A numerical method for deriving shape functions of nanoparticles for pair distribution function refinements
Abstract
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.more »
- Authors:
-
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Neutron Scattering Division
- Publication Date:
- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES)
- OSTI Identifier:
- 1474606
- Grant/Contract Number:
- AC05-00OR22725; AC02-06CH11357
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Acta Crystallographica. Section A, Foundations and Advances (Online)
- Additional Journal Information:
- Journal Name: Acta Crystallographica. Section A, Foundations and Advances (Online); Journal Volume: 74; Journal Issue: 4; Journal ID: ISSN 2053-2733
- Publisher:
- International Union of Crystallography
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 36 MATERIALS SCIENCE; nanoparticles; shape function; pair distribution function; total scattering
Citation Formats
Usher, Tedi-Marie, Olds, Daniel P., Liu, Jue, and Page, Katharine L. A numerical method for deriving shape functions of nanoparticles for pair distribution function refinements. United States: N. p., 2018.
Web. doi:10.1107/S2053273318004977.
Usher, Tedi-Marie, Olds, Daniel P., Liu, Jue, & Page, Katharine L. A numerical method for deriving shape functions of nanoparticles for pair distribution function refinements. United States. https://doi.org/10.1107/S2053273318004977
Usher, Tedi-Marie, Olds, Daniel P., Liu, Jue, and Page, Katharine L. Wed .
"A numerical method for deriving shape functions of nanoparticles for pair distribution function refinements". United States. https://doi.org/10.1107/S2053273318004977. https://www.osti.gov/servlets/purl/1474606.
@article{osti_1474606,
title = {A numerical method for deriving shape functions of nanoparticles for pair distribution function refinements},
author = {Usher, Tedi-Marie and Olds, Daniel P. and Liu, Jue and Page, Katharine L.},
abstractNote = {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.},
doi = {10.1107/S2053273318004977},
journal = {Acta Crystallographica. Section A, Foundations and Advances (Online)},
number = 4,
volume = 74,
place = {United States},
year = {2018},
month = {6}
}
Figures / Tables:

Works referenced in this record:
Multiple Wurtzite Twinning in CdTe Nanocrystals Induced by Methylphosphonic Acid
journal, January 2006
- Carbone, Luigi; Kudera, Stefan; Carlino, Elvio
- Journal of the American Chemical Society, Vol. 128, Issue 3
Electrochemical Photolysis of Water at a Semiconductor Electrode
journal, July 1972
- Fujishima, Akira; Honda, Kenichi
- Nature, Vol. 238, Issue 5358, p. 37-38
Efficient algorithms for calculating small-angle scattering from large model structures
journal, May 2014
- Olds, Daniel P.; Duxbury, Phillip M.
- Journal of Applied Crystallography, Vol. 47, Issue 3
The Problem with Determining Atomic Structure at the Nanoscale
journal, April 2007
- Billinge, S. J. L.; Levin, I.
- Science, Vol. 316, Issue 5824
Combined fitting of small- and wide-angle X-ray total scattering data from nanoparticles: benefits and issues
journal, March 2014
- Gagin, Anton; Allen, Andrew J.; Levin, Igor
- Journal of Applied Crystallography, Vol. 47, Issue 2
Highly Crystalline Multimetallic Nanoframes with Three-Dimensional Electrocatalytic Surfaces
journal, February 2014
- Chen, C.; Kang, Y.; Huo, Z.
- Science, Vol. 343, Issue 6177
For high-pressure experiments using total scattering spectrometer NOVA at J-PARC
journal, March 2010
- Hattori, T.; Katayama, Y.; Machida, A.
- Journal of Physics: Conference Series, Vol. 215
Molecular arrangement in water: random but not quite
journal, March 2012
- Petkov, V.; Ren, Y.; Suchomel, M.
- Journal of Physics: Condensed Matter, Vol. 24, Issue 15
Solvothermal synthesis and controlled self-assembly of monodisperse titanium-based perovskite colloidal nanocrystals
journal, January 2015
- Caruntu, Daniela; Rostamzadeh, Taha; Costanzo, Tommaso
- Nanoscale, Vol. 7, Issue 30
Fast synthesis and refinement of the atomic pair distribution function
journal, May 2015
- Coelho, A. A.; Chater, P. A.; Kern, A.
- Journal of Applied Crystallography, Vol. 48, Issue 3
Three-dimensional controlled growth of monodisperse sub-50 nm heterogeneous nanocrystals
journal, January 2016
- Liu, Deming; Xu, Xiaoxue; Du, Yi
- Nature Communications, Vol. 7, Issue 1
DISCUS : a program for diffuse scattering and defect-structure simulation
journal, April 1997
- Proffen, Th.; Neder, R. B.
- Journal of Applied Crystallography, Vol. 30, Issue 2
Complex modeling: a strategy and software program for combining multiple information sources to solve ill posed structure and nanostructure inverse problems
journal, September 2015
- Juhás, Pavol; Farrow, Christopher L.; Yang, Xiaohao
- Acta Crystallographica Section A Foundations and Advances, Vol. 71, Issue 6, p. 562-568
Single Crystal Manganese Oxide Multipods by Oriented Attachment
journal, November 2005
- Zitoun, David; Pinna, Nicola; Frolet, Nathalie
- Journal of the American Chemical Society, Vol. 127, Issue 43
Quantitative Analysis of the Morphology of {101} and {001} Faceted Anatase TiO 2 Nanocrystals and Its Implication on Photocatalytic Activity
journal, June 2017
- Liu, Jue; Olds, Daniel; Peng, Rui
- Chemistry of Materials, Vol. 29, Issue 13
GEM — General materials diffractometer at ISIS
journal, December 1997
- Williams, W. G.; Ibberson, R. M.; Day, P.
- Physica B: Condensed Matter, Vol. 241-243
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
journal, October 2009
- Petkov, Valeri; Cozzoli, P. Davide; Buonsanti, Raffaella
- Journal of the American Chemical Society, Vol. 131, Issue 40
On the True Photoreactivity Order of {001}, {010}, and {101} Facets of Anatase TiO2 Crystals
journal, January 2011
- Pan, Jian; Liu, Gang; Lu, Gao Qing Max
- Angewandte Chemie International Edition, Vol. 50, Issue 9
Anatase TiO2 single crystals with a large percentage of reactive facets
journal, May 2008
- Yang, Hua Gui; Sun, Cheng Hua; Qiao, Shi Zhang
- Nature, Vol. 453, Issue 7195
Real-space calculation of powder diffraction patterns on graphics processing units
journal, April 2010
- Gelisio, Luca; Azanza Ricardo, Cristy Leonor; Leoni, Matteo
- Journal of Applied Crystallography, Vol. 43, Issue 3
Complexity at mesoscopic lengthscale
journal, August 2015
- Egami, T.
- IUCrJ, Vol. 2, Issue 5, p. 479-480
Nanoscale Ni/Mn Ordering in the High Voltage Spinel Cathode LiNi 0.5 Mn 1.5 O 4
journal, September 2016
- Liu, Jue; Huq, Ashfia; Moorhead-Rosenberg, Zachary
- Chemistry of Materials, Vol. 28, Issue 19
Insights into the Local Structure of Tb-Doped KY 3 F 10 Nanoparticles from Synchrotron X-ray Diffraction
journal, August 2017
- Ichikawa, Rodrigo U.; Linhares, Horacio M. S. M. D.; Peral, Inma
- ACS Omega, Vol. 2, Issue 8
Resolving the Structure of Ti 3 C 2 T x MXenes through Multilevel Structural Modeling of the Atomic Pair Distribution Function
journal, December 2015
- Wang, Hsiu-Wen; Naguib, Michael; Page, Katharine
- Chemistry of Materials, Vol. 28, Issue 1
Precise implications for real-space pair distribution function modeling of effects intrinsic to modern time-of-flight neutron diffractometers
journal, June 2018
- Olds, Daniel; Saunders, Claire N.; Peters, Megan
- Acta Crystallographica Section A Foundations and Advances, Vol. 74, Issue 4
Structural coherence and ferroelectricity decay in submicron- and nano-sized perovskites
journal, August 2008
- Petkov, V.; Buscaglia, V.; Buscaglia, M. T.
- Physical Review B, Vol. 78, Issue 5
Building a high resolution total scattering powder diffractometer - upgrade of NPD at MLNSC
journal, December 2002
- Proffen, T.; Egami, T.; Billinge, S. J. L.
- Applied Physics A: Materials Science & Processing, Vol. 74, Issue 0
Building and refining complete nanoparticle structures with total scattering data
journal, February 2011
- Page, Katharine; Hood, Taylor C.; Proffen, Thomas
- Journal of Applied Crystallography, Vol. 44, Issue 2
Structure and energetics of stoichiometric anatase surfaces
journal, March 2001
- Lazzeri, Michele; Vittadini, Andrea; Selloni, Annabella
- Physical Review B, Vol. 63, Issue 15
Relationship between the atomic pair distribution function and small-angle scattering: implications for modeling of nanoparticles
journal, April 2009
- Farrow, Christopher L.; Billinge, Simon J. L.
- Acta Crystallographica Section A Foundations of Crystallography, Vol. 65, Issue 3, p. 232-239
The Polaris powder diffractometer at ISIS
journal, June 1992
- Hull, S.; Smith, R. I.; David, W. I. F.
- Physica B: Condensed Matter, Vol. 180-181
Pair distribution function and structure factor of spherical particles
journal, March 2006
- Howell, Rafael C.; Proffen, Thomas; Conradson, Steven D.
- Physical Review B, Vol. 73, Issue 9
DShaper : an approach for handling missing low- Q data in pair distribution function analysis of nanostructured systems
journal, October 2015
- Olds, Daniel; Wang, Hsiu-Wen; Page, Katharine
- Journal of Applied Crystallography, Vol. 48, Issue 6
Ferroelectric order in individual nanometre-scale crystals
journal, July 2012
- Polking, Mark J.; Han, Myung-Geun; Yourdkhani, Amin
- Nature Materials, Vol. 11, Issue 8
Titanium Dioxide Nanomaterials: Synthesis, Properties, Modifications, and Applications
journal, July 2007
- Chen, Xiaobo; Mao, Samuel S.
- Chemical Reviews, Vol. 107, Issue 7
Percolating bulk heterostructures from neutron reflectometry and small-angle scattering data
journal, December 2012
- Olds, Daniel P.; Duxbury, Phillip M.; Kiel, Jonathan W.
- Physical Review E, Vol. 86, Issue 6
The Morphology of TiO 2 (B) Nanoparticles
journal, October 2015
- Hua, Xiao; Liu, Zheng; Bruce, Peter G.
- Journal of the American Chemical Society, Vol. 137, Issue 42
Anisotropic phase segregation and migration of Pt in nanocrystals en route to nanoframe catalysts
journal, August 2016
- Niu, Zhiqiang; Becknell, Nigel; Yu, Yi
- Nature Materials, Vol. 15, Issue 11
Manganese(II) Oxide Nanohexapods: Insight into Controlling the Form of Nanocrystals
journal, April 2006
- Ould-Ely, Teyeb; Prieto-Centurion, Dario; Kumar, A.
- Chemistry of Materials, Vol. 18, Issue 7
Reactivity of Anatase TiO 2 Nanoparticles: The Role of the Minority (001) Surface
journal, October 2005
- Gong, Xue-Qing; Selloni, Annabella
- The Journal of Physical Chemistry B, Vol. 109, Issue 42
The Nanoscale Ordered MAterials Diffractometer NOMAD at the Spallation Neutron Source SNS
journal, September 2012
- Neuefeind, Jörg; Feygenson, Mikhail; Carruth, John
- Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, Vol. 287
Ferroelectric Barium Titanate Nanocubes as Capacitive Building Blocks for Energy Storage Applications
journal, October 2014
- Parizi, Saman Salemizadeh; Mellinger, Axel; Caruntu, Gabriel
- ACS Applied Materials & Interfaces, Vol. 6, Issue 20
Reciprocal-space instrumental effects on the real-space neutron atomic pair distribution function
journal, January 2004
- Qiu, Xiangyun; Božin, Emil S.; Juhas, Pavol
- Journal of Applied Crystallography, Vol. 37, Issue 1
Structure and Stability of SnO 2 Nanocrystals and Surface-Bound Water Species
journal, April 2013
- Wang, Hsiu-Wen; Wesolowski, David J.; Proffen, Thomas E.
- Journal of the American Chemical Society, Vol. 135, Issue 18
PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals
journal, July 2007
- Farrow, C. L.; Juhas, P.; Liu, J. W.
- Journal of Physics: Condensed Matter, Vol. 19, Issue 33
Monopod, Bipod, Tripod, and Tetrapod Gold Nanocrystals
journal, December 2003
- Chen, Sihai; Wang, Zhong Lin; Ballato, John
- Journal of the American Chemical Society, Vol. 125, Issue 52, p. 16186-16187
Uncovering the intrinsic geometry from the atomic pair distribution function of nanomaterials
journal, July 2009
- Lei, Ming; de Graff, Adam M. R.; Thorpe, M. F.
- Physical Review B, Vol. 80, Issue 2
Local and average structures of BaTiO3-Bi(Zn1/2Ti1/2)O3
journal, November 2016
- Usher, Tedi-Marie; Iamsasri, Thanakorn; Forrester, Jennifer S.
- Journal of Applied Physics, Vol. 120, Issue 18
Effects of Bragg peak profiles and nanoparticle sizes on the real-space pair distribution function
journal, January 2005
- Jeong, Il-Kyoung; Graf, M. J.; Heffner, R. H.
- Journal of Applied Crystallography, Vol. 38, Issue 1
Nanoparticulate TiO2(B): An Anode for Lithium-Ion Batteries
journal, January 2012
- Ren, Yu; Liu, Zheng; Pourpoint, Frédérique
- Angewandte Chemie International Edition, Vol. 51, Issue 9
Colloidal Synthesis and Characterization of Tetrapod-Shaped Magnetic Nanocrystals
journal, September 2006
- Cozzoli, P. Davide; Snoeck, Etienne; Garcia, Miguel Angel
- Nano Letters, Vol. 6, Issue 9
Finite size effects on the real-space pair distribution function of nanoparticles
journal, May 2008
- Gilbert, Benjamin
- Journal of Applied Crystallography, Vol. 41, Issue 3
Total scattering atomic pair distribution function: new methodology for nanostructure determination
journal, May 2016
- Masadeh, Ahmad S.
- Journal of Experimental Nanoscience, Vol. 11, Issue 12
Ferroelectric properties of composites containing BaTiO 3 nanoparticles of various sizes
journal, January 2014
- Adam, Jens; Lehnert, Tobias; Klein, Gabi
- Nanotechnology, Vol. 25, Issue 6
Structure determination of molecular nanocomposites by combining pair distribution function analysis and solid-state NMR
journal, January 2015
- Bendeif, E. -E.; Gansmuller, A.; Hsieh, K. -Y.
- RSC Advances, Vol. 5, Issue 12
Unit-cell scale mapping of ferroelectricity and tetragonality in epitaxial ultrathin ferroelectric films
journal, December 2006
- Jia, Chun-Lin; Nagarajan, Valanoor; He, Jia-Qing
- Nature Materials, Vol. 6, Issue 1
Crystal Structure and the Paraelectric-to-Ferroelectric Phase Transition of Nanoscale BaTiO 3
journal, June 2008
- Smith, Millicent B.; Page, Katharine; Siegrist, Theo
- Journal of the American Chemical Society, Vol. 130, Issue 22
Probing Local Dipoles and Ligand Structure in BaTiO 3 Nanoparticles
journal, August 2010
- Page, Katharine; Proffen, Thomas; Niederberger, Markus
- Chemistry of Materials, Vol. 22, Issue 15
Nonaqueous Synthesis of TiO 2 Nanocrystals Using TiF 4 to Engineer Morphology, Oxygen Vacancy Concentration, and Photocatalytic Activity
journal, April 2012
- Gordon, Thomas R.; Cargnello, Matteo; Paik, Taejong
- Journal of the American Chemical Society, Vol. 134, Issue 15
The interpretation of real-space information from small-angle scattering experiments
journal, April 1979
- Glatter, O.
- Journal of Applied Crystallography, Vol. 12, Issue 2
Characteristics of Barium Titanate Nanocube Ordered Assembly Thin Films Fabricated by Dip-Coating Method
journal, September 2013
- Mimura, Ken-ichi; Kato, Kazumi
- Japanese Journal of Applied Physics, Vol. 52, Issue 9S1
Nanoparticulate TiO2(B): An Anode for Lithium-Ion Batteries
journal, January 2012
- Ren, Yu; Liu, Zheng; Pourpoint, Frédérique
- Angewandte Chemie, Vol. 124, Issue 9
Titanium Dioxide Nanomaterials: Synthesis, Properties, Modifications, and Applications
journal, October 2007
- Chen, Xiaobo; Mao, Samuel S.
- ChemInform, Vol. 38, Issue 41
On the True Photoreactivity Order of {001}, {010}, and {101} Facets of Anatase TiO2 Crystals
journal, January 2011
- Pan, Jian; Liu, Gang; Lu, Gao Qing Max
- Angewandte Chemie, Vol. 123, Issue 9
Structure determination of molecular nanocomposites by combining pair distribution function analysis and solid-state NMR
journal, August 2015
- Schaniel, Dominik; Bendeif, El-Eulmi; Gansmuller, Axel
- Acta Crystallographica Section A Foundations and Advances, Vol. 71, Issue a1
Single Crystal Manganese Oxide Multipods by Oriented Attachment.
journal, January 2006
- Zitoun, David; Pinna, Nicola; Frolet, Nathalie
- ChemInform, Vol. 37, Issue 3
Relationship between the atomic pair distribution function and small angle scattering: implications for modeling of nanoparticles
text, January 2008
- Farrow, Christopher L.; Billinge, Simon J. L.
- arXiv
Pair distribution function and structure factor of spherical particles
text, January 2005
- Howell, Rafael C.; Proffen, Thomas; Conradson, Steven D.
- arXiv
Works referencing / citing this record:
Reverse Monte Carlo modeling for low-dimensional systems
journal, August 2019
- Zhang, Yuanpeng; McDonnell, Marshall; Liu, Wei
- Journal of Applied Crystallography, Vol. 52, Issue 5
A simple correction for the parallax effect in X-ray pair distribution function measurements
journal, September 2019
- Marlton, Frederick; Ivashko, Oleh; Zimmerman, Martin v.
- Journal of Applied Crystallography, Vol. 52, Issue 5
A simple correction for the parallax effect in X-ray pair distribution function measurements
text, January 2019
- Marlton, Frederick; Ivashko, Oleh; Von Zimmermann, Martin
- Deutsches Elektronen-Synchrotron, DESY, Hamburg
Figures / Tables found in this record: