Article Cite This: Inorg. Chem. 2018, 57, 15123−15132
pubs.acs.org/IC
Simulating Powder X‑ray Diffraction Patterns of Two-Dimensional Materials Yibin Jiang,† Lingyun Cao,† Xuefu Hu,† Zikun Ren,‡ Cankun Zhang,† and Cheng Wang*,†,‡ †
Collaborative Innovation Center of Chemistry for Energy Materials, State Key Laboratory of Physical Chemistry of Solid Surfaces, Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, P.R. China ‡ College of Atmospheric Science, Nanjing University of Information Science and Technology, Nanjing 210000, P.R.China
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ABSTRACT: Powder X-ray diffraction (PXRD) is widely used to study atomic arrangements in ordered materials. The Bragg equation, which describes diffraction of a three-dimensional crystal, fails in two-dimensional (2D) cases. Complete integration of diffraction signals from a continuum instead of discrete directions in the Bragg equation is thus required for proper data interpretation of 2D materials. Furthermore, modeling of preferred orientation of the 2D crystals as well as geometric disorders are also of vital importance. Here, we present a complete integration method in real space (CIREALS) for PXRD simulation of monolayer or multilayer 2D crystals, especially 2D metal−organic layers and 2D covalent organic frameworks. By working in real space instead of reciprocal space, we can readily capture the 2D geometry and preferred orientation of these materials. The predicted PXRD patterns by CIREALS facilitates structure analysis of these new types of 2D material.
1. INTRODUCTION Ultrathin two-dimensional (2D) crystalline materials are of great interest because of their promise in revolutionizing technologies in electronics, nanomedicine, catalysis, and many other fields.1−8 For example, electronic devices based on graphene and other 2D materials together with their van der Waals heterojunctions exhibited extraordinary performances.9−13 In addition to these well-known members of 2D materials, recently we and others investigated a new type of 2D material, metal−organic layer (MOL),14−22 which is a 2D version of the metal−organic framework (MOF)23−36 constructed from organic linkers and metal-connecting nodes. These ultrathin MOLs contain 2D networks that either stand alone as monolayers or stack onto each other in the third dimension to form multilayers. Similar to 2D MOLs, covalent organic frameworks (COFs)37−42 that are linked from organic building units by covalent bonds often exhibited 2D networks that stack in the third dimension.43−45 These new types of 2D materials have versatile structures that present challenges for structure analysis. Imaging techniques such as transmission electron microscopy,46 atomic force microscopy,20 and scanning tunneling microscopy47 can provide information of morphology, thickness, and even lateral arrangement of atoms in the 2D materials. However, more cost-effective characterization methods are still of interest to serve as routines for characterizing 2D crystalline materials. Powder X-ray diffraction (PXRD) is widely used to study crystal structures. A simple comparison between experimental © 2018 American Chemical Society
and simulated PXRD patterns quickly tells the phase purity and crystallinity of the sample. Positions of PXRD peaks of a threedimensional (3D) crystal are described by the Bragg equation, which give diffractions in discrete directions because of the grating-like selection rule of a 3D periodic lattice. However, 2D crystals have a limited number of repeats in the third dimension. The lack of grating-like structure in this dimension invalidates the Bragg equation. Consequently, PXRD signals of a 2D material are no longer along discrete directions as measured in 2θ; instead, finite intensity should be observed at all 2θ angles (Figure 1). There are generally two strategies in simulating PXRD patterns of 2D materials. The first one is to sum up X-ray scattering signals of every atom in a solid to obtain the PXRD intensities in every 2θ. This strategy using a Debye equation48 ignores the periodic structure and treats the 2D materials the same way as an amorphous material. The Debye method is thus very demanding for computational resources, because of the necessity to address every atom individually. Practically, the Debye method is more suitable for small nanoparticles instead of crystalline materials with a reasonable crystal domain size.49 The time cost is most severe in simulating 2D MOLs or 2D COFs which contain several atoms in a unit cell. We performed a test to simulate the PXRD pattern of a monolayer of a Zr-based 2D MOL (hxl-UiO-67)20 of 200 × 200 × 1 unit cells in size using the Debye method in the DISCUS program,50 but could not Received: August 17, 2018 Published: November 28, 2018 15123
DOI: 10.1021/acs.inorgchem.8b02315 Inorg. Chem. 2018, 57, 15123−15132
Article
Inorganic Chemistry
Figure 1. PXRD can provide rich structure information for 3D crystals through the Bragg equation, which is limited for 2D materials.
with σ ranging from 0 to π and ω ranging from 0 to 2π. Although the obtained mathematical expressions in real space are more complicated than those obtained in reciprocal space, the real space formulation is more compatible with digital computation, thanks to the fixed integration area in CIREALS that facilitates the Gaussian quadrature method for fast integration.79,80 The real space parameters can directly associate with crystal size and orientation, which makes CIREALS an attractive alternative to explore many structural details of a 2D monolayer and multilayers through studying diffraction patterns. CIREALS is a general method that can be applied to a variety of 2D materials such as MOFs, COFs, molybdenum disulfide, and graphene. (Typical examples can be found in Sections S1 and S3). In this article, we describe the CIREALS algorithm in PXRD simulation for 2D materials, especially 2D-MOLs and 2D-COFs. Using this tool for simulation, we will discuss: (1) PXRD patterns of 2D crystals vs 3D crystals; (2) PXRD patterns of eclipsed stacking vs staggered stacking for multilayers; (3) the intrinsic asymmetrical PXRD peak shapes of ultrathin 2D materials; (4) the influence of preferred orientation and facial defects on PXRD of 2D materials (Figure 2). A few MOLs and
finish the simulation after 240 CPU hours. It is worth noting that other software using the Debye method (https://code.google. com/archive/p/debyer/) can be faster than DISCUS, especially when codes for graphic processing units (GPUs) are used.51 Another strategy to simulate PXRD of a 2D crystal is the complete integration method, which integrates the diffraction intensities contributed from all possible orientations of the 2D nanosheets. Key to this method is to construct a pair of variables that describe a specific orientation of a crystal and the corresponding contribution to the overall diffraction intensity in a given 2θ direction. One then obtains the total diffraction intensity by integrating the fractional intensity in the space of this set of variables. Earlier researchers chose to perform complete integration in reciprocal space and constructed the orientational descriptors from the h, k, or l indexes of the diffractions.52 However, the intrinsic nonlinear relationship between the reciprocal space and the geometric parameters in real space leads to an irregular integration area as a penalty for the efficiency of the overall algorithm. Moreover, modeling of preferred orientations that is of importance to 2D materials is cumbersome in the reciprocal space. The complete integration method is used in cases50 to simulate PXRD patterns of crystals with defects and limited sizes of the coherently scattering domain (
