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Characterization of Shape and Monodispersity of Anisotropic Nanocrystals through Atomistic X-ray Scattering Simulation Thomas R. Gordon, Benjamin T. Diroll, Taejong Paik, Vicky V. T. Doan-Nguyen, E. Ashley Gaulding, and Christopher B. Murray Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/cm5047676 • Publication Date (Web): 19 Mar 2015 Downloaded from http://pubs.acs.org on March 29, 2015
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Characterization of Shape and Monodispersity of Anisotropic Nanocrystals through Atomistic X-ray Scattering Simulation Thomas R. Gordon,1† Benjamin T. Diroll,1† Taejong Paik,1 Vicky V. T. Doan-Nguyen,2 E. Ashley Gaulding,2 and Christopher B. Murray1,2 1
Department of Chemistry and 2Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA KEYWORDS X-ray Scattering, Debye Formula, Nanocrystals, Anisotropic Shape, Monodispersity
ABSTRACT: Nanocrystals with anisotropic shape and high uniformity are now commonly produced as a result of significant advances in synthetic control. In most cases, the morphology of such materials is characterized only by electron microscopy, which makes the extraction of statistical information laborious and subject to bias. In this work, we describe how X-ray scattering patterns in conjunction with Debye formula simulations can be used to provide accurate atomisitic models for ensembles of anisotropic nanocrystals to complement and extend microscopic studies. Methods of sample preparation and measurement conditions are also discussed to provide appropriate experimental data. The scripts written to implement the Debye function are provided as a tool to allow researchers to obtain atomisitic models of nanocrystals.
Newly developed synthetic methods have allowed for an incredible expansion in the variety of shapes available in nanocrystal (NC) systems. Monodisperse spheres,1 cubes,2 octahedra,3 tetrapods,4 bipyramids,5 rods,6,7 plates,8 and many other shapes have been prepared for a variety of materials with high reproducibility and uniformity. On the other hand, characterization of NC shape is primarily done through analysis of the two-dimensional projections provided by transmission electron microscopy (TEM). Although tomography9 or exit-wave reconstruction10 help to visualize NCs three-dimensionally, these techniques are inherently non-statistical and, due to simplicity, most researchers rely on the analysis of multiple NC projections to determine NC shape. Although an indispensible tool for any shape characterization, electron microscopy alone can yield errors in shape description, particularly when NCs lie preferentially on one surface. X-ray scattering in conjunction with simulation can act as a compliment to electron microscopy for the analysis of NC phase and composition and provides an ensemble confirmation of microscopic findings.11,12 The basis for this simulation is the Debye formula, described by the formula’s namesake in 1915, which allows calculation of the X-ray scattering intensity at arbitrary angle for any set of atomistic coordinates.13,14 This formula has since been utilized for the analysis of X-ray scattering from nano-sized crystallites,12,15–20 as well as the analysis of scattering from proteins, polymers, and other macromolecules.21–23 In this work, we illustrate the utility of X-ray scattering simulation, coupled with electron microscopy, in elucidating the morphology and monodispersity of anisotropic NCs. X-ray scattering provides statistical descriptions of size, shape, and polydispersity for an ensemble. Proper methods, including some which are specifically useful for colloidal nanocrystals, are described for the collection of the X-ray scattering pattern which ensure that the NCs are
arranged isotropically during the measurement, a critical aspect to retrieve interpretable data. The script used to calculate scattering patterns has been included in the Supporting Information, which allows for common NC shapes to be simulated for any given unit cell. Previous software packages have provided access to Debye function calculations, but none have been specifically tailored for the simulation of X-ray scattering from nanocrystals of uniform shape. The scripts provided along with this publication provide a more targeted, user friendly approach for accomplishing this task for a variety of commonly observed particle shapes. Scripts provided in the Supporting Information, can perform crystal cutting and Debye simulation for crystal structures forming spheres; prolate and oblate ellipsoids; cuboctahedra; octahedra; cubes; bipyramids; rods with rhombic, square, rectangular, and circular cross sections; and circular, hexagonal, and rhombic plates. EXPERIMENTAL Synthesis. Details of experimental protocols, where different than in the literature, can be found in the Supporting Information. GdF3 plates,24 PbTe cuboctahedra,25 Bi spheres,26 CdSe ellipsoids and rods,27,28 TiO2 rods and bipyramids,5,29 CdO octahedra,30 and PbSe rods31,32 and spheres33,34 were synthesized by literature procedures with little or no modifications. PbSe cubes were isolated from a slightly-modified reaction which also produces PbSe nanowires.35 X-ray Diffraction Characterization. All necessary data collection can be performed on typical laboratory X-ray diffraction equipment keeping in mind procedures for appropriate data collection.14,36,37 Wide-angle X-Ray scattering (WAXS) patterns were measured using a Rigaku Smartlab diffractometer equipped with a Cu Kα source in Braggs-Brentano (transmission) or parallel beam (reflection) geometry. For transmission WAXS measurements, NCs were placed as highly concentrated toluene or hexanes solu
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Figure 1. SAXS and WAXS data (black circles) fitted with atomistic X-ray scattering simulation data (red lines) for (a) PbSe spheres, (b) CdSe nanorods, (c) PbTe cuboctahedra, (d), anatase TiO2 bipyramids, (e) PbSe cubes, (f) rhombic GdF3 Plates, (g) CdO octahedra, (h) prolate CdSe ellipsoids, and (i) PbSe nanorods. Cartoons of the structures are inset.
tions into 1.0 mm glass capillaries (Charles Supper) or dispersed into polyvinylbutyral (Butvar® B-98) by co-dissolving NCs and Butvar in chloroform and allowing the chloroform to evaporate. (Details can be found in the Supporting Information.) Dispersed samples were also used for small-angle X-ray scattering (SAXS) measurements. For reflection WAXS measurements, NCs were either drop-cast or spin-coated (500 rpm) on to glass slides from concentrated hexanes solutions. SAXS patterns were recorded on three instruments: 1) a Rigaku Smartlab laboratory diffractometer using 10 mm slits; 2) a multi-angle X-ray scattering instrument (150 cm, 54 cm, or 11 cm sample-detector distances) with Osmic flux optics, pinhole collimation, and a Bruker Hi-Star multiwire detector; and 3) the X9 beamline at the National Synchrotron Light Source (NSLS I) at Brookhaven National Laboratory using 13.5 keV X-rays and a Mar CCD detector at a 3 m distance. X-ray Diffraction Simulation. The q-dependent X-ray diffraction intensity, I(q), is calculated using the Debye equation:11 sin where I0 is the incident intensity, q = 4π sinθ / λ is the scattering parameter for X-rays of wavelength λ diffracted through the angle 2θ, rmn is the distance between atoms m and n, with atomic form factors Fm and Fn, respectively. Atomic form factors are calculated
from Cromer-Mann coefficients.38–40 To improve calculation time, the Debye equation is discretized by binning identical distances to give the following equation:41 sin where ρ(rmn) is the multiplicity of each unique distance rmn in the structure. Thermal vibrations are known to distort the diffraction pattern due to uncertainty in the atomic positions. We simulate this through multiplication of the atomic form factors by a temperature factor, which has the Debye-Waller factor as the input: where B is the Debye-Waller factor, and ai, bi and c are the CromerMann coefficients.39 Particle size distribution is incorporated using a probability distribution, where each final pattern is typically the Gaussian weighted sum of 15 or more patterns from individual NCs spanning at least two standard deviations of the characteristic length. In its current form, the distribution is applicable to periodic, monocrystalline, non-epitaxial structures. RESULTS AND DISCUSSION There are two angular regimes of X-ray scattering which may be used to characterize dispersed species, such as NCs, biomolecules
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and polymers, which are categorized as SAXS and WAXS. SAXS encompasses the q range of roughly < 1 Å-1. In assemblies of NCs, this angular range is useful for probing superlattice structures or nearest neighbor distances.11,42 In a dilute homogeneous solution, where the structure factor does not contribute strongly to scattering, SAXS is a measurement of the form factor of the NC, although the signal at very small angles is associated with interparticle potentials in solution.36,43,44 This particle form factor represents the Fourier transform of the particle shape, and has been solved analytically for some morphologies (e.g. ellipsoid, infinitely long rod, etc.), assuming that the particle has uniform electron density.36 On the other hand, this is a clear oversimplification, as colloidal particles are composed of arrangements of atoms with non-uniform electron density and truncation at defined lattice sites, especially prominent for small NCs, which do not conform to ideal shapes. Both phenomena result in smearing of the form factor.11 In addition, most colloid morphologies are not solved analytically, forcing researchers to use a similar morphology in analysis (e.g. ellipsoid for rod, sphere for cube, etc.), limiting the accuracy of the particle dimensions and distribution parameters retrieved. WAXS covers the larger q and 2θ angular range corresponding to the interatomic distances present in a material. WAXS is an indispensable method to identify the crystalline structure of NCs. Although the peaks are broad at small particle size due to Scherrer broadening, patterns are typically indexed against bulk crystal patterns. Less appreciated is that WAXS patterns are quite sensitive to NC morphology, as the intensity and breadth of each peak are highly dependent on the number of each contributing atomic plane.7,45 Although not the focus of this work, WAXS data and atomistic simulations can also describe the diffraction from defective or strained nanocrystalline materials.46,47 On the other hand, WAXS patterns, particularly of anisotropic NCs, are subject to the effects of preferred orientation, in which NCs organize anisotropically upon deposition onto a flat substrate, complicating interpretation of the particle shape. The Debye equation provides a method by which a selfconsistent atomisitic model can be constructed for NCs by simulating both WAXS and SAXS simultaneously. The NC sample is first characterized through transmission electron microscopy (TEM) to identify the likely morphology, average particle size, and approximate size distribution. Then, an atomistic model is developed based on the crystal structure identified by a WAXS measurement and the appropriate morphology. Finally, the Debye function is used to generate both a SAXS and a WAXS pattern for the atomistic model along with a distribution. The advantages of this approach include the capacity of SAXS and WAXS to quickly provide a bulk measurement of average NC morphology, which overcomes the potential biases of microscopy, and the ability to easily simulate the scattering pattern of materials with arbitrary shape. Nine SAXS/WAXS patterns for NCs (black circles) along with Debye function simulations (red lines) are presented in Figure 1. As expected, spherical PbSe NCs in Figure 1a exhibit a classic pattern in the SAXS regime indicative of a spherical Bessel function. In addition to simulating the mean diameter of the NC, the SAXS pattern of an ensemble must be simulated using an ensemble distribution, as particles of different size dampen the pattern. To accomplish this, simulations are performed for many NC diameters spanning the mean value, typically by at least two standard devia-
tions, which are then Gaussian-weighted and summed. Although a Gaussian distribution is used here, any distribution function that accurately models the particle size distribution can be used. Many common distributions (e.g. log-normal, gamma) result in very similar patterns at small distributions in particle size. In addition to spherical NCs, which may be fit using analytical solutions, Figure 1 also depicts a diverse array of NC shapes which have be simulated using our method: 12.8±1.3 nm x 4.3±0.4 nm CdSe rods (Figure 1b), 11.3±0.7 nm PbTe cuboctahedra (Figure 1c) 18.0±2.7 nm long x 10.0±1.5 nm wide truncated anatase bipyramids (Figure 1d), 12.1±1.1 nm PbSe cubes (Figure 1e), 34.5±1.6 nm tip-to-tip x 2.1±0.1 nm thick rhombic plates of GdF3 (Figure 1f), 61.0±3.4 nm octahedra of CdO (Figure 1g), 4.4±0.7 nm prolate CdSe ellipsoids with aspect ratio 1.4 (Figure 1h), and 14.6±1.3 nm long x 5.3±0.5 nm diameter PbSe rods (Figure 1i). An analytical solution for each of these shapes is unavailable, but they are reasonably simulated with atomistic scattering models. The monodispersity of the samples varies from ~5% (CdO octahedra) to 15% (TiO2 truncated bipyramids). For the samples presented in Figure 1, percent deviation in size is applied isotropically along all axes of the structure, which approximates the distributions well. However, some samples, such as atomically-flat sheets,8 may show uncorrelated dispersions along different growth directions and these can be simulated by applying direction-specific dispersions. Distinguishing particular shapes of NCs from the X-ray scattering patterns alone is not recommended; X-ray simulation should be used to provide a statistically-valid estimate of size, shape, and monodisperisty. However, certain shapes, such as the nanorods shown in Figure 1b are relatively easily distinguished from the Xray scattering pattern because the peak intensities and line broadening of the diffraction peaks are quite distinct from the bulk. The nanorods in Figure 1i are further distinguished as anisotropic by the two distinct contributions to broadening of the signal at 29.1° in 2θ. (Supporting Information Figure S10) However, similar shapes such as spheres and cuboctahedra, as in Figures 1a and 1c, show very similar scattering patterns in the 10 nm size range; TEM is important to distinguish between these shapes. (see Supporting Information Figure S11) In all cases except for the PbSe nanorods, the NCs are shaped in a manner commensurate with the underlying symmetry of the atomic crystal. For example, wurtzite CdSe grows preferentially along the hexagonal c-axis to form elongated rods or cubic NCs (e.g. PbE, CdO) show varying truncation along (100) and (111) facets to generate shapes from cubes to cuboctahedra to octahedra. PbSe nanorods, which form by oriented attachment of smaller crystallites,31 represent a rarer situation in which the underlying symmetry of the crystal structure is not preserved in the particle shape. This is also apparent in the powder and simulated X-ray patterns from the distinctive (200) reflection at 29.1° in 2θ, which shows both a sharp reflection from the nanorod long axis and a broader reflection from the short axes. The sharp (200) reflection unambiguously identifies the (100) as the direction of oriented attachment in PbSe nanorods. Atomistic X-ray simulation is a powerful tool for NC size and shape determination, but it cannot be applied without appropriate data. Both Guinier and Klug and Alexander provide extensive information on sample preparation for the best measurements.14,36,37 Resolution, especially important in the ringing patterns of SAXS
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measurements, is strongly affected by the X-ray beam size according to well-known slit functions.36 Supporting Information Figure S12 and related discussion demonstrates the significant of slit functions. Often, the simulated monodispersity necessary to simulate data at higher angles generates a sharper first valley feature (see particularly Figure 1c) than obtained in experimental data, indicating that experimental data is likely resolution-limited.
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appropriate null signal. Using the atomistic simulation pattern, we estimate that the vertical fraction of nanorods is negligible for the spincoated film and roughly 40% less than expected for an isotropic solution in the drop-cast film. Using the bulk scattering peak intensities would misleadingly suggest that the film has a preference for vertical orientation.
CONCLUSIONS. We have developed an accessible code for atomistic simulation of X-ray scattering based on the Debye equation. This procedure was applied to NCs of several shapes and compositions to generate simulated X-ray patterns that closely match experimental data. We have also explained important considerations of both experimental apparatus and sample preparation which are necessary to obtain useful data for X-ray simulations.
ASSOCIATED CONTENT Supporting Information. Materials and methods, TEM images, instructions and source code for X-ray scattering simulation (in C++), and script for building and running simulations (perl) can be found in the supporting information. This material is available free of charge via the Internet at http://pubs.acs.org.
AUTHOR INFORMATION Corresponding Author *
[email protected] Author Contributions †These authors contributed equally. Figure 2. WAXS scattering data collected for a sample of 12 nm x 1.8 nm anatase TiO2 nanorods prepared in a capillary solution (black), drop-cast (green), and spin-coated (blue) with cartoons showing demonstrating the particle distribution. Overlaid on the data is an atomistic simulation of the ensemble (red line)
Figure 2 shows a common problem which complicates the interpretation of WAXS data from anisotropic particles: preferential orientation. Samples prepared by drop-casting (green), by far the most common method, or spin-coating (blue) show dramatically different intensity ratios between the most prominent peaks. Neither is wholly representative of the scattering of the ensemble. It is critical to ensure that the sample analyzed in X-ray measurements represents an isotropic distribution. This is occasionally done by spinning the sample, which still ignores anisotropic alignment along the spinning axis, but a simpler solution is to disperse the sample in a solvent or low-background matrix (e.g. polymers). Background scattering from glassy matrices, either liquid or solid, is low, reproducible, and easily subtracted. Data taken from solution (black) is readily simulated by the simulated X-ray pattern for anatase TiO2 nanorods, which shows an intense (004) peak at 38.4° in 2θ. Consistent with similar results in Figure 1, the atomistic simulation of anatase nanorods diverges from the peak intensities of bulk anatase powders, reflecting the elongated c-axis of the nanorod. In bulk powders, the (101) peak at 25.7° in 2θ is the most intense reflection with the (004) peak having only 14% of the (101) intensity. Due to the difficulty of collecting pole figures from thin NC films,48 single reflection measurements have been used to quantify orientation of thin films.49 To obtain quantitative orientation information, the solution pattern and its matching simulation, not the bulk powder data, is the
ACKNOWLEDGMENT The authors would like to thank P. Heiney for assistance with acquisition of SAXS patterns and S. Hershman for assistance in writing the C++ code. This work was supported by the National Science Foundation through the University of Pennsylvania’s Nano/Bio Interface Center Grant No. DMR08-32802, which supported the development of modeling, and the MRSEC award No. DMR-1120901, which supported materials synthesis and measurement. Work performed at the Center for Functional Nanomaterials and the National Synchrotron Light Source, Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-SC0012704 C.B.M. acknowledges the Richard Perry University Professorship.
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