Common volume functions and diffraction line profiles of polyhedral domains

Efficient solution of particle shape functions for the analysis of powder total scattering data

Structural characterization of powder samples via total scattering methods, in either real or reciprocal space, must take into account the effect of particle shape. Here, the shape contribution of a set of ideally isolated particles to the small-angle scattering (SAS) component of the intensity profile is modelled using the shape function [Svergun & Koch (2003). Rep. Prog. Phys. 66, 1735-1782]. The shape function is obtained by orientational averaging of common volume functions (CVFs) for a discrete set of directions. The effects of particle size and size dispersity are accounted for via scaling of the CVFs and their convolution with the underlying probability distribution. The method is applied to shapes with CVFs expressed analytically or by using discrete tables. The accurate calculation of SAS particle shape contributions up to large momentum transfer demonstrates the reliability and flexibility of modelling shape functions from sets of CVFs. The algorithm presented here is computationally efficient and can be directly incorporated into existing routines for analysis of powder total scattering data.

X-ray powder diffraction in education. Part I. Bragg peak profiles

A collection of scholarly scripts dealing with the mathematics and physics of peak profile functions in X-ray powder diffraction has been written using the Wolfram language in Mathematica. Common distribution functions, the concept of convolution in real and Fourier space, instrumental aberrations, and microstructural effects are visualized in an interactive manner and explained in detail. This paper is the first part of a series dealing with the mathematical description of powder diffraction patterns for teaching and education purposes.

Diffraction line profiles from polydisperse crystalline systems. Corrigenda

Equation (16) and some entries in Table 1 in the article by Scardi & Leoni [(2001), Acta Cryst. A57, 604–613] are corrected.

Whole pair distribution function modeling: the bridging of Bragg and Debye scattering theories

Microstructure-based design of materials requires an atomic level understanding of the mechanisms underlying structure-dependent properties. Methods for analyzing either the traditional diffraction profile or the pair distribution function (PDF) differ in how the information is accessed and in the approximations usually applied. Any variation of structural and microstructural features over the whole sample affects the Bragg peaks as well as any diffuse scattering. Accuracy of characterization relies, therefore, on the reliability of the analysis methods. Methods based on Bragg's law investigate the diffraction peaks in the intensity plot as distinct pieces of information. This approach reaches a limitation when dealing with disorder scenarios that do not conform to such a peak-by-peak basis. Methods based on the Debye scattering equation (DSE) are, otherwise, well suited to evaluate the scattering from a disordered phase but the structure information is averaged over short-range distances usually accessed by experiments. Moreover, statistical reliability is usually sacrificed to recover some of the computing-efficiency loss compared with traditional line-profile-analysis methods. Here, models based on Bragg's law are used to facilitate the computation of a whole PDF and then model powder-scattering data via the DSE. Models based on Bragg's law allow the efficient solution of the dispersion of a crystal's properties in a powder sample with statistical reliability, and the PDF provides the flexibility of the DSE. The whole PDF is decomposed into the independent directional components, and the number of atom pairs separated by a given distance is statistically estimated using the common-volume functions. This approach overcomes the need for an atomistic model of the material sample and the computation of billions of pair distances. The results of this combined method are in agreement with the explicit solution of the DSE although the computing efficiency is comparable with that of methods based on Bragg's law. Most importantly, the method exploits the strengths and different sensitivities of the Bragg and Debye theories.

Complex modeling for the quantification of nanoscale disorder using genetic algorithms, density functional theory and line-profile analysis

A new, computationally efficient, complex modeling approach is presented for the quantification of the local and average atomic structure, nanostructure and microstructure of an Au0.25Cu0.75 alloy. High-resolution X-ray powder diffraction and whole pattern fitting show that the sample is phase pure, with isotropic lattice strain and a distribution of equiaxed crystallites of mean size 144 (11) nm, where each crystallite has on average four twin boundaries and an average of three deformation faults per four crystallites. Both small- and large-box model optimizations were used to extract local and long-range information from the pair distribution function. The large-box, 640 000-atom-ensemble optimization approach applied herein relies on differential evolution optimization and shows that the alloy has chemical short-range ordering, with correlation parameters of −0.26 (2) and 0.36 (8) in the first and second correlation shells, respectively. Locally, there is a 1.45 (8)% tetragonal distortion which on average results in a cubic atomic structure. The isotropic lattice strain is a result of atom-pair-dependent bond lengths, following the trend d Au—Au > d Au—Cu > d Cu—Cu, highlighted by density functional theory calculations. This approach is generalizable and should be extensible to other disordered systems, allowing for quantification of localized structure deviations.

Understanding Powder X-ray Diffraction Profiles from Layered Minerals: The Case of Kaolinite Nanocrystals

Powder X-ray diffraction (PXRD) techniques are widely used to characterize the nature of stacking of submicrometer-wide nanometer-thick layers that form layered mineral nanocrystals, but application of these methods to infer the in-plane configuration of the layers is difficult. Line-profile-analysis algorithms based on the Bragg equation cannot describe the broken periodicity in the stacking direction. The Debye scattering equation is an alternative approach, but it is limited by the large-scale atomistic models required to capture the multiscale nature of the layered systems. Here, we solve the Debye scattering equation for kaolinite nanocrystals to understand the contribution of different layer-stacking defects to PXRD profiles. We chose kaolinite as a case study because its approximately constant composition and lack of interlayer expansion ensure that interstitial cations and/or molecules and substitutional ions can be ignored. We investigated the structure factor change as a function of crystal structural and microstructural features such as layer structure in-plane misorientation and shift (in or out of the 2D plane) and the diameter, number, and lateral indentation of the layers. Perfect and turbostratic stacking configurations bounded the range of intensity variation for hkl and 00l reflections, as well as for any scattering angle. A unique degree of disorder was computed by the average deviation from such limiting cases, and multivariate analysis was used to interpret the observed diffraction profiles. Analysis of the data for KGa-1, KGa-2, and API-9 standard kaolinites demonstrated that the estimated densities of different stacking defects are highly correlated. In addition, analysis of API-9 particle-size fractions revealed a dispersion of four or more components in the standard sample. The results illustrate that the use of a distribution of sizes, defects, and even individual kaolinite components is necessary to accurately characterize any sample of natural kaolinite.

Vibrational Properties of Pd Nanocubes

The atomic disorder and the vibrational properties of Pd nanocubes have been studied through a combined use of X-ray diffraction and molecular dynamics simulations. The latter show that the trend of the mean square relative displacement as a function of the radius of the coordination shells is characteristic of the nanoparticle shape and can be described by a combined model: A correlated Debye model for the thermal displacement and a parametric expression for the static disorder. This combined model, supplemented by results of line profile analysis of the diffraction patterns collected at different temperatures (100, 200, and 300 K) can explain the observed increase in the Debye–Waller coefficient, and shed light on the effect of the finite domain size and of the atomic disorder on the vibrational properties of metal nanocrystals.

Enhancing Electrocatalytic Water Splitting by Strain Engineering

Electrochemical water splitting driven by sustainable energy such as solar, wind, and tide is attracting ever-increasing attention for sustainable production of clean hydrogen fuel from water. Leveraging these advances requires efficient and earth-abundant electrocatalysts to accelerate the kinetically sluggish hydrogen and oxygen evolution reactions (HER and OER). A large number of advanced water-splitting electrocatalysts have been developed through recent understanding of the electrochemical nature and engineering approaches. Specifically, strain engineering offers a novel route to promote the electrocatalytic HER/OER performances for efficient water splitting. Herein, the recent theoretical and experimental progress on applying strain to enhance heterogeneous electrocatalysts for both HER and OER are reviewed and future opportunities are discussed. A brief introduction of the fundamentals of water-splitting reactions, and the rationalization for utilizing mechanical strain to tune an electrocatalyst is given, followed by a discussion of the recent advances on strain-promoted HER and OER, with special emphasis given to combined theoretical and experimental approaches for determining the optimal straining effect for water electrolysis, along with experimental approaches for creating and characterizing strain in nanocatalysts, particularly emerging 2D nanomaterials. Finally, a vision for a future sustainable hydrogen fuel community based on strain-promoted water electrolysis is proposed.

Whole powder pattern modelling macros for TOPAS

Macros implementing the main concepts of the whole powder pattern modelling approach have been written for TOPAS. Size and strain broadening components of the diffraction line profiles can be convolved with the instrumental profile already available among the standard commands of TOPAS. Specific macros are presented with examples of applications including plastically deformed powders and atomistic simulations. A macro is presented for the modelling of surface relaxation effects in spherical nanocrystals.

Simulating the diffraction line profile from nanocrystalline powders using a spherical harmonics expansion

An accurate description of the diffraction line profile from nanocrystalline powders can be obtained by a spherical harmonics expansion of the profile function. The procedure outlined in this work is found to be computationally efficient and applicable to the line profile for any crystallite shape and size. Practical examples of the diffraction pattern peak profiles resulting from cubic crystallites between 1 and 100 nm in size are shown.

Influence of atomic site-specific strain on catalytic activity of supported nanoparticles

Heterogeneous catalysis is an enabling technology that utilises transition metal nanoparticles (NPs) supported on oxides to promote chemical reactions. Structural mismatch at the NP–support interface generates lattice strain that could affect catalytic properties. However, detailed knowledge about strain in supported NPs remains elusive. We experimentally measure the strain at interfaces, surfaces and defects in Pt NPs supported on alumina and ceria with atomic resolution using high-precision scanning transmission electron microscopy. The largest strains are observed at the interfaces and are predominantly compressive. Atomic models of Pt NPs with experimentally measured strain distributions are used for first-principles kinetic Monte Carlo simulations of the CO oxidation reaction. The presence of only a fraction of strained surface atoms is found to affect the turnover frequency. These results provide a quantitative understanding of the relationship between strain and catalytic function and demonstrate that strain engineering can potentially be used for catalyst design.

Detailed knowledge of how strain influences catalytic reactions remains elusive. Here, the authors experimentally measure the strain in supported Pt nanoparticles on alumina and ceria with atomic resolution and computationally explore how the strain affects the CO oxidation reaction.

Debye-Waller coefficient of heavily deformed nanocrystalline iron

Extensive deformation of an iron alloy powder increases the static disorder contribution to the thermal factor, with an increase of ∼20% in the Debye–Waller coefficient observed by both X-ray diffraction and extended X-ray absorption fine structure. Molecular dynamics simulations shed light on the underlying mechanisms, confirming the major role played by the grain boundary.

Synchrotron radiation X-ray diffraction (XRD) patterns from an extensively ball-milled iron alloy powder were collected at 100, 200 and 300 K. The results were analysed together with those using extended X-ray absorption fine structure, measured on the same sample at liquid nitrogen temperature (77 K) and at room temperature (300 K), to assess the contribution of static disorder to the Debye–Waller coefficient (B_iso). Both techniques give an increase of ∼20% with respect to bulk reference iron, a noticeably smaller difference than reported by most of the literature for similar systems. Besides good quality XRD patterns, proper consideration of the temperature diffuse scattering seems to be the key to accurate values of the Debye–Waller coefficient. Molecular dynamics simulations of nanocrystalline iron aggregates, mapped on the evidence provided by XRD in terms of domain size distribution, shed light on the origin of the observed B_iso increase. The main contribution to the static disorder is given by the grain boundary, while line and point defects have a much smaller effect.

High-performance powder diffraction pattern simulation for large-scale atomistic models via full-precision pair distribution function computation

A new full-precision algorithm to solve the Debye scattering equation has been developed for high-performance computing of powder diffraction line profiles from large-scale atomistic models of nanomaterials. The Debye function was evaluated using a pair distribution function computed with high accuracy, exploiting the series expansion of the error between calculated and equispace-sampled pair distances of atoms. The intensity uncertainty (standard deviation) of the computed diffraction profile was estimated as a function of the algorithm-intrinsic approximations and coordinate precision of the atomic positions, confirming the high accuracy of the simulated pattern. Based on the propagation of uncertainty, the new algorithm provides a more accurate powder diffraction profile than a brute-force calculation. Indeed, the precision of floating-point numbers employed in brute-force computations is worse than the estimated accuracy provided by the new algorithm. A software application, ROSE-X, has been implemented for parallel computing on CPU/GPU multi-core processors and distributed clusters. The computing performance is directly proportional to the total processor speed of the devices. An average speed of ∼30 × 109 computed pair distances per second was measured, allowing simulation of the powder diffraction pattern of an ∼23 million atom microstructure in a couple of hours. Moreover, the pair distribution function was recorded and reused to evaluate powder diffraction profiles of the same system with different properties (i.e. Q rather than 2θ range, step and wavelength), avoiding additional pair distance computations. This approach was used to investigate a large collection of monoatomic and polyatomic microstructures, isolating the contribution from atoms belonging to different moieties (e.g. different species or crystalline domains).

On the reliability of powder diffraction Line Profile Analysis of plastically deformed nanocrystalline systems

An iron-molybdenum alloy powder was extensively deformed by high energy milling, so to refine the bcc iron domain size to nanometer scale (~10 nm) and introduce a strong inhomogeneous strain. Both features contribute to comparable degree to the diffraction peak profile broadening, so that size and strain contributions can be easily separated by exploiting their different dependence on the diffraction angle. To assess the reliability of Line Profile Analysis, results were compared with evidence from other techniques, including scanning and transmission electron microscopy and X-ray small angle scattering. Results confirm the extent of the size broadening effect, whereas molecular dynamics simulations provide insight into the origin of the local atomic, inhomogeneous strain, pointing out the role of dislocations, domain boundaries and interactions among crystalline domains.

Structure and morphology of shape-controlled Pd nanocrystals

Pd nanocrystals were produced with uniform truncated-cube shape and a narrow size distribution, yielding controlled surface area fractions from low Miller index ({100}, {110}, {111}) crystalline facets. Details on the structure and morphology of the nanocrystals were obtained by combining X-ray powder diffraction line profile analysis, high-resolution transmission electron microscopy and surface electrochemistry based on Cu underpotential deposition.

Building up strain in colloidal metal nanoparticle catalysts

The focus on surface lattice strain in nanostructures as a fundamental research topic has gained momentum in recent years as scientists investigated its significant impact on the surface electronic structure and catalytic properties of nanomaterials. Researchers have begun to tell a more complete story of catalysis from a perspective which brings this concept to the forefront of the discussion. The nano-'realm' makes the effects of surface lattice strain, which acts on the same spatial scales, more pronounced due to a higher ratio of surface to bulk atoms. This is especially evident in the field of metal nanoparticle catalysis, where displacement of atoms on surfaces can significantly alter the sorption properties of molecules. In part, the concept of strain-engineering for catalysis opened up due to the achievements that were made in the synthesis of a more sophisticated nanoparticle library from an ever-expanding set of methodologies. Developing synthesis methods for metal nanoparticles with well-defined and strained architectures is a worthy goal that, if reached, will have considerable impact in the search for catalysts. In this review, we summarize the recent accomplishments in the area of surface lattice-strained metal nanoparticle synthesis, framing the discussion from the important perspective of surface lattice strain effects in catalysis.