the BET method is not applicable for microporous adsorbents. However, the determination of the surface area by nitrogen physisorption is generally ac-cepted in the literature as an adequate expression for the surface area if certain quality criteria are applied.[227,228]
1. The empirical constantCmust be positive.
2. The term n·(p/p0) must increase monotonically in the chosen range of p/p0.
3. The endpoint of monolayer formation(p/p0)nm = √1
C+1must lie within the chosen range ofp/p0.
BET plots of adsorptions isotherms of nitrogen physisorption may exhibit linearity in more than one range ofp/p0. The application of these quality criteria provides reproducibility by excluding certain pressure ranges.
2.4 Crystal Structure Determination of Metal-Organic
imposed by the difficulty to obtain crystals that are suitable for single crystal diffraction experiments. Crystals that are too small may not provide enough lat-tice planes for satisfactory diffraction and a broadening of the reflections is ob-served because destructive inteference is incomplete close to the Bragg angles.
Crystals from small molecules can be recrystallized until the conditions are de-termined that yield appropriate single crystals. This process is not applicable to MOFs, where crystal growths occurs simultaneously to the irreversible gen-eration of an extended framework structure, which is insoluble. Furthermore, the modification of the synthetical conditions can result in the formation of polymorphs.[231–240]Therefore, these conditions have to be optimized to obtain the desired phase and to control the size and quality of the crystals. Addition-ally, the size of the crystals can be enlarged by so-called modulators that com-pete with linkers at the coordination sites and thereby decelerate crystal growth, which can yield high quality crystals.[30,241,242]
The minimum size of a crystal in a diffraction experiment depends on the scattering power of its components as well as intensity of the X-ray beam. If very powerful beam is needed to measure small crystals, diffraction experi-ments may be performed with synchrotron radiation at specialized facilities.[234]
For the analysis of MOFs on a laboratory scale, copper anodes are commonly chosen over molybdenum anodes as radiation sources, because of the greater diffraction power of the copper KαX-ray beam. Another advantage of copper anodes over molybdenum anodes is an increase in the separation of diffraction spots. MOFs are often associated with large unit cells that exhibit cell dimen-sions greater than 30 Å. This leads to d spacings of the lattice planes at which constructive diffraction occurs. According to Bragg’s law the diffraction angles for the corresponding peaks are relatively small and they may overlap. When using copperKαradiation this effect is less pronounced compared to molybde-numKαradiation because of the difference in wavelengths (λCu Kα = 1.5406 Å, λMo Kα= 0.7107 Å), which directly influence the diffraction angles according to Bragg’s law. An illustration of this relationship is depicted in Figure 2.5. How-ever, with longer wavelengths less reflections are available for structure refine-ment because of an reduction of the observable reciprocal space. Furthermore, the application of molybdenum Kαradiation allows the measurement of sam-ples that show strong X-ray absorption.
Another method to analyze even very small crystals is electron crystallogra-phy. In these methods, X-ray beams are replaced by electron beams, which inter-act more strongly with the analyte and exhibit significantly smaller wavelenghts compared to, for example, copperKαradiation.[243]Kolb and co-workers have applied electron crystallography to determine the structure of MOF crystals ranging from 200 nm to 300 nm in length.[244]
short wavelength long wavelength
Figure 2.5– Illustration of two hypothetical X-ray diffraction patterns of a compound with a large unit cell; short wavelengths lead to small diffraction angles that cause overlap of diffraction peaks; long wavelengths lead to separation of diffraction peaks.
In many cases, suitable single crystals for diffraction experiments are inac-cessible. However, structural information can also be acquired from powder X-ray diffraction (PXRD). This procedure is not trivial because initial solutions are hard to find due to the difficulties in recovering the phase information from PXRD data. A successfulab initioapproach has been demonstrated, for example, in the elucidation of the crystal structure of UiO-66 by applying direct methods to powder diffraction data obtained from a synchrotron source.[24]These meth-ods have been implemented in the EXPO software.[245]
Another approach to solve crystal structures from PXRD data is based on the identification of regions with high and low electron density. In the case of a MOF an obvious distinction can be made between the framework structure and the void space. Brenner and co-workers reported a procedure that defines a surface between high and low electron density areas as a structure envelope.[246,247]This allows the reduction of space where atoms that belong to the framework can be located. These restrictions can be used to improve charge-flipping calculations to solve the crystallographic phase problem. An adaption of the structural enve-lope method to MOFs has been reported by Zhou and co-workers on the basis of HKUST-1 as a model structure.[248]A representation of a structural envelope that has been determined for HKUST-1 is shown in Figure 2.6
In some cases, certain structural information about the crystal is already known. This is the case, for example, for certain rigid linkers that are employed in the synthesis of MOFs. This allows the application of so-called direct space methods, which are based on the localization of the linkers in the unit cell by using systematic or random moves of these building blocks and comparision
Figure 2.6– Three-dimensional representation of the unit cell of HKUST-1 overlapped with a structural envelope determined from PXRD data; the volume with a high electron density lies within the boundaries of the green surface and corresponds to the framework structure (green surface: struc-tural envelope, grey: carbon, red: oxygen, brown: copper, white: hydrogen, reprinted from reference [248]).
of the calculated and observed diffraction pattern.[249]This approach has been demonstrated for a pyrazolyl-based MOF by Bordiga and co-workers..[250]
In contrast to structure determination of small molecules, preconceived as-sumptions about the framework topology are possible for MOFs, which may help in finding structural solutions. In principle, it is possible to limit the num-ber of possible networks from which crystal structures can be derived by such a homology modeling. This means it is possible to create structural models that are related to existing compounds with known crystal structures. One of the ear-liest homology modeling of a MOF structure has been reported by Ferey and co-workers.[251]Furthermore, this technique has been applied to determine the structure of the series of MOF-74-I to MOF-74-XI on the basis of MOF-74 (see 1.24c on page 40 for a representative).[55] To obtain crystal structures, models were constructed by adding phenylene units with aliphatic substituents to the organic SBU under the retention of the inorganic SBU of MOF-74. The struc-tural models were then subjected to a Rietveld refinement against the powder diffraction data. This procedure is generally applicable for MOFs that exhibit an isoreticular relationship to each other.