Nanostructured Drugs Embedded into a Polymeric Matrix: Vinpocetine/PVP Hybrids Investigated by Debye Function Analysis.

Microcrystalline vinpocetine, coground with cross-linked polyvinylpyrrolidone, affords hybrids containing nanosized drug nanocrystals, the size and size distributions of which depend on milling times and drug-to-polymer weight ratios. Using an innovative approach to microstructural characterization, we analyzed wide-angle X-ray total scattering data by the Debye function analysis and demonstrated the possibility to characterize pharmaceutical solid dispersions obtaining a reliable quantitative view of the physicochemical status of the drug dispersed in an amorphous carrier. The microstructural properties derived therefrom have been successfully employed in reconciling the enigmatic difference in behavior between in vitro and in vivo solubility tests performed on nanosized vinpocetine embedded in a polymeric matrix.

Site-occupancy factors in the Debye scattering equation. A theoretical discussion on significance and correctness.

The Debye scattering equation (DSE) [Debye (1915). Ann. Phys. 351, 809-823] is widely used for analyzing total scattering data of nanocrystalline materials in reciprocal space. In its modified form (MDSE) [Cervellino et al. (2010). J. Appl. Cryst. 43, 1543-1547], it includes contributions from uncorrelated thermal agitation terms and, for defective crystalline nanoparticles (NPs), average site-occupancy factors (s.o.f.'s). The s.o.f.'s were introduced heuristically and no theoretical demonstration was provided. This paper presents in detail such a demonstration, corrects a glitch present in the original MDSE, and discusses the s.o.f.'s physical significance. Three new MDSE expressions are given that refer to distinct defective NP ensembles characterized by: (i) vacant sites with uncorrelated constant site-occupancy probability; (ii) vacant sites with a fixed number of randomly distributed atoms; (iii) self-excluding (disordered) positional sites. For all these cases, beneficial aspects and shortcomings of introducing s.o.f.'s as free refinable parameters are demonstrated. The theoretical analysis is supported by numerical simulations performed by comparing the corrected MDSE profiles and the ones based on atomistic modeling of a large number of NPs, satisfying the structural conditions described in (i)-(iii).

Effects of Structural and Microstructural Features on the Total Scattering Pattern of Nanocrystalline Materials.

Atomic- and nanometer-scale features of nanomaterials have a strong influence on their chemical and physical properties and a detailed description of these elements is a crucial step in their characterization. Total scattering methods, in real and reciprocal spaces, have been established as fundamental techniques to retrieve this information. Although the impact of microstructural features, such as defectiveness of different kinds, has been extensively studied in reciprocal space, disentangling these effects from size- and morphology-induced properties, upon downsizing, is not a trivial task. Additionally, once the experimental pattern is Fourier transformed to calculate the pair distribution function, the direct fingerprint of structural and microstructural features is severely lost and no modification of the histogram of interatomic distances derived therefrom is clearly discussed nor considered in the currently available protocols. Hereby, starting from atomistic models of a prototypical system (cadmium selenide), we simulate multiple effects on the atomic pair distribution function, obtained from reciprocal space patterns computed through the Debye scattering equation. Size and size dispersion effects, as well as different structures, morphologies, and their interplay with several kinds of planar defects, are explored, aiming at identifying the main (measurable and informative) fingerprints of these features on the total scattering pattern in real and reciprocal spaces, highlighting how, and how much, they become evident when comparing different cases. The results shown herein have general validity and, as such, can be further extended to other classes of nanomaterials.

Small-angle X-ray scattering: characterization of cubic Au nanoparticles using Debye's scattering formula.

A versatile software package in the form of a Python extension, named CDEF (computing Debye's scattering formula for extraordinary form factors), is proposed to calculate approximate scattering profiles of arbitrarily shaped nanoparticles for small-angle X-ray scattering (SAXS). CDEF generates a quasi-randomly distributed point cloud in the desired particle shape and then applies the open-source software DEBYER for efficient evaluation of Debye's scattering formula to calculate the SAXS pattern (https://github.com/j-from-b/CDEF). If self-correlation of the scattering signal is not omitted, the quasi-random distribution provides faster convergence compared with a true-random distribution of the scatterers, especially at higher momentum transfer. The usage of the software is demonstrated for the evaluation of scattering data of Au nanocubes with rounded edges, which were measured at the four-crystal monochromator beamline of PTB at the synchrotron radiation facility BESSY II in Berlin. The implementation is fast enough to run on a single desktop computer and perform model fits within minutes. The accuracy of the method was analyzed by comparison with analytically known form factors and verified with another implementation, the SPONGE, based on a similar principle with fewer approximations. Additionally, the SPONGE coupled to McSAS3 allows one to retrieve information on the uncertainty of the size distribution using a Monte Carlo uncertainty estimation algorithm.

Structure and Surface Relaxation of CeO2 Nanoparticles Unveiled by Combining Real and Reciprocal Space Total Scattering Analysis.

We present a combined real and reciprocal space structural and microstructural characterization of CeO2 nanoparticles (NPs) exhibiting different crystallite sizes; ~3 nm CeO2 NPs were produced by an inverse micellae wet synthetic path and then annealed at different temperatures. X-ray total scattering data were analyzed by combining real-space-based Pair Distribution Function analysis and the reciprocal-space-based Debye Scattering Equation method with atomistic models. Subtle atomic-scale relaxations occur at the nanocrystal surface. The structural analysis was corroborated by ab initio DFT and force field calculations; micro-Raman and electron spin resonance added important insights to the NPs' defective structure. The combination of the above techniques suggests a core-shell like structure of ultrasmall NPs. These exhibit an expanded outer shell having a defective fluorite structure, while the inner shell is similar to the bulk structure. The presence of partially reduced O2-δ species testifies to the high surface activity of the NPs. On increasing the annealing temperature, the particle dimensions increase, limiting disorder as a consequence of the progressive surface-to-volume ratio reduction.

An Atomistic Model Describing the Structure and Morphology of Cu-Doped C-S-H Hardening Accelerator Nanoparticles.

Calcium silicate hydrate (C-S-H) is the main binding phase in Portland cement. The addition of C-S-H nanoparticles as nucleation seeds has successfully been used to accelerate the hydration process and the precipitation of binding phases either in conventional Portland cement or in alternative binders. Indeed, the modulation of the hydration kinetics during the early-stage dissolution-precipitation reactions, by acting on the nucleation and growth of binding phases, improves the early strength development. The fine-tuning of concrete properties in terms of compressive strength and durability by designed structural modifications can be achieved through the detailed description of the reaction products at the atomic scale. The nano-sized, chemically complex and structurally disordered nature of these phases hamper their thorough structural characterization. To this aim, we implement a novel multi-scale approach by combining forefront small-angle X-ray scattering (SAXS) and synchrotron wide-angle X-ray total scattering (WAXTS) analyses for the characterization of Cu-doped C-S-H nanoparticles dispersed in a colloidal suspension, used as hardening accelerator. SAXS and WAXTS data were analyzed under a unified modeling approach by developing suitable atomistic models for C-S-H nanoparticles to be used to simulate the experimental X-ray scattering pattern through the Debye scattering equation. The optimization of atomistic models against the experimental pattern, together with complementary information on the structural local order from 29Si solid-state nuclear magnetic resonance and X-ray absorption spectroscopy, provided a comprehensive description of the structure, size and morphology of C-S-H nanoparticles from the atomic to the nanometer scale. C-S-H nanoparticles were modeled as an assembly of layers composed of 7-fold coordinated Ca atoms and decorated by silicate dimers and chains. The structural layers are a few tens of nanometers in length and width, with a crystal structure resembling that of a defective tobermorite, but lacking any ordering between stacking layers.

Signature of antiphase boundaries in iron oxide nanoparticles.

Iron oxide nanoparticles find a wide variety of applications, including targeted drug delivery and hyperthermia in advanced cancer treatment methods. An important property of these particles is their maximum net magnetization, which has been repeatedly reported to be drastically lower than the bulk reference value. Previous studies have shown that planar lattice defects known as antiphase boundaries (APBs) have an important influence on the particle magnetization. The influence of APBs on the atomic spin structure of nanoparticles with the γ-Fe2O3 composition is examined via Monte Carlo simulations, explicitly considering dipole-dipole interactions between the magnetic moments that have previously only been approximated. For a single APB passing through the particle centre, a reduction in the magnetization of 3.9% (for 9 nm particles) to 7.9% (for 5 nm particles) is found in saturation fields of 1.5 T compared with a particle without this defect. Additionally, on the basis of Debye scattering equation simulations, the influence of APBs on X-ray powder diffraction patterns is shown. The Fourier transform of the APB peak profile is developed to be used in a whole powder pattern modelling approach to determine the presence of APBs and quantify them by fits to powder diffraction patterns. This is demonstrated on experimental data, where it could be shown that the number of APBs is related to the observed reduction in magnetization.

On the amorphous layer in bone mineral and biomimetic apatite: A combined small- and wide-angle X-ray scattering analysis.

The occurrence of an amorphous calcium phosphate layer covering the crystalline apatite core has been suggested to be an intrinsic feature of both bone mineral and synthetic biomimetic analogs. However, an exahustive quantitative picture of the amorphous-crystalline relationship in these materials is still missing. Here, we present a multiple scale modelling that combines small-angle X-ray scattering (SAXS) and synchrotron wide-angle X-ray total scattering (WAXTS) analyses to investigate the amorphous-crystalline spatial interplay in bone sample and biomimetic carbonated nano-apatites. SAXS analysis indicates the presence of a single morphology consisting of tiny nanoplates (NPLs) and provides a measure of their thickness (falling in the 3-5 nm range). WAXTS analysis was performed by developing atomistic models of apatite NPLs incorporating lattice strain, mostly attributed to the carbonate content, and calculating the X-ray patterns using the Debye Scattering Equation. Upon model optimization, the size and strain parameters of the crystalline platelets were derived and the amorphous component, co-existing with the crystalline one, separated and quantified (in the 23-33 wt% range). Notably, the thickness of the apatite core was found to exhibit nearly null (bone) or minor (< 0.5 nm, biomimetic samples) deviations from that of the entire NPLs, suggesting that the amorphous material remains predominantly distributed along the lateral sides of the NPLs, in a core-crown-like arrangement. The lattice strain analysis indicates a significant stiffness along the c axis, which is comparable in bone and synthetic samples, and larger deformations in the other directions. STATEMENT OF SIGNIFICANCE: Current models of bone mineral and biomimetic nanoapatites suggest the occurrence of an amorphous layer covering the apatitic crystalline nanoplates in a core-shell arrangement. By combining X-ray scattering techniques in the small and wide angle regions, we propose a joint atomic-to-nanometre scale modelling to investigate the amorphous-crystalline interplay within the nanoplates. Estimates are extracted for the thickness of the entire nanoplates and the crystalline core, together with the quantification of the amorphous fraction and apatite lattice strain. Based on the thickness matching, the location of the amorphous material mostly along the edges of the nanoplates is inferred, with a vanishing or very thin layer in the thickness direction, suggesting a core-crown-like arrangement, with possible implications on the mineral surface reactivity.

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.

Extending the Debye scattering equation for diffraction from a cylindrically averaged group of atoms: detecting molecular orientation at an interface.

The Debye scattering equation is now over 100 years old and has been widely used to interpret diffraction patterns from randomly oriented groups of atoms. The present work develops and applies a related equation that calculates diffraction intensity from groups of atoms randomly oriented about a fixed axis, a scenario that occurs when molecules are oriented at an interface by the presentation of a binding motif as in antibody binding. Using an example biomolecule, the high level of sensitivity of the diffraction pattern to the orientation of the molecule and to the direction of the incident beam is shown. The use of the method is proposed not only for determining the orientation of molecules in biosensors and at membrane interfaces, but also for determining molecular conformation without the need for crystallization.

Texture corrections for total scattering functions.

Many functional materials are today synthesized in the form of nanoparticles displaying preferred orientation effects to some small or large extent. The analysis of diffraction data of such kinds of systems is best performed in the framework of the total scattering approach that prescinds from translation symmetry assumptions. Therefore modified expressions were derived for the most common total scattering functions, in particular the Debye scattering equation (DSE) which yields the texture-averaged differential cross section as a function of atomic coordinates and texture parameters. The modified DSE encodes higher-order even spherical Bessel functions which account for the texture effect. Selection rules arising from experimental geometries and symmetries are discussed. In addition the duality of the texture effect is introduced showing the effects of texture on both the I(Q) and {\cal{G}}(r). The paper includes several definitions and appendices which are meant to be useful for those involved in the development of crystallographic computing.

Structure, Morphology, and Faceting of TiO2 Photocatalysts by the Debye Scattering Equation Method. The P25 and P90 Cases of Study.

Characterization of functional nanocrystalline materials in terms of quantitative determination of size, size dispersion, type, and extension of exposed facets still remains a challenging task. This is particularly the case of anisotropically shaped nanocrystals (NCs) like the TiO2 photocatalysts. Here, commercially available P25 and P90 titania nanopowders have been characterized by wide-angle X-ray total scattering techniques. Synchrotron data were modelled by the reciprocal space-based Debye scattering equation (DSE) method using atomistic models of NC populations (simultaneously carrying atomic and nanoscale structural features) for both anatase and rutile phases. Statistically robust descriptors are provided of size, morphology, and {101} vs. {001} facet area of truncated tetragonal bipyramids for anatase, jointly to polymorph quantification. The effects of using the proper NC shape on the X-ray diffraction pattern are analyzed in depth through DSE simulations by considering variable bipyramid aspect ratios (resulting in different {101} vs. {001} surface) and relative dispersion in a bivariate manner. We demonstrate that using prismatic NCs having equal volume and aspect ratio as bipyramids provides reasonably accurate sizes and {101} and {001} surface areas of the parent morphology.

Crystal Structure, Morphology, and Surface Termination of Cyan-Emissive, Six-Monolayers-Thick CsPbBr3 Nanoplatelets from X-ray Total Scattering.

Highly anisotropic colloidal CsPbBr3 nanoplatelets (NPLs) represent an appealing class of colloidal quantum wells with enhanced light emissivity. Strong quantum confinement imposed by the small platelet thickness and atomic flatness gives rise to enhanced oscillator strength, higher exciton binding energy, and narrow emission linewidth. While discrete thicknesses manifest themselves in discrete bandgap energies, fine-tuning of the emission energy can be achieved by compositional modulations. Here we address one of the most debated aspects of perovskite nanoplatelets: their crystal structure. Starting with the direct imaging by high-resolution electron microscopy (providing a clue on the pseudocubic faceting of the NPLs), we focus the study on X-ray total scattering techniques, based on the Debye scattering equation (DSE) approach, to obtain better atomistic insight. The nanoplatelets are six-monolayers thick and exhibit an orthorhombic structure. A thorough structure-morphology characterization unveils a specific orientation of the axial and equatorial bromides of the PbBr6 octahedra versus the NPLs thickness; we found that {010} and {101} planes of the orthorhombic CsPbBr3 lattice (Pnma space group) correspond to the six facets of the NPL, with basal planes being of {101} type. The NPLs undergo a lattice relaxation in comparison to cuboidal CsPbBr3 NCs; the major deformation is observed in the axial direction, which suggests a structural origin of the higher compliance along the b axis. The DSE-based analysis also supports a CsBr surface termination model, with half Cs sites and a half (or slightly more) Br sites vacant.

Algorithm for distance list extraction from pair distribution functions.

An algorithm is presented to extract the distance list from atomic pair distribution functions in a highly automated way. The algorithm is constructed via curve fitting based on a Debye scattering equation model. Because of the non-convex nature of the resulting optimization problem, a number of techniques are developed to overcome various computational difficulties. A key ingredient is a new approach to obtain a reasonable initial guess based on the theoretical properties of the mathematical model. Tests on various nanostructured samples show the effectiveness of the initial guess and the accuracy and overall good performance of the extraction algorithm. This approach could be extended to any spectrum that is approximated as a sum of Gaussian functions.

Size-Dependent Fault-Driven Relaxation and Faceting in Zincblende CdSe Colloidal Quantum Dots.

Surface chemistry and core defects are known to play a prominent role in governing the photophysical properties of nanocrystalline semiconductors. Nevertheless, investigating them in small nanocrystals remains a complex task. Here, by combining X-ray scattering techniques in the wide- and small-angle regions and using the Debye scattering equation (DSE) method of analysis, we unveil a high density of planar defects in oleate-terminated zincblende (ZB) CdSe colloidal quantum dots (QDs) and size-dependent faceting within a square-cuboid morphology. Atomistic models of faulted ZB nanocrystals, based on the probabilistic stacking of CdSe layers in cubic and hexagonal sequences, and data analysis point to the preferential location of faults near the center of nanocrystals. By finely modeling faulting and morphological effects on the X-ray scattering pattern, a relaxation of the Cd-Se bond distance parallel to the stacking direction, up to +3% (2.71 Å) with respect to the reference bulk value (2.63 Å), is detected, at the cubic/hexagonal transitions. The smallest nanocrystals show cubic {100} facets; {111} facets appear above 4 nm and progressively extend at larger sizes. These structural and morphological features likely vary depending on the synthesis conditions; nevertheless, since planar defects are nearly ubiquitous in CdSe QDs, the modeling approach here presented has a general validity. This work also points to the great potential of combining small- and wide-angle X-ray scattering and DSE-modeling techniques in gaining important knowledge on atomic-scale defects of semiconductor nanocrystals, underpinning the comprehension of the impact of structural defectiveness on the exciting properties of these QDs.

Coherent Nanotwins and Dynamic Disorder in Cesium Lead Halide Perovskite Nanocrystals.

Crystal defects in highy luminescent colloidal nanocrystals (NCs) of CsPbX3 perovskites (X = Cl, Br, I) are investigated. Here, using X-ray total scattering techniques and the Debye scattering equation (DSE), we provide evidence that the local structure of these NCs always exhibits orthorhombic tilting of PbX6 octahedra within locally ordered subdomains. These subdomains are hinged through a two-/three-dimensional (2D/3D) network of twin boundaries through which the coherent arrangement of the Pb ions throughout the whole NC is preserved. The density of these twin boundaries determines the size of the subdomains and results in an apparent higher-symmetry structure on average in the high-temperature modification. Dynamic cooperative rotations of PbX6 octahedra are likely at work at the twin boundaries, causing the rearrangement of the 2D or 3D network, particularly effective in the pseudocubic phases. An orthorhombic, 3D γ-phase, isostructural to that of CsPbBr3 is found here in as-synthesized CsPbI3 NCs.

Celebrating 100 years of the Debye scattering equation.

One hundred years of the Debye scattering equation are celebrated with a series of articles arising from the DSE 2015 conference.

A total scattering Debye function analysis study of faulted Pt nanocrystals embedded in a porous matrix.

Faulted face-centred cubic platinum nanocrystals, grown within a nanoporous silica matrix, have been extensively characterized by the Debye function analysis method applied to wide-angle synchrotron X-ray total scattering data. A method for building databases of sampled interatomic distances of weakly faulted materials is proposed, maintaining statistical significance and allowing complete populations of differently sized and shaped nanocrystals to be used within the DEBUSSY approach. This study suggests that anisotropic Pt nanoclusters are formed in the presence of a shape-directing (templating) agent, and tentatively describes the effects of post-synthetic temperature treatments on fault probability, size, shape and dispersion of the nanocrystal populations. Surface relaxation effects are also observed in the smallest particles.

Peter J. W. Debye - a whole life devoted to science.

In 1915 P. Debye, one of the most prominent scientists in the field of condensed-matter physics and physical chemistry, published an X-ray scattering equation for randomly oriented scattering sites. This formula, since then used for describing the structure of powders, liquids and gases, has become a model for material analysis at the nanoscale. This paper re-examines briefly Debye's works on the origin and evolution of the scattering equation and its first uses. The career of the great scientist and some of his other numerous and diverse contributions to science are also reviewed. Additionally the paper addresses aspects of his life as a teacher, as a science manager and as a man, including the recent controversy about his conduct during the Third Reich regime.

100 years of Debye's scattering equation.

Debye's scattering equation (DSE) has spanned a century of scientific development, from the dawn of quantum mechanics and the investigation of the structure of atoms and molecules to the era of nanotechnology, paving the way to total scattering methods. The formulation offers the most accurate representation of the intensity scattered by randomly oriented atomic aggregates, constructed by superimposing the signal from each atomic distance in the molecule. The present paper reviews some of the milestone applications, from the interpretation of the intensity curves from gases and vapours, to aggregates of increasing size and more extended order. Important developments, aimed at mitigating the prohibitive computational complexity of the DSE, and state-of-the-art methods for the characterization of static and dynamic displacements are also discussed.