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# Isotherms of individual pores by gas adsorption crystallography

## Abstract

Accurate measurements and assessments of gas adsorption isotherms are important to characterize porous materials and develop their applications. Although these isotherms provide knowledge of the overall gas uptake within a material, they do not directly give critical information concerning the adsorption behaviour of adsorbates in each individual pore, especially in porous materials in which multiple types of pore are present. Here we show how gas adsorption isotherms can be accurately decomposed into multiple sub-isotherms that correspond to each type of pore within a material. Specifically, two metal–organic frameworks, PCN-224 and ZIF-412, which contain two and three different types of pore, respectively, were used to generate isotherms of individual pores by combining gas adsorption measurements with in situ X-ray diffraction. This isotherm decomposition approach gives access to information about the gas uptake capacity, surface area and accessible pore volume of each individual pore, as well as the impact of pore geometry on the uptake and distribution of different adsorbates within the pores.

## Main

Gas adsorption isotherm studies are an essential and reliable characterization tool for porous materials1,2,3,4,5,6,7,8. They can be coupled with coincident X-ray and neutron diffraction to probe the overall gas adsorption behaviour in porous materials, such as zeolite, mesoporous silica and metal–organic frameworks (MOFs), by taking snapshots of the gas distribution at certain points of an isotherm9,10,11,12,13,14,15,16,17,18 or monitoring the structure transition in flexible MOF crystals during adsorption18,19,20. However, the gas adsorption behaviour in individual pores throughout the entire pore filling, which is critical for the design and application of porous materials, has remained largely unexplored. Recently, we showed that in situ X-ray scattering profiles along the entire pore filling led to the discovery of an extra-adsorption domain and superlattice formation in mesoporous MOFs with single pore type21. This method, named ‘gas adsorption crystallography’, allows for the access to knowledge, such as the subtle changes in the lattice constant, strain of the framework, quantitative and spatial distribution of gases, that reflects the gas behaviour at any stage in the isotherm.

## Results and discussion

### Isotherm of individual pores

The argon isotherm generally serves as a benchmark for the assessment of porosity, due to the simplicity of the Ar atom and its lack of multipole moment30. For two known MOFs, PCN-224 (Fig. 1) and ZIF-412 (Fig. 2), the isotherm decomposition was demonstrated first on Ar isotherms at 87 K (Fig. 3) and then extended to those of CO2 at 194 K and N2 at 77 K (Fig. 4). The PCN-224 crystal (space group of Im$$\bar3$$m), which is composed of Zr6 clusters and square planar porphyrin linkers22, comprises two distinct pore types that form a bi-continuous channel shaped pore at the edges and face centres of the cubic unit cell, and another type at the body centre where six channels intersect (Fig. 1). The channel pores (yellow polyhedra in Fig. 1a,b) are covered by porphyrin walls that connect the adjacent intersection pores (green polyhedron in Fig. 1a,d). In situ X-ray diffraction (XRD) was collected simultaneously at each adsorption point along both the adsorption and desorption process of the Ar isotherm (Fig. 3a,c).

The electron density maps within both pores were generated from the integrated intensities of 176 reflections (ten independent) in the in situ XRD patterns and those of PCN-224 under vacuum using the difference maximum entropy method (MEM)31,32. Sufficient spatial and quantitative accuracy were achieved in these electron density maps to show the Ar adsorption behaviour without the need to make any assumptions. The precise control of the temperature and pressure of the specimen in the instrumental set-up guaranteed the reproducibility of the measurement. The reliability of MEM analysis was demonstrated by comparing refinements on the XRD data at various q ranges with different number of diffraction peaks, with the electron density maps and the amount of adsorbate essentially the same within the individual pores (Supplementary Section 6).

Integration of the electron density map for an individual pore gives the average density of Ar, on the basis of which the exact amount of Ar in the pore was extracted from the total Ar uptake (Fig. 3c–e). In this way, an isotherm for each individual pore was generated by connecting the Ar uptake at each pressure (the sub-isotherm) (Fig. 3d,e). The exact contribution of the Ar uptake in each pore (Nchannel and Nintersection) to the total uptake of the MOF (NPCN-224) can be represented as:

$$N_{{\mathrm{PCN}}{\mbox{-}} 224} = \frac{{6V_{{\mathrm{channel}}}N_{{\mathrm{channel}}} + 2V_{{\mathrm{intersection}}}N_{{\mathrm{intersection}}}}}{{V_{{\mathrm{PCN}}{\mbox{-}} 224}}}$$

where Nchannel and Nintersection represent the number of adsorbates per volume of the corresponding pores and Vchannel, Vintersection and VPCN-224 represent the geometric volumes of the channel pore, intersection pore and PCN-224, respectively. In the crystal structure of PCN-224, one unit cell contains six channel pores and two intersection pores (Supplementary Fig. 3 and Supplementary Table 2).

Distinct differences can be observed between the sub-isotherms of the channel and intersection pores (Fig. 3d,e, respectively). Although the different pores shown on Figs. 1 and 2 are discussed below (Comparison of adsorptive behaviour within and between pores), some trends can first be observed. Ar uptake mostly accumulated from 2 to 11 kPa in the sub-isotherm of the channel pore, whereas in that of the intersection pore a steep Ar uptake was observed from 11 to 15 kPa. This clearly reflects the size difference between these two types of pore (1.5 nm and 2.5 nm, respectively (Supplementary Fig. 2 and Supplementary Tables 23 and 24)). The exact pore sizes were determined by the full-width at half-maximum of the electron density profiles of these pores (Supplementary Fig. 2). In contrast, only one broad peak was observed in the pore diameter distribution generated in the fitting of the overall Ar isotherm with conventional quenched solid density function theory33,34 (Supplementary Fig. 25). The uncovering of two distinct and sequential events in each individual pore was made possible by gas adsorption crystallography, otherwise these remained hidden under the overall isotherm of PCN-224.

In the same way, the sub-isotherms of Ar at 87 K were obtained for ZIF-412, a hierarchical porous MOF composed of Zn and imidazolate linkers (space group of Fm$$\bar3$$m) with three distinct types of pore, lta, fau and ucb17 (Fig. 2a–c,f). The sub-isotherm of each pore was generated on the basis of 536 reflections (26 independent) in the in situ XRD data of ZIF-412 (Fig. 3f–i). Similar to that of PCN-224, the detailed allocation of Ar uptake in each pore (Nlta, Nfau and Nucb) from the total uptake is:

$$N_{{\mathrm{ZIF}}{\mbox{-}} 412} = \frac{{4V_{lta}N_{lta} + 8V_{fau}N_{fau} + 4V_{ucb}N_{ucb}}}{{V_{{\mathrm{ZIF}}{\mbox{-}} 412}}}$$

where Nlta, Nfau and Nucb represent the number of adsorbates per volume of the corresponding pores and Vlta, Vfau, Vucb and VZIF-412 represent the geometric volumes of the three pores and the MOF, ZIF-412, respectively. One unit cell of ZIF-412 contains four lta pores, eight fau pores and four ucb pores (Supplementary Fig. 3 and Supplementary Table 2).

Three distinct and sequential events can be clearly identified in the sub-isotherms of the pores in ZIF-412 attributed to the difference in their pore parameters (Fig. 3). Specifically, pore filling occurred from 0.5 to 4 kPa in the lta pore with a relatively small pore opening (0.8 nm) and pore diameter (2.1 nm) and from 6 to 10 kPa in the fau pore with a larger pore opening (1.5 nm) and only a slightly larger pore diameter (2.5 nm) than that of the lta pore. Such a subtle difference was hidden in the overall isotherm by the analysis using the conventional quenched solid density function theory method33. In the case of the largest pore in ZIF-412, the ucb pore, with two types of pore opening (0.8 and 1.5 nm) and an ultralarge pore diameter of 3.9 nm, the pore filling did not begin until 28 kPa and ended at 31 kPa. We note that the unit cell parameters decreased during the pore filling of the fau and ucb pores, but not in that of the lta pore (Supplementary Fig. 38). This indicates that the Ar adsorption behaviour in both the fau and ucb pores with relative large pore diameters (2.5 and 3.9 nm, respectively) exhibited a capillary condensation mechanism, which is typical for mesopores9,35. In contrast, the Ar adsorption behaviour in the lta pore was similar to micropore filling, although the size of the pore diameter, 2.1 nm, was in the mesopore regime according to the International Union of Pure and Applied Chemistry36. The explicit information above was hardly accessible in the general analysis of the overall isotherms.

### Metrics of individual pores

Surface area and pore volume are the two most important pore metrics for porous materials37,38,39. Here we show that these values can be derived from the Ar sub-isotherm for each pore (Supplementary Tables 25 and 38). The surface area of PCN-224 is 28.0 nm2 per unit cell, derived from the overall Ar isotherm based on the monolayer adsorption. The overall surface area can be divided into six channel pores and two intersection pores in one unit cell, with the contribution from each channel and intersection pore identified as 4.55 nm2 and 0.35 nm2, respectively. This is further validated in their corresponding Ar sub-isotherms. Accordingly, the total pore volume per unit cell is 27.6 nm3 for PCN-224, with those of the channel and intersection pores added up and each contribute 3.13 and 4.39 nm3, respectively. Similarly, the surface area of ZIF-412 per unit cell is 255 nm2 based on the overall Ar isotherm, in which each lta, fau and ucb pore contributes 3.55, 8.06 and 44.2 nm2, respectively. The total pore volume per unit cell is 212 nm3 for ZIF-412 according to the overall isotherm, with the contribution from each lta, fau and ucb pore identified as 3.27, 6.28 and 37.3 nm3, respectively. This isotherm decomposition enabled by gas adsorption crystallography unveiled the pore metrics of individual pores.

### Comparison of adsorptive behaviour within and between pores

The overall isotherm and the sub-isotherms of these MOFs can be accurately divided into several stages by sudden changes in the unit cell parameters and the full-width at half-maximum of the diffraction peaks (Supplementary Figs. 1012, 2224 and 3840). This allowed us to directly compare the uptake of adsorbate between different individual pores at each stage. For example, the Ar isotherm of PCN-224 can be divided into five stages (1 (0–2 kPa), 2 (2–7 kPa), 3 (7–11 kPa), 4 (11–15 kPa) and 5 (15–100 kPa)), which represent monolayer adsorption, multilayer adsorption, pore filling in the channel pores, capillary condensation in the intersection pores and pore saturation, respectively] (Fig. 3c–e and Supplementary Fig. 19). Similarly, for ZIF-412, the overall isotherm and the sub-isotherms of Ar can be divided into seven stages (1 (0–0.5 kPa), 2 (0.5–4 kPa), 3 (4–6 kPa), 4 (6–10 kPa), 5 (10–28 kPa), 6 (28–31 kPa) and 7 (31–100 kPa)), which represent monolayer adsorption in all three pores, pore filling in the lta pore, multilayer adsorption in the fau and ucb pores, capillary condensation in the fau pores, multilayer adsorption in the ucb pore, capillary condensation in the ucb pore and saturation, respectively) (Fig. 3f–i and Supplementary Figs. 2931). The CO2 and N2 overall isotherm and the sub-isotherms were divided in the same way on both PCN-224 (Supplementary Figs. 20 and 21) and ZIF-412 (Supplementary Figs. 3237). We previously studied IRMOF-74-V-hex, a honeycomb-like structure constructed by a Mg oxide unit with a terphenylene organic linker functionalized with a hexyl chain; the channel pore of this material provides an additional type of individual pore for comparison21,40 (Supplementary Figs. 716 and Supplementary Tables 612).

In multilayer adsorption stage, correlations were found between the densities of adsorbates and the geometry of each individual pore. In the pore geometry, pore diameter describes the internal space of the pore, and pore opening describes the size of the pathways between adjacent pores. Comparison of electron density maps among different individual pores for the same adsorbate revealed the relation between the pore opening and the average density of CO2, where a much higher CO2 density was found in pores with larger pore openings. Specifically, the CO2 density increased in the order ucb pore of ZIF-412 (0.8 nm), channel pore of PCN-224 (1.5 nm), fau pore of ZIF-412 (1.5 nm) and channel pore of IRMOF-74-V-hex (3.5 nm), as the pore opening increased (Fig. 5). In contrast, the densities of Ar and N2 correlated with the pore diameter, instead of the pore opening. The pores with larger pore diameters, such as the ucb pore of ZIF-412 (3.9 nm) and the hexagonal channel pore of IRMOF-74-V-hex (3.5 nm), exhibited higher Ar and N2 densities than those with smaller pore diameters, such as the fau pore of ZIF-412 (2.5 nm) and the channel pore of PCN-224 (1.5 nm). These correlations were further confirmed by plotting the sizes of pore diameters and pore openings against the densities of Ar and CO2 in these individual pores (Fig. 5a,b, respectively). Here, another MOF, IRMOF-74-IV, a single pore type with a smaller pore diameter and opening (2.8 nm for both) than those of IRMOF-74-V-hex, was added as an additional point in the comparison. The results are consistent with the analysis above (Fig. 5a,b).

## Conclusion

Extension of gas adsorption crystallography from MOFs with a single type of pore to those with multiple types of pore allowed us to uncover the hidden adsorption stages under the overall isotherm, and to analyse gas adsorption in each individual pore in a continuous manner along the entire isotherm. This led to the disclosure of previously inaccessible information concerning the subtle difference in pore metrics between the pores, the adsorbate framework interactions and the impact of pore geometry on the multilayer adsorption of different adsorbates. We believe this approach is applicable to a large variety of MOFs as well as other types of porous materials, such as zeolite and mesoporous silica, with hierarchically arranged pores. The capability to elucidate gas adsorption at the individual pore level will be of great interest to scientists who work on computation and modelling as well as chemists for the rational design of porous materials.

## Methods

### MOF synthesis

The MOFs with multiple types of pore used for gas adsorption crystallography in this study, which include PCN-224 and ZIF-412, were synthesized on the basis of reported procedures with slight modifications to ensure purity22,23. Before the gas adsorption crystallography experiments, all the MOFs were prepared and confirmed in solvent-free form. Complete activation was achieved by extensive solvent exchange with methanol, followed by outgassing under vacuum. The XRD profiles of both MOFs under vacuum matched well with those derived from their single-crystal structure, which ensures both the phase purity and the absence of solvent molecules in the pores.

The set-up for gas adsorption crystallography has two essential components. One is the gas adsorption instrument, BELSORP-max, equipped with a customized sample cell on a cryostat21. The other is XRD synchronized with the adsorption instrument to take in situ measurements; we used a Rigaku BioSAXS-1000 equipped with a rotating anode, FR-E+ SuperBright. The MOF samples (~0.03 g) were precisely weighed before they were mounted in the sample cell, followed by on-site activation at 373 K for 6 h under vacuum (~0.01 Pa) for the removal of any moisture introduced during the sample transfer. The Ar, CO2 and N2 isotherms were measured at 87 K, 77 K and 194 K, respectively. It took approximately 30 min for the system to reach equilibrium for each pressure point (fluctuation of pressure in the sample cell <1 Pa for 5 min). Each XRD pattern was collected at each equilibrium point of the sorption isotherms with an exposure time of 30 min, during which no pressure change was observed. Before the in situ XRD was applied for data collection, to verify the quality of MOF sample gas adsorption isotherms were taken on site without the involvement of XRD measurement. These isotherms also provided guidance for the set-up of pressure points in the follow-up in situ XRD measurements. The stages in the isotherms of ZIF-412 and PCN-224 (seven and five, respectively) were divided by the difference in the slopes of these isotherms (Supplementary Figs. 4–6). At least five equilibrium points were collected in each stage of the adsorption and desorption process, with an XRD profile taken for each equilibrium point, including the initial point under vacuum. Note that the samples did not undergo any structural transformation after in situ gas adsorption XRD measurement. This was confirmed by the reproducible adsorption data and the unaltered XRD profiles under vacuum.

### Structural analysis

Le Bail refinements47 were performed on each XRD profile using the JANA program48 over the full q range given by the detector in the instrument. The space groups used for the refinement of PCN-224 and ZIF-412 were Im$$\bar3$$m and Fm$$\bar3$$m, respectively. The XRD profiles of MOFs at the vacuum condition were used as references. The reflection peaks in each experimental XRD profile were modelled by a Pseudo-Voigt peak-shape function modified for asymmetry using six refinement coefficients. The background of the XRD profile was treated using a Legendre polynomial with six refinement parameters. The structure factor amplitudes were calculated from the integrated intensities of reflections in the XRD profile with corrections for the polarization factor, Lorentz factor and scale factor. The scale factor used here was determined by comparing the intensities observed in the experimental XRD profile at the vacuum condition with the model structure factors derived from single-crystal XRD data of the corresponding MOF. Phases of the model structure factors were adopted from the initial phases set of structure factors at the vacuum condition. Along the adsorption process, the change in phases of the structure factor was carefully monitored by plotting the structure factor amplitude of each reflection against gas pressure. Sudden alteration in the slope of the structure factor amplitude curve, which indicated the presence of a phase flip except for special events, such as condensation. MEM was applied on the experimentally derived structure factors to determine the electron density distributions in the pores of the MOF structures at each gas pressure. The program Dysnomia32 for MEM analysis was applied with the L-BFGS algorithm and the so-called ‘G-constraint’ for exactly overlapping reflections. The accumulated distribution of gases at each stage was visualized by subtracting electron densities at the beginning of the stage from that of the end. The gradient of electron density (e Å–3) in each electron density map is reflected in the BGR colour code. The relative position of the gases in each individual pores was visualized by displaying the atomic coordinates of the pore structure in the electron distribution maps using the program VESTA49. The gravimetric volume of each pore of the MOFs in which we counted the number of adsorbates was defined by a polyhedron, where its vertex is the centre of the metal clusters (Supplementary Fig. 3). Detailed methods and analysis are provided in Supplementary Section 2. Electron distributions in the pores are given in the Supplementary Sections 35.

## Data availability

The data that support the findings in this study are available within the article and its Supplementary Information and/or from the corresponding authors on reasonable request.

## Code availability

The program VESTA was coded by K.M. and is available free of charge via public domain at http://jp-minerals.org/vesta/en/.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## Acknowledgements

We acknowledge financial support from BK21+ program, the Center for Hybrid Interface Materials (2013M3A6B1078884) and the National Research Foundation of Korea (2017M2A2A6A01070673) (H.S.C., J.K.K. and O.T.), CEM, SPST of ShanghaiTech University (no. EM02161943) (H.S.C. and O.T.), Foreign 1000 Talents Plan, China (O.T.), King Abdulaziz City for Science and Technology, Riyadh, Saudi Arabia (O.M.Y.), National Natural Science Foundation of China (no. 21471118, 91545205 and 91622103) and National Key Basic Research Program of China (no. 2014CB239203) (X.G. and H.D.), NSFC 21522105 (Y.B.Z.) and an Advanced European Research Grant (ERC, no. 321140) (H.S.C. and B.M.W.). We also thank R. Flaig for proofreading the manuscript, and X. Cai for providing the three-dimensional structure illustration of each individual cage.

## Author information

### Author notes

1. These authors contributed equally: Hae Sung Cho, Jingjing Yang, Xuan Gong.

### Affiliations

1. #### Graduate School of EEWS, KAIST, Daejeon, Republic of Korea

• Hae Sung Cho
• , Jeung Ku Kang
•  & Osamu Terasaki
2. #### School of Physical Science and Technology, ShanghaiTech University, Shanghai, China

• Hae Sung Cho
• , Yue-Biao Zhang
•  & Osamu Terasaki
3. #### Inorganic Chemistry and Catalysis group, Debye Institute for Nanomaterials Science, Faculty of Science, Utrecht University, Utrecht, Netherlands

• Hae Sung Cho
•  & Bert M. Weckhuysen
4. #### UC Berkeley-Wuhan University Joint Innovative Center, Institute for Advanced Studies, Wuhan University, Luojiashan, Wuhan, China

• Jingjing Yang
• , Xuan Gong
• , Hexiang Deng
•  & Omar M. Yaghi
5. #### Department of Chemistry, University of California, Berkeley , Berkeley, CA, USA

• Jingjing Yang
•  & Omar M. Yaghi
6. #### Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, USA

• Jingjing Yang
•  & Omar M. Yaghi
7. #### Kavli Energy NanoSciences Institute, Berkeley, CA, USA

• Jingjing Yang
•  & Omar M. Yaghi
8. #### Key Laboratory of Biomedical Polymers-Ministry of Education, College of Chemistry and Molecular Sciences, Wuhan University, Wuhan, China

• Xuan Gong
•  & Hexiang Deng
9. #### National Museum of Nature and Science, Tsukuba, Ibaraki, Japan

• Koichi Momma
10. #### King Abdulaziz City for Science and Technology, Riyadh, Saudi Arabia

• Omar M. Yaghi
11. #### Department of Materials and Environmental Chemistry, Stockholm University, Stockholm, Sweden

• Osamu Terasaki

### Contributions

O.T. and O.M.Y. conceived the idea. O.T., O.M.Y. and H.D. led the project. H.S.C. performed the in situ XRD experiments and analysis. J.Y., X.G., Y.-B.Z. and H.D. prepared the samples, and K.M. coded the computer program VESTA. O.T, H.S.C. and J.K.K. contributed to set up the experimental system. H.S.C., J.Y., X.G., H.D., O.M.Y. and O.T. prepared the first version of the manuscript and all the authors contributed to the final version.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Hexiang Deng or Omar M. Yaghi or Osamu Terasaki.

## Supplementary information

1. ### Supplementary Information

Supplementary data, discussion and methods, Supplementary Figs. 1–52 and Supplementary references 1–11.