A Statistical Representation of Stacking Disorder in Layered Covalent Organic Frameworks

Covalent organic frameworks (COFs) are among the fastest growing classes of materials with an almost unlimited number of achievable structures, topologies, and functionalities. The exact structure of layered COFs is, however, hard to determine due to often significant mismatch between experimental powder X-ray diffraction pattern (PXRD) and predicted geometries. We attribute these discrepancies to an inherent disorder in the stacking of layered COFs, invalidating


INTRODUCTION
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).
Although some COF structures can be precisely identified by single crystal X-Ray diffraction 7,8 .
However, the great challenge in the synthesis of the COF single crystal, especially LCOFs, renders the revealing of most of the COFs crystal structure still heavily depends on the powder XRD together with the theoretical simulation. There is no unambiguous knowledge of the atomistic LCOF structures as powder X-ray diffraction (PXRD) data itself does not contain sufficient 3 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). [9][10][11] 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. 12 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 13 and molecular crystals. 14 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.

RESULTS AND DISCUSSION
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. 10   Due to the large discrepancy between PXRD patterns of as-synthesized and activated COF-1 and COF-5, we have modeled the presence of solvent by adding mesitylene molecules ("+mesi"), considering one mesitylene molecule per pore, the same number as the original experimental paper (= full coverage in ABC, Figure 4). 10 The solvent changes the ABCe/s structure, preventing layer corrugation. This affects stacking energetics considerably; in COF-1, ABCe and ABe become isoenergetic, while being more stable than AAs, whereas in COF-5 AAs is preferred (Table 1)  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 10,15 ). 7 We then construct the bulk structures layer-by-layer, using Markov chain random walk algorithm with different stacking modes. We have verified the long-range shift direction independence optimizing all COF-1 trilayer AAs structures, (Table SI-6). 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 .

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  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 lowintensity 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. 9 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 that the needed 1D pores for solvent diffusion 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   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-16).
ZnPc-pz energetic profile favors the ABCs-axis structure (Table 1), and the PXRD patterns suggest either AAstat or AB(C) axis stat stacking 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-17) 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.
Beyond the materials mentioned above, we have applied the statistical method to other LCOFs using already published energetic and structural data ( Figure SI-18). 18,19 The results show an improvement of the match of experimental and predicted PXRD when using the statistical model.   Dresden, Germany; orcid.org/ 0000-0003-2135-3799; Email: Yingying.zhang@tu-dresden.de

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.