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# Reminiscent capillarity in subnanopores

## Abstract

Fluids in large and small pores display different behaviors with a crossover described through the concept of critical capillarity. Here we report experimental and simulation data for various siliceous zeolites and adsorbates that show unexpected reminiscent capillarity for such nanoporous materials. For pore sizes D exceeding the fluid molecule size, the filling pressures p are found to follow a generic behavior kBT ln pγ/ρD where γ and ρ are the fluid surface tension and density. This result is rationalized by showing that the filling chemical potential for such ultra-small pores is the sum of an adsorption energy and a capillary energy that remains meaningful even for severe confinements. A phenomenological model, based on Derjaguin’s formalism to bridge macroscopic and molecular theories for condensation in porous materials, is developed to account for the behavior of fluids confined down to the molecular scale from simple parameters.

## Introduction

Confinement of fluids in porous media involves a diversity of phenomena such as physical adsorption, chemical reactions, and solubilisation which give rise to complex behaviors including wetting/adhesion, nucleation, slippage, surface diffusion, surface reconstruction, etc. Owing to their ultra-small pore size D and large surface area, nanoporous (D ~ nm) and subnanoporous (D < nm) solids such as zeolites, active carbons, and metal organic frameworks constitute a critical subfamily of the broad class of porous materials with specific applications in phase separation, catalysis, etc.1,2,3. Yet, despite their increasing role in fundamental and applied science, the behavior of fluids in these materials still remains only partially explored by many aspects. Predicting the thermodynamical equilibrium of a given fluid in one of these nano/subnanoporous hosts requires to carry out on purpose specific experiments or molecular simulations as there is no general macroscopic theory relying on a simple set of known experimental parameters (e.g., density, surface tension, etc.). From a fundamental standpoint, confinement in porous solids with subnanometric to nanometric cavities was first described by extending classical adsorption and capillarity theories4,5. In practice, while physical models such as the celebrated Langmuir, BET or Kelvin equations were found to be qualitatively valid for such small pores, it was soon realized that their use in this specific context remains essentially empirical (for instance, the concept of independent adsorption sites and adsorbed layers in such ultraconfining media is clearly inconsistent with their geometry and atomic structure).

The advent of molecular theories such as the Density Functional Theory (DFT) in statistical mechanics6,7 and atom-scale simulations has allowed describing the physics of fluids in nanoporous media and establishing a bridge with the classical thermodynamics for large pores8,9,10. From this new era, a unified picture has emerged with a crossover between irreversible capillary condensation—i.e., first order transition—for large pores and reversible, continuous pore filling—i.e., second order transition—for small pores (Fig. 1)6,10,11,12,13. On the one hand, for large pores D 10σ (D and σ are, respectively, the sizes of the pore and of the confined molecules), pore filling first involves the formation of an adsorbed film followed by hysteretic capillary condensation at a pressure smaller than the bulk saturating vapor pressure p0. On the other hand, for small pores D ~ σ, pore filling does not involve a well-defined confined gas/liquid interface and occurs through a continuous and progressive density increase of the confined fluid. The crossover between these two asymptotic limits is described through the concept of the capillary critical temperature Tcc which is shifted with respect to the bulk critical temperature Tc (Fig. 1); for a given pore size D, capillary condensation occurs for T < Tcc(D) while pore filling is reversible and continuous for T > Tcc(D). Reciprocally, for a given T, capillary condensation occurs for pore sizes D > Dc while filling is reversible and continuous for D < Dc where Dc ~ 4σTc/(Tc − Tcc)10,12. Despite the comprehensive picture above, a macroscopic theory capable to predict the quantitative thermodynamic behavior (typically the pore filling pressure) of fluids in ultraconfining media is still missing. Such a theory is highly desirable as available molecular tools suffer from the following limitations. On the one hand, atom-scale simulations are efficient but time-consuming and, more importantly, forcefield-dependent so that large departure with respect to experiments cannot be ruled out. On the other hand, Density Functional Theory is more practical but, being a mean-field theory, it requires to calibrate interaction parameters against some already existing available experiments.

Here, we use experimental and molecular simulation data to establish a simple macroscopic model that allows predicting the behavior of any confined fluid in subnanoporous and nanoporous solids using simple parameters such as density and surface tension. Non-polar and polar fluids with drastically different bulk properties are used to obtain a fluid-independent, general picture when confined in such solids. Siliceous zeolites (the so-called zeosils) are considered as they exhibit pores of a regular geometry combined with a simple surface chemistry and therefore allow developing a macroscopic phenomenological model. We first show that, unless pores are smaller than the confined fluid size D < σ, all our data follow a master behavior in which the chemical potential is proportional to a capillary energy, μ ~ γ/ρD, but is shifted by an offset corresponding to an adsorption energy. This important finding, which suggests a reminiscent capillary behavior even in ultra-confining pores, allows bridging the gap between molecular theories relevant to small pores and capillary concepts relevant to macroscopic pores. From this observation, we build on Derjaguin’s theory for adsorption and capillary condensation14,15,16,17 to extend its framework to subnanoporous media. We show that its unexpected applicability to such materials arises because the following asymptotic limit is reached. Upon decreasing D, the filling pressure tends to p ~ 0 with a corresponding chemical potential μ ~ kBT ln p that is given by the sum of a capillary contribution and an adsorption contribution. The first contribution μcap ~ γ/ρD corresponds to the chemical potential predicted from the surface tension and the surface to volume ratio using the capillarity theory taken in the limit of vanishing adsorption (i.e., the film thickness is vanishingly small). The second contribution μads ~ f(t, ξ) is an adsorption contribution, dependent on the surface interaction range ξ and film thickness t. As expected, for nanoporous and subnanoporous media, this adsorption contribution is by no means negligible compared to the capillary contribution. Yet, its variations when considering the different fluids, temperatures, etc. used here remain small compared to the capillary contribution so that the latter mostly governs the chemical potential at pore filling. A large body of experimental data taken from the literature confirm the simple picture underlying the proposed model. This phenomenological approach presents some limitations (for instance, when solids with significant surface heterogeneity are considered as it leads to far more scattered adsorption contributions in the chemical potential at pore filling) but its simplicity and robustness makes it a very promising tool in fields where ultra-confining media are relevant (phase separation, catalysis, membrane science in chemistry/chemical engineering, nanofluidics and energy storage/conversion in physics, depollution and fluid transfer in earth, and soil science, etc.).

## Results

Different molecules were considered to probe a broad range of zeolite/fluid interactions (Fig. 2): acetone (a highly polar oxygenated hydrocarbon), n-hexane (a weakly polar aliphatic hydrocarbon), p-xylene (a non-polar aromatic hydrocarbon with aliphatic substitution) at room temperature and nitrogen at 77 K (a simple molecular probe used for routine characterization of porous materials). Details of the experimental measurements can be found in the Methods section. Some important physico-chemical properties of these molecular probes are summarized in Supplementary Table 1. Similarly, several representative zeolites were considered in this work (Fig. 2): beta zeolite (*BEA)18, silicalite-1 (MFI)19, chabazite (CHA)20, and STT21. The three-dimensional channel networks in *BEA, MFI, and CHA are delimited by 12, 10, and 8 membered-rings, respectively, while the two-dimensional channel network in STT is made up of odd openings formed by 7 and 9 membered-rings. The main structural parameters of these zeolites are reported in Supplementary Table 2 while an additional view of their porous network structure is shown in Supplementary Fig. 1.

Gas adsorption in these siliceous zeolites was assessed for the four adsorbates using Monte Carlo simulations in the Grand Canonical Monte Carlo ensemble (GCMC)22. Like in a real experimental adsorption set-up, this statistical mechanics technique allows considering a host porous medium having a constant volume V in equilibrium with a bulk reservoir of gas molecules that imposes its temperature T and chemical potential μ. Once equilibrium is reached, the adsorbed amount n(μ, T) at a given μ and T is readily obtained as an ensemble average of the number of molecules in the zeolite. The adsorption isotherm n(p, T) is then plotted by converting μ into gas pressure p. For each gas, the relationship μ(p) was determined using the Widom insertion method in the course of NpT Monte Carlo simulations. Such simulations are efficient for bulk phases as considered here (since the relationship μ(p) has to be determined for the bulk fluids only). As for the adsorption simulations using the GCMC algorithm, especially for complex fluids such as some of the fluids considered here, it is known that proper equilibration to reach the physical density of the confined fluid suffers from technical limitations. More in detail, due to the intrinsic difficulty in inserting molecules into the small pores of nanoporous solids, GCMC can be very slowly converging to reach the equilibrium density. To circumvent poor sampling efficiency and guarantee convergence towards equilibrium, our GCMC molecular simulations were performed using the Configurational-Bias algorithm23. All details regarding the applied computational technique and the molecular models used for the different gases and zeolites are described in the Methods section. A detailed analysis of the simulated data can be found in Supplementary Figs. 35 and Supplementary Tables 4 and 5 together with a thorough comparison with available experimental and simulation data in the Supplementary Discussion (including a comparison of adsorbed amounts, Henry constants, isosteric heats of adsorption). The supplementary information also contains the structure file for each of the 4 zeolite structures (in .car format).

The adsorption isotherms for the different gases and zeolite structures are shown in Fig. 3. As expected for subnanoporous materials such as zeolites, all adsorption isotherms are of type I—Langmuir-like—according to IUPAC classification; the adsorbed amount increases rapidly at very low pressure in a continuous and reversible fashion (note the use in Fig. 3 of a log scale for the pressure axis to better display the very low pressure range). As shown in Supplementary Fig. 6, in agreement with the experimental data, for each fluid and zeolite, the plateau reached at saturation can be correctly predicted from the known porous volume of the zeolite if the density of the confined adsorbate is taken equal to the bulk density. This relation, known as Gurvich’s rule5, suggests that the molecular simulations have reached equilibrium and that the configuration-bias employed in the framework of GCMC simulations allows efficient phase space sampling. From a practical viewpoint, such a simple scaling provides a simple and rapid means to estimate adsorption capacities for any adsorbate/adsorbent couple.

### Reminiscent capillarity

The capillarity dependence of pore filling in nanopores/subnanopores as evidenced in Fig. 4 might appear as a surprising result. Yet, as will be shown in the remaining of this paper, such a scaling can be rationalized through simple thermodynamic arguments. Typically, the data shown in Fig. 4 suggest that a classical, macroscopic behavior remains meaningful at least in an effective way. While this result is rather unexpected for such ultraconfining pores, it is consistent with results from molecular simulation and classical DFT for simple pores which suggest that the scaling predicted from Laplace equation remains appropriate at least qualitatively24. In fact, rather than a true capillarity effect, such capillarity dependence should be referred to reminiscent capillarity since pore filling in nanoporous and subnanoporous media exhibits specificity that departs from the well-established capillary regime. In particular, as explained in the introduction, for ultra-confining media, the absence upon adsorption of a well-defined gas/liquid interface within the porosity renders the concept of surface tension ambiguous. For such blurred interfaces, pore filling becomes reversible and continuous (second order transition) in stark contrast with the capillary-driven regime which corresponds to a discontinuous and irreversible process (first order transition). As a result, the pseudo-capillarity observed in Fig. 4 should rather be considered as reminiscent capillarity relevant to the strong surface to volume ratio in these systems with an associated surface tension that remains physical and meaningful down to pore sizes that are comparable with the granularity of the confined fluid.

As shown in what follows, such reminiscent capillarity can be predicted using simple classical thermodynamic modeling. Motivated by the simple capillarity scaling observed in Fig. 4, we attempt to predict vapor adsorption in zeosils using Derjaguin’s formalism which allows describing both adsorption and condensation in porous media. In the framework of the Gibbs dividing surface concept, the grand potential Ω of a pore with a diameter D and the fluid film of a thickness t adsorbed at the vapor pressure pV and temperature T writes15,16:

$$\Omega = - p_{\mathrm{V}}V_{\mathrm{V}} - p_{\mathrm{L}}V_{\mathrm{L}} + \gamma _{{\mathrm{SL}}}A_{{\mathrm{SL}}} + \gamma _{{\mathrm{LV}}}A_{{\mathrm{LV}}} + A_{{\mathrm{LV}}}W_{{\mathrm{SLV}}}(t)$$
(1)

where pV, pL, VV, VL, and VS are the pressure and volume of the vapor and adsorbed phases, respectively. γLV, γSL, ALV, and ASL are the liquid/gas and liquid/solid surface tensions and surface areas, respectively. The interface potential WSLV(t) in the above equation, which allows describing adsorption at the solid surface, accounts for the interaction between the liquid/solid and adsorbate/gas interfaces. WSLV(t) is linked to the disjoining pressure Π(t) = −dWSLV(t)/dt = pV − pL for planar interfaces25,26. Throughout this study, only bulk values are used for the liquid density and surface tension. However, the disjoining pressure can be seen as a correction to the bulk surface tensions due to interface coupling, i.e., Π(t) = (γSL(t) + γLV(t))/∂t, so that the approach above does account for confinement. In order to derive an expression for the surface potential WSLV(t), we write that it must verify the following condition: as t becomes much larger than the characteristic interaction range ξ, the interactions between the gas/adsorbate and adsorbate/solid interfaces vanish—i.e., WSLV(t) → 0 for t → ∞. On the other hand, for t → 0, the adsorbate vanishes and the surface contribution in the grand potential given in Eq. (1) reduces to γSVASV so that WSLV(0) → S where S is the spreading coefficient defined by27:

$$S = \gamma _{\rm{SV}} - \gamma _{\rm{SL}} - \gamma _{\rm{LV}}$$
(2)

Among possible functions, WSLV(t) = S exp(−t/ξ) verifies the above asymptotic limits and has been shown to allow reproducing adsorption of different fluids on various pore surfaces28. For a van der Waals fluid, WSLV(t) HSLV/t2 where HSLV is the so-called Hamaker constant—which is representative of the interaction strength between the solid/liquid and gas/liquid interfaces—is often proposed in the literature. In practice, such a power law scaling is valid for a finite film thickness t only as it displays a non-physical divergence in the limit of vanishing films. To go beyond this specific van der Waals model and account for the non divergence of WSLV(t) when t vanishes, several works suggest that a generic scaling WSLV(t)  exp(−t/ξ) is more accurate and suitable to describe experimental/simulation data16. In capturing the so-called interface coupling, the disjoining pressure Π(t) and the underlying surface potential WSLV(t) are effective parameters which also account for the microscopic details of nanoconfined fluids—such as the strong layering observed in classical DFT and molecular simulation—in a continuum behavior picture. In this sense, a generic and empirical exponentially decaying behavior for WSLV(t), with a prefactor and a lengscale corresponding to the strength and range of the interface coupling, is justified.

For a given gas pressure pV, the stable solution predicted using Derjaguin’s model is obtained by determining the minimum in the grand potential Ω(t)—defined in Eq. (1)—upon varying t. For cylindrical mesopores, in agreement with previous works, the model is found to be in excellent agreement with the experimental data obtained for MCM-41 silica of a diameter D = 4.7 nm as shown in Fig. 529. At low pressures, the only stable solution of the model corresponds to an adsorbed film of a thickness t. At the transition pressure $$p_{\mathrm{V}}^{\mathrm{e}}$$, the grand potential Ω(t) displays two minima which correspond to the same grand potential; These two solutions correspond to a configuration with an adsorbed film of finite thickness, i.e., t < D/2, coexisting with a configuration where the pore is completely filled with the liquid, i.e., t = D/2. At pressures above $$p_{\mathrm{V}}^{\mathrm{e}}$$, the film can remain stable in a metastable fashion until it collapses at the pressure where the corresponding grand potential minimum disappears. A single set of parameters in Derjaguin’s model (S = 0.069 J m−2 and ξ = 0.24 nm), fitted against a single experimental adsorption isotherm shown in Fig. 5a, allows reproducing, without any additional fitting and yet with a very good agreement, the experimental data obtained with other pore sizes (Fig. 5b). Note that both the parameters S and ξ are obtained from a single fit as there is only one combination that allows capturing quantitatively the adsorption (low pressure) and capillary condensation (high pressure) regimes. Clearly, the value ξ ~ 0.24 shows that the correction from the adsorbed film and the interface coupling is far from being negligible for nanopores. In particular, in agreement with experimental data on regular porous silica (for which D is simply a few times ξ), the correction due to the disjoining term—which decays over a typical lengthscale ξ — leads to quantitative departures from the conventional Kelvin equation. By comparing Derjaguin’s model for the cylindrical and spherical geometries with the predictions of the corresponding Kelvin equation, Fig. 5b clearly shows that the chemical potential at pore filling is the sum of a capillary contribution, μcap, given by the Kelvin equation derived for each pore geometry, and an adsorption contribution μads, which depends on the surface potential WSLV(t) = S exp(−t/ξ) through the characteristic surface interaction range ξ and spreading parameter S.

Figure 4 suggests that all the microporous structures considered in this work can be modeled using an effective spherical geometry as the slope ~6 of the master curve coincides with the expected value for spherical pores. The factor ~6 is far from being a trivial result as some of the zeolites considered here present pores closer to the cylindrical geometry. It is believed that, due to the severe confinement experienced by the fluid in these ultra-small pores, the confinement is equivalent in each of the 3 directions (x, y, z). In other words, when calculating the interaction field for a confined fluid, the direction along the pore axis is almost as confining as the other directions because of the vicinity of all zeolite atoms at the pore surface. Coming back to the thermodynamic model, for the spherical pore geometry, the vapor and liquid volumes expressed in terms of pore diameter D and film thickness t are VV = 1/6π(D − 2t)3 and VL = 1/6π[D3 − (D − 2t)3]. The surface areas of the liquid/vapor and solid/liquid interfaces are, respectively, ALV = π(D − 2t)2 and ASL = πD2. For a given pore diameter D, the gas pressure $$p_{\mathrm{V}}^{\mathrm{e}}(D)$$ at which pore filling occurs can be determined by writing the following condition. The grand free energy Ω(t) of the configuration corresponding to the low density phase equals that of the filled configuration Ω(t = D/2) =  −pLVL + γSLASL with VL = πD3/6. After a little algebra, one arrives at:

$$(D - 2t)(p_{\mathrm{L}} - p_{\mathrm{V}}^{\mathrm{e}}(D)) + 6\gamma _{{\mathrm{LV}}} + 6S\exp ( - t/\xi ) = 0$$
(3)

Using the Gibbs–Duhem relation, i.e., dp = ρdμ, for both the adsorbed and gas phases, it can be shown that $$p_{\mathrm{L}} - p_{\mathrm{V}}^{\mathrm{e}}(D) = RT\rho _{\mathrm{L}}\ln [p_{\mathrm{V}}^{\mathrm{e}}(D)/p_0]$$ where p0 is the bulk saturating vapor pressure while R is the ideal gas constant (note that in deriving this relation we neglect the density of the gas phase before that of the adsorbed phase which is taken equal to the liquid phase density ρL). Upon inserting the latter expression in Eq. (3), one arrives at the filling pressure for a spherical pore of a diameter D in which adsorption occurs prior to capillary filling:

$$RT\ln \frac{{p_{\mathrm{V}}^{\mathrm{e}}(D)}}{{p_0}} = - \frac{{6\gamma _{{\mathrm{LV}}}}}{{\rho _{\mathrm{L}}(D - 2t)}}\left[ {1 + \frac{{S\exp ( - t/\xi )}}{{\gamma _{{\mathrm{LV}}}}}} \right]$$
(4)

The equilibrium configuration Ω(t) is a thermodynamic minimum, i.e., dΩ(t)/dt = 0, which leads after derivation of Eq. (1) to the following expression: $$S\exp ( - t/\xi )(D - 2t + 4\xi ) =- \rho _{\mathrm{L}}RT\ln [p_{\mathrm{V}}^{\mathrm{e}}(D)/p_0](D - 2t)\xi - 4\gamma _{{\mathrm{LV}}}\xi$$. Upon inserting this expression into Eq. (4), it is straightforward to show that:

$$RT\ln \frac{{p_{\mathrm{V}}^{\mathrm{e}}(D)}}{{p_0}} = - \frac{{6\gamma _{{\mathrm{LV}}}}}{{\rho _{\mathrm{L}}(D - 2t - 2\xi )}} = - \frac{{6\gamma _{{\mathrm{LV}}}}}{{\rho _{\mathrm{L}}D}}\left[ {1 + \frac{{2t + 2\xi }}{{D - 2t - 2\xi }}} \right]$$
(5)

This equation rigorously predicts the slope, μ ~ −6γ/ρD, observed in the linear master curve shown in Fig. 4. That capillarity remains relevant to pore filling in such small nanoporous materials is clearly an unexpected result. Yet, it is fully consistent with already available experimental, theoretical and molecular simulation data which all converge to show the following results. On the one hand, the pore condensation pressure is always found to be lower than the capillary condensation pressure predicted using simple surface to volume ratio arguments (Kelvin equation and any extension that includes ad hoc the adsorbed film thickness such as the well-known BJH method). On the other hand, such condensation pressures always follow the expected capillary scaling with a quantitative departure that vanishes as pores get large enough to recover the conventional macroscopic behavior. These results are fully consistent with the picture emerging from the present work with a chemical potential at pore filling that includes both capillary and adsorption energy contributions.

## Discussion

The agreement between the simulated/experimental data and the theoretical modeling supports the proposed idea of reminiscent capillarity in subnanoporous media. The underlying picture consists of a chemical potential at pore filling that can be subdivided into an adsorption energy and a capillary energy. For such ultra-small pores, these two terms are equally important but capillarity imposes a pore size dependence which is reminiscent of the macroscopic theory. Some remarks are in order regarding the use of the word “reminiscent” in “reminiscent capillarity”. On the one hand, while our approach supports the idea that capillary concepts still apply in some effective and quantitative fashion, the fact that pore filling becomes reversible and continuous in very small pores (i.e., above the so-called capillary critical temperature Tcc) indicates that first-order capillary condensation does not occur. On the other hand, while capillary condensation does not apply stricto sensu to the situations considered here, the pore filling observed “does remind of capillary condensation” especially since the use of the corresponding quantitative parameters (surface tension, Laplace pressure, etc.) seems appropriate.

Far from being a trivial result, our genuine finding is very important by many practical and fundamental aspects. The unraveled reminiscent capillarity dependence goes well beyond a simple asymptotic limit as an important crossover between capillary condensation in large pores (first order phase transition) and continuous and reversible filling in small pores (second order phase transition) still persists behind the apparent unifying capillary behavior established in the present work. Indeed, despite the unexpected robustness of capillarity as revealed in the present work, it is known that there is for a given temperature a critical pore diameter below which pore filling no longer proceeds through capillary condensation but corresponds to a continuous and reversible process. Equivalently, for a given pore size, there is a so-called capillary critical temperature above which capillary condensation is suppressed and replaced by continuous and reversible pore filling. Our findings are consistent with this picture and the apparent capillarity observed here suggests a reminiscent behavior of the physics in large pores rather than a simple mathematical extension. The applicability down to the molecular scale of macroscopic concepts such as those involved in capillarity (surface tension, Laplace pressure, etc.) is somewhat analog to the extension of hydrodynamics to fluids in nanoporous media. Indeed, while hydrodynamics in these ultraconfined environments must be corrected to account for novel phenomena such as slippage and memory effects, its underlying physics remains valid.

In any case, by bringing fundamental insights into the crossover between small and large nanopores (i.e., the conventional frontier between microporous solids D < 2 nm and mesoporous solids D ~ 2–50 nm), these findings help close—even if only partially—the gap between the classical/macroscopic thermodynamics of porous media and the nanophysics that has emerged with the advent of nanoporous materials. More in detail, while the filling/emptying kinetics for microporous and mesoporous solids are expected to be different, the idea of underlying capillary concepts that remain meaningful even in ultraconfined environments allows reconciling these two regimes. In the light of these results, the nanoscale appears as a particular lengthscale where different physical effects compete: adsorption, confinement, surface interactions, etc. From a practical viewpoint, while conventional techniques are usually valid for a given fluid in a specific type of porous solids, our results also pave the way for extended characterization techniques that would apply to any system. However, by many aspects, the simple model used to account for such reminiscent capillarity must be improved as it remains at this stage limited to homogeneous porous solids. For instance, the theoretical approach reported here does not apply to solids exhibiting surface chemistry heterogeneity where adsorption sites lead to a complex adsorption contribution in the chemical potential at pore filling. This includes for instance processes involving adsorption of polar molecules on very strong adsorption sites but also adsorption in cationic zeolites. Moreover, as in the case of porous solids with important morphological (pore shape) disorder, we do not expect the simple approach used in the present work to apply for highly disordered materials although the concept of capillarity and surface free energy minimization with respect to the free energy volume should remain key ingredients. Finally, in recent years, an increasing amount of effort has been devoted to the coupling of adsorption and deformation in nanoporous solids. The thermodynamic formalism to capture such effects is intrinsically more complex than for non-compliant materials (typically, one has to use a hybrid ensemble with a thermodynamic potential that includes both the grand free energy of the fluid and the free energy or free enthalpy of the host solid). Yet, as far as the fluid contribution is concerned, we expect the results reported in the present work to remain meaningful.

## Methods

Nitrogen adsorption/desorption isotherms were measured using a Micromeritics ASAP 2420 apparatus. Prior to the adsorption measurements, the calcined samples were out-gassed at 300 °C overnight under vacuum. For the other fluids considered in this work, dynamic adsorption measurements were performed under different atmospheres of volatile organic compounds (n-hexane, p-xylene, acetone) and controlled values of relative pressure p/p0 = 0.5 (p is the pressure and p0 the saturation vapor pressure at a given temperature T of the considered organic compound) using a thermogravimetric balance Setaram TG92 instrument30. The experiments were done under flow. The relative pressure p/p0 = 0.5 was obtained by setting the pressures of auxiliary gas and carrier gas to 1.5 bar at the inlet of the oven and controlled by measuring the gas flow rate at the outlet of the oven. The gas flow rate was found to be stable (114 ml.min−1). Prior to the adsorption experiment, a preliminary activation phase was accomplished which consisted in heating up the zeosil to 350 °C with the aim to remove all adsorbate traces. Subsequently, the sample was cooled back to T = 25 °C and the organic compound was then introduced in the system to saturate the zeosil. The adsorbed amount was reported every 20 s. Experiments were performed on 100 mg of zeosil.

### Computational models

All investigated zeolite structures (Supplementary Fig. 1) were simulated under their purely silicate form and were maintained rigid during the simulation, with framework atoms fixed to their crystallographic positions. For CHA, STT, and BETAPA type zeolites the atomic positions were taken from the IZA database. For the zeolite beta, whose crystal structure is formed by an inter-growth of two crystallographic forms, designed, respectively, A and B, we have focused uniquely on the A polymorph in our simulation work. The silicalite-1 is known to exist in three distinct forms: a monoclinic one with Pnma space group and two orthorhombic forms with Pnma and P212121 space groups, designed respectively as MONO, ORTHO, and PARA. silicalite-1 is usually synthesized in the ORTHO form and passes into the low temperature MONO system after calcination. Further, this low temperature form (MONO) experiences a reversible phase transition into the ORTHO structure at about 350 K, as well as under adsorption of various molecules. Finally, upon adsorption of certain adsorbates such as nitrogen at 77 K or p-xylene the ORTHO structure moves into the PARA one. Thus, in order to reproduce in a basic manner the experimental isotherms, we have considered separately the low pressure (ORTHO) and the high pressure (PARA) systems as previously done by Snurr et al.31. The atomic positions for the ORTHO silicalite-1 structure were taken from the IZA database32, whereas those for the PARA silicalite-1 were taken from van Koningsveld et al.33.

The investigated molecules are described through a “united atom” model that has been employed successfully for the investigation of their adsorption behavior in zeolites34 and MOFs35. In this model, each –CHx– (with 0 ≥ x ≤ 3) and (=O) group is treated as a single interaction site. Such “united atoms” are connected by bonds maintained at fixed distances. In addition, several interaction sites bear a partial charge, contributing to the interaction energy through the Coulombic term. The nitrogen molecule is described via an explicit model: each nitrogen atom of the rigid molecule constitutes a single interaction center bearing a negative partial charge. In order to compensate the negative charges on the nitrogen atoms, there is a positive partial charge bearing no-interacting site in the middle of the nitrogen-nitrogen bond. Such three sites model allows reproducing the experimentally measured quadrupole moment of the nitrogen molecule. The intermolecular interactions between the adsorbate molecules were modeled using a sum of repulsion-dispersion potential term expressed as the Lennard–Jones interaction and the Coulombic interaction. The cross LJ terms were calculated applying the Lorentz–Berthelot combination rules. The bond distances, the partial charges and the interatomic potential parameters for all investigated molecules are summarized in Supplementary Table 3. Within the frame of the selected models, the nitrogen and p-xylene molecules are considered as rigid; therefore no intramolecular interactions are taken into account. While the intramolecular interactions for the acetone molecules are described solely via an harmonic bending term, for the n-hexane an additional dihedral torsion angle term is considered expressed by a cosine series potential. The parameters corresponding to those terms have been taken from the Transferable Potential for the Phase Equilibrium (TraPPE) forcefield, respectively, for n-hexane36, p-xylene37, acetone38, and nitrogen39 (initially fitted to reproduce the liquid/vapor coexistence curves of various fluid molecules and summarized in Supplementary Table 3).

The absolute adsorption isotherms of n-hexane, acetone, p-xylene at 298 K and nitrogen at 77 K were computed in each zeosil using the Monte Carlo simulation within the Grand Canonical ensemble implemented within the code MCCCS Towhee40. These simulations consisted of evaluating the average number of adsorbate molecules whose chemical potential equals those of the bulk phase for given chemical potential and temperature. The chemical potential values were calculated by the test particle Widom insertion method from the NpT ensemble Monte Carlo simulation. The conventional scheme of the GCMC simulation for flexible, long chain molecules is expensive in computational time as the fraction of successful insertion moves is too low. Moreover, it does not fully explore the conformation part of the configuration space and thus does not allow achieving a proper distribution of bending and dihedral angles. The configurational-biased algorithm overcomes such shortcomings by sampling more efficiently the configuration space23. We have applied the coupled – decoupled biased selection scheme developed by Martin and Siepmann, performing a coupled biased selection for Lennard-Jones and torsion angles selection steps, while decoupling the angle bending energy into split biased selections. The detail description of the particular algorithm employed in our simulation can be found in Martin and Siepmann41,42. The structures of considered zeosils were treated as rigid and the periodic conditions were applied. A typical Monte Carlo run consisted of 3 × 106 steps. Each step corresponded to a single MC move, including a center of mass translation, center of mass rotation, insertion of a new molecule, deletion of a randomly selected existing molecule, partial or complete regrowth of the adsorbate. The Ewald summation technique was used in the calculation of the long-range electrostatic interactions.

## Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding authors on reasonable request. All simulations were performed using the software TOWHEE: MCCCS Towhee-Version 7.0.6 (July 27, 2013).

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## Acknowledgements

I.D. thanks the Pôle HPC and Equipex Equip@Meso at the University of Strasbourg. This work was supported by the French Research Agency (ANR TAMTAM 15-CE08-0008 and LyStEn 15-CE06-0006).

## Author information

Authors

### Contributions

I.D. and B.C. designed the work. I.D. performed and analyzed the molecular simulations while T.J.D. carried out the adsorption experiments. B.C. developed the theoretical model with the help of C.P. B.C wrote the paper with input from all the authors.

### Corresponding authors

Correspondence to Irena Deroche or Benoit Coasne.

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Deroche, I., Daou, T.J., Picard, C. et al. Reminiscent capillarity in subnanopores. Nat Commun 10, 4642 (2019). https://doi.org/10.1038/s41467-019-12418-9

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• ### Bridging scales in disordered porous media by mapping molecular dynamics onto intermittent Brownian motion

• Colin Bousige
• , Pierre Levitz
•  & Benoit Coasne

Nature Communications (2021)