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J. Phys. Chem. C 2007, 111, 15350-15356
Calculating Geometric Surface Areas as a Characterization Tool for Metal-Organic Frameworks Tina Du1 ren,† Franck Millange,‡ Ge´ rard Fe´ rey,‡ Krista S. Walton,§ and Randall Q. Snurr| Institute for Materials and Processes, School of Engineering and Electronics, UniVersity of Edinburgh, King’s Buildings, Edinburgh, EH9 3JL, United Kingdom, Institut LaVoisier Versailles, UMR8180, UniVersite´ de Versailles St-Quentin-en-YVelines, 45 AVenue des Etats-Unis, 78035 Versailles Cedex, France, Department of Chemical Engineering, Kansas State UniVersity, 1005 Durland Hall, Manhattan, Kansas 66506, and Department of Chemical and Biological Engineering, Northwestern UniVersity, 2145 Sheridan Road, EVanston, Illinois 60208 ReceiVed: June 18, 2007; In Final Form: August 2, 2007
Metal-organic frameworks (MOFs) synthesized in a building-block approach from organic linkers and metal corner units offer the opportunity to design materials with high surface areas for adsorption applications by assembling the appropriate building blocks. In this paper, we show that the surface area calculated in a geometric fashion from the crystal structure is a useful tool for characterizing MOFs. We argue that the accessible surface area rather than the widely used Connolly surface area is the appropriate surface area to characterize crystalline solids for adsorption applications. The accessible surface area calculated with a probe diameter corresponding to the adsorbate of interest provides a simple way to screen and compare adsorbents. We investigate the effects of the probe molecule diameter on the accessible surface area and discuss the implications for increasing the surface area of metal-organic frameworks by the use of catenated structures. We also demonstrate that the accessible surface area provides a useful tool for judging the quality of a synthesized sample. Experimental surface areas can be adversely affected by incomplete solvent removal during activation, crystal collapse, or interpenetration. The easily calculated accessible surface area provides a benchmark for the theoretical upper limit for a perfect crystal.
Introduction One of the most important properties for characterization of an adsorbent is the surface area. In the low-coverage range, the amount adsorbed is often strongly correlated to the heat of adsorption, and at high loading, the adsorption capacity is clearly limited by the pore volume. However, there is a substantial intermediate coverage range where the surface area dominates adsorption.1 Linear relationships between the amount adsorbed and the surface area have been found for different classes of porous materials such as carbons,2 zeolites,3 and metal-organic frameworks.4 Synthesizing materials with high specific surface areas therefore seems to be a viable strategy to create materials with high uptake for gas-storage applications. Metal-organic frameworks (MOFs) are a relatively new class of materials synthesized in a building-block approach from corner and linker units5-8 with potential applications in gas storage4,9-17 and gas separation.18-27 Because of their modular synthesis, they offer the opportunity to rationally design materials with high surface areas by choosing appropriate building blocks, resulting in many examples of MOFs with exceptionally high surface areas.13,28-30 Experimental surface areas are reported as either the Langmuir surface area or the BET surface area derived from nitrogen adsorption isotherms measured at 77 K. Both methods assume a close packing of molecules in a monolayer (Langmuir) or in multilayers (BET).31 For materials with pores large enough to support multiple layers, use of the Langmuir isotherm to obtain * Corresponding author. E-mail:
[email protected]. † University of Edinburgh. ‡ Institut Lavoisier Versailles. § Kansas State University. | Northwestern University.
the surface area is clearly inappropriate, and the BET model is more suitable. It should be kept in mind that the surface area is not a direct experimental observable; it must be calculated from the nitrogen isotherm assuming a model such as BET. BET surface areas for a particular MOF material reported in the literature, even by the same group, vary widely as demonstrated in Table 1. There are two possible reasons for these variations: the pressure range chosen for the BET analysis and the quality of sample. The standard BET analysis is performed for a pressure range of 0.05 < P/P0 < 0.3. However, microporous MOFs show saturation well below the standard range. To choose the pressure range appropriate for a particular adsorbent and to avoid ambiguity when reporting the BET surface area, two consistency criteria have been proposed:31,38 (1) the straight line fitted to the BET plot must have a positive intercept, and (2) the pressure range should be chosen so that Vads(1 - P/P0) is always increasing with P/P0. Synthesizing and activating a MOF sample (i.e., removing any unreacted reactants and solvent molecules from the cavities) is not straightforward.33 A recent study of IRMOF-1 (MOF-5) showed that depending on the synthesis route the surface area is reduced by cavities occupied by Zn(OH)2 or solvent molecules and by the presence of a minor phase of doubly interpenetrated IRMOF-1 frameworks.39 Looking at the variations and ambiguities in the BET surface areas, it becomes apparent that a reliable way of determining the surface area that avoids the assumptions regarding the adsorption mechanism would be beneficial. For many porous materials, analysis of adsorption isotherms provides the only possibility for determining their surface areas. However, for
10.1021/jp074723h CCC: $37.00 © 2007 American Chemical Society Published on Web 10/02/2007
Calculating Geometric Surface Areas
Figure 1. Definition of the accessible surface area (green line), the Connolly surface area (red line), and the van der Waals surface area (blue line). Note that the van der Waals surface area is illustrated only on the lower surface for clarity.
crystalline materials, the surface area can also be calculated directly from the crystal structure by using geometric methods.40 A popular method to characterize porous solids geometrically33,41,42 is the Connolly surface area,43 which is widely used in protein science and is implemented in commercial software packages such as Cerius.2 It is calculated from the bottom of a probe molecule rolling across the surface as illustrated in Figure 1. By looking at the Connolly surface areas of different fragments of a graphene sheet, Yaghi and co-workers derived a theoretical upper limit for the specific surface area of a MOF and concluded that having single rather than condensed rings leads to increased specific surface area because of the number of exposed ring faces and edges.41 The accessible surface area is a different method of calculating the surface area. As shown in Figure 1, it is calculated from the center of a probe molecule rolling along the surface.40 In this paper, we argue that the accessible surface area rather than the Connolly surface area is more appropriate to characterize porous solids for assessing their adsorption performance. Furthermore, we present results showing the effect of probe molecule diameter on the accessible surface area and discuss implications for increasing the surface area of metal-organic frameworks by the use of catenated structures.44 Finally, by comparing the accessible surface area to the BET surface areas calculated from simulated and experimental nitrogen adsorption isotherms, we demonstrate that the accessible surface area presents a quick and easy way to judge whether the pores in the adsorbent samples are accessible and not blocked by solvent molecules or partially collapsed or catenated frameworks.
J. Phys. Chem. C, Vol. 111, No. 42, 2007 15351 the UFF force field (H ) 2.57 Å, C ) 3.43 Å, O ) 3.12 Å, Cr ) 2.69 Å, Cu ) 3.11 Å).47 In the literature, values for the Lennard-Jones parameter, σ, of nitrogen vary between 3.575 48 and 3.798 Å,49 resulting in a variation of the accessible surface area for all of the MOFs studied in this work of less than 4%. All values reported in this paper for nitrogen were calculated with a probe diameter of 3.681 Å.50 Connolly surface areas were calculated with Materials Studio (version 4.0, Accelrys, San Diego, CA) using a nitrogen-sized probe molecule (diameter 3.68 Å) and a grid interval of 0.25 Å. Grand canonical Monte Carlo simulations were used to calculate nitrogen adsorption isotherms at 77 K with full simulation details given previously.51 The BET surface areas calculated from simulated isotherms allow a more consistent comparison with the accessible surface area because both of them are determined in a perfect crystal, whereas issues such as incomplete solvent removal, sample purity, or crystal defects can complicate the interpretation of experimental measurements. The simulated isotherms were analyzed in the same way as experimental isotherms to determine BET surface areas. We report two BET surface areas: one calculated with the standard BET range (0.05 < P/P0 < 0.3) and the other with the appropriate pressure range determined using the consistency criteria as explained above. Experimental Work. Two HKUST-1 samples (A and B) were prepared by solvothermal methods closely following the procedure described in ref 33. Benzene-1,3,5-tricarboxylic acid (2.10 g, 10 mmol, Aldrich) and copper (II) nitrate hemipentahydrate (2.41 g, 10 mmol, Aldrich) were stirred for 15 min in 50 mL of solvent consisting of equal parts N,N-dimethylformamide (Fluka), ethanol (Fluka), and deionized water in a 250-L wide Teflon jar. Hydrofluoric acid (10 mmol) was added to the mixture. The solution was refluxed at 110 °C (sample A) or 85 °C (sample B) for 12 h to yield octahedral crystals of the desired phase, providing initial optical confirmation of product purity. The blue as-synthesized materials were scrubbed in N,N-dimethylformamide at room temperature to remove unreacted reagents and immersed in the appropriate activation solvent (dichloromethane) to exchange the occluded solvent for 3 d, during which the activation solvent was decanted and freshly replenished three times. After activation, the solvent was evacuated in situ under vacuum at 180 °C in the adsorption apparatus prior to sorption measurements. Nitrogen adsorption isotherms were measured at 77 K using a Micromeritics ASAP 2010. Results and Discussions
Methods Simulations. The accessible surface areas were calculated from a simple Monte Carlo integration technique where the probe molecule is “rolled” over the framework surface. For this, a probe molecule was randomly inserted around each of the framework atoms in turn and checked for overlap with other framework atoms. The fraction of the probe molecules that did not overlap with other framework atoms was then used to calculate the accessible surface area. The source code of the FORTRAN program including instructions on how to use it can be found on our webpage.45 The diameters of the framework atoms of the IRMOFs were taken from the Dreiding force field (H ) 2.85 Å, C ) 3.47 Å, O ) 3.03 Å, Zn ) 4.04 Å).46 Because the Dreiding force field does not contain parameters for chromium and copper and to demonstrate that the results are not unique to the Dreiding force field, the diameters of the framework atoms for HKUST-1 and MIL-53 were taken from
First, let us consider why the accessible surface area is more appropriate than the Connolly surface area to characterize porous solids for adsorption applications. Fundamentally, the accessible surface area is closely related to the volume accessible to molecular centers. This accessible volume is related to the entropy of the adsorbed molecules and is important in statistical mechanical theories.52 Another reason for preferring the accessible surface area over the Connolly surface area as a means to characterize porous materials is illustrated in Figure 2 a and b. In two dimensions, consider the simple example of a square surface on which circular molecules can adsorb, either on the outside (Figure 2a) or on the inside (Figure 2b). The Connolly surface area is almost identical for the two cases, whereas the accessible surface area is considerably smaller for the case where molecules adsorb on the inside. If the surface area divided by the area of the adsorbate molecule is taken to get a simple estimate for the adsorption capacity of the material, then using
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TABLE 1: Variation of Reported BET Surface Areas (m2/g) for MOFs and the Corresponding Pressure Ranges Used in the BET Analysis experimental (0.02 < P/P0 < 0.1)32
experimental (0.02 < P/P0 < 0.3)33
experimental single-point BET
experimental (no range given)
IRMOF-1
3534
2833
283334
IRMOF-6
2804
2476
251634
IRMOF-11
1984
209634
191135
1944
36
69237 96527 11544 133327
HKUST-1
1507
1264
57217 22934 336235
the Connolly surface area would lead to the same adsorption capacity for both cases. This is clearly wrong, as demonstrated by Figure 2a and b. In contrast, the accessible surface area is much better suited to give an estimate for the adsorption capacity and therefore to characterize porous solids for adsorption applications. However, as illustrated in Figure 2c, the accessible surface area depends on the size of the probe molecule, a point we will revisit later. Although this is a very simple example, this idea also holds true in real three-dimensional materials. In particular, it should be recognized that the scaffold structures of MOFs exhibit a combination of areas where molecules can adsorb on the outside curvature (e.g., around the edges of exposed ring faces) and on the inside curvature (e.g., corners of cavities). The theoretical upper limit for the surface area of MOFs derived by Yaghi and co-workers focused on the outside curvature of isolated sixmembered rings and provides useful insight into the design of new materials.41 For real materials, of course, the individual rings must be linked together. This necessarily results in corners, where the inside curvature reduces the accessible surface area below that of a simple planar surface. There is, therefore, a compromise between exposed edges and corners.
To explore this compromise further, we investigated the effect of the probe diameter on the accessible surface area and the resulting implications for the use of catenated MOF structures to increase the surface area in a series of isoreticular MOFs (IRMOFs). These materials all have the same basic framework topology.9,53 They consist of oxide-centered Zn4O tetrahedra linked by dicarboxylate molecules to form extended cubic networks as shown in Figure 3. By using linker molecules with different lengths, one can synthesize materials with different cavity sizes.9 (Note that the linkers are orientated in such a way that each of the noncatenated IRMOFs exhibits two different cavities: one where all of the aromatic groups are pointing in and one where they are pointing out as shown in Figure 3c.) If the linker molecules are large enough, then materials consisting of two catenated frameworks can also form,13 as shown in Figure 3c. A list of all of the materials studied in this work, their linkers, and corresponding pore sizes is given in Table 2. Figure 4 shows the accessible surface area per volume as a function of the probe diameter for a series of IRMOFs and illustrates the tradeoff between exposed edges, which increase the accessible surface area, and corners, which reduce it. Here, a probe diameter of 0 Å results in the van der Waals surface
Figure 2. (a) Two-dimensional schematic illustration of molecules adsorbed on the outside of a box and the corresponding surface areas (Connolly surface: red, dotted line; accessible surface: green, dashed line). (b) Two-dimensional schematic illustration of molecules adsorbed inside the same box. (c) Two-dimensional schematic illustration of the accessible surface area measured on the inside and outside of the box with a small (dark blue, dash-dotted line) and large (light blue, dashed line) probe molecule.
Figure 3. (a) Schematic representation of the self-assembly process of a metal-organic framework from corner and linker units. (b) Building blocks for IRMOF-13 and IRMOF-14. IRMOF-13 is the catenated form of IRMOF-14. (c) The resulting materials. The transparent sphere was added to demonstrate the size of the cavities. For clarity, the two frameworks in the catenated IRMOF-13 are shown in different shades.
Calculating Geometric Surface Areas
J. Phys. Chem. C, Vol. 111, No. 42, 2007 15353
TABLE 2: Chemical Formula, Linker Molecule, and Pore Diameters of the MOFs Studied in This Worka
a Pore diameters were calculated according to the method of Gelb and Gubbins.54
Figure 4. Accessible surface area, Sacc, as function of the probe diameter (a) for four noncatenated structures (], IRMOF-1; 9, IRMOF6; b, IRMOF-10; ×, IRMOF-16), (b) for two catenated structures (O, IRMOF-9; ∆, IRMOF-13) and their noncatenated counterparts (b, IRMOF-10; 2, IRMOF-14).
area (compare Figure 1), and probe sizes of 2.96, 3.73, and 5.13 Å correspond to the Lennard-Jones diameters commonly used in molecular simulations for hydrogen,57 methane,58 and SF6,59 respectively. Figure 4a shows that the van der Waals surface area per volume (Sacc) is largest for the smaller IRMOFs. In addition, IRMOF-1 and IRMOF-6 have similar cavity sizes, but the additional surface group on IRMOF-6 leads to an increased van der Waals surface area (Figure 4a). With increasing probe size, the accessible surface area of all noncatenated materials is increasing at first. In this regime, the increase of the surface area at the exposed edges dominates over the decrease in the corners (compare Figure 2c). All curves show a maximum where this relation is reversed, and Sacc decreases with increasing probe size. This maximum occurs at smaller probe diameters for the smaller IRMOFs because they have a smaller proportion of exposed edges to corners. Figure 4b shows the same relation for two catenated frameworks and their noncatenated counterparts. As expected, the van der Waals surface area per volume of the catenated frameworks is about twice as large as that of the noncatenated structures. Because the catenated frameworks contain more corners, the maximum is shifted toward smaller probe sizes or in the case of IRMOF-13 no maximum is observed. Furthermore, both catenated frameworks have cavities smaller than 5 Å (compare Table 2). These cavities are therefore inaccessible for larger molecules, resulting in a steep decrease in Sacc. Overall, this leads to the, at first sight, counterintuitive fact that the accessible surface area per volume can be smaller for catenated frameworks than for their noncatenated counterparts, depending on the size of the molecular probe. This clearly shows that for calculating the accessible surface area and comparing different materials, the probe diameter should be that of the adsorbate of interest. How do these theoretical findings relate to experiments? Comparing the calculated accessible surface areas directly with BET values obtained from experimental nitrogen isotherms is complicated by several issues. Experimental samples inevitably contain defects and will deviate to some degree from the perfect crystal structures used in calculating the accessible surface area.
Differences between the two surface areas could thus be due to the quality of the sample. But different pressure ranges employed for the BET analysis and inherent limitations of the BET theory could also be responsible. In previous work, we got around this problem by using molecular simulation to predict nitrogen isotherms (from the perfect crystal structure) and then used the simulated isotherms as pseudo-experimental data.51 The BET surface areas calculated from the simulated isotherms agree very well with the accessible surface area calculated in a geometric fashion from the crystal structure, validating the use of the BET theory for MOFs. To investigate the influence of the pressure range used in the BET analysis, we determined the BET surface areas from the simulated adsorption isotherms for the standard range as well as a range that takes the two consistency criteria into account. Table 3 shows a comparison between the accessible surface area, the Connolly surface area, and the BET surface areas determined from experimental and simulated nitrogen adsorption isotherms. The corresponding pressure ranges are given for the BET results. For the two noncatenated IRMOFs for which experimental data are available (IRMOF-1 and IRMOF-6), the BET surface areas from experimental and simulation data agree very well. This is especially true for the experimental BET surface areas determined for a pressure range of 0.02 < P/P0 < 0.1, which is closest to the pressure ranges for the simulated isotherms determined with the two consistency criteria. The values determined from the GCMC simulations are slightly higher than the experimental values, reflecting that the simulations are carried out in a perfect crystal. A full discussion about the simulation results including explanations why the BET model works surprisingly well for these microporous materials is given in ref 51. For the simulated data, we observed the same variation of the surface area with the pressure range as for the experimental data where the surface area is larger if a consistent pressure range is chosen. This emphasizes that BET surface areas should always be given with the range used for analysis and that the consistency of the chosen pressure range has to be checked carefully in order to allow reproducibility and comparison.
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TABLE 3: Simulated and Experimental Surface Areas (m2/g) acc. surface area IRMOF-1 IRMOF-6 IRMOF-9 IRMOF-10 IRMOF-11 IRMOF-13 IRMOF-14 IRMOF-16 IRMOF-18 HKUST-1 MIL-53ht(Cr) a
3580 3050 4017 4970 2702 2998 4926 6166 1869 2153 1504
Connolly surface area 3396 3770 3957 3722 3561 3457 3775 4097 3722
sim. (cons. criteria)
pressure range
sim. (standard BET range)
exp. 353432 280432
3619 2956
0.004-0.013 0.001-0.012
2830 2309
4972
0.005-0.017
4991 198432
4625 6146 1842
0.001-0.025 0.07-0.098 0.0002-0.0005
pressure range 0.02-0.1 0.02-0.1 0.02-0.1
4805 6146 1481 194432 135056
0.02-0.1 0.0001-0.03c
exp.
pressure range
283333 247633 190433
0.02-0.3 0.02-0.3
209634 155133 14539 123936
a b
0.02-0.3 0.02-0.3 a
No pressure range given. b Single-point BET. c Determined using the consistency criteria.
How does the accessible surface area correspond to these values? The accessible surface area shows very good agreement with the simulated best-fit values and therefore the experimental values. Yet, because it takes only minutes to calculate the accessible surface area rather than days to calculate a complete nitrogen adsorption isotherm by GCMC simulation, the accessible surface area provides a convenient way to determine a theoretical limiting value for the BET surface area. In contrast, the Connolly surface areas do not correlate with the BET areas, as has been reported before.33 This is a further indication that apart from the theoretical reasons outlined above, the Connolly surface area is less suitable to characterize porous solids for adsorption applications. For the noncatenated IRMOFs consisting of flat linkers (IRMOF-1, -10, -14, and -16), the Connolly surface area is smaller than the accessible surface area because for these materials the outside edges on the linkers dominate (compare Figure 2a). For IRMOF-6 and -18 where the additional surface groups in the linkers lead to additional corners, the Connolly surface area is larger than the accessible surface area (compare Figure 2b). The situation is slightly more complex for the catenated structures. To understand why the Connolly surface area is smaller than the accessible surface area for IRMOF-9 but larger for IRMOF-11 and IRMOF-13, one has to keep in mind that IRMOF-9 is an interpenetrated structure where the two frameworks are maximally displaced from each other, whereas IRMOF-11 and -13 are interwoven structures where the frameworks are minimally displaced and are in close contact. A closer look at the pore size distribution (not shown) for the catenated IRMOF structures reveals that the number of larger pores is larger in IRMOF-9, resulting in a predominance of the exposed edges and therefore yielding an accessible surface area that is larger than the Connolly surface area. Table 3 also demonstrates that the accessible surface area provides an easy way to detect when there is something wrong with experimental samples. The experimentally measured surface area for IRMOF-14 is 1453 m2/g, which is considerably smaller than the calculated accessible surface area (4926 m2/g). A comparison with the BET surface areas calculated from the simulated nitrogen isotherms supports the accessible surface area.51 Because the PDC linker in IRMOF14 is longer and contains more carbon atoms than the BDC linker in IRMOF-1 (see Table 2), the surface area of IRMOF14 should be larger as demonstrated by the accessible surface area and the BET surface area determined from the simulated isotherm. This indicates that there might have been problems with the experimental sample such as pores blocked by incomplete solvent removal, partial collapse, or catenation of the sample. Similarly, the discrepancy between the experimental
BET surface area and the accessible surface area in the catenated frameworks is most likely due to impartial solvent removal and partial collapse of the framework (e.g., IRMOF-9 shows a pronounced loss of crystallinity upon solvent removal33). To demonstrate that the good agreement between the accessible surface area and experimental BET surface areas determined for an appropriate pressure range is not restricted to the cubic IRMOF frameworks comprising three-dimensional pores (see Figure 3), we studied two more materials with different pore structures: HKUST-137 and MIL-53ht.56 HKUST-1 (sometimes also called CuBTC) comprises a complex threedimensional channel system of interconnected smaller and larger cavities, whereas MIL-53ht comprises straight, one-dimensional, diamond-shaped pores. As shown in Table 3, the agreement between the accessible surface area and the experimental BET surface area is good, proving the broad applicability of our concept and pointing toward an important application of the accessible surface area: providing a lead about the quality of an experimental sample. Obtaining a pure sample and finding a suitable activation procedure for MOFs, that is, removing solvent and unreacted reactants completely from the pore space, is challenging.33,39,56,60 An example of how the activation procedure can influence the maximum amount adsorbed (and therefore the pore volume) as well as the surface area is demonstrated in Figure 5, which shows the nitrogen isotherms for two HKUST-1 samples synthesized at two different temperatures (110 °C for sample A and 85 °C for sample B). It is obvious that the different synthesis conditions strongly affect the isotherms. The corresponding BET surface areas determined by taking into account the two consistency criteria are 1680 m2/g for sample A and 1850 m2/g for sample B, respectively. The synthesis procedures used for the preparation of sample B (1850 m2/g) and the sample previously reported by Wong-Foy and co-workers32 (1944 m2/
Figure 5. Low-pressure nitrogen adsorption isotherms for HKUST-1 at 77 K (∆, sample A; 9, sample B).
Calculating Geometric Surface Areas g) only differ by the use of HF although it is not clear what role HF could play at this stage. It has to be noted that the BET surface area is derived by completely different means than the accessible surface area. Yet, as we have shown, the experimental surface area carefully determined by taking the two consistency criteria into account compares very well with the accessible surface area determined for an ideal crystal. Therefore, this comparison can shed instant light if something went completely wrong with the experimental sample. An example is MIL-68, which has an accessible surface area of 3333 m2/g, whereas the reported experimental surface area is only 603 m2/g because of the free terephthalic acid and DMF molecules still present inside the pores.61 However, the final proof that the framework is pure and does not contain any solvent molecules can only be provided by advanced experimental techniques such as careful analysis of powder XRD patterns39 or NMR.62 Yet, the ease of its calculation makes the accessible surface area a useful property to provide a quick benchmark to judge the quality of the synthesized material and should be used as one of several methods for characterization. Conclusions One of the most important characteristics for a metal-organic framework, or any other porous material, used for adsorption applications is the surface area. We have shown that the accessible surface area rather than the Connolly surface area should be used to characterize porous, crystalline solids. For screening and comparing porous materials for adsorption applications, the accessible surface area calculated with a probe diameter corresponding to the diameter of the adsorbate of interest is an easily calculated property that can give valuable information about the expected adsorption performance. When trying to design materials with large accessible surface areas, it must be remembered that the surface area is controlled by two opposing effects: exposed edges lead to an increase, and corners to a decrease compared to a flat plane. The accessible surface area depends on the diameter of the probe molecule. For small probe molecules, the increase due to the exposed edges dominates, whereas for larger probe molecules the decrease due to the corners dominates, leading to a maximum in the accessible surface area with increasing probe diameter (or adsorbate diameter). This leads to the at-first-sight counterintuitive fact that the accessible surface area per volume of a catenated framework can be smaller than that of its noncatenated counterpart. Another practical application of the accessible surface area is that it provides a benchmark for synthesized materials. The accessible surface area compares well with BET surface areas determined from GCMC simulations of nitrogen adsorption isotherms as well as experimentally measured values for goodquality samples. It therefore provides a quick and easy way to get some information about the quality of a synthesized sample: if the experimentally determined BET surface area is much smaller than the accessible surface area, then it is an indication that the cavities of the sample are not completely empty but are partially blocked by, for example, solvent, unreacted reactants, partial framework collapse, or catenation. Comparing BET surface areas from experimental nitrogen isotherms with calculated accessible surface areas could be useful as a routine characterization method for MOFs with known crystal structures. Acknowledgment. This work was partially supported by the Nuffield Foundation, the EU (DeSANNS-FP6-SES6-020133),
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