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Title: 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 » 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.« less

Authors:
ORCiD logo [1]; ORCiD logo [1]; ORCiD logo [1]; ORCiD logo [1]
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  1. 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. https://doi.org/10.1107/S2053273318004977.
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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
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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.
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@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}
}
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Journal Article:
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https://doi.org/10.1107/S2053273318004977
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Figures / Tables:

Figure 1 Figure 1: Schematic describing the steps in Section 2.1 to generate a numerical approximation for the shape function of a given shape. Starting at the left, Step 1. is construction of a training set of nanocrystal models; shown here are three representative cubes. The colors of the cubes in Stepmore » 1 correspond to the line colors in Steps 2 and 3. Step 2. is calculation of PDFs from the finite nanocrystal models. Step 3. is extraction of the shape functions from the PDFs using DShaper and conversion of the shape functions to $γ$($r$) . Step 4. is fitting $γ$($r$) with a numerical approximation. The result in Step 4. is shown for a set of nine nanocube $γ$($r$) (black dots) fit with the two-Gaussian $γ$$num$($r$) (color lines). The difference curves are shown offset below and demonstrate good agreement between $γ$($r$) and $γ$$num$($r$) )« less
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