A numerical method for deriving shape functions of nanoparticles for pair distribution function refinements

Usher, Tedi-Marie; Olds, Daniel P.; Liu, Jue; Page, Katharine L.

doi:10.1107/S2053273318004977

Title: A numerical method for deriving shape functions of nanoparticles for pair distribution function refinements

Journal Article · Wed Jun 06 00:00:00 EDT 2018 · Acta Crystallographica. Section A, Foundations and Advances (Online)

DOI:https://doi.org/10.1107/S2053273318004977· OSTI ID:1474606

^[1];

^[1]

Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Neutron Scattering Division

We report that in the structural refinement of nanoparticles, discrete atomistic modeling can be used for small nanocrystals (< 15 nm), but becomes computationally unfeasible at larger sizes, where instead unit-cell-based small-box modeling is usually employed. However, the effect of the nanocrystal's shape is often ignored or accounted for with a spherical model regardless of the actual shape due to the complexities of solving and implementing accurate shape effects. Recent advancements have provided a way to determine the shape function directly from a pair distribution function calculated from a discrete atomistic model of any given shape, including both regular polyhedra (e.g. cubes, spheres, octahedra) and anisotropic shapes (e.g. rods, discs, ellipsoids) [Olds et al. (2015). J. Appl. Cryst. 48, 1651–1659], although this approach is still limited to small size regimes due to computational demands. In order to accurately account for the effects of nanoparticle size and shape in small-box refinements, a numerical or analytical description is needed. This article presents a methodology to derive numerical approximations of nanoparticle shape functions by fitting to a training set of known shape functions; the numerical approximations can then be employed on larger sizes yielding a more accurate and physically meaningful refined nanoparticle size. Lastly, the method is demonstrated on a series of simulated and real data sets, and a table of pre-calculated shape function expressions for a selection of common shapes is provided.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES)

Grant/Contract Number:: AC05-00OR22725; AC02-06CH11357

OSTI ID:: 1474606

Journal Information:: Acta Crystallographica. Section A, Foundations and Advances (Online), Vol. 74, Issue 4; ISSN 2053-2733

Publisher:: International Union of CrystallographyCopyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 22 works

Citation information provided by
Web of Science

References (69)

Multiple Wurtzite Twinning in CdTe Nanocrystals Induced by Methylphosphonic Acid Carbone, Luigi; Kudera, Stefan; Carlino, Elvio Journal of the American Chemical Society, Vol. 128, Issue 3 https://doi.org/10.1021/ja054893c	journal	January 2006
Electrochemical Photolysis of Water at a Semiconductor Electrode Fujishima, Akira; Honda, Kenichi Nature, Vol. 238, Issue 5358, p. 37-38 https://doi.org/10.1038/238037a0	journal	July 1972
Efficient algorithms for calculating small-angle scattering from large model structures Olds, Daniel P.; Duxbury, Phillip M. Journal of Applied Crystallography, Vol. 47, Issue 3 https://doi.org/10.1107/S1600576714005925	journal	May 2014
Diffuse Scattering and Defect Structure Simulations Neder, Reinhard B.; Proffen, Thomas Oxford University Press https://doi.org/10.1093/acprof:oso/9780199233694.001.0001	book	July 2008
The Problem with Determining Atomic Structure at the Nanoscale Billinge, S. J. L.; Levin, I. Science, Vol. 316, Issue 5824 https://doi.org/10.1126/science.1135080	journal	April 2007
Combined fitting of small- and wide-angle X-ray total scattering data from nanoparticles: benefits and issues Gagin, Anton; Allen, Andrew J.; Levin, Igor Journal of Applied Crystallography, Vol. 47, Issue 2 https://doi.org/10.1107/S1600576714001046	journal	March 2014
Highly Crystalline Multimetallic Nanoframes with Three-Dimensional Electrocatalytic Surfaces Chen, C.; Kang, Y.; Huo, Z. Science, Vol. 343, Issue 6177 https://doi.org/10.1126/science.1249061	journal	February 2014
For high-pressure experiments using total scattering spectrometer NOVA at J-PARC Hattori, T.; Katayama, Y.; Machida, A. Journal of Physics: Conference Series, Vol. 215 https://doi.org/10.1088/1742-6596/215/1/012024	journal	March 2010
Molecular arrangement in water: random but not quite Petkov, V.; Ren, Y.; Suchomel, M. Journal of Physics: Condensed Matter, Vol. 24, Issue 15 https://doi.org/10.1088/0953-8984/24/15/155102	journal	March 2012
Solvothermal synthesis and controlled self-assembly of monodisperse titanium-based perovskite colloidal nanocrystals Caruntu, Daniela; Rostamzadeh, Taha; Costanzo, Tommaso Nanoscale, Vol. 7, Issue 30 https://doi.org/10.1039/C5NR00737B	journal	January 2015
Fast synthesis and refinement of the atomic pair distribution function Coelho, A. A.; Chater, P. A.; Kern, A. Journal of Applied Crystallography, Vol. 48, Issue 3 https://doi.org/10.1107/S1600576715007487	journal	May 2015
Three-dimensional controlled growth of monodisperse sub-50 nm heterogeneous nanocrystals Liu, Deming; Xu, Xiaoxue; Du, Yi Nature Communications, Vol. 7, Issue 1 https://doi.org/10.1038/ncomms10254	journal	January 2016
DISCUS : a program for diffuse scattering and defect-structure simulation Proffen, Th.; Neder, R. B. Journal of Applied Crystallography, Vol. 30, Issue 2 https://doi.org/10.1107/S002188989600934X	journal	April 1997
Complex modeling: a strategy and software program for combining multiple information sources to solve ill posed structure and nanostructure inverse problems Juhás, Pavol; Farrow, Christopher L.; Yang, Xiaohao Acta Crystallographica Section A Foundations and Advances, Vol. 71, Issue 6, p. 562-568 https://doi.org/10.1107/S2053273315014473	journal	September 2015
Single Crystal Manganese Oxide Multipods by Oriented Attachment Zitoun, David; Pinna, Nicola; Frolet, Nathalie Journal of the American Chemical Society, Vol. 127, Issue 43 https://doi.org/10.1021/ja0555926	journal	November 2005
Quantitative Analysis of the Morphology of {101} and {001} Faceted Anatase TiO ₂ Nanocrystals and Its Implication on Photocatalytic Activity Liu, Jue; Olds, Daniel; Peng, Rui Chemistry of Materials, Vol. 29, Issue 13 https://doi.org/10.1021/acs.chemmater.7b01172	journal	June 2017
GEM — General materials diffractometer at ISIS Williams, W. G.; Ibberson, R. M.; Day, P. Physica B: Condensed Matter, Vol. 241-243 https://doi.org/10.1016/S0921-4526(97)00561-9	journal	December 1997
Size, Shape, and Internal Atomic Ordering of Nanocrystals by Atomic Pair Distribution Functions: A Comparative Study of γ-Fe ₂ O ₃ Nanosized Spheres and Tetrapods Petkov, Valeri; Cozzoli, P. Davide; Buonsanti, Raffaella Journal of the American Chemical Society, Vol. 131, Issue 40 https://doi.org/10.1021/ja9067589	journal	October 2009
On the True Photoreactivity Order of {001}, {010}, and {101} Facets of Anatase TiO2 Crystals Pan, Jian; Liu, Gang; Lu, Gao Qing Max Angewandte Chemie International Edition, Vol. 50, Issue 9 https://doi.org/10.1002/anie.201006057	journal	January 2011
Anatase TiO2 single crystals with a large percentage of reactive facets Yang, Hua Gui; Sun, Cheng Hua; Qiao, Shi Zhang Nature, Vol. 453, Issue 7195 https://doi.org/10.1038/nature06964	journal	May 2008
Real-space calculation of powder diffraction patterns on graphics processing units Gelisio, Luca; Azanza Ricardo, Cristy Leonor; Leoni, Matteo Journal of Applied Crystallography, Vol. 43, Issue 3 https://doi.org/10.1107/S0021889810005133	journal	April 2010
Complexity at mesoscopic lengthscale Egami, T. IUCrJ, Vol. 2, Issue 5, p. 479-480 https://doi.org/10.1107/S2052252515014670	journal	August 2015
Nanoscale Ni/Mn Ordering in the High Voltage Spinel Cathode LiNi _0.5 Mn _1.5 O ₄ Liu, Jue; Huq, Ashfia; Moorhead-Rosenberg, Zachary Chemistry of Materials, Vol. 28, Issue 19 https://doi.org/10.1021/acs.chemmater.6b02946	journal	September 2016
Insights into the Local Structure of Tb-Doped KY ₃ F ₁₀ Nanoparticles from Synchrotron X-ray Diffraction Ichikawa, Rodrigo U.; Linhares, Horacio M. S. M. D.; Peral, Inma ACS Omega, Vol. 2, Issue 8 https://doi.org/10.1021/acsomega.7b00668	journal	August 2017
Resolving the Structure of Ti ₃ C ₂ T _x MXenes through Multilevel Structural Modeling of the Atomic Pair Distribution Function Wang, Hsiu-Wen; Naguib, Michael; Page, Katharine Chemistry of Materials, Vol. 28, Issue 1 https://doi.org/10.1021/acs.chemmater.5b04250	journal	December 2015
Precise implications for real-space pair distribution function modeling of effects intrinsic to modern time-of-flight neutron diffractometers Olds, Daniel; Saunders, Claire N.; Peters, Megan Acta Crystallographica Section A Foundations and Advances, Vol. 74, Issue 4 https://doi.org/10.1107/S2053273318003224	journal	June 2018
Structural coherence and ferroelectricity decay in submicron- and nano-sized perovskites Petkov, V.; Buscaglia, V.; Buscaglia, M. T. Physical Review B, Vol. 78, Issue 5 https://doi.org/10.1103/PhysRevB.78.054107	journal	August 2008
Building a high resolution total scattering powder diffractometer - upgrade of NPD at MLNSC Proffen, T.; Egami, T.; Billinge, S. J. L. Applied Physics A: Materials Science & Processing, Vol. 74, Issue 0 https://doi.org/10.1007/s003390201929	journal	December 2002
Building and refining complete nanoparticle structures with total scattering data Page, Katharine; Hood, Taylor C.; Proffen, Thomas Journal of Applied Crystallography, Vol. 44, Issue 2 https://doi.org/10.1107/S0021889811001968	journal	February 2011
Structure and energetics of stoichiometric ${TiO}_{2}$ anatase surfaces Lazzeri, Michele; Vittadini, Andrea; Selloni, Annabella Physical Review B, Vol. 63, Issue 15 https://doi.org/10.1103/PhysRevB.63.155409	journal	March 2001
Relationship between the atomic pair distribution function and small-angle scattering: implications for modeling of nanoparticles Farrow, Christopher L.; Billinge, Simon J. L. Acta Crystallographica Section A Foundations of Crystallography, Vol. 65, Issue 3, p. 232-239 https://doi.org/10.1107/S0108767309009714	journal	April 2009
The Polaris powder diffractometer at ISIS Hull, S.; Smith, R. I.; David, W. I. F. Physica B: Condensed Matter, Vol. 180-181 https://doi.org/10.1016/0921-4526(92)90533-X	journal	June 1992
Pair distribution function and structure factor of spherical particles Howell, Rafael C.; Proffen, Thomas; Conradson, Steven D. Physical Review B, Vol. 73, Issue 9 https://doi.org/10.1103/PhysRevB.73.094107	journal	March 2006
DShaper : an approach for handling missing low- Q data in pair distribution function analysis of nanostructured systems Olds, Daniel; Wang, Hsiu-Wen; Page, Katharine Journal of Applied Crystallography, Vol. 48, Issue 6 https://doi.org/10.1107/S1600576715016581	journal	October 2015
Ferroelectric order in individual nanometre-scale crystals Polking, Mark J.; Han, Myung-Geun; Yourdkhani, Amin Nature Materials, Vol. 11, Issue 8 https://doi.org/10.1038/nmat3371	journal	July 2012
Titanium Dioxide Nanomaterials: Synthesis, Properties, Modifications, and Applications Chen, Xiaobo; Mao, Samuel S. Chemical Reviews, Vol. 107, Issue 7 https://doi.org/10.1021/cr0500535	journal	July 2007
Percolating bulk heterostructures from neutron reflectometry and small-angle scattering data Olds, Daniel P.; Duxbury, Phillip M.; Kiel, Jonathan W. Physical Review E, Vol. 86, Issue 6 https://doi.org/10.1103/PhysRevE.86.061803	journal	December 2012
The Morphology of TiO ₂ (B) Nanoparticles Hua, Xiao; Liu, Zheng; Bruce, Peter G. Journal of the American Chemical Society, Vol. 137, Issue 42 https://doi.org/10.1021/jacs.5b08434	journal	October 2015
Anisotropic phase segregation and migration of Pt in nanocrystals en route to nanoframe catalysts Niu, Zhiqiang; Becknell, Nigel; Yu, Yi Nature Materials, Vol. 15, Issue 11 https://doi.org/10.1038/nmat4724	journal	August 2016
Manganese(II) Oxide Nanohexapods: Insight into Controlling the Form of Nanocrystals Ould-Ely, Teyeb; Prieto-Centurion, Dario; Kumar, A. Chemistry of Materials, Vol. 18, Issue 7 https://doi.org/10.1021/cm052492q	journal	April 2006
Reactivity of Anatase TiO ₂ Nanoparticles: The Role of the Minority (001) Surface Gong, Xue-Qing; Selloni, Annabella The Journal of Physical Chemistry B, Vol. 109, Issue 42 https://doi.org/10.1021/jp055311g	journal	October 2005
The Nanoscale Ordered MAterials Diffractometer NOMAD at the Spallation Neutron Source SNS Neuefeind, Jörg; Feygenson, Mikhail; Carruth, John Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, Vol. 287 https://doi.org/10.1016/j.nimb.2012.05.037	journal	September 2012
Ferroelectric Barium Titanate Nanocubes as Capacitive Building Blocks for Energy Storage Applications Parizi, Saman Salemizadeh; Mellinger, Axel; Caruntu, Gabriel ACS Applied Materials & Interfaces, Vol. 6, Issue 20 https://doi.org/10.1021/am502547h	journal	October 2014
Reciprocal-space instrumental effects on the real-space neutron atomic pair distribution function Qiu, Xiangyun; Božin, Emil S.; Juhas, Pavol Journal of Applied Crystallography, Vol. 37, Issue 1 https://doi.org/10.1107/S0021889803026670	journal	January 2004
Structure and Stability of SnO ₂ Nanocrystals and Surface-Bound Water Species Wang, Hsiu-Wen; Wesolowski, David J.; Proffen, Thomas E. Journal of the American Chemical Society, Vol. 135, Issue 18 https://doi.org/10.1021/ja312030e	journal	April 2013
PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals Farrow, C. L.; Juhas, P.; Liu, J. W. Journal of Physics: Condensed Matter, Vol. 19, Issue 33 https://doi.org/10.1088/0953-8984/19/33/335219	journal	July 2007
Monopod, Bipod, Tripod, and Tetrapod Gold Nanocrystals Chen, Sihai; Wang, Zhong Lin; Ballato, John Journal of the American Chemical Society, Vol. 125, Issue 52, p. 16186-16187 https://doi.org/10.1021/ja038927x	journal	December 2003
Uncovering the intrinsic geometry from the atomic pair distribution function of nanomaterials Lei, Ming; de Graff, Adam M. R.; Thorpe, M. F. Physical Review B, Vol. 80, Issue 2 https://doi.org/10.1103/PhysRevB.80.024118	journal	July 2009
Local and average structures of BaTiO3-Bi(Zn1/2Ti1/2)O3 Usher, Tedi-Marie; Iamsasri, Thanakorn; Forrester, Jennifer S. Journal of Applied Physics, Vol. 120, Issue 18 https://doi.org/10.1063/1.4967222	journal	November 2016
Effects of Bragg peak profiles and nanoparticle sizes on the real-space pair distribution function Jeong, Il-Kyoung; Graf, M. J.; Heffner, R. H. Journal of Applied Crystallography, Vol. 38, Issue 1 https://doi.org/10.1107/S0021889804025841	journal	January 2005
Nanoparticulate TiO2(B): An Anode for Lithium-Ion Batteries Ren, Yu; Liu, Zheng; Pourpoint, Frédérique Angewandte Chemie International Edition, Vol. 51, Issue 9 https://doi.org/10.1002/anie.201108300	journal	January 2012
Colloidal Synthesis and Characterization of Tetrapod-Shaped Magnetic Nanocrystals Cozzoli, P. Davide; Snoeck, Etienne; Garcia, Miguel Angel Nano Letters, Vol. 6, Issue 9 https://doi.org/10.1021/nl061112c	journal	September 2006
Finite size effects on the real-space pair distribution function of nanoparticles Gilbert, Benjamin Journal of Applied Crystallography, Vol. 41, Issue 3 https://doi.org/10.1107/S0021889808007905	journal	May 2008
Total scattering atomic pair distribution function: new methodology for nanostructure determination Masadeh, Ahmad S. Journal of Experimental Nanoscience, Vol. 11, Issue 12 https://doi.org/10.1080/17458080.2016.1184769	journal	May 2016
Ferroelectric properties of composites containing BaTiO ₃ nanoparticles of various sizes Adam, Jens; Lehnert, Tobias; Klein, Gabi Nanotechnology, Vol. 25, Issue 6 https://doi.org/10.1088/0957-4484/25/6/065704	journal	January 2014
Structure determination of molecular nanocomposites by combining pair distribution function analysis and solid-state NMR Bendeif, E. -E.; Gansmuller, A.; Hsieh, K. -Y. RSC Advances, Vol. 5, Issue 12 https://doi.org/10.1039/C4RA11470A	journal	January 2015
Unit-cell scale mapping of ferroelectricity and tetragonality in epitaxial ultrathin ferroelectric films Jia, Chun-Lin; Nagarajan, Valanoor; He, Jia-Qing Nature Materials, Vol. 6, Issue 1 https://doi.org/10.1038/nmat1808	journal	December 2006
Crystal Structure and the Paraelectric-to-Ferroelectric Phase Transition of Nanoscale BaTiO ₃ Smith, Millicent B.; Page, Katharine; Siegrist, Theo Journal of the American Chemical Society, Vol. 130, Issue 22 https://doi.org/10.1021/ja0758436	journal	June 2008
Probing Local Dipoles and Ligand Structure in BaTiO ₃ Nanoparticles Page, Katharine; Proffen, Thomas; Niederberger, Markus Chemistry of Materials, Vol. 22, Issue 15 https://doi.org/10.1021/cm100440p	journal	August 2010
Nonaqueous Synthesis of TiO ₂ Nanocrystals Using TiF ₄ to Engineer Morphology, Oxygen Vacancy Concentration, and Photocatalytic Activity Gordon, Thomas R.; Cargnello, Matteo; Paik, Taejong Journal of the American Chemical Society, Vol. 134, Issue 15 https://doi.org/10.1021/ja300823a	journal	April 2012
The interpretation of real-space information from small-angle scattering experiments Glatter, O. Journal of Applied Crystallography, Vol. 12, Issue 2 https://doi.org/10.1107/S0021889879012139	journal	April 1979
Characteristics of Barium Titanate Nanocube Ordered Assembly Thin Films Fabricated by Dip-Coating Method Mimura, Ken-ichi; Kato, Kazumi Japanese Journal of Applied Physics, Vol. 52, Issue 9S1 https://doi.org/10.7567/JJAP.52.09KC06	journal	September 2013
Nanoparticulate TiO2(B): An Anode for Lithium-Ion Batteries Ren, Yu; Liu, Zheng; Pourpoint, Frédérique Angewandte Chemie, Vol. 124, Issue 9 https://doi.org/10.1002/ange.201108300	journal	January 2012
Titanium Dioxide Nanomaterials: Synthesis, Properties, Modifications, and Applications Chen, Xiaobo; Mao, Samuel S. ChemInform, Vol. 38, Issue 41 https://doi.org/10.1002/chin.200741216	journal	October 2007
On the True Photoreactivity Order of {001}, {010}, and {101} Facets of Anatase TiO2 Crystals Pan, Jian; Liu, Gang; Lu, Gao Qing Max Angewandte Chemie, Vol. 123, Issue 9 https://doi.org/10.1002/ange.201006057	journal	January 2011
Structure determination of molecular nanocomposites by combining pair distribution function analysis and solid-state NMR Schaniel, Dominik; Bendeif, El-Eulmi; Gansmuller, Axel Acta Crystallographica Section A Foundations and Advances, Vol. 71, Issue a1 https://doi.org/10.1107/s2053273315094292	journal	August 2015
Single Crystal Manganese Oxide Multipods by Oriented Attachment. Zitoun, David; Pinna, Nicola; Frolet, Nathalie ChemInform, Vol. 37, Issue 3 https://doi.org/10.1002/chin.200603216	journal	January 2006
Relationship between the atomic pair distribution function and small angle scattering: implications for modeling of nanoparticles Farrow, Christopher L.; Billinge, Simon J. L. arXiv https://doi.org/10.48550/arxiv.0811.1140	text	January 2008
Pair distribution function and structure factor of spherical particles Howell, Rafael C.; Proffen, Thomas; Conradson, Steven D. arXiv https://doi.org/10.48550/arxiv.cond-mat/0510679	text	January 2005

Cited By (3)

Reverse Monte Carlo modeling for low-dimensional systems Zhang, Yuanpeng; McDonnell, Marshall; Liu, Wei Journal of Applied Crystallography, Vol. 52, Issue 5 https://doi.org/10.1107/s160057671901080x	journal	August 2019
A simple correction for the parallax effect in X-ray pair distribution function measurements Marlton, Frederick; Ivashko, Oleh; Zimmerman, Martin v. Journal of Applied Crystallography, Vol. 52, Issue 5 https://doi.org/10.1107/s1600576719011580	journal	September 2019
A simple correction for the parallax effect in X-ray pair distribution function measurements Marlton, Frederick; Ivashko, Oleh; Von Zimmermann, Martin Deutsches Elektronen-Synchrotron, DESY, Hamburg https://doi.org/10.3204/pubdb-2019-03307	text	January 2019