metastable behavior. In the case of CH₄ and CO₂ adsorption, we observed pronounced fluctuations between high- and low-density pore-filling states, governed by the height of the free energy barriers separating them. For CO₂ in IRMOF-1, strong intermolecular CO₂–CO₂ interactions and the high pore symmetry stabilize two distinct adsorption states, effectively suppressing intermediate configurations. In contrast, CH₄ exhibits transitions between competing low-density states, influenced by the spatial distribution of adsorption sites and the underlying framework symmetry [ 9 ]. Water adsorption represents an even more intricate case [ 14 – 17 , 21 – 23 ]. Unlike simple gases, water’s behavior is governed by directional hydrogen bonding, which promotes the formation of stable molecular clusters and results in multiple distinct free energy minima. These metastable states arise from strong electrostatic interactions and lead to exceptionally slow equilibration in simulations, especially at low temperatures. Therefore, conventional grand canonical Monte Carlo approaches often fail to fully capture the adsorption process, resulting in hysteresis and under-sampling of relevant configurations. Understanding metastable mechanisms is crucial for the rational design of materials for applications such as water harvesting and thermal energy storage. Yet accurately simulating water adsorption under confinement remains a major challenge. No existing force field fully captures all of water’s essential properties—particularly hydrogen bonding, polarizability, and molecular flexibility—when restricted to a nanoscale environment. Furthermore, the intrinsically rugged free energy landscape significantly hampers sampling efficiency and imposes computational cost. In conclusion, this study highlights the pivotal role of metastability in governing adsorption processes under confinement. Our findings demonstrate how pore geometry, fluid–fluid and fluid–framework interactions, and temperature jointly shape sorption dynamics across different adsorbates. To achieve realistic and predictive modeling of these systems, particularly for complex molecules like water, future efforts must explicitly account for metastability through advanced sampling techniques and the development of more accurate, physically grounded force fields. These improvements are essential to ensure the rational design of nanoporous materials for practical applications. Finally, it is noteworthy that an adsorption mechanism similar to this observed in IRMOFs has also been reported in a simplified model of ordered pores [ 24 , 25 ]. This underscores the need for further in-depth investigations to fully understand the role of metastability in the adsorption process across both realistic and idealized systems. By its nature, metastability is the field of research that must be strongly assisted and supported by numerical modeling, using methodologies that allow one to calculate free energy of the studied systems. Methods Experimental setup Synthesis IRMOF-1 was prepared in N,N-diethylformamide (DEF) solution following the procedure described in the literature [ 26 ]. Due to the absence of product after the initially postulated 8 h reaction time, the synthesis duration was extended to a total of 24 h. All other reagents and solvents were of analytical grade (Sigma Aldrich, TokyoChemical Industry) and were used as received, without further purification. Powder X-ray diffraction (PXRD) measurement The PXRD pattern of desolvated IRMOF-1 was recorded at room temperature using a STOE STADI P diffractometer equipped with Cu-Kα 1 radiation (λ = 1.54059 Å) and a 1D Mythen detector (Dectris). Data were collected in transmission mode using a rotating flatbed sample holder. The measurements were performed using a steps size of 6° in 2Θ and an exposure time of 20 s per step. Prior to measurement, IRMOF-1 was mounded in the glovebox and sealed under parafilm to prevent exposure to air. Gas adsorption measurements Prior to the physisorption measurements, the as-synthesized IRMOF-1 was washed eight times with 20–30 mL of anhydrous DMF, allowing the solid soak for 1–3 h during each washing step. Subsequently, the DMF was decanted, and the sample was washed seven times with 20–30 ml of anhydrous CH2Cl2, again with 1–3 soak intervals per cycle. Following solvent exchange, the sample was dried using the Schlenk technique under a flow of Argon to remove the excess of solvent. Final desolvation was performed under reduced pressure (~ 10 − 3 kPa) at 363 K for ~ 16 h. The disolvated IRMOF-1 was then transferred in the Schlenk tube into glovebox (MBRAUN) ) under an inert atmosphere for storage and handling. Gas adsorption measurements for CO 2 (99.999% purity) and CH 4 (99.999% purity) were conducted using a BELSORP-max adsorption apparatus (MicrotracBEL Corp.). BELSORP-max was connected to the home-built adsorption cell, connected to the closed cycle helium cryostat DE-202AG (ARS). The adsorption temperature was precisely controlled using LS-336 temperature controller (LAKE SHORE), and the excess heat generated by the cryostat was dissipated via water-cooled helium compressor ARS-2HW. For each experiment 32.2 mg of desolvated IRMOF-1 was loaded into the adsorption cell, which was sealed externally with a copper dome using a copper gasket. The system was thermally isolated under dynamic vacuum ( p < 10 − 4 kPa) and connected to the BELSORP-max adsorption instrument using a 1/8 inch stainless steel capillary. Following assembly the sample was degassed under a dynamic ultra-high vacuum of 0.01 Pa at 298 K for 12 h to ensure complete removal of residual adsorbates. Adsorption equilibrium was defined as the conditions under which pressure variations remained within 1% over a period of 500 s. Numerical setup Methodology of free energy calculations – Transition Matrix Grand Canonical Monte Carlo (TM-GCMC) The free energy profile, Ω( N ), where N denotes the number of adsorbed molecules, is a fundamental descriptor for identifying true thermodynamic equilibrium states, metastable configurations, and assessing the overall metastability of a system. However, this profile (Fig. 1, central panels) cannot be directly obtained from the conventional Metropolis grand canonical Monte Carlo (GCMC) simulations, as the method’s inherent limitations prevent adequate exploration of the full configuration space. To overcome this, the Transition Matrix Monte Carlo (TMMC), has been successfully employed to reconstruct the adsorption free energy landscape. This approach provides valuable insight into both equilibrium and non-equilibrium aspects of the adsorption process. Moreover, kinetic information can be inferred from the energy barriers separating local minima of Ω( N ), to estimate transition rates between metastable and stable states. The fundamental principles of the Transition Matrix Grand Canonical Monte Carlo (TM-GCMC) method are extensively documented in the literature [ 4 , 18 , 19 ] and are described in detail in our previous paper [ 8 ]. This advanced simulation technique offers enhanced sampling efficiency compared to conventional GCMC methods and enables the calculations of free energy profiles as a function of the number of system’s macroscopic states quantified by the number of adsorbed molecules N , at specified external pressure or chemical potential. Declarations Funding The authors declare no competing interests. Acknowledgements: M.S., B.M. B.K. and L.F are supported by the National Science Centre (NCN), Poland (grant no. 2022/45/B/ST8/02028). Monte Carlo numerical simulation results are created using resources provided by Wroclaw Centre for Networking and Supercomputing (https://wcss.pl). A.D. and C.W. are supported by the National Science Foundation Grant No. IIP-2044726, the University of Missouri Research Council and Materials Science and Engineering Institute. Author contributions: Conceptualization: B.K., B.M., Methodology: B.M., B.K., Investigation: M.S., A.D., V.B., B.M., K.R., Visualization: M.S., B.M., B.K. Funding acquisition: B.K., C.W., Project administration: B.K., Supervision: B.K., S.K., C.W., L.F. Writing – original draft: B.K., L.F. Writing – review & editing: B.K., K.R., B.M., C.W, M.S., L.F. The authors declare no competing interests. References Langmuir I (1918) The adsorption of gases on plane surfaces of glass, mica and platinum. J Am Chem Soc 40(9):1361–1403 Rouquerol J et al (2013) Adsorption by powders and porous solids: principles, methodology and applications. Academic Thommes M et al (2015) Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report). Pure and applied chemistry, 87(9–10): pp. 1051–1069 Li Z et al (2025) Understanding Pore Filling Processes and Adsorption/Desorption Hysteresis in Nanoporous Metal–Organic Frameworks: Insights from Grand Canonical Monte Carlo Simulations and Free Energy Calculations. Langmuir Canivet J et al (2014) Water adsorption in MOFs: fundamentals and applications. Chem Soc Rev 43(16):5594–5617 Cho HS et al (2019) Isotherms of individual pores by gas adsorption crystallography. Nat Chem 11(6):562–570 Bon V et al (2019) Insights into the water adsorption mechanism in the chemically stable zirconium-based MOF DUT-67–a prospective material for adsorption-driven heat transformations. J Mater Chem A 7(20):12681–12690 Mazur B et al (2022) Quasicontinuous Cooperative Adsorption Mechanism in Crystalline Nanoporous Materials. J Phys Chem Lett 13(30):6961–6965 Kuchta B et al (2017) Adsorption-Induced Structural Phase Transformation in Nanopores. Angew Chem 129(51):16461–16464 Furukawa H et al (2014) Water adsorption in porous metal–organic frameworks and related materials. J Am Chem Soc 136(11):4369–4381 Gupta A et al (2021) Improved computational method to generate properly equilibrated atomistic microstructures. MethodsX 8:101217 Walton KS et al (2008) Understanding inflections and steps in carbon dioxide adsorption isotherms in metal-organic frameworks. J Am Chem Soc 130(2):406–407 Fairen-Jimenez D, Seaton NA, Duren T (2010) Unusual adsorption behavior on metal – organic frameworks. Langmuir 26(18):14694–14699 Sarkisov L, Centineo A, Brandani S (2017) Molecular simulation and experiments of water adsorption in a high surface area activated carbon: Hysteresis, scanning curves and spatial organization of water clusters. Carbon 118:127–138 Ghosh P, Kim KC, Snurr RQ (2014) Modeling water and ammonia adsorption in hydrophobic metal–organic frameworks: single components and mixtures. J Phys Chem C 118(2):1102–1110 Paranthaman S, Coudert F-X, Fuchs AH (2010) Water adsorption in hydrophobic MOF channels. Phys Chem Chem Phys 12(28):8124–8130 Zhang H, Snurr RQ (2017) Computational study of water adsorption in the hydrophobic metal–organic framework ZIF-8: adsorption mechanism and acceleration of the simulations. J Phys Chem C 121(43):24000–24010 Siderius DW, Hatch HW, Shen VK (2024) Flat-Histogram Monte Carlo Simulation of Water Adsorption in Metal–Organic Frameworks. J Phys Chem B 128(19):4830–4845 Datar A, Witman M, Lin L-C (2021) Improving computational assessment of porous materials for water adsorption applications via flat histogram methods. J Phys Chem C 125(7):4253–4266 Greathouse JA, Allendorf MD (2006) The interaction of water with MOF-5 simulated by molecular dynamics. J Am Chem Soc 128(33):10678–10679 Alkhatib N, Naleem N, Kirmizialtin S (2024) How Does MOF-303 Achieve High Water Uptake and Facile Release Capacity? J Phys Chem C 128(20):8384–8394 Algara-Siller G et al (2015) Square ice in graphene nanocapillaries. Nature 519(7544):443–445 Burtch NC, Jasuja H, Walton KS (2014) Water stability and adsorption in metal–organic frameworks. Chem Rev 114(20):10575–10612 Kuchta B, Firlej L, Maurin G (2005) Mechanism of adsorption in cylindrical nanopores: The roles of fluctuations and correlations in stabilizing the adsorbed phase. J Chem Phys, 123(17) Neimark AV, Vishnyakov A (2006) Phase transitions and criticality in small systems: vapor – liquid transition in nanoscale spherical cavities. J Phys Chem B 110(19):9403–9412 Kaye SS et al (2007) Impact of preparation and handling on the hydrogen storage properties of Zn4O (1, 4-benzenedicarboxylate) 3 (MOF-5). J Am Chem Soc 129(46):14176–14177 Additional Declarations There is NO Competing Interest. Supplementary Files Adsorptionmechanism2025July16NatCommSI.docx Supplementary INFO: Adsorption mechanism in crystalline micropores: multimodal fluctuations, phase coexistence and phase transformations in nanoconfinement Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7148852","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":490413151,"identity":"1d357685-e385-4ff5-8f81-fa82b68c073b","order_by":0,"name":"Bogdan Kuchta","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/UlEQVRIiWNgGAWjYBACAyCWgNA8QKoCIgoSkSFGC2MDwxmEFh7itDC2EaHFnL394o2fOxiMDY6fPf7g57w7idsbmA/e5mG4g1OLZc+ZYsveMwxmBmfyEht7tz1LnHOALdmah+EZbofdyEmT4G1jsDG4wWPYwLvtcO4MBh4zaR6Gw3i1SP6Famn8Owekhf8bAS3px6SBtpiBtDTzNoBtYcOrBegXZmvZNgljyTM5hrNljj2rn8HMZmw5xwC3X4Ah9vDm2zYbw77jZww+vqm5YyzB3vzwxpuKO3K4tADDHxY1YHCAgYEZ7OADuHUwsD9A5h3AYIyCUTAKRsEoAACoOVdXW+8jTAAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-8635-4154","institution":"Wroclaw University of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Bogdan","middleName":"","lastName":"Kuchta","suffix":""},{"id":490413152,"identity":"11a8fcb5-5010-43b1-bab2-24906e2d23f4","order_by":1,"name":"Małgorzata Stankiewicz","email":"","orcid":"","institution":"Wroclaw University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Małgorzata","middleName":"","lastName":"Stankiewicz","suffix":""},{"id":490413153,"identity":"f4ea1639-e6b4-43c7-8bd8-3111aa3715ae","order_by":2,"name":"Anthony Dorhauer","email":"","orcid":"","institution":"University of Missouri Columbia,","correspondingAuthor":false,"prefix":"","firstName":"Anthony","middleName":"","lastName":"Dorhauer","suffix":""},{"id":490413154,"identity":"11fb68ba-1a4f-44e7-a059-f893525cf232","order_by":3,"name":"Carlos Wexler","email":"","orcid":"","institution":"University of Missouri","correspondingAuthor":false,"prefix":"","firstName":"Carlos","middleName":"","lastName":"Wexler","suffix":""},{"id":490413155,"identity":"eb59eb76-509a-4401-beb7-4b184504280b","order_by":4,"name":"Kornel Roztocki","email":"","orcid":"","institution":"Adam Mickiewicz University,","correspondingAuthor":false,"prefix":"","firstName":"Kornel","middleName":"","lastName":"Roztocki","suffix":""},{"id":490413156,"identity":"cefffec6-d417-4ab9-87cc-19f008dac71d","order_by":5,"name":"Bartosz Mazur","email":"","orcid":"","institution":"Wroclaw University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Bartosz","middleName":"","lastName":"Mazur","suffix":""},{"id":490413157,"identity":"5187b8a2-d89a-405e-afab-459a8e35aae3","order_by":6,"name":"Lucyna Firlej","email":"","orcid":"","institution":"Universite de Montpellier","correspondingAuthor":false,"prefix":"","firstName":"Lucyna","middleName":"","lastName":"Firlej","suffix":""},{"id":490413158,"identity":"f50558c3-034c-45e8-9ab1-6452476809d6","order_by":7,"name":"Stefan Kaskel","email":"","orcid":"https://orcid.org/0000-0003-4572-0303","institution":"TU Dresden","correspondingAuthor":false,"prefix":"","firstName":"Stefan","middleName":"","lastName":"Kaskel","suffix":""},{"id":490413159,"identity":"cdd70d0d-8438-44a9-af8f-d39201adf923","order_by":8,"name":"Volodymyr Bon","email":"","orcid":"https://orcid.org/0000-0002-9851-5031","institution":"Technische Universität Dresden","correspondingAuthor":false,"prefix":"","firstName":"Volodymyr","middleName":"","lastName":"Bon","suffix":""}],"badges":[],"createdAt":"2025-07-17 12:05:38","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7148852/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7148852/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":87564491,"identity":"f7fe51d8-2dc3-41a6-b3e1-b02ff019575e","added_by":"auto","created_at":"2025-07-25 09:11:52","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":80217,"visible":true,"origin":"","legend":"\u003cp\u003eTypical metastability observed for the CO\u003csub\u003e2\u003c/sub\u003e and CH\u003csub\u003e4\u003c/sub\u003e adsorption in IRMOFs. Numerical adsorption isotherms (left), free energy profiles (center), and uptake fluctuations (right) at pressures near pore filling. Top row: two-state system (CO₂ adsorption in IRMOF-1 at 214 K). Bottom row: three-state system (CH₄ adsorption in IRMOF-8 at 85 K). These results illustrate the presence of metastable states and distinct adsorption pathways. A more detailed analysis of these behaviors is provided in the following sections.\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/3431733425aad70eaa41ea1a.jpg"},{"id":87564492,"identity":"403d743e-bba5-4752-868c-6740bc7da016","added_by":"auto","created_at":"2025-07-25 09:11:52","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":67272,"visible":true,"origin":"","legend":"\u003cp\u003eTemperature-dependent evolution of experimental adsorption isotherms for CO\u003csub\u003e2\u003c/sub\u003e (left) and CH\u003csub\u003e4\u003c/sub\u003e (right) in IRMOF-1. Adsorption and desorption points are practically indistinguishable. The CH\u003csub\u003e4\u003c/sub\u003e data are reproduced from our previous work [8]. The uptake units ‘molecules/u.c’ are natural for numerical results and they are used in this paper. They can be transferred into ‘ml/g’ by multiplying by 3.63946 (We provide the AIF files with experimental data).\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/a4f950ae816a1ea173c6dccf.jpg"},{"id":87564498,"identity":"9128856d-3ec4-4048-90f2-3c593414c216","added_by":"auto","created_at":"2025-07-25 09:11:52","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":144184,"visible":true,"origin":"","legend":"\u003cp\u003eAdsorption\u003csub\u003e \u003c/sub\u003eisotherms (left), free energy (right), and bimodal fluctuations (center columns) of the CO\u003csub\u003e2 \u003c/sub\u003euptake in IRMOF-1 at three temperatures: \u0026nbsp;(A) \u003cem\u003eT\u003c/em\u003e = 214 K, (B) 220 K, and (C) 230 K. \u0026nbsp;Colored mark points on the isotherms indicate pressures at which the free energy profiles and uptake fluctuations in high density (red) and low-density (black) states are shown.\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/04aca151f1a1c88028b522b2.jpg"},{"id":87564494,"identity":"5dbc6294-3c6a-4964-979c-1d0f284d0eff","added_by":"auto","created_at":"2025-07-25 09:11:52","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":49898,"visible":true,"origin":"","legend":"\u003cp\u003eCO\u003csub\u003e2\u003c/sub\u003e adsorption isotherms in IRMOF-1, IRMOF-8, IRMOF-10 and IRMOF-14 at T=220 K and 230 K. The shift of the pressure associated with the sharp step-like reflects dependence on pore volume and linker structure.\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/67278f9c8dcc4f5690076380.jpg"},{"id":87564874,"identity":"9198f578-1c9e-452d-9498-b86680b9fca2","added_by":"auto","created_at":"2025-07-25 09:19:52","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":23575,"visible":true,"origin":"","legend":"\u003cp\u003eEstimated fluctuation probabilities (p) for CO₂ adsorption in IRMOF-X (X = 1, 8, 10, 14) at equilibrium pressure (i.e., when the high- and low-uptake states have equal free energy, E\u003csub\u003eH\u003c/sub\u003e = E\u003csub\u003eL\u003c/sub\u003e). As p approaches zero, observing fluctuations becomes increasingly rare and strongly dependent on the simulation length (typically requiring significantly more than ~10⁶ Monte Carlo cycles).\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/c212a02c9ef804a12f83a2cd.jpg"},{"id":87564507,"identity":"48bfdcc2-ceeb-4d8c-9a8c-8c2cbba42209","added_by":"auto","created_at":"2025-07-25 09:11:52","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":144608,"visible":true,"origin":"","legend":"\u003cp\u003eFrom left to right: adsorption isotherms, uptake fluctuations, and free energy profiles for CH\u003csub\u003e4\u003c/sub\u003e adsorption in IRMOF-8, at \u003cem\u003eT\u003c/em\u003e = 85 K, 92 K, 102 K and 110 K. The uptake fluctuations are shown at pressures: 19.9 Pa, 79.5 Pa, 440 Pa and 1370 Pa, respectively for each temperature.\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/7cff408ad998413b007480f7.jpg"},{"id":87564878,"identity":"7a4a337f-0948-41af-b8ba-5817a152d79f","added_by":"auto","created_at":"2025-07-25 09:19:52","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":73536,"visible":true,"origin":"","legend":"\u003cp\u003eAdsorption isotherms, equilibrium uptake fluctuations and free energy profiles for CH\u003csub\u003e4\u003c/sub\u003e adsorption in IRMOF-14 at \u003cem\u003eT\u003c/em\u003e = 115 K and 125 K. Colored points on isotherms indicate the pressures at which the free energy was calculated.\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/33073040ebb4dd4d7cb26f6e.jpg"},{"id":87564510,"identity":"68d4f3e2-2c34-42e9-82e7-58bdbc0bd6ed","added_by":"auto","created_at":"2025-07-25 09:11:52","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":31668,"visible":true,"origin":"","legend":"\u003cp\u003eFree energy profiles for CH\u003csub\u003e4\u003c/sub\u003e adsorption in IRMOF-8 at T = 110K and \u003cem\u003eP\u003c/em\u003e = 1385 Pa, simulated using MC boxes composed of (1x1x1), (2x2x1), and (3x3x1) IRMOF-1’s unit cells. The free energy barriers height (right panel) is shown as a function of the number of unit cells included in the MC simulation box. Instantaneous snapshots of fluctuations in a 3×3×1 unit cell are presented in SI, Fig.7S.\u003c/p\u003e","description":"","filename":"8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/22278cb7ffae620a566da6e8.jpg"},{"id":87564499,"identity":"33efa403-f51c-4290-aa6f-0c36dfc3fad2","added_by":"auto","created_at":"2025-07-25 09:11:52","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":63300,"visible":true,"origin":"","legend":"\u003cp\u003eThe adsorption energy distributions and spatial distribution of CH\u003csub\u003e4\u003c/sub\u003e in IRMOF-1. Left two panels: adsorption energy maps for CH₄ in the smaller (left) and larger (middle-left) pore of IRMOF-1. The strongest adsorption sites are in the corners of the larger pore. Middle-right panel: methane distribution at low uptake, closely matching the geometry of the most favorable adsorption sites. Right panel: at higher loading, CH₄ density distribution becomes more localized due to packing effects and enhanced molecule–pore interactions. Methane density is represented by grey areas.\u003c/p\u003e","description":"","filename":"9.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/b19e6c3c01358b046f17f0b5.jpg"},{"id":87564522,"identity":"1aa99f00-541a-454b-9371-87a89f639981","added_by":"auto","created_at":"2025-07-25 09:11:53","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":67268,"visible":true,"origin":"","legend":"\u003cp\u003eFree energy profile and representative example of metastable configuration of water adsorption in Al(OH)(1,4-ndc), modeled using the TIP4P-Ew water potential. Water adsorption was not simulated in IRMOFs due to their known instability in the presence of water [20].\u003c/p\u003e","description":"","filename":"10.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/eedf83fad6722b6f4cfbb37d.jpg"},{"id":89496250,"identity":"41f65c62-31b7-495a-a516-77e27b9b2f18","added_by":"auto","created_at":"2025-08-20 14:54:18","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1601349,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/46e35758-3db6-4923-9b11-d90525fef0b0.pdf"},{"id":87564883,"identity":"9cfac5d7-4927-448f-ba88-3504c62069d0","added_by":"auto","created_at":"2025-07-25 09:19:52","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":4642727,"visible":true,"origin":"","legend":"Supplementary INFO: Adsorption mechanism in crystalline micropores: multimodal fluctuations, phase coexistence and phase transformations in nanoconfinement","description":"","filename":"Adsorptionmechanism2025July16NatCommSI.docx","url":"https://assets-eu.researchsquare.com/files/rs-7148852/v1/cce9a7e92dedb265a4d1f95b.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Adsorption mechanism in crystalline micropores: multimodal fluctuations, phase coexistence and phase transformations in nanoconfinement","fulltext":[{"header":"Introduction","content":"\u003cp\u003eAdsorption of gases on the surfaces is often perceived as a conceptually simple phenomenon. Its fundamental principles were formulated by Irving Langmuir in 1918, who wrote: \u0026ldquo;\u003cem\u003ewhen gas molecules impinge against any solid or liquid surface, they do not in general rebound elastically, but condense on the surface, being held by the field of force of the surface atoms. These molecules may subsequently evaporate from the surface. The length of time that elapses between the condensation of a molecule and its subsequent evaporation depends on the intensity of the surface forces. Adsorption is the direct result of this time lag\u003c/em\u003e\u0026rdquo; [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. This description intuitively captures the dynamic nature of the adsorbed phase, whose composition continuously fluctuates and is consistent with both thermodynamic and kinetic stability. While this definition remains valid across a wide range of environments, adsorption under nanoconfinement reveals additional, intriguing behaviors. One of them is the existence of metastability.\u003c/p\u003e\u003cp\u003eThe analysis of sorption isotherms remains a fundamental and widely applied method for characterizing adsorption/desorption phenomena in a broad range of materials. This approach relies on the analysis of the isotherms\u0026rsquo; shapes and their classification according to established schemes, such as those proposed by IUPAC [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Although the IUPAC classification provides a valuable phenomenological framework, it offers only limited insight into the microscopic mechanisms driving sorption processes. These mechanisms are often complex and system-specific, emerging from the interplay of multiple factors, including structural heterogeneity of the adsorbent surface, a broad distribution of pore sizes and geometries, and spatial variations in the adsorption energy landscape. This complexity is particularly pronounced in crystalline microporous materials. This aspect has also been emphasized in the recent paper discussing adsorption hysteresis in nanoporous MOFs (Metal\u0026thinsp;\u0026minus;\u0026thinsp;Organic Frameworks) [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], revealing two different mechanisms of rapid changes in the adsorption uptake: one resembling capillary condensation (gas-to-liquid transition) and the other more akin to liquid-to-solid transition.\u003c/p\u003e\u003cp\u003eAccording to IUPAC classification, microporous systems \u0026ndash; defined by pore sizes below 2 nm - are associated with Type I adsorption isotherms, characterized by a steep, near-vertical uptake at very low relative pressures, indicative of rapid pore filling. However, both experimentally measured and computationally derived adsorption isotherms frequently show significant deviations from this idealized Type I behavior in such systems [\u003cspan additionalcitationids=\"CR5 CR6 CR7 CR8 CR9\" citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. These deviations are particularly pronounced in materials with crystallographic symmetry and/or well-ordered pore-wall structures. They primarily result from heterogeneities in the adsorption energy landscape and from the complex interplay between different interaction components, modulated by the thermal energy of the surrounding environment. In particular, strong fluid-fluid interactions can compete with fluid-framework interactions, potentially giving rise to metastable states [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] and specific pore filling mechanisms, including structural transitions [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eMetastability plays a critical role in molecular simulations. Systems that become trapped within local - rather than global-thermodynamic minima are described as kinetically stable or persistent (long-lived). Although such metastable states may be unobservable in a macroscopic experimental viewpoint, where observation timescales typically range from minutes to hours, they are often essential for interpreting microscopic behavior. In molecular simulations, which are generally limited to nanosecond timescales, metastability can significantly hinder reaching thermodynamic equilibrium, increase statistical uncertainty, and substantially increase the computational cost required to sample representative equilibrium ensembles [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eIn this study, we investigate the emergence of metastable states during gas adsorption in microporous materials. We show that this phenomenon is common across ordered or crystalline microporous systems, including metal-organic frameworks (MOFs) and zeolites. The formation of metastable states is attributed to a confluence of factors: the spatial distribution and crystallographic symmetry of adsorption sites, the restricted pore dimensions (typically 1\u0026ndash;2 nm), and the competitive interplay between gas\u0026ndash;gas and gas\u0026ndash;framework interactions. Together, these factors create a complex free energy landscape characterized by multiple local minima, leading to non-trivial adsorption pathways and mechanisms [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThe formation of metastable states during gas adsorption in microporous systems is analyzed using methane and carbon dioxide adsorption in selected model IRMOF structures. Our results demonstrate how the structural features and crystallographic symmetry of pore walls influence the pore-filling process, frequently leading to metastable configurations that can temporarily trap the system. The lifetimes of these states are found to be temperature-dependent, reflecting a delicate balance between thermodynamic driving forces and kinetic barriers. These findings underscore the intricate and dynamic nature of adsorption in nanoporous crystalline materials and provide new insights into the mechanisms governing molecular confinement and transport in such systems.\u003c/p\u003e\u003cp\u003e\u003cb\u003eMetastability: fundamental element of multimodal adsorption mechanism\u003c/b\u003e\u003c/p\u003e\u003cp\u003eIn \u003cem\u003emesoporous\u003c/em\u003e materials (pore dimensions\u0026thinsp;\u0026gt;\u0026thinsp;2 nm), a well-documented example of a metastability-driven phenomenon is the hysteresis observed during capillary condensation, consisting in rapid pore filling. In contrast, \u003cem\u003emicroporous\u003c/em\u003e materials exhibit different filling mechanisms [\u003cspan additionalcitationids=\"CR9\" citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. While capillary condensation is primarily governed by fluid-fluid interaction and often displays hysteresis, micropore filling is controlled by a more nuanced balance between fluid-fluid and fluid-framework interactions and is typically reversible. Moreover, micropore filling can induce structural transformations in the adsorbed phase [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], potentially leading to the formation of new stable phases or metastable intermediate states. This stands in contrast to capillary condensation, where pores filling proceeds rapidly without altering the structure of the already adsorbed fluid layers.\u003c/p\u003e\u003cp\u003ePrevious experimental and computational studies have reported atypical features in the adsorption behavior of methane and carbon dioxide in microporous MOF-5 (IRMOF-1) framework [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. Similarly, water adsorption isotherms in microporous materials have been shown to adopt a variety of shapes [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan additionalcitationids=\"CR15 CR16 CR17 CR18\" citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] and are well-known to involve long living metastable states. In many crystalline microporous systems, adsorption isotherms deviate from the classical IUPAC Type I profile, characterized by rapid pore filling at very low pressures. Specifically, at low temperatures and pressures, the isotherms often exhibit an initial monolayer-like adsorption, followed by an S-shaped profile, and ultimately a sharp, step-like uptake [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. In our previous work [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], this abrupt pore filling was interpreted as the nano-scale coexistence of low- and high-density adsorbate states, driven by large-amplitude, correlated fluctuations between these configurations. Such a mechanism requires the presence of at least one metastable state [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e], with transitions between phases governed by a relatively low (and depending on temperature) free energy barrier separating the macroscopic low- and high-density phases (Fig.\u0026nbsp;1). As a result, adsorption proceeds not through gradual, incremental accumulation of adsorbate molecules, but rather via a collective, synchronized, and cooperative structural transformation of the adsorbed phase. At higher temperatures, the free energy barrier between states of different density diminishes, leading to more frequent fluctuations and a gradual transition to a continuous adsorption profile.\u003c/p\u003e\u003cp\u003eMetastability plays a critical role in shaping the dynamics and kinetics of the adsorption process. In our previous paper [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e], we demonstrated how metastability between low-density and high-density methane phase structures within IRMOF-1 pores governs the adsorption mechanism of methane. Building on these results, the present study explores how metastability evolves as a function of pore size and temperature in a series of IRMOF-type frameworks (IRMOF-X, where X = 8, 10, 14), for both methane (CH₄) and carbon dioxide (CO₂).\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003e\u003cstrong\u003eCO\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003e2\u003c/strong\u003e\u003c/sub\u003e \u003cstrong\u003eadsorption in IRMOF-X materials (X\u0026thinsp;=\u0026thinsp;1, 8, 10, 14)\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe begin our analysis of adsorption mechanisms by examining CO\u003csub\u003e2\u003c/sub\u003e adsorption in IRMOF-type structures. Figure 2 presents the experimental isotherms of CO\u003csub\u003e2\u003c/sub\u003e and CH\u003csub\u003e4\u003c/sub\u003e adsorption in IRMOF-1. These results serve as a reference for understanding the distinct adsorption behaviors observed in this class of materials and guide the interpretation of corresponding simulation results.\u003c/p\u003e\n\u003cp\u003eAt temperatures near the bulk triple point of CO₂ (216.58 K, NIST database), the adsorption isotherms display a fully reversible, step-like character (Fig.\u0026nbsp;2, left panel). A similar behavior is observed for CH\u003csub\u003e4\u003c/sub\u003e adsorption at temperatures approaching its bulk triple point (90.67 K), as shown in Fig. 2 (right panel) [\u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e]. In both cases, increasing the temperature leads to a progressive transformation of the sharp, quasi-discrete transition into a smoother, S-shaped adsorption profile, indicating a more continuous adsorption process. The comparison of CO₂ and CH\u003csub\u003e4\u003c/sub\u003e adsorption isotherms suggests a shared underlying mechanism governing transitions between low- and high-density nanophases. This resemblance persists despite differences in molecular symmetry of guest molecules, adsorption conditions (temperature and pressure), and the absence of electrostatic interactions in CH\u003csub\u003e4\u003c/sub\u003e adsorption. These findings point to \u003cem\u003ea degree of universality\u003c/em\u003e in the adsorption mechanism that transcends specific structural and chemical properties of the adsorbate. This observation raises a fundamental question: what physicochemical parameters dictate the onset and nature of transitions between nanophases during pore filling in microporous materials?\u003c/p\u003e\n\u003cp\u003eTM-GCMC calculations of the free energy landscapes (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e, right column) offer critical insight into this question [\u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e]. The calculated free energy reveals distinct minima over a specific pressure range, corresponding to low- and high-density adsorption states, separated by a finite energy barrier. When this barrier is relatively low (here, below 10 kJ/mol for CO₂), thermal fluctuations enable spontaneous transitions between the two states, at pressures corresponding to rapid pore filling (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e, middle columns). Within the pressure range defining the step on the adsorption isotherm, the free energy consistently shows at least one metastable state. However, this state remains macroscopically undetectable, as the experimentally measured uptake represents a time-average value, weighted by the residence time in each of the coexisting states [\u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e]. As the temperature increases, the energy barrier progressively diminishes, and the two minima ultimately coalesce into a single, broader minimum. This marks the disappearance of distinct nanophases, and the onset of a continuous adsorption regime. We define the corresponding temperature as the \u003cem\u003epore critical temperature T\u003c/em\u003e\u003csub\u003e\u003cem\u003epc\u003c/em\u003e\u003c/sub\u003e.\u003c/p\u003e\n\u003cp\u003eAs an initial step toward generalization, we extended the analysis to CO₂ adsorption in larger-pore IRMOF structures: IRMOF-8, IRMOF-10, and IRMOF-14. Structural parameters for all investigated frameworks are provided in the Supplementary Information (Table \u003cspan class=\"InternalRef\"\u003eS1\u003c/span\u003e in the Supplementary Information).\u003c/p\u003e\n\u003cp\u003eCO\u003csub\u003e2\u003c/sub\u003e adsorption isotherms for all studied IRMOF-X materials at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;220 K and 230 K are presented in Fig. 4. Qualitatively, the isotherms exhibit similar features across the series. However, the pressure range associated with the sharp, step-like adsorption in IRMOF-X (X\u0026thinsp;\u0026gt;\u0026thinsp;1) is systematically shifted to higher values relative to IRMOF-1. This shift primarily reflects the variation of the pore volume resulting from the incorporation of longer organic linkers. Interestingly, the shift between IRMOF-8 and IRMOF-14 isotherms is relatively small, despite the larger pore volume of IRMOF-14 (see Fig. 2S in the Supplementary Information). This behavior can be attributed to a compensating effect: the longer and more massive linker in IRMOF-14 introduces additional strong adsorption sites, which facilitate adsorption at lower pressures. For the same reason, the pressure shift between IRMOF-8 and IRMOF-10 is larger than that between IRMOF-8 and IRMOF-14, even though the unit cell volumes of IRMOF-10 and IRMOF-14 are comparable. In the case of IRMOF-10, the linker structure results in weaker adsorption forces, allowing the low-density phase to persist over a broader pressure range. Figure \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eS-\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003eS in the Supplementary Information show the results of free energies and the corresponding bimodal fluctuations of the CO\u003csub\u003e2\u003c/sub\u003e uptake in IRMOF-8, IRMOF-10 and IRMOF-14\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eBimodal behavior: the limit of reversible pore filling.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe calculated free energy profiles (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e) indicate that the bimodal behavior observed at pressures corresponding to step-like adsorption can be captured in Monte Carlo simulations when the energy barrier \u003cem\u003eE\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e separating the low- and high-density states remains sufficiently low (typically \u003cem\u003eE\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e \u0026lt; 10 kJ/mol for CO\u003csub\u003e2\u003c/sub\u003e). In this regime, the transition probability is high enough to allow fluctuations between the two states within the simulation timescale. To quantify this effect, we explicitly calculated the probabilities of crossing the barrier, assuming the transition probability between the low-density state (with energy \u003cem\u003eE\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e) and the high-density state (with energy \u003cem\u003eE\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e) as a two-step process. The total transition probability was expressed as the product of two subsequent transition probabilities, with the intermediate state defined by the barrier height \u003cem\u003eE\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e:\u003c/p\u003e\n\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e$$\\:p\\left({E}_{L}\\to\\:{E}_{H}\\right)=p\\left({E}_{L}\\to\\:{E}_{B}\\:\\right)p\\left({E}_{B}\\to\\:{E}_{H}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e,\u003c/p\u003e\n\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e$$\\:p\\left({E}_{H}\\to\\:{E}_{L}\\right)=p\\left({E}_{H}\\to\\:{E}_{B}\\:\\right)p\\left({E}_{B}\\to\\:{E}_{L}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e,\u003c/p\u003e\n\u003cp\u003eproportional to the Boltzmann factors when the final state has higher energy, and equal to 1 when the final state has lower energy:\u003c/p\u003e\n\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e$$\\:p\\left({E}_{L}\\to\\:{E}_{B}\\:\\right)=\\text{e}\\text{x}\\text{p}(-\\frac{{E}_{BL}}{{k}_{B}T})\\le\\:1\\:\\text{w}\\text{h}\\text{e}\\text{r}\\text{e}\\:{\\Delta\\:}EBL\\:=\\:{E}_{B}-{E}_{L}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e$$\\:p\\left({E}_{H}\\to\\:{E}_{B}\\:\\right)=\\text{exp}\\left(-\\frac{{E}_{BH}}{{k}_{B}T}\\right)\\le\\:1\\:\\:\\:\\text{w}\\text{h}\\text{e}\\text{r}\\text{e}\\:{\\Delta\\:}EBH\\:={\\:E}_{B}-{E}_{H}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e$$\\:p\\left({E}_{B}\\to\\:{E}_{H}\\right)=\\:p\\left({E}_{B}\\to\\:{E}_{L}\\right)=1$$\u003c/div\u003e\u003c/div\u003e\u003cp\u003eThese transition probabilities satisfy the detailed balance (microscopic reversibility condition), which leads to the following relation:\u003c/p\u003e\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equf\" name=\"EquationSource\"\u003e$$\\:p\\left({E}_{L}\\right)p\\left({E}_{L}\\to\\:{E}_{H}\\right)=p\\left({E}_{H}\\right)p\\left({E}_{H}\\to\\:{E}_{L}\\right)$$\u003c/div\u003e\u003c/div\u003e\u003cp\u003eTherefore, we can estimate the transition probabilities \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:p\\left({E}_{L}\\to\\:{E}_{H}\\right)\\:\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:p\\left({E}_{H}\\to\\:{E}_{L}\\right)\\:\\)\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equg\" name=\"EquationSource\"\u003e$$\\:p\\left({E}_{L}\\to\\:{E}_{H}\\right)=p\\left({E}_{L}\\to\\:{E}_{B}\\:\\right)p\\left({E}_{B}\\to\\:{E}_{H}\\right)\\:=\\:p\\left({E}_{L}\\to\\:{E}_{B}\\:\\right)$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equh\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equh\" name=\"EquationSource\"\u003e$$\\:p\\left({E}_{H}\\to\\:{E}_{L}\\right)=p\\left({E}_{H}\\to\\:{E}_{B}\\:\\right)p\\left({E}_{B}\\to\\:{E}_{L}\\right)=p\\left({E}_{H}\\to\\:{E}_{B}\\:\\right)$$\u003c/div\u003e\u003c/div\u003e\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e shows the estimated transition probabilities for IRMOF-X (X\u0026thinsp;=\u0026thinsp;1, 8, 10, 14) at equilibrium pressures, i.e. when the free energies of the low- and high-density states are equal \u003cem\u003eE\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e = \u003cem\u003eE\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e, plotted as a function of temperature. As the transition probability approaches \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.6, the distinction between the two well defined nanophases disappears. We define the corresponding temperature at which this occurs as the pore critical temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003epc\u003c/sub\u003e.\u003c/p\u003e\u003cp\u003eThe pore critical temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003epc\u003c/sub\u003e defines the upper temperature limit at which bimodal adsorption behavior can be observed. This condition corresponds to the point where the free energy barrier separating low- and high-density states is comparable to, or smaller than, the thermal energy \u003cem\u003ek\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e\u003cem\u003eT\u003c/em\u003e. Our results indicate that \u003cem\u003eT\u003c/em\u003e\u003csub\u003epc\u003c/sub\u003e is governed primarily by two factors: pore size and linker topology. We observe an increase in the \u003cem\u003eT\u003c/em\u003e\u003csub\u003ecp\u003c/sub\u003e from IRMOF-1 to both IRMOF-8 and IRMOF-10, consistent with their larger pore volume. However, the \u003cem\u003eT\u003c/em\u003e\u003csub\u003ecp\u003c/sub\u003e of IRMOF-14 is approximately equal to that of IRMOF-8, despite IRMOF-14 requiring higher pressure for pore filling (Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e). This suggests that, beyond pore size, the linker structure also plays a critical role in stabilizing specific adsorbed states.\u003c/p\u003e\u003cp\u003eFor CO\u003csub\u003e2\u003c/sub\u003e adsorption, all studied IRMOFs exhibit qualitatively similar free energy profiles, each characterized by two metastable states. In contrast, CH\u003csub\u003e4\u003c/sub\u003e adsorption reveals more complex behavior in some frameworks, with free energies featuring multiple distinct minima indicative of additional metastable configurations.\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eTriple point (\u003cem\u003eT\u003c/em\u003e\u003csub\u003etr\u003c/sub\u003e) and critical (\u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e) temperatures for bulk (CH\u003csub\u003e4\u003c/sub\u003e and CO\u003csub\u003e2\u003c/sub\u003e), and pore critical (\u003cem\u003eT\u003c/em\u003e\u003csub\u003epc\u003c/sub\u003e) temperatures for CO\u003csub\u003e2\u003c/sub\u003e and CH\u003csub\u003e4\u003c/sub\u003e adsorbed in IRMOF-X (X\u0026thinsp;=\u0026thinsp;1, 8, 10, 14). \u003cem\u003eT\u003c/em\u003e\u003csub\u003ecp\u003c/sub\u003e (IRMOF-X) is the lowest temperature where continuous transformation from the low density to high density adsorption state is observed (estimated accuracy of \u0026plusmn; 4 K).\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\u003cth align=\"left\"\u003e\u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003etr\u003c/sub\u003e (bulk)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\"\u003e\u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003ecp\u003c/sub\u003e(IRMOF-1)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\"\u003e\u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003ecp\u003c/sub\u003e (IRMOF-8)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\"\u003e\u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003ecp\u003c/sub\u003e (IRMOF-10)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\"\u003e\u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003ecp\u003c/sub\u003e (IRMOF-14)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\"\u003e\u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e (bulk)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003epore volume\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e---\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e9.38 nm\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e27.25 nm\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e40.29 nm\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e40.64 nm\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e---\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eCH\u003csub\u003e4\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u003cstrong\u003e90.67 K\u003c/strong\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u0026sim; 110 K\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u0026sim; 125 K\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u0026sim; 120 K\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u0026sim; 125 K\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e190.6 K\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eCO\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u003cstrong\u003e216.58 K\u003c/strong\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u0026sim; 230 K\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u0026sim; 240 K\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u0026sim; 255 K\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u0026sim; 240 K\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e304.13 K\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003cstrong\u003eCH\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003e4\u003c/strong\u003e\u003c/sub\u003e \u003cstrong\u003eadsorption in IRMOF-X materials (X\u0026thinsp;=\u0026thinsp;1, 8, 10, 14)\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eThe mechanism of CH\u003csub\u003e4\u003c/sub\u003e adsorption in IRMOF-1 has been recently investigated and reported in detail [\u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e]. Its free energy profile is qualitatively like that discussed above for CO\u003csub\u003e2,\u003c/sub\u003e exhibiting two well-defined metastable states. However, in larger pore frameworks, specifically IRMOF-8, IRMOF-10, and IRMOF-14, the adsorption behavior becomes more complex, with free energy landscape changes significantly displaying multiple minima (Fig. 1). This suggests that CH\u003csub\u003e4\u003c/sub\u003e adsorption in these systems involves a series of structural transformations as a function of adsorption uptake. As the temperature increases, the adsorption landscape progressively smooths: the discrete minima begin to merge, and the energy barriers separating them decrease, ultimately leading to a more continuous adsorption process.\u003c/p\u003e\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e illustrates the key features of CH4 adsorption in IRMOF-8. At \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;85 K and 92 K, the free energy profile shows a multi-minima structure within a very narrow pressure range, revealing the presence of intermediate configurations at the uptake of \u0026sim; 150 molecules per unit cell. These intermediate states are not thermodynamically stable and can only be inferred from the evolution of uptake fluctuations observed during the equilibration phase of simulations. Around pressures 53\u0026ndash;54 Pa, the system becomes temporarily trapped in a low-density configuration (analogous to monolayer adsorption) before reaching a stable, high-density state. The pressure range associated with these metastable states is extremely narrow and is not resolved in the corresponding adsorption isotherms. As temperature increases, the isotherms\u0026rsquo; shape evolves. Metastable low-density states disappear, and uptake increases continuously with pressure. At higher pressure, where the pores are partially filled pores and the accessible volume is reduced, only equilibrium fluctuations are observed (Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e, bottom row: \u003cem\u003eP\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1365 Pa, \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;110 K).\u003c/p\u003e\u003cp\u003eIn IRMOF-14, CH₄ uptake increases smoothly at low pressures until approximately half of the pore volume is filled (Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e). Within the temperature range of 85\u0026ndash;125 K, two high-density metastable adsorption states are observed, accompanied by large-amplitude uptake fluctuations in the upper part of the adsorption step. The adsorption mechanism resembles that of capillary condensation, where adsorption of initial layers is followed by more abrupt pore filling. However, the nature of the filling process varies with temperature. At lower temperatures (e.g., \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;115), pore filling proceeds via more scattered events, while at higher temperatures (e.g., \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;125 K), the process becomes more continuous (Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e).\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eInfluence of the size of Monte Carlo simulation box on barriers between metastable states\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eTwo primary input parameters govern the outcome of the adsorption simulation. The first is the intermolecular potential model, which reflects the underlying physical interactions. The critical role of the model is well established and widely recognized (Fig.\u0026nbsp;1S).\u003c/p\u003e\u003cp\u003eThe second parameter concerns the simulation methodology: the size of the simulation box. To evaluate the impact of box dimensions and geometry on the system\u0026rsquo;s tendency to become trapped in metastable, yet physically unrealistic states, we performed CH₄ adsorption simulations in IRMOF-8 using Monte Carlo boxes of varying sizes. Three configurations were considered, corresponding to the following three-dimensional (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e, \u003cem\u003ez\u003c/em\u003e) unit cell arrangements: (1) 1\u0026times;1\u0026times;1, (2) 2\u0026times;2\u0026times;1, and (3) 3\u0026times;3\u0026times;1. These simulations were performed at a temperature of 110 K and a pressure of 1385 Pa.\u003c/p\u003e\u003cp\u003eThe free energy profile for CH\u003csub\u003e4\u003c/sub\u003e in the smallest simulation box (1\u0026times;1\u0026times;1) is shown in Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e (right column), while those for larger boxes (2\u0026times;2\u0026times;1, and 3\u0026times;3\u0026times;1) are presented in Fig. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e. In the smallest box, the equilibrium free energy landscape exhibits a barrier of approximately 3 kJ/mol and the system displays fluctuations between a medium density state (\u003cem\u003eN\u003c/em\u003e \u0026sim; 150) and a high-density state (\u003cem\u003eN\u003c/em\u003e \u0026sim; 300) during the simulation. As the box size increases, the free energy barrier also rises, reaching approximately 15 kJ/mol for the biggest box. According to the relation \u003cem\u003ep\u003c/em\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;exp(-\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{E}_{B}\\)\u003c/span\u003e\u003c/span\u003e/\u003cem\u003ek\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e\u003cem\u003eT\u003c/em\u003e), the probability of observing fluctuations over such a barrier at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;110K is of the order of p \u0026asymp; 10\u003csup\u003e\u0026minus;\u0026thinsp;8\u003c/sup\u003e. This vanishingly small probability effectively precludes the observation of fluctuations within standard MC simulations comprising \u0026sim; 10\u003csup\u003e6\u003c/sup\u003e steps. For comparison, in molecular dynamics simulations where the simulation timestep is usually of the order of 0.5 fs, 10\u003csup\u003e6\u003c/sup\u003e steps correspond to simulated times shorter than 1 ns. This limitation is purely technical and could be mitigated in longer simulations or with more powerful computer resources. In contrast, real experiments are not constrained by such restrictions and can probe systems over macroscopic timescales (minutes to hours), which are many orders of magnitude longer than those accessible in molecular simulations (nanoseconds to microseconds).\u003c/p\u003e\u003cp\u003eInterestingly, increasing the size of the simulation box alone does not lead to the emergence of additional intermediate metastable states (i.e., additional local minima) in the free energy landscape of CH\u003csub\u003e4\u003c/sub\u003e adsorption. This observation supports the interpretation of the transformations as a discontinuous, first-order process, in which intermediate macrostates represent weighted (statistical) averages over the two coexisting metastable microscopic states, corresponding to either fully filled or fully empty pores. A similar conclusion was reached in our previous study on CH\u003csub\u003e4\u003c/sub\u003e adsorption in IRMOFs [\u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e]. These results reinforce the hypothesis that pore filling proceeds via a first-order transformation, independent of the Monte Carlo simulation box size. At the same time, they show that the metastability observed in this system originates from a mechanism fundamentally different than that reported for water adsorption [\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e19\u003c/span\u003e], where the metastable states originate from the formation and rearrangement of molecular clusters. In the case of water, the number of metastable states may depend on the size of the simulated sample (MC box), and their presence is associated with extremely slow equilibration dynamics [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e18\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e19\u003c/span\u003e].\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eOrigin of the metastability\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eA fundamental question in the study of adsorption is the origin of metastability. Simulations of CO₂ adsorption reveal that at low temperatures, certain macrostates are effectively inaccessible\u0026mdash;neither thermodynamically stable nor kinetically reachable. Rather than stabilizing in a single well-defined macrostate (which would correspond to lower entropy), the system establishes a dynamic equilibrium between two metastable states, each statistically populated over time. These states are separated by a finite free energy barrier, making the probability of transitions between the states dependent on temperature.\u003c/p\u003e\u003cp\u003ePore symmetry plays a crucial role in governing adsorption behavior. Adsorbate configurations that preserve the symmetry of the framework are generally lower in energy and thus thermodynamically more favored. In contrast, intermediate macrostates that break this symmetry tend to be both higher in energy and lower in entropy, which renders them thermodynamically inaccessible. As a result, the system avoids these asymmetric configurations, and metastability emerges because of the combined enthalpic and entropic penalties associated with symmetry-breaking intermediate states.\u003c/p\u003e\u003cp\u003eAt higher temperatures, the behavior of the system changes. Increased thermal energy introduces greater disorder into the adsorbed phase, enhancing the entropy and allowing the system to explore a broader region of configuration space. As a result, the free energy barriers separating macrostates become easier to overcome (or disappear), and the adsorption process becomes more continuous in nature.\u003c/p\u003e\u003cp\u003eA complementary perspective emerges from the analysis of CH₄ adsorption, where additional metastable states are observed, particularly at low adsorbate densities. In this case, the spherical shape of the methane molecule and the symmetric distribution of adsorption sites within the pore play a critical role. In our previous work [\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e], we identified a distinct metastable configuration that disappeared at elevated temperatures. This structure was stabilized by the geometric symmetry of the pore walls and corresponded to a low-density gas molecules arrangement aligned with shallow adsorption sites (Fig. \u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e). At very low loading, methane molecules preferentially populate these shallow, yet energetically favorable, sites.\u003c/p\u003e\u003cp\u003eAs the adsorbate density increased, the distribution of CH₄ molecules became more localized, giving rise to a configuration that was also symmetric, but distinct from the low-density state. This newly formed structure\u0026mdash;though metastable\u0026mdash;was stabilized by the interplay of molecule\u0026ndash;pore interactions and packing constraints. It occupied a region of the free energy landscape near the barrier separating macrostates, illustrating how geometric symmetry and molecular shape contribute to the emergence and persistence of metastable states in adsorption processes.\u003c/p\u003e\u003cp\u003eIn summary, at low temperatures and within highly symmetric pores, the emergence of two metastable states corresponding to low- and high-density states appears to be a natural consequence of the system\u0026rsquo;s energetic and geometric constraints. At low adsorbate densities, the ordered pore environment energetically favors only symmetric molecular arrangements. As the density increases, the balance between the gas-gas and gas-pore interactions begins to influence the stability and structure of the final configuration. With increasing temperature, thermal energy introduces disorder and raises the system\u0026rsquo;s entropy, enabling stabilization of a broader ensemble of macrostates. Additional metastable states may arise when specific distributions of adsorption sites provide sufficient energetic stabilization (see also Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003eS).\u003c/p\u003e\u003cp\u003eFinally, we compare the metastability observed in CH₄ and CO₂ adsorption with water adsorption in micropores of Al(OH)(1,4-ndc) MOF [\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e], where strong water-water interactions play a central role. These interactions promote the formation of multiple free energy minima [\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e], corresponding to clusters of water molecules that assemble within the pores into energetically stable networks resembling the structure of bulk liquid water [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e]. These macrostates represent metastable configurations of the system (Fig. \u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e). The number of free energy minima increases with the size of the Monte Carlo simulation box, suggesting that in real (macroscopic) systems, the transition between the low- and high-density states of adsorbed water may be continuous. In the thermodynamic limit, where system size approaches infinity, the number of metastable minima likewise becomes infinite [\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e]. In this regime, water adsorption proceeds via sequential filling of pores, with each occupancy level corresponding to a distinct metastable state. This behavior is consistent with the previously reported simulation studies [\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThe nature of metastability in water adsorption differs from that observed for CO\u003csub\u003e2\u003c/sub\u003e and CH\u003csub\u003e4\u003c/sub\u003e in IRMOF structures. Water clusters are stabilized by strong intermolecular hydrogen bonding, resulting in relatively deep energy minima. This makes the addition or removal of individual molecules energetically unfavorable, leading to discrete cooperative adsorption events. Such a collective behavior is absent in the gas-like adsorption of CO₂ and CH₄. Consequently, water fills the pores sequentially, each step corresponding to the formation of a stable cluster. The number of resulting metastable states scales with the system size.\u003c/p\u003e\u003cp\u003eIn contrast, for CO\u003csub\u003e2\u003c/sub\u003e and CH\u003csub\u003e4\u003c/sub\u003e in IRMOFs, intermediate metastable states emerge only under specific symmetry conditions and are stabilized by the delicate competition between fluid\u0026ndash;fluid and fluid\u0026ndash;framework interactions. These states are typically observed at low temperatures and in highly ordered pore environments. In the absence of such symmetry constraints or when thermal energy introduces a structural disorder, these intermediate, partially filled configuration states become thermodynamically unstable, and no metastable states persist. The systems then transition smoothly between low- and high-density phases, resulting in continuous adsorption behavior.\u003c/p\u003e\u003cp\u003eIn conclusion, water represents a unique case among adsorbates due to its pronounced hydrogen bonding, which results in strong, directional electrostatic interactions. These interactions substantially slow down adsorption kinetics and promote the formation of stable molecular clusters within the pores. In narrow or hydrophilic pores, water often undergoes phase-like transitions (such as pore filling or emptying) that result in metastable states with long residence times, posing significant challenges for numerical simulations.\u003c/p\u003e\u003cp\u003eFor the same reasons, water adsorption is often cooperative: the initial adsorption of a few water molecules reduces the energetic barriers for subsequent uptake. Furthermore, confinement imposes substantial constraints on both rotational and translational degrees of freedom, leading to notable entropy losses. These constraints, in combination with directional bonding, contribute to the complexity and distinct behavior of water under nanoconfinement. Metastability and the mechanism of water adsorption need more fundamental studies.\u003c/p\u003e\u003cp\u003eTable \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e below provides a comparative summary of the main features associated with the adsorption and metastability of water and CO\u003csub\u003e2\u003c/sub\u003e/CH\u003csub\u003e4\u003c/sub\u003e in crystalline microporous materials.\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable id=\"Tab2\" border=\"1\" class=\"fr-table-selection-hover\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eKey differences between the mechanisms of adsorption: water versus CO₂ / CH₄\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\"\u003e\u003cp\u003eFeature\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\"\u003e\u003cp\u003eWater\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\"\u003e\u003cp\u003eCO₂ / CH₄\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eIntermolecular interactions\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003estrong, directional hydrogen bonds, which drive cluster formation inside pores.\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eweak, non-directional van der Waals or quadrupolar interactions.\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eAdsorption behavior\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003ecooperative adsorption: once a few molecules adsorb, it becomes energetically favorable for more to follow rapidly.\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003egradual, smoother isotherms with sharp transitions only at temperature close to the triple point temperature\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eFree energy surface\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003erugged, with deep multi-minima\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003esmooth and continuous, with a limited number of minima\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eGCMC acceptance rate\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eoften very low: inserting new molecules into a dense, hydrogen-bonded cluster has a very low acceptance rate.\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003emoderate to high: insertion energies fluctuate mildly, and moves are more often accepted.\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eMetastability\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003epronounced (e.g., pore filling): it leads to metastable states with long residence times.\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eonly at very low temperature (below the triple point)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eSampling bottlenecks\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003esevere\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eonly at very low temperature (below the triple point)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eMetastability plays a fundamental role in molecular simulations of adsorption, particularly in systems characterized by complex energy landscapes or undergoing rare events such as phase transitions. In this study, we examined the adsorption mechanisms of CO₂, CH₄, and H₂O in crystalline microporous materials, with a focus on how metastable states influence dynamic equilibrium and sorption behavior. We show that the presence of free energy barriers, separating the metastable states, can significantly hinder transitions within simulation-accessible timescales. These barriers—and the corresponding rare fluctuations—are key to understanding adsorption in highly structured porous materials, where conventional interpretations based on equilibrium thermodynamics may be insufficient.\u003c/p\u003e\u003cp\u003eWhile many microporous materials display IUPAC Type I isotherms, this classification oversimplifies the behavior observed in highly ordered frameworks [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Our simulations reveal that, particularly near the adsorbate’s triple point, adsorption proceeds through a low-density phase followed by a sharp pore-filling transition. As the temperature increases, this behavior shifts toward smoother, Type V-like isotherms.\u003c/p\u003e\u003cp\u003eThe geometry of micropores, in combination with framework symmetry and fluid–fluid interactions, gives rise to complex dynamics and unexpected metastable behavior. In the case of CH₄ and CO₂ adsorption, we observed pronounced fluctuations between high- and low-density pore-filling states, governed by the height of the free energy barriers separating them. For CO₂ in IRMOF-1, strong intermolecular CO₂–CO₂ interactions and the high pore symmetry stabilize two distinct adsorption states, effectively suppressing intermediate configurations. In contrast, CH₄ exhibits transitions between competing low-density states, influenced by the spatial distribution of adsorption sites and the underlying framework symmetry [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eWater adsorption represents an even more intricate case [\u003cspan additionalcitationids=\"CR15 CR16\" citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e–\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e, \u003cspan additionalcitationids=\"CR22\" citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e–\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. Unlike simple gases, water’s behavior is governed by directional hydrogen bonding, which promotes the formation of stable molecular clusters and results in multiple distinct free energy minima. These metastable states arise from strong electrostatic interactions and lead to exceptionally slow equilibration in simulations, especially at low temperatures. Therefore, conventional grand canonical Monte Carlo approaches often fail to fully capture the adsorption process, resulting in hysteresis and under-sampling of relevant configurations.\u003c/p\u003e\u003cp\u003eUnderstanding metastable mechanisms is crucial for the rational design of materials for applications such as water harvesting and thermal energy storage. Yet accurately simulating water adsorption under confinement remains a major challenge. No existing force field fully captures all of water’s essential properties—particularly hydrogen bonding, polarizability, and molecular flexibility—when restricted to a nanoscale environment. Furthermore, the intrinsically rugged free energy landscape significantly hampers sampling efficiency and imposes computational cost.\u003c/p\u003e\u003cp\u003eIn conclusion, this study highlights the pivotal role of metastability in governing adsorption processes under confinement. Our findings demonstrate how pore geometry, fluid–fluid and fluid–framework interactions, and temperature jointly shape sorption dynamics across different adsorbates. To achieve realistic and predictive modeling of these systems, particularly for complex molecules like water, future efforts must explicitly account for metastability through advanced sampling techniques and the development of more accurate, physically grounded force fields. These improvements are essential to ensure the rational design of nanoporous materials for practical applications.\u003c/p\u003e\u003cp\u003eFinally, it is noteworthy that an adsorption mechanism similar to this observed in IRMOFs has also been reported in a simplified model of ordered pores [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. This underscores the need for further in-depth investigations to fully understand the role of metastability in the adsorption process across both realistic and idealized systems. By its nature, metastability is the field of research that must be strongly assisted and supported by numerical modeling, using methodologies that allow one to calculate free energy of the studied systems.\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003e\u003cb\u003eExperimental setup\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eSynthesis\u003c/b\u003e\u003c/p\u003e\u003cp\u003eIRMOF-1 was prepared in N,N-diethylformamide (DEF) solution following the procedure described in the literature [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. Due to the absence of product after the initially postulated 8 h reaction time, the synthesis duration was extended to a total of 24 h. All other reagents and solvents were of analytical grade (Sigma Aldrich, TokyoChemical Industry) and were used as received, without further purification.\u003c/p\u003e\u003cp\u003e\u003cb\u003ePowder X-ray diffraction (PXRD) measurement\u003c/b\u003e\u003c/p\u003e\u003cp\u003eThe PXRD pattern of desolvated IRMOF-1 was recorded at room temperature using a STOE STADI P diffractometer equipped with Cu-Kα\u003csub\u003e1\u003c/sub\u003e radiation (λ = 1.54059 Å) and a 1D Mythen detector (Dectris). Data were collected in transmission mode using a rotating flatbed sample holder. The measurements were performed using a steps size of 6° in 2Θ and an exposure time of 20 s per step. Prior to measurement, IRMOF-1 was mounded in the glovebox and sealed under parafilm to prevent exposure to air.\u003c/p\u003e\u003cp\u003e\u003cb\u003eGas adsorption measurements\u003c/b\u003e\u003c/p\u003e\u003cp\u003ePrior to the physisorption measurements, the as-synthesized IRMOF-1 was washed eight times with 20–30 mL of anhydrous DMF, allowing the solid soak for 1–3 h during each washing step. Subsequently, the DMF was decanted, and the sample was washed seven times with 20–30 ml of anhydrous CH2Cl2, again with 1–3 soak intervals per cycle. Following solvent exchange, the sample was dried using the Schlenk technique under a flow of Argon to remove the excess of solvent. Final desolvation was performed under reduced pressure (~ 10\u003csup\u003e− 3\u003c/sup\u003e kPa) at 363 K for ~ 16 h. The disolvated IRMOF-1 was then transferred in the Schlenk tube into glovebox (MBRAUN) ) under an inert atmosphere for storage and handling.\u003c/p\u003e\u003cp\u003eGas adsorption measurements for CO\u003csub\u003e2\u003c/sub\u003e (99.999% purity) and CH\u003csub\u003e4\u003c/sub\u003e (99.999% purity) were conducted using a \u003cem\u003eBELSORP-max\u003c/em\u003e adsorption apparatus (MicrotracBEL Corp.).\u003c/p\u003e\u003cp\u003e\u003cem\u003eBELSORP-max\u003c/em\u003e was connected to the home-built adsorption cell, connected to the closed cycle helium cryostat DE-202AG (ARS). The adsorption temperature was precisely controlled using LS-336 temperature controller (LAKE SHORE), and the excess heat generated by the cryostat was dissipated via water-cooled helium compressor ARS-2HW.\u003c/p\u003e\u003cp\u003eFor each experiment 32.2 mg of desolvated IRMOF-1 was loaded into the adsorption cell, which was sealed externally with a copper dome using a copper gasket. The system was thermally isolated under dynamic vacuum (\u003cem\u003ep\u003c/em\u003e \u0026lt; 10\u003csup\u003e− 4\u003c/sup\u003e kPa) and connected to the \u003cem\u003eBELSORP-max\u003c/em\u003e adsorption instrument using a 1/8 inch stainless steel capillary. Following assembly the sample was degassed under a dynamic ultra-high vacuum of 0.01 Pa at 298 K for 12 h to ensure complete removal of residual adsorbates. Adsorption equilibrium was defined as the conditions under which pressure variations remained within 1% over a period of 500 s.\u003c/p\u003e\u003cp\u003e\u003cb\u003eNumerical setup\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eMethodology of free energy calculations – Transition Matrix Grand Canonical Monte Carlo (TM-GCMC)\u003c/b\u003e\u003c/p\u003e\u003cp\u003eThe free energy profile, Ω(\u003cem\u003eN\u003c/em\u003e), where \u003cem\u003eN\u003c/em\u003e denotes the number of adsorbed molecules, is a fundamental descriptor for identifying true thermodynamic equilibrium states, metastable configurations, and assessing the overall metastability of a system. However, this profile (Fig.\u0026nbsp;1, central panels) cannot be directly obtained from the conventional Metropolis grand canonical Monte Carlo (GCMC) simulations, as the method’s inherent limitations prevent adequate exploration of the full configuration space. To overcome this, the Transition Matrix Monte Carlo (TMMC), has been successfully employed to reconstruct the adsorption free energy landscape. This approach provides valuable insight into both equilibrium and non-equilibrium aspects of the adsorption process. Moreover, kinetic information can be inferred from the energy barriers separating local minima of Ω(\u003cem\u003eN\u003c/em\u003e), to estimate transition rates between metastable and stable states.\u003c/p\u003e\u003cp\u003eThe fundamental principles of the Transition Matrix Grand Canonical Monte Carlo (TM-GCMC) method are extensively documented in the literature [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] and are described in detail in our previous paper [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. This advanced simulation technique offers enhanced sampling efficiency compared to conventional GCMC methods and enables the calculations of free energy profiles as a function of the number of system’s macroscopic states quantified by the number of adsorbed molecules \u003cem\u003eN\u003c/em\u003e, at specified external pressure or chemical potential.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eFunding\u003c/h2\u003e\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eAcknowledgements:\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eM.S., B.M. B.K. and L.F are supported by the National Science Centre (NCN), Poland (grant no. 2022/45/B/ST8/02028). \u0026nbsp;Monte Carlo numerical simulation results are created using resources provided by Wroclaw Centre for Networking and Supercomputing (https://wcss.pl). A.D. and C.W. are supported by the National Science Foundation Grant No. IIP-2044726, the University of Missouri Research Council and Materials Science and Engineering Institute. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions: \u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eConceptualization: B.K., B.M.,\u003c/p\u003e\n\u003cp\u003eMethodology: B.M., B.K.,\u003c/p\u003e\n\u003cp\u003eInvestigation: M.S., A.D., V.B., B.M., K.R.,\u003c/p\u003e\n\u003cp\u003eVisualization: M.S., B.M., B.K.\u003c/p\u003e\n\u003cp\u003eFunding acquisition: B.K., C.W.,\u003c/p\u003e\n\u003cp\u003eProject administration: B.K.,\u003c/p\u003e\n\u003cp\u003eSupervision: B.K., S.K., C.W., L.F.\u003c/p\u003e\n\u003cp\u003eWriting \u0026ndash; original draft: B.K., L.F.\u003c/p\u003e\n\u003cp\u003eWriting \u0026ndash; review \u0026amp; editing: B.K., K.R., B.M., C.W, M.S., L.F.\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eLangmuir I (1918) The adsorption of gases on plane surfaces of glass, mica and platinum. J Am Chem Soc 40(9):1361\u0026ndash;1403\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eRouquerol J et al (2013) Adsorption by powders and porous solids: principles, methodology and applications. Academic\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eThommes M et al (2015) \u003cem\u003ePhysisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report).\u003c/em\u003e Pure and applied chemistry, 87(9\u0026ndash;10): pp. 1051\u0026ndash;1069\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLi Z et al (2025) \u003cem\u003eUnderstanding Pore Filling Processes and Adsorption/Desorption Hysteresis in Nanoporous Metal\u0026ndash;Organic Frameworks: Insights from Grand Canonical Monte Carlo Simulations and Free Energy Calculations.\u003c/em\u003e Langmuir\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eCanivet J et al (2014) Water adsorption in MOFs: fundamentals and applications. Chem Soc Rev 43(16):5594\u0026ndash;5617\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eCho HS et al (2019) Isotherms of individual pores by gas adsorption crystallography. Nat Chem 11(6):562\u0026ndash;570\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBon V et al (2019) Insights into the water adsorption mechanism in the chemically stable zirconium-based MOF DUT-67\u0026ndash;a prospective material for adsorption-driven heat transformations. J Mater Chem A 7(20):12681\u0026ndash;12690\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eMazur B et al (2022) Quasicontinuous Cooperative Adsorption Mechanism in Crystalline Nanoporous Materials. J Phys Chem Lett 13(30):6961\u0026ndash;6965\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKuchta B et al (2017) Adsorption-Induced Structural Phase Transformation in Nanopores. Angew Chem 129(51):16461\u0026ndash;16464\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eFurukawa H et al (2014) Water adsorption in porous metal\u0026ndash;organic frameworks and related materials. J Am Chem Soc 136(11):4369\u0026ndash;4381\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eGupta A et al (2021) Improved computational method to generate properly equilibrated atomistic microstructures. MethodsX 8:101217\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWalton KS et al (2008) Understanding inflections and steps in carbon dioxide adsorption isotherms in metal-organic frameworks. J Am Chem Soc 130(2):406\u0026ndash;407\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eFairen-Jimenez D, Seaton NA, Duren T (2010) Unusual adsorption behavior on metal\u0026thinsp;\u0026ndash;\u0026thinsp;organic frameworks. Langmuir 26(18):14694\u0026ndash;14699\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSarkisov L, Centineo A, Brandani S (2017) Molecular simulation and experiments of water adsorption in a high surface area activated carbon: Hysteresis, scanning curves and spatial organization of water clusters. Carbon 118:127\u0026ndash;138\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eGhosh P, Kim KC, Snurr RQ (2014) Modeling water and ammonia adsorption in hydrophobic metal\u0026ndash;organic frameworks: single components and mixtures. J Phys Chem C 118(2):1102\u0026ndash;1110\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eParanthaman S, Coudert F-X, Fuchs AH (2010) Water adsorption in hydrophobic MOF channels. Phys Chem Chem Phys 12(28):8124\u0026ndash;8130\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eZhang H, Snurr RQ (2017) Computational study of water adsorption in the hydrophobic metal\u0026ndash;organic framework ZIF-8: adsorption mechanism and acceleration of the simulations. J Phys Chem C 121(43):24000\u0026ndash;24010\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSiderius DW, Hatch HW, Shen VK (2024) Flat-Histogram Monte Carlo Simulation of Water Adsorption in Metal\u0026ndash;Organic Frameworks. J Phys Chem B 128(19):4830\u0026ndash;4845\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eDatar A, Witman M, Lin L-C (2021) Improving computational assessment of porous materials for water adsorption applications via flat histogram methods. J Phys Chem C 125(7):4253\u0026ndash;4266\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eGreathouse JA, Allendorf MD (2006) The interaction of water with MOF-5 simulated by molecular dynamics. J Am Chem Soc 128(33):10678\u0026ndash;10679\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eAlkhatib N, Naleem N, Kirmizialtin S (2024) How Does MOF-303 Achieve High Water Uptake and Facile Release Capacity? J Phys Chem C 128(20):8384\u0026ndash;8394\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eAlgara-Siller G et al (2015) Square ice in graphene nanocapillaries. Nature 519(7544):443\u0026ndash;445\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBurtch NC, Jasuja H, Walton KS (2014) Water stability and adsorption in metal\u0026ndash;organic frameworks. Chem Rev 114(20):10575\u0026ndash;10612\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKuchta B, Firlej L, Maurin G (2005) Mechanism of adsorption in cylindrical nanopores: The roles of fluctuations and correlations in stabilizing the adsorbed phase. J Chem Phys, 123(17)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNeimark AV, Vishnyakov A (2006) Phase transitions and criticality in small systems: vapor\u0026thinsp;\u0026ndash;\u0026thinsp;liquid transition in nanoscale spherical cavities. J Phys Chem B 110(19):9403\u0026ndash;9412\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKaye SS et al (2007) Impact of preparation and handling on the hydrogen storage properties of Zn4O (1, 4-benzenedicarboxylate) 3 (MOF-5). J Am Chem Soc 129(46):14176\u0026ndash;14177\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"metastability, adsorption mechanism, nano-pores, phase coexistence, MOFs, water, methane, carbon dioxide","lastPublishedDoi":"10.21203/rs.3.rs-7148852/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7148852/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eUnderstanding molecular adsorption in microporous materials is key to advancing gas separation, storage, and catalysis. Here, we study CO₂ and CH₄ adsorption in crystalline metal-organic frameworks (IRMOF-1, 8, 10, and 14), emphasizing the emergence of metastable states. Molecular simulations reveal that adsorption is governed by a fine balance between fluid\u0026ndash;fluid and fluid\u0026ndash;framework interactions, leading to transitions between low- and high-density pore-filling states. These metastable features are highly sensitive to pore geometry and thermodynamic conditions, especially near the adsorbate\u0026rsquo;s triple point. In contrast, water adsorption displays more complex behavior: strong hydrogen bonding induces stable clusters, multiple free energy minima, and exceptionally slow equilibration. These features often escape conventional simulations. Our results provide fundamental insight into adsorption mechanisms and underscore the importance of metastability in accurately modeling and designing advanced nanoporous materials for practical applications.\u003c/p\u003e","manuscriptTitle":"Adsorption mechanism in crystalline micropores: multimodal fluctuations, phase coexistence and phase transformations in nanoconfinement","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-25 09:11:47","doi":"10.21203/rs.3.rs-7148852/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"5e2df3fc-0a54-44fc-8655-f20bbcd46d98","owner":[],"postedDate":"July 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":52072441,"name":"Physical sciences/Materials science/Theory and computation/Atomistic models"},{"id":52072442,"name":"Physical sciences/Nanoscience and technology/Nanoscale materials/Structural properties"}],"tags":[],"updatedAt":"2025-08-20T14:46:10+00:00","versionOfRecord":[],"versionCreatedAt":"2025-07-25 09:11:47","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7148852","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7148852","identity":"rs-7148852","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}