There's no place like real-space: elucidating size-dependent atomic structure of nanomaterials using pair distribution function analysis

Abstract

The development of new functional materials builds on an understanding of the intricate relationship between material structure and properties, and structural characterization is a crucial part of materials chemistry. However, elucidating the atomic structure of nanomaterials remains a challenge using conventional diffraction techniques due to the lack of long-range atomic order. Over the past decade, Pair Distribution Function (PDF) analysis of X-ray or neutron total scattering data has become a mature and well-established method capable of giving insight into the atomic structure in nanomaterials. Here, we review the use of PDF analysis and modelling in characterization of a range of different nanomaterials that exhibit unique atomic structure compared to the corresponding bulk materials. A brief introduction to PDF analysis and modelling is given, followed by examples of how essential structural information can be extracted from PDFs using both model-free and advanced modelling methods. We put an emphasis on how the intuitive nature of the PDF can be used for understanding important structural motifs, and on the diversity of applications of PDF analysis to nanostructure problems.

Fig. 1. G(r) functions calculated for hypothetical 1D atomic structures. A–E illustrate different aspects of the PDF and how they are interpreted in relation to atomic structure: (A) the impact of atomic coordinates on peak position, (B) the impact of atomic vibration on peak broadening, (C) the impact of crystallite size on the extent of the PDF, (D) the impact of atomic identity (as seen when replacing Au with Na, an element with lower scattering power) on the peak intensity and (E) the effects of disorder on the PDF. The blue line is the simulated PDF under consideration, while the dotted line is the PDF simulated in A.

Fig. 2. Illustration of the stacking sequence in the wurtzite structure (A) and the zinc-blende structure (B). Cations are represented as purple dots, and anions are represented as green dots. The tetrahedrally coordinated Cd cations are shown to emphasize the layering of the structure.

Fig. 3. Three real-space Rietveld refinements of a PDF from CdSe nanoparticles (3.5 nm). The PDF is fit using a zinc-blende model (top), a wurtzite model (middle) or a two-phase model using both structures (bottom). The blue line is experimental data and the red line is a simulated PDF. Reproduced from Yang et al., Phys. Chem. Chem. Phys., 2013, 15, 8480–8486 (ref. 89) with permission from the PCCP Owner Societies.

Fig. 4. (a) Gaussian fits to the first peak in experimental PDFs of CdSe nanoparticles of various sizes. Data are represented as black dots, and the fit as a red line. The green dotted line indicates the peak r value from a bulk sample. (b) The relative percentage change in position as obtained from the Gaussian fits, plotted as function of nanoparticle size. (c) Changes in peak width obtained from the Gaussian fits, plotted as function of nanoparticle size. The green dotted line indicates the value of peak width from a bulk sample. Reproduced from Yang et al., Phys. Chem. Chem. Phys., 2013, 15, 8480–8486 (ref. 89) with permission from the PCCP Owner Societies.

Fig. 5. (A) Experimental PDF of ZnS nanoparticles (black line) compared to a simulated PDF from a 3.4 nm ZnS particle with a zinc-blende structure (grey line) and a simulated PDF from bulk zinc-blende (dashed line). (B) Fit (grey line) to the experimental PDF of ZnS nanoparticles (black line), taking into account particle size and using atomic displacement parameters (MSRD) from a bulk ZnS sample. The r dependence of the additional mean square displacement parameter and the short-range order parameter included in the model is shown in the insert. Reproduced from Gilbert et al., Science, 2004, 305, 651–654. Reprinted with permission from AAAS.

Fig. 6. Different structures seen in metal nanoparticles. (A) The face-centered cubic (fcc) structure; (B) the tetrahedrally close-packed (tcp) Frank–Casper phase, (C) a 561-atom icosahedral cluster core, and (D) an 811-atom decahedral cluster core. The structures in C and D are generated using the ASE module..

Fig. 7. Fit residuals from real-space Rietveld refinements using an fcc model to several experimental PDFs of different metallic nanoparticles. The difference curve has been scaled for easier visual comparison. Reprinted with permission from Banerjee et al., J. Phys. Chem. C, 2018, 122, 29498–29506. Copyright 2018 American Chemical Society.

Fig. 8. R _w values obtained for a number of discrete non-twinned nanoparticle models, fitted to the experimental PDF of Pd nanoparticles. The R_w values are plotted as a function of the number of atoms in the structure. The teal circle labelled AC corresponds to the R_w obtained from a real-space Rietveld refinement of the data using an fcc model. Symbols for the discrete nanoparticle models with different geometries can be seen in the bottom right of the figure. Figure from Banerjee et al., Acta Crystallogr., Sect. A: Found. Adv., 2020, 76, 24–31.

Fig. 9. Characterization of three NiPd samples using PDF and TEM. (a) Fits of PDFs using an fcc structure. The experimental PDF is shown in blue, the calculated PDF in red, and difference curve in grey. (b) TEM images obtained for the three NiPd samples, scale bar is 20 nm. Reprinted with permission from Doan-Nguyen et al., ACS Nano, 2014, 8, 6163–6170. Copyright 2014 American Chemical Society.

Fig. 10. Fits to experimental PDFs from Au₁₄₄(SR)₆₀ samples. (a) Fit of an icosahedral structure model to data collected from Au₁₄₄(SC)₆₀. (b) Fit of an icosahedral structure model, (c) of an fcc/hcp model, (d) a 114 atom decahedral model and (e) a decahedral model with ‘staples’ to the PDF obtained from Au₁₄₄(p-MBA)₆₀. From Jensen et al., Nat. Commun., 2016, 7, 11859.

Fig. 11. Fits to PDFs from nanostructured molybdenum oxide using different structural models. (A) Fit of the data with a bulk rutile structure and (B) with an ‘interwoven’ rutile model. (C) High-resolution annular dark field TEM image obtained from the sample, overlaid with the rutile structure. The image shows intensity in the unoccupied sites in the rutile structure. Reprinted with permission from Lindahl Christiansen et al., ACS Nano, 2019, 13, 8725–8735. Copyright 2019 American Chemical Society.

Fig. 12. Fits to PDF from nanostructured cobalt oxide. (A) Fit of the PDF from model 1 (red) to the experimental PDF (black). (B) Fit of the PDF from model 2 (red) and model 3 (green), compared to the experimental PDF in black. R_w is 0.27 for structure 1 and 0.19 for structures 2 and 3. Adapted with permission from Du et al., J. Am. Chem. Soc., 2012, 134, 11096–11099. Copyright 2012 American Chemical Society.

Troels Lindahl Christiansen

Susan R. Cooper

Kirsten M. Ø. Jensen