Atomistically Resolved Structure of Layered Covalent Organic Frame- works

Covalent organic frameworks (COFs) are among the fastest growing classes of materials with an almost unlimited number of achievable structures, topologies, and functionalities. Their exact structure remains, however, unknown to date as reflected by the significant mismatch between experimental powder X-ray diffraction pattern (PXRD) and predicted geometries. We attribute these discrepancies to an overlooked, inherent disorder in the stacking of layered COFs, invalidating standard theoretical 3D models. We have built models of COF-1, COF-5 and ZnPc-pz by stacking layers following the Maxwell-Boltzmann energy distribution of their stacking modes. Simulated PXRD patterns of these model structures are close experiments, featuring an agreement in peak intensity, width and asymmetry that has never been obtained before. The rarely considered ABC stacking mode plays an important role in layered COFs, and solvent molecules have a major effect on the stacking. As our model holds for two main layered COF lattice types (honeycomb/kagome and square), generalization is expected for all layered COFs and MOFs. Covalent organic frameworks (COFs) are crystalline, porous networks built from organic linkers via covalent bond formation.1–3 Most COFs to date are layered materials composed of atomically thin two-dimensional (2D) sheets, which create stacked crystals via non-bonding interactions.2,4,5 Commonly, these materials are referred as 2D COFs, a somewhat misleading nomenclature as most representatives are layered 3D bulk materials. Although we will show that in most layered COFs there is no periodicity normal to the 2D crystal plane and thus a common definition of a 2D periodic material (translational symmetry only in two dimensions) fits, we will, following the traditional nomenclature of related layered materials,6 employ here the term layered COFs (LCOFs). There is no unambiguous knowledge of the atomistic LCOF structures as powder X-ray diffraction (PXRD) data does not contain sufficient information to resolve the structure in atomistic resolution, and predicted structures do not closely match experimental PXRD patterns. LCOF PXRD patterns commonly have signs of structural disorder with wide, diffused peaks, while those simulated from predicted structures feature many narrow peaks (Figure 1).7–9 Recently, a total scattering study by Pütz et al. has shown that LCOFs with a well-defined local structure exhibit a long-range stacking disorder.10 This disorder systematically causes false interpretation of experimental results, which commonly is mapped to an average structure of high symmetry in stacking direction. Similar disorder was identified layered zeolites11 and molecular crystals.12 Figure 1. A sketch of common differences of experimental and theoretical PXRD patterns in LCOFs. Here, we present a general approach to simulate the crystal disorder in LCOFs, by statistically stacking COF monolayers in a sequence following a Maxwell-Boltzmann distribution of their relative stacking energies. This model yields unprecedented agreement of PXRD patterns between experiment and theory, to the point of being able to identify the actual ratios of stacking modes present in the samples.

Covalent organic frameworks (COFs) are crystalline, porous networks built from organic linkers via covalent bond formation. [1][2][3] Most COFs to date are layered materials composed of atomically thin two-dimensional (2D) sheets, which create stacked crystals via non-bonding interactions. 2,4,5 Commonly, these materials are referred as 2D COFs, a somewhat misleading nomenclature as most representatives are layered 3D bulk materials. Although we will show that in most layered COFs there is no periodicity normal to the 2D crystal plane and thus a common definition of a 2D periodic material (translational symmetry only in two dimensions) fits, we will, following the traditional nomenclature of related layered materials, 6 employ here the term layered COFs (LCOFs).
There is no unambiguous knowledge of the atomistic LCOF structures as powder X-ray diffraction (PXRD) data does not contain sufficient information to resolve the structure in atomistic resolution, and predicted structures do not closely match experimental PXRD patterns. LCOF PXRD patterns commonly have signs of structural disorder with wide, diffused peaks, while those simulated from predicted structures feature many narrow peaks ( Figure 1). [7][8][9] Recently, a total scattering study by Pütz et al. has shown that LCOFs with a well-defined local structure exhibit a long-range stacking disorder. 10 This disorder systematically causes false interpretation of experimental results, which commonly is mapped to an average structure of high symmetry in stacking direction. Similar disorder was identified layered zeolites 11 and molecular crystals. 12 Figure 1. A sketch of common differences of experimental and theoretical PXRD patterns in LCOFs.
Here, we present a general approach to simulate the crystal disorder in LCOFs, by statistically stacking COF monolayers in a sequence following a Maxwell-Boltzmann distribution of their relative stacking energies. This model yields unprecedented agreement of PXRD patterns between experiment and theory, to the point of being able to identify the actual ratios of stacking modes present in the samples.

LCOF stacking
We have chosen the two most common LCOF framework lattice types; namely honeycomb (hcb) (more correctly kagome (kgm), for simplicity we refer to the building block centers which form hcb lattices) represented by COF-1 and COF-5, and square (sql) lattices, represented by ZnPc-pz. 8,13 These LCOFs have flat, atom-thin layers with only ZnPc-pz featuring bulky tert-butyl side groups reaching out of the layer plane (Figure 2e). A search for all possible stacking modes in the three LCOFs using bulk structural models employing the Density-Functional based Tight-Binding (DFTB) method resulted in five distinct types ( Figure 2, Tables 1, SI-2): i) AA, (hcb and sql); with layers directly above each other. The "eclipsed" stacking mode (AAe) is energetically highly improbable, so the layers shift laterally by about 1.5 Å to AAs. This shift is confirmed theoretically and experimentally. 10 ii) AB (hcb); with top layer shifted laterally to middle of the hexagonal pore, i.e. by [⅓,⅓] unit cell (UC) and [-⅓,-⅓] in the next layer. ABe and ABs (same shift types as AAs) are almost isoenergetic and can be of low energy. iii) ABC (hcb); layers shifted laterally by [⅓,⅓] UC as in AB and again [⅓,⅓] in the next layer. The interlayer shift is identical to ABs, but a different superlattice appears in projection along layer normal. ABC is known in other hcb COFs. 14,15 Due to layer corrugation ( Figure SI-3) it is the most stable structure for COF-1 and COF-5. iv) ABdiag (sql); layers shifted laterally by [0.5,0.5] UC.
Only structures with small additional shift ABs-diag exist in ZnPc-pz, all of them high in energy. v) ABaxis (sql); layers shifted laterally by [0.
UC. The shift changes to [±⅓,0] or [0,±⅓] for ZnPcpz due to the bulky side chains (ABCs-axis). The most stable structure of ZnPc-pz. Due to the large discrepancy between PXRD patterns of assynthesized and activated COF-1 and COF-5, 8 we have modeled the presence of solvent by adding mesitylene molecules ("+mesi"), considering one mesitylene molecule per pore; (= full coverage in ABC, Figure 4). The solvent changes the ABCe/s structure, preventing layer corrugation. This affects stacking energetics considerably; in COF-1, ABCe remains the most stable structure, whereas in COF-5 AAs is preferred (Table 1).

Statistical COF Models
The presence of interlayer shifts breaks the LCOF symmetry, so the interlayer shifts split into ensembles of isoenergetic non-equivalent variants. The two shift directions in COF-1 and COF-5 result in6 and 12 shifts, respectively, and in ZnPc-pz there are two pairs of shifts ( Figure 2d). Thus, due to configurational entropy, any LCOF must have a disordered structure with statistical set of these shift directions ( Figure   To obtain realistic LCOF structure models, we first create a set of all possible interlayer shifts using Maxwell-Boltzmann distribution of their relative energies (at the synthesis temperature 8,13 ). We then construct the bulk structures layer-by-layer, using Markov chain random walk algorithm of the interlayer stackings. We have verified the long-range shift direction independence optimizing all COF-1 trilayer AAs structures, (Table SI-4). The resulting structures are denoted AA/AB/ABC stat . Generally, energetics allows combining multiple stacking types (AAs, ABCe/s,…) in the same crystal. We denote such combinations AAs:AB(C) stat X:Y, specifying their proportion from statistical distribution. In the special case for ABe/s and ABCe/s, which have the same interlayer shift, we denote them AB(C) stat .
We have also simulated individual AA stat , AB/ABC stat and AB(C) stat structures, as well as their mixtures to identify their characteristic features.

COF-1, COF-5 and ZnPc-pz structures and their PXRD patterns
We have verified our model by simulating PXRD patterns and comparing them with experiment ( Figure 3, peak positions Table SI The PXRD patterns of the disordered crystals show several common features. In both hcb and sql LCOFs, the AA stat structures have a pronounced main peak in the range of 2θ=4°…9°. Only hcb AB stat and sql AB(C) axis stat patterns feature this peak too, though, whereas hcb AB(C) stat and sql AB diag stat have only low-intensity peak in this range. Both these structures feature a new super-lattice motif forming in projection along the layer normal ( Figure SI-2), which is likely the reason for the distinct pattern. After adding mesitylene (solvent) molecules in the simulation, the statistical model predicts an occurrence of 92% of the ABCe stacking mode for as-synthesized COF-1 ( Table 1). The resulting PXRD pattern, AAs+AB(C)+mesi stat 1:9 ( Figure  3a), matches the experiment well, with a low-intensity peak at 6-8°, higher intensity peak at 11-13° and an intensive peak at 27°. Removal of mesitylene molecules shifts the last peak position to higher angles ( Figure 4) due to layer corrugation.
The COF-1 experimental PXRD pattern changes considerably after solvent evacuation (Figure 3b). The 6-10° peak becomes the most intensive one and a wide peak forms around 2θ=27°. This matches well our simulated AA stat PXRD pattern, which only lacks asymmetry of the main peak. Thus, COF-1 transforms from ABCe/s to AAs stacking during solvent removal, likely because for solvent diffusion 1D pores are needed that are lacking in ABCe/s stacked COF-1 ( Figure SI-3).
Stacking change due to evacuation is inherently a kinetic process, so the resulting stacking order will not follow Maxwell-Boltzmann statistics. We have therefore sampled AAs+AB(C)+mesi stat structures with varying ratios ( Figure  5), assuming that all AB(C) stat layers still contain mesitylene. The 6-8° peak intensity decreases together with the AAs probability. This peak also becomes asymmetric towards higher 2θ angles when ABCe/s content exceeds 25 %. This asymmetry is also seen in the experimental PXRD pattern, with best agreement in 3:7 AAs+AB(C)+mesi stat structure. Both AA stat and AB(C) stat modes are present in the same crystals, as the peak asymmetry disappears, if the same proportion of isolated AA stat and AB(C) stat crystallites is used (Figure SI-15).  The most probable stacking mode of COF-5 is AAs (Table 1). The corresponding AA stat pattern directly matches the experimental pattern (Figure 3c). The wide peak around 2θ=27° observed in experiment suggests there is some structural disorder. This can originate from a small proportion (<10%) of AB(C) stat ( Figure SI-13).
ZnPc-pz energetic profile favors the ABCs-axis structure (Table 1), and the PXRD patterns suggest either AA stat or AB(C) axis stat staking mode due to high intensity of 4° peak. The experimental powder pattern shows low crystallinity, though, so we have also sampled the stacking mode mixing as a potential disorder source. The combination of three main stacking modes ( Figure SI-14) shows features seen in both hcb LCOFs, i.e. low-intensity of AB diag stat 4° peak and asymmetrical peaks forming in mixed-type crystals. This suggests that the peak asymmetry is in general a distinguishing feature for reading LCOF PXRD data.

Electronic Structure of disordered LCOF
Electronic structure in LCOFs is sensitive to the layer stacking order. For example, it was shown for COF-5 that different small-UC models, AA s inc , AA s serr yield electronic structures of significant differences in band gap and dispersion. 16 Therefore, we have evaluated the electronic structure of COF-1 and COF-5 in statistical stacking order. As a compromise between good statistics and computational costs, we have built multi-layered bulk "snapshots" of disordered LCOF structure. Due to the low symmetry the band structures become crowded, so we focus on the densities-ofstates (DOS) differences. The AA s serr model of both COF-1 and COF-5 yields larger bandgaps than corresponding inclined structures ( Figure 6, Figure SI-1). The AA stat bandgap is close to that of serrated structure, or even higher, although it also depends on the local structure ( Figure SI-16). A sharp peak in DOS forms at the edge of both valence and conduction bands of the statistical structure. The solvent-imposed structure change has severe consequences for the electronic structure, which has more diffuse bands and narrower bandgap ( Figure SI-17,18).

Summary
We have shown that the layered COFs, despite having crystalline layers, exhibit disorder along third axis, making them by definition true 2D materials. Statistical stacking order thus needs to be considered to properly describe their structure and properties. We have developed a procedure to create atomistic models of COF-1, COF-5 and ZnPC-pz with a statistical sequence of stacking modes with the ratio of stacking types determined by Maxwell-Boltzmann energy distribution. This allowed us to simulate PXRD patterns that closely match the experimental data, both in peak intensities and shapes. The rarely considered ABC stacking mode turns out to be crucial for LCOFs, being the most stable mode of COF-1 and ZnPc-pz. The presence of solvent results in different LCOF structures, e.g. in as-synthesized COF-1 with small pores. We have described an experimentally observed change of COF-1 stacking order accompanying solvent evacuation, which leads to structures with a mixed AAs/ABC stacking order. While we have only studied hcb and sql LCOFs, we expect that our method is transferable to other layered framework materials including LCOFs and LMOFs.

Supporting Information
The Supporting Information is available free of charge on the ACS Publications website at DOI: Details of the DFTB and DFT calculations, stacking modes and corresponding relative energies, details of the statistical model, more results of PXRD patterns and electronic properties.

Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes
The authors declare no competing financial interest.