Integrated Computational and Experimental Structure Refinement for Nanoparticles Min Yu,* Andrew B. Yankovich, Amy Kaczmarowski, Dane Morgan, and Paul M. Voyles* Department of Materials Science and Engineering, University of WisconsinMadison, Madison, Wisconsin 53706, United States S Supporting Information *
ABSTRACT: Determining the three-dimensional (3D) atomic structure of nanoparticles is critical to identifying the structures controlling their properties. Here, we demonstrate an integrated genetic algorithm (GA) optimization tool that refines the 3D structure of a nanoparticle by matching forward modeling to experimental scanning transmission electron microscopy (STEM) data and simultaneously minimizing the particle energy. We use the tool to create a refined 3D structural model of an experimentally observed ∼6000 atom Au nanoparticle. KEYWORDS: structure optimization, genetic algorithm, scanning transmission electron microscopy, nanoparticles, nanoclusters
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tomography is currently restricted to a limited range of problems and a few highly specialized laboratories. Purely experimentally driven structural refinement approaches typically rely on minimizing the error between forward simulation from atomic models and the experiment data. This approach works well for systems with a limited number of unique atomic positions, such as small unit cell crystals. However, for nanostructures, metastable materials, and other systems with many unique positions, such optimizations are difficult because the available experimental data often do not adequately constrain the structure. This work joins experimental and computational techniques to obtain the best of both methods and provide a tool for creating structural models of nanoparticles. This tool will enable much more complete and rapid refinement of 3D structures from limited data, helping determine structure− property relationships and providing critical input for molecular simulations of nanoparticle systems. Previous efforts to combine experimental and computational techniques to study structure have addressed amorphous materials,16,17 the average structure of populations of nanoparticles,18,19 simultaneous GA optimization of experimental X-ray diffraction data and DFT energies for crystal structure refinement,20 and GA optimization of the orientation of Au clusters to match with model STEM images.21 Serial, by-hand iteration between microscopy experiments and simulations is also a common approach.22 Here, we define a fitness function and mutation schemes for GA optimization to simultaneously and automatically combine experimental STEM image matching and energy minimization
tomic details of nanostructures are important to materials performance for catalysis, solar energy, optoelectronics, sensing, and many other fields. For example, in catalysis, active sites are often surface defects or step edges, and understanding their structure is critical to understanding and predicting their catalytic properties.1 However, solving the three-dimensional (3D) structure of nanoscale materials at the atomic level is challenging, especially for metastable systems (materials that do not occupy the state at the global minimum of the free energy).2 In this work, we develop a tool that refines the 3D structure of individual nanostructures using genetic algorithms (GAs). GAs are a robust global search mechanism that has proven to be extremely effective in structure prediction for a wide range of complex structures, including clusters,3−6 crystals,7−9 grain boundaries,10 and embedded defects.11 Purely computational energy minimization approaches generally make no initial assumptions about a material’s structure and reliably locate lowenergy structures. However, nanomaterials are often metastable, and it is not generally expected that a computational search for the minimal energy will find the correct metastable configuration out of all the possible structures with very similar energy. On the other hand, purely experimental techniques, such as scanning transmission electron microscopy (STEM), provide accurate structural images of materials at atomic resolution. However, a single image generally provides only a two-dimensional (2D) projection of the structure. 3D tomographic STEM at atomic resolution has been demonstrated using prior knowledge in the form of discrete tomography reconstruction from a few carefully selected zone axis images12,13 and very recently using more general tomography methods on very small, highly scattering samples.14,15 However, although the methods are developing rapidly, atomic resolution © XXXX American Chemical Society
Received: September 11, 2015 Accepted: April 11, 2016
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Figure 1. (a) Top view of the Mackay icosahedron 309-atom structure. (b and c) Simulated STEM images of the target structure and the GA optimized structure, respectively. (d) Fitness functions vs generation for GA optimization to minimize energy and discrepancy of STEM image with the reference model Mackay icosahedron. (e) The atomic displacement of individual atoms comparing the GA optimized structure with the target structure shows better than 0.5 pm fidelity.
surface. The details of the GA operations and parameters can be found in the Methods section.
to generate realistic, experimentally based 3D models of nanoparticles. To improve the algorithms speed, we utilize a simple and efficient approximate method for STEM image simulation. We validate this algorithm on Au309 “magic number” nanoclusters with stable Mackay icosahedral and metastable Ino-decahedral geometries, and then apply it to refine the 3D structure of an experimentally measured Au nanoparticle prepared on an amorphous carbon substrate.23 The GA24 fitness function determines whether a population member is selected for mating and survival from one generation to the next. In this work, we define a fitness function as f=
E + αχ 2 atom
RESULTS AND DISCUSSION Our GA optimization is validated using two known “magic number” Au309 nanocluster structures, the Mackay icosahedral and Ino-decahedral geometries.30−32 The stable Mackay icosahedron model is shown in Figures 1a and S1 and displays 5-fold atomic symmetry, indicating non-fcc internal structure, while the simulated STEM image shown in Figure 1b displays 10-fold symmetry due to rotation reflections. We use this simulated STEM image as a reference image for the GA optimization. The optimization starts with random distribution of atoms in a sphere with a diameter of 23 Å, and each generation consists of a population size of 60 structures. The evolution of the fitness function and its component terms is shown in Figure 1d. The whole simulation takes less than 24 h with 60 AMD Opteron 2.2 GHz model 2427 processors, or about 1400 CPU hours. The energy term starts at a high value due to the randomly distributed structure, and when combined with the STEM discrepancy term, the fitness function decays monotonically. All three terms reach convergence after about 2200 generations, and the GA optimized STEM image (Figure 1c) is shown to agree perfectly with reference image (Figure 1b). Furthermore, because we have the exact 3D model structure of the reference image, we can perform one-to-one mapping of 3D atomic positions to verify the agreement between the GA optimized structure and the target structure. Figure 1e shows atomic displacements for individual atoms with a maximum value of 0.5 pm and an
(1)
where E/atom is the energy per atom, χ2 measures the discrepancy between experimental and simulated STEM data, and α is a weighting parameter that is chosen empirically to make variations in the E term and the χ2 term the same order of magnitude. The energy term in eq 1 can be evaluated by either empirical potentials or ab initio methods. In this work, we focus on Au nanoclusters and nanoparticles with hundreds to thousands of atoms and use the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)25 and embedded atom method (EAM) potentials26,27 to calculate the structures’ energies. We use the GA code StructOpt,11 which is open source and distributed as part of the MAterials Simulation Toolkit.28,29 The GA includes both established operations for cluster optimization3 as well as special operations associated with optimizing column intensity matches and creating a stable B
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Figure 2. (a) Top view of the metastable Ino-decahedron 309-atom structure. (b and c) Simulated STEM images of the target structure and the GA optimized structure, respectively. (d) Fitness functions vs generation for GA optimization to minimize energy and discrepancy of STEM image with the reference of the model Ino-decahedron. (e) The atomic displacement of individual atoms comparing the GA optimized structure with the target structure show better than 2 pm fidelity.
Figure 3. (a) Experimental HAADF STEM image of the Au nanoparticle. (b) Multislice simulated STEM image of the GA optimized structure. (c) GA energy evolution of the Au nanoparticle to match the experimental STEM data. (d) Comparison of the projected atomic positions normal to beam direction in the two images.
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Figure 4. (a) Multislice simulated STEM image of the GA optimized structure. (b) Magnified region of the experimental STEM image with the yellow circles representing the atomic column positions that are automatically located in the simulated STEM image but not in the experimental image because they are too diffuse. The green markers are the atomic column positions automatically located in both the simulation and experimental STEM images.
average value of 0.1 pm. Thus, GA optimization reproduces the target stable Mackay icosahedron structure along the designed orientation. The Mackay icosahedron is a stable structure, so optimization against only the energy will find the right structure, although not the right orientation. Figure 2a (and Figure S2) shows the Ino-decahedral structure, which is 12 meV/atom higher in EAM energy than the Mackay icosahedral structure, and also does not have an fcc atomic arrangement. With the use of only energy optimization, the GA optimization would never converge to the metastable structure. However, when the energy and STEM optimization are performed simultaneously, the metastable structure along the correct orientation can be found, as shown in Figure 2b,c. As shown in Figure 2d, the fitness function reaches convergence after about 2500 generations, while the energy term of the GA structure converges to the model Ino-decahedron energy and the STEM discrepancy reduces to 10−6. Atomic displacements analysis for individual atoms is shown in Figure 2e with a maximum value of 2 pm and an average value of 1 pm. This analysis confirms that the GA optimization successfully locates the target metastable structure along the designed orientation. Figure 3 demonstrates 3D structure refinement of an experimentally observed Au nanoparticle. Figure 3a is the experimental STEM image of an ∼8 nm diameter Au particle consisting of ∼6000 atoms oriented with the electron beam direction along [110], containing two crystal domains separated by a (1̅12) twin boundary. The particle is metastable in the sense that it does not occupy the global free energy minimum configuration for its size, since it does not have flat faceted surfaces or the most compact shape, and it contains a defect. To speed convergence of the optimization, we chose as the initial population for the GA a group of single crystalline [110] oriented Au nanoparticles, rather than sets of random atom positions. A population size of 60 was used for each generation. The initial number of atoms was chosen to be 5000 atoms, within the estimated range of the standardless atom counting, and the GA optimization was free to add and remove atoms randomly during evolution. Because the initial population contains low-energy structures, the energy term fluctuates as the optimization proceeds. Nevertheless, the fitness function
decays monotonically through the GA evolution (Figure 3c). At 3000 generations, we introduced new mutation schemes directed toward optimizing the STEM column intensity matches. These mutations (discussed in Supporting Information) randomly select, move, remove, or add surface atoms from or to atomic columns based on the intensity difference between the simulated and the experimental STEM image. These schemes reduce the STEM discrepancy and speed up the final convergence. After 4000 generations, the GA optimization finds the 4998 atom structure shown in Figure 3b and Figure S3, which displays the same ellipsoid shape, twin boundary, and surfaces as the experiment. The GA optimization takes on average 2.5 min per generation with the same 60 AMD Opteron 2.2 GHz model 2427 processors, or about 10 000 CPU hours. A multislice simulated STEM image of the GA optimized structure (Figure 4a) shows good agreement with both the experimental and convolution simulated STEM images. The projected 2D atomic column positions within the simulated and experimental STEM images yield are in excellent agreement, showing the ability of the method to fill in for coarsely sampled experimental data. The atomic column positions in both the experimental and multislice simulated images were extracted by using particle analysis methods to find the columns, then refining the position by fitting to 2D Gaussians. In the multislice simulated image, the analysis found 733 atomic columns, 41 of which are not found in the experimental image. Figure 4b shows a magnified region of the experimental image where example atomic columns that are located in the simulated image but not located in the experimental image are marked by yellow circles. There is intensity in the experimental image consistent with the presence of atoms, but the column images are too diffuse to fit even by hand. These atomic columns are right at the nanoparticle surface, and likely experienced motion during the 240 image series acquisition, causing their diffuse appearance. This motion could be caused by the electron beam or by enhanced thermal vibrations of surface atoms. The presence of these columns in the simulated STEM image shows the ability of the GA optimization to develop a physically reasonable model, informed by the available but intrinsically limited D
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Figure 5. (a) Simulated STEM images using the multislice (top) and convolution (bottom) methods at various model thicknesses. The overestimation of intensity by the convolution method is non-negligible for a thicker sample. (b) Pixel-by-pixel comparison of the 2D STEM intensity calculated using multislice and convolution simulations, fit to a polynomial (red line).
3D atomic structures of individual nanostructures by matching forward modeling to experimental STEM data and simultaneously minimizing the energy. The tool incorporates a fast approximate simulation of STEM images and associated mutation schemes within the GA. We validated the accuracy of this tool by finding both stable and metastable 309-atom Au nanocluster target structures along particular orientations by using simulated STEM images as references. The refined structures match the original input structures to within 2 pm for every atom. We then used the tool to create a refined 3D atomic structural model of an experimentally observed metastable Au nanoparticle prepared on an amorphous carbon support. Given the experimental STEM image, GA optimization converges to a structure with an averaged atomic column displacement of 0.12 Å compared to the experimental STEM image and better information about the position of critical surface atoms than is available from experiment alone. This tool enables practical refinement of 3D chemical and topological structures of individual nanostructures from intrinsically limited experimental data, and will provide critical, experimentally validated starting structures for simulation studies of materials’ properties and developing structure−property relationships. It is available at no cost as part of the Materials Simulation Toolkit.29
experimental data for these potentially important atom positions despite their being very weakly localized in the experiments. On average, the 2D atomic column positions in the experimental and simulated STEM images differ by 0.12 Å, as shown in Figure 3d. To account for pixel nonsquareness in the STEM scan, the experimental pixel sizes in the x and y directions were allowed to scale independently to minimize the root-mean-square displacement between the simulated and experimental atomic column positions. The remaining differences in atomic column positions between simulation and experiment are likely due to the experimental coarse pixel size and potentially inadequate amorphous carbon and ligand background subtraction. The resulting GA optimized structure (Figure S3) is pancake-shaped, rather than round, indicating potentially strong selection bias in the STEM experimental search for a [110]-oriented particle. In principle, GA optimization does not require a zone-axis oriented particle if the image has sufficiently high signal-to-noise ratio to capture contrast variations off zone, potentially allowing us to escape this selection bias in the future. The GA optimized structure also has a relatively rough surface, with significant divots across the nanoparticle, rather than the smoother surfaces one might expect to minimize the energy. Further energy-only optimization starting from Figure S3 where the atomic columns only were allowed to shift up or down (a change intended to maintain the STEM image while allowing some smoothing described in the Supporting Information) led to a structure with smoother surfaces and 13.1 meV/atom lower energy (Figure S4). However, this structure increased the αχ2 term in eq 1 by 1.35. This structure may also arise from large intensity variations in the background support and ligand scattering at unexpected short, interatomic length scales, and inadequacies of the approximate convolution model. Further study on different particles and supports and implementation of full multislice simulations within the GA, potentially with GPU-based computational advancement,33 will be valuable for exploring these effects. Overall, the integrated GA tool successfully optimized the experimentally observed Au nanoparticle structure by matching the experimental and simulated STEM image while simultaneously achieving a structure occupying a local minimum in the energy.
METHODS STEM Experiments. STEM experiments were conducted on a FEI Titan microscope equipped with a CEOS probe aberration corrector. High angle annular dark field (HAADF) STEM images were collected at 200 keV, a 24.5 mrad probe semiangle, a 24.5 pA probe current, a resolution of ∼0.9 Å, and a HAADF detector range of 54−270 mrad. The STEM images were acquired with known probe current and HAADF detector gain to allow the data to be placed on an absolute intensity scale,34,35 as discussed in detail in the Methods section of ref 34. The STEM sample was created by dispersing colloidal Au nanoparticles synthesized using the phosphorus method suspended in water onto a nonpourous 5 nm thick Si membrane window grid. The STEM sample was annealed under vacuum at 200 °C for 48 h, then plasma cleaned in a Fischione plasma cleaner in 25% oxygen−75% argon mixture for ∼5 min. The STEM sample was searched for a nanoparticle near the [011] zone axis, and then minor stage tilting was done to fine-tune the zone axis alignment. To achieve better image signal-to-noise ratio and remove image distortions caused by instabilities in the probe and sample during image acquisition, we acquired an image series of the Au nanoparticle, aligned them using nonrigid (NR) registration,36,37 and then averaged the align series. We acquired 240 images, each having 256 × 256 pixels and recorded with a 5 μs pixel dwell time, for a total dose of ∼850 C/cm2. To fit the entire nanoparticle in the field of view, the atomic lattice of the particle
CONCLUSION In conclusion, we have developed an integrated GA optimization tool to enable accurate and fast refinement of E
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ACS Nano is coarsely sampled, with ∼(60 pm)2 pixels, so each atomic column is sampled by ∼4 × 4 pixels. The intensity from the amorphous carbon support and the ligand material surrounding the nanoparticle, modeled as a 2D Gaussian function plus a constant, was subtracted from the experimental image. The resulting corrected intensity arises to a good approximation only from the nanoparticle. STEM Simulations. Frozen-phonon multislice simulations38 are a general method to simulate STEM images that quantitatively agree with experimental STEM images.34 However, these simulations are too computationally expensive to be applied to every structure during the GA optimization. A more approximate but computationally efficient method to simulate STEM images is the convolution method,39 I(r) = R(r, Z) ⊗ PSF(r)
atom method (EAM) potential,27 evaluated using LAMMPS.25 Whenever there was a change in structure (crossover or mutation), the new model was relaxed to the nearest local energy minimum using conjugate gradient minimization in LAMMPS. Thus, the full fitness function was only ever evaluated for structures that were at least a local minimum in energy. Without human intervention, optimizing the STEM intensity term in the fitness function automatically aligns the Au309 nanoclusters to agree with the reference STEM orientations and the Au nanoparticle along the [110] orientation to agree with experimental STEM image. During evolution, parents are selected to crossover using the tournament selection scheme as described in ref 3. Crossover is carried out on 80% of the population using two-point crossover method3 for Au309 nanoclusters and cut-splice crossover method24 for Au nanoparticles. To maintain population diversity, 20% of the population undergoes atom move and rotate mutation operations. In addition to the standard mutations described in refs 3 and 11, we introduce a number of mutations for optimizing column intensity matches and establishing a stable surface. To optimize STEM intensity matches, we first evaluate the intensity difference between simulated and experimental STEM images, then randomly select and move surface atoms from atomic columns that have higher intensity to atomic columns that have lower intensity compared to the experimental image. For the experimental Au nanoparticle, the exact number of atoms is not known a priori, so we randomly add or remove surface atoms onto or from atomic columns with lower or higher intensity compared to the experimental image. We set intensity difference cutoffs of moving atoms to be greater than 50% of positive maximum or lower than 50% of negative maximum, and apply these schemes to optimize the Au nanoparticle after 3000 generations. These schemes quickly reduce the STEM discrepancy and speed up the convergence. To establish a stable Au nanoparticle surface, we introduce a move mutation scheme to simultaneously shift a number of neighboring atomic columns along the column direction. This scheme can lead to a smoother and more stable surface with a lower energy without significantly impacting the simulated STEM intensity. During each generation, after all the individuals in a population have been locally minimized and their fitness functions have been evaluated, we then use a combination of energy predator and the fixed uniform selection scheme (FUSS) natural selection scheme as described in ref 43 to determine which individuals survive to the next generation.
(2)
where I(r) is the simulated STEM intensity for each point on a 2D grid, R(r, Z) includes the structural information on the atomic coordinates (x,y) and the atomic numbers (Z) of all atoms, and PSF(r) is the point spread function of the electron probe accounting for coherent aberrations and the incoherent electron source size. The simulated intensity using the convolution method is not dependent on the atoms’ z-coordinates; therefore, it gives a linear relationship between intensity and sample thickness. This approximation is qualitatively adequate for thin sample along beam direction, but it overestimates the on-column intensity and underestimates the between-column intensity in thicker samples because it does not account for probe channeling. Figure 5a compares simulated images of an Au [110] unit cell with the frozen-phonon multislice and convolution methods. In the multislice simulation, images of a [110] Au atom column of varying thickness were calculated using 16 phonon configurations and a rootmean-square displacement of 0.0917 Å40 using the Kirkland code.38 The [110] Au supercell had dimensions 34.53 × 32.56 Å2 and was sampled with a 2048 × 2048 pixel wave function using typical measured aberrations for the UWMadison probe-corrected Titan STEM. The simulated images were convolved with a 88.2 pm fullwidth at half-maximum Gaussian function to account for incoherent source broadening.34,42 The atom column peak intensity at 20 layers of atoms is 0.315 from the convolution simulation compared to 0.266 from the multislice simulation. We have developed an approximate correction to convolution intensity from eq 2 that reduces these errors sufficiently for successful structure optimization for materials with known composition. Before GA optimization, we precompute frozen phonon multislice and convolution intensities as a function of sample thickness, shown in part in Figure 5a. The intensity of all the image pixels calculated with multislice as a function of the corresponding pixels calculated with the convolution method is shown in Figure 5b. We fit a high order polynomial to this data set, then use that relationship to scale the convolution-simulated intensities at every step inside the GA to accurately and quickly imitate multislice results. The nonlinear scaling factor needs to be re-evaluated for every different material or change in imaging conditions. The multislice simulations were also used for standardless atom counting41 to obtain an initial estimate of the number of atoms in the experimental nanoparticle, which consists of 6000 ± 1500 atoms. The large uncertainty of this measurement is due to large variation of the background intensity and the coarse image pixel size. Genetic Algorithm on Au Nanoclusters and Nanoparticles. We validated the GA algorithm on Au309 “magic number” nanoclusters with stable Mackay icosahedral and metastable Ino-decahedral geometries, and applied it to refine the 3D structure of an experimentally measured metastable Au nanoparticle prepared on an amorphous carbon substrate. The initial population is generated randomly for Au309 nanoclusters and is chosen as a group of single crystalline [110] oriented and various x−y plane oriented Au nanoparticles based on the selected orientation of the experimental STEM image. A population size of 60 is used in all calculations. The fitness function in eq 1 contains the system energy and matching between experimental and simulated STEM data. The weighting parameter α is chosen as 10 for Au309 nanoclusters and 100 for the Au nanoparticle. The energy was calculated using an embedded
ASSOCIATED CONTENT S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.5b05722. Mackay icosahedral structure of 309-atom Au nanocluster; ino-decahedral structure of 309-atom Au nanocluster; structure of GA optimized and restricted energyonly optimized smooth Au nanoparticle (PDF) original Mackay icosahedron (CIF) Crystallographic data for GA refined Mackay icosahedron (CIF) Crystallographic data for original Ino-decahedron (CIF) Crystallographic data for GA refined Ino-decahedron (CIF) Crystallographic data for GA refined Au nanoparticle (CIF)
AUTHOR INFORMATION Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. Author Contributions
M.Y. developed the GA optimization method based on StructOpt by A.K., performed the optimizations, analyzed the F
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results, and wrote the manuscript. A.B.Y. acquired the STEM images and performed multislice simulations. P.M.V. and D.M. conceived and supervised the project. Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS This work was primary supported by the NSF (DMR1332851), including GA optimization development and application. STEM image acquisition and processing was supported by the Department of Energy, Office of Basic Energy Sciences (DE-FG02−08ER46547) using instrumentation in the UWMadison Materials Science Center, which is supported in part by the UW Materials Research Science and Engineering Center (DMR-1121288). Development and distribution of MAST is support by the NSF Software Infrastructure for Sustained Innovation (SI2), Award No. 1148011. REFERENCES (1) Zaera, F. New Challenge in Heterogeneous Catalysis for the 21st century. Catal. Lett. 2012, 142, 501. (2) Billinge, S. J. L.; Levin, I. The Problem with Determining Atomic Structure at the Nanoscale. Science 2007, 316, 561. (3) Johnston, R. L. Evolving Better Nanoparticles: Genetic Algorithms for Optimising Cluster Geometries. Dalton Trans. 2003, 4193. (4) Hartke, B. Application of Evolutionary Algorithms to Global Cluster Geometry Optimization. Struct. Bonding (Berlin, Ger.) 2004, 110, 33. (5) Dieterich, J. M.; Hartke, B. OGOLEM: Global Cluster Structure Optimisation for Abritrary Mixtures of Flexible Molecules. A Multiscaling, Object-Oriented Approach. Mol. Phys. 2010, 108, 279− 291. (6) Dugan, N.; Erkoc, S. Genetic Algorithm − Monte Carlo Hybrid Geometry Optimization Method for Atomic Clusters. Comput. Mater. Sci. 2009, 45, 127. (7) Revard, B. C.; Tipton, W. W.; Hennig, R. G. Structure and Stability Prediction of Compounds and Evolutionary Algorithms. Top. Curr. Chem. 2014, 345, 181. (8) Glass, C. W.; Oganov, A. R.; Hansen, N. USPEXEvolutionary Crystal Structure Prediction. Comput. Phys. Commun. 2006, 175, 713. (9) Woodley, S. M. Prediction of Crystal Structures Using Evolutionary Algorithms and Related Techniques. Struct. Bonding (Berlin, Ger.) 2004, 110, 95. (10) Zhang, J.; Wang, C. Z.; Ho, K. M. Finding the Low-Energy Structure of Si[001] Symmetric Tilted Grain Boundaries with a Genetic Algorithm. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 174102. (11) Kaczmarowski, A.; Yang, S.; Szlufarska, I.; Morgan, D. Genetic Algorithm Optimization of Defect Clusters in Crystalline Materials. Comput. Mater. Sci. 2015, 98, 234. (12) Van Aert, S.; Batenburg, K. J.; Rossell, M. D.; Erni, R.; Van Tendeloo, G. Three-Dimensional Atomic Imaging of Crystalline Nanoparticles. Nature 2011, 470, 374−377. (13) Goris, B.; De Backer, A.; Van Aert, S.; Gómez-Graña, S.; LizMarzán, L. M.; Van Tendeloo, G.; Bals, S. Three-Dimensional Elemental Mapping at the Atomic Scale in Bimetallic Nanocrystals. Nano Lett. 2013, 13, 4236−4241. (14) Scott, M. C.; Chen, C.-C.; Mecklenburg, M.; Zhu, C.; Xu, R.; Ercius, P.; Dahmen, U.; Regan, B. C.; Miao, J. Electron Tomography at 2.4-ångström Resolution. Nature 2012, 483, 444−447. (15) Chen, C.-C.; Zhu, C.; White, E. R.; Chiu, C.-Y.; Scott, M. C.; Regan, B. C.; Marks, L. D.; Huang, Y.; Miao, J. Three-Dimensional Imaging of Dislocations in a Nanoparticle at Atomic Resolution. Nature 2013, 496, 74−77. G
DOI: 10.1021/acsnano.5b05722 ACS Nano XXXX, XXX, XXX−XXX
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DOI: 10.1021/acsnano.5b05722 ACS Nano XXXX, XXX, XXX−XXX
