Atomistic X-ray Simulation using the Debye Equation

X-ray scattering in conjunction with simulation can act as a compliment to electron microscopy for the analysis of nanocrystal phase and composition.109,110 The basis for this simulation is the Debye formula, described by the formula’s namesake in 1915, which allows calculation of the X-ray scattering intensity at arbitrary angle for any set of atomistic coordinates.111,112 This formula has since been utilized for the analysis of X-ray scattering from nanosized crystallites,110,113–118 as well as the analysis of scattering from proteins, polymers, and other macromolecules.119–121

The q-dependent X-ray diffraction intensity, I(q), is calculated using the Debye equation 2.1:109

sin (2.1)

where I0 is the incident intensity, q = 4π sinθ / λ is the scattering parameter for X-rays of wavelength λ diffracted through the angle θ, rmn is the distance between atoms m and n, with atomic form factors Fm and Fn, respectively. Atomic form factors are calculated

*_{This work (sections 2.4.2 and 2.4.3) was developed over several publications with the X-ray simulation}

code and methodology submitted for publication as: T.R. Gordon, B.T. Diroll, T. Paik, V.V.T. Doan- Nguyen, C.B. Murray. Charaterization of Shape and Monodispersity of Anisotropic Nanocrystals through Atomistic X-ray Scattering Simulation. Under review.

from Cromer-Mann coefficients.122–124 To improve calculation time, the Debye equation is discretized by binning identical distances to give the following equation 2.2:125

sin (2.2)

where ρ(rmn) is the multiplicity of each unique distance rmn in the structure. Thermal vibrations distort the diffraction pattern due to uncertainty in the atomic positions and these are simulated by multiplication of the atomic form factors by a temperature factor, which has the Debye-Waller factor B as the input:

(2.3)

where and ai, bi and c are the Cromer-Mann coefficients.123 Particle size distribution is incorporated using a probability distribution of particle sizes which is reasonable for the sample, typically Gaussian or log-normal, which appear very similar at small size dispersions. The simulated X-ray pattern of each atomistic model size is then summed in a Gaussian distribution to obtain an ensemble simulation. A code which can implement the Debye function on a series of atomically-defined nanocrystals with a given shape and composition was developed over many publications and several years.50,92,126,127

There are two angular regimes of X-ray scattering which may be used to characterize dispersed species, such as NCs, biomolecules and polymers, which are categorized as small-angle X-ray scattering (SAXS) and wide-angle X-ray scattering (WAXS). SAXS encompasses the q range of roughly < 1 Å-1. In assemblies of NCs, this angular range is useful for probing superlattice structures or nearest neighbor distances.109,128 In a dilute homogeneous solution, where the structure factor does not contribute strongly to scattering, SAXS is a measurement of the form factor of the NC, although the signal at very small angles is associated with interparticle potentials in solution.129–131 This particle form factor represents the Fourier transform of the particle shape, and has been solved analytically for some morphologies (e.g. ellipsoid, infinitely long rod, etc.), assuming that the particle has uniform electron density.129 On the other hand, this is a clear oversimplification, as colloidal particles are composed of

arrangements of atoms with non-uniform electron density and truncation at defined lattice sites, especially prominent for small NCs, which do not conform to ideal shapes. Both phenomena result in smearing of the form factor.109 In addition, most colloid morphologies are not solved analytically, forcing researchers to use a similar morphology in analysis (e.g. ellipsoid for rod, sphere for cube, etc.), limiting the accuracy of the particle dimensions and distribution parameters retrieved.

WAXS covers the larger q and 2θ angular range corresponding to the interatomic distances present in a material. WAXS is an indispensable method to identify the crystalline structure of NCs. Although the peaks are broad at small particle size due to Scherrer broadening, patterns are typically indexed against bulk crystal patterns. Less appreciated is that WAXS patterns are quite sensitive to NC morphology, as the intensity and breadth of each peak are highly dependent on the number of each contributing atomic plane.62,64 Although not the focus of this work, WAXS data and atomistic simulations can also describe the diffraction from defective or strained nanocrystalline materials.132,133 On the other hand, WAXS patterns, particularly of anisotropic NCs, are subject to the effects of preferred orientation, in which NCs organize anisotropically upon deposition onto a flat substrate, complicating interpretation of the particle shape.

The Debye equation provides a method by which a self-consistent atomisitic model can be constructed for NCs by simulating both WAXS and SAXS simultaneously. The NC sample is first characterized through transmission electron microscopy (TEM) to identify the likely morphology, average particle size, and approximate size distribution. Then, an atomistic model is developed based on the crystal structure identified by a WAXS measurement and the appropriate morphology. Finally, the Debye function is used to generate both a SAXS and a WAXS pattern for the atomistic model along with a distribution. The advantages of this approach include the capacity of SAXS and WAXS to quickly provide a bulk measurement of average NC morphology, which overcomes the potential biases of microscopy, and the ability to easily simulate the scattering pattern of materials with arbitrary shape.

Figure 2.9 SAXS and WAXS (open circles) patterns with atomistic X-ray simulation

(solid red) for (a) PbSe spheres, (b) CdSe nanorods, (c) PbTe cuboctahedra, (d) anatase bipyramids, (e) PbSe cubes, (f) GdF3 plates, (g) CdO octahedral, (h) prolate CdSe ellipsoids, and (i) PbSe nanorods. A TEM image of the PbSe nanorod sample appears above.

Nine SAXS/WAXS patterns for NCs (black circles) along with Debye function simulations (red lines) are presented in Figure 2.9. As expected, spherical PbSe NCs in Figure 1a exhibit a classic pattern in the SAXS regime indicative of a spherical Bessel function. In addition to simulating the mean diameter of the NC, the SAXS pattern of an ensemble must be simulated using an ensemble distribution, as particles of different size dampen the pattern. To accomplish this, simulations are performed for many NC diameters spanning the mean value, typically by at least two standard deviations, which are then Gaussian-weighted and summed. Although a Gaussian distribution is used here,

any distribution function that accurately models the particle size distribution can be used. Many common distributions (e.g. log-normal, gamma) result in very similar patterns at small distributions in particle size.

In addition to spherical NCs, which may be fit using analytical solutions, Figure 1 also depicts a diverse array of NC shapes which have be simulated using our method: 12.8±1.3 nm x 4.3±0.4 nm CdSe rods (Figure 1b), 11.3±0.7 nm PbTe cuboctahedra (Figure 1c) 18.0±2.7 nm long x 10.0±1.5 nm wide truncated anatase bipyramids (Figure 1d), 12.1±1.1 nm PbSe cubes (Figure 1e), 34.5±1.6 nm tip-to-tip x 2.1±0.1 nm thick rhombic plates of GdF3 (Figure 1f), 61.0±3.4 nm octahedra of CdO (Figure 1g), 4.4±0.7 nm prolate CdSe ellipsoids with aspect ratio 1.4 (Figure 1h), and 14.6±1.3 nm long x 5.3±0.5 nm diameter PbSe rods (Figure 1i). An analytical solution for each of these shapes is unavailable, but they are reasonably simulated with atomistic scattering models. The monodispersity of the samples varies from ~5% (CdO octahedra) to 15% (TiO2 truncated bipyramids). For the samples presented in Figure 1, percent deviation in size is applied isotropically along all axes of the structure, which approximates the distributions well. However, some samples, such as atomically-flat sheets,134 may show uncorrelated dispersions along different growth directions and these can be simulated by applying direction-specific dispersions.

Distinguishing particular shapes of NCs from the X-ray scattering patterns alone is not recommended; X-ray simulation should be used to provide a statistically-valid estimate of size, shape, and monodisperisty. However, certain shapes, such as the nanorods shown in Figure 2.9b are relatively easily distinguished from the X-ray scattering pattern because the peak intensities and line broadening of the diffraction peaks are quite distinct from the bulk. The nanorods in Figure 1i are further distinguished as anisotropic by the two distinct contributions to broadening of the signal at 29.1° in 2θ. However, similar shapes such as spheres and cuboctahedra, as in Figures 2.9a and 2.9c, show very similar scattering patterns in the 10 nm size range; TEM is important to distinguish between these shapes.

In all cases except for the PbSe nanorods, the NCs are shaped in a manner commensurate with the underlying symmetry of the atomic crystal. For example, wurtzite

CdSe grows preferentially along the hexagonal c-axis to form elongated rods or cubic NCs (e.g. PbE, CdO) show varying truncation along (100) and (111) facets to generate shapes from cubes to cuboctahedra to octahedra.

PbSe nanorods, which form by oriented attachment of smaller crystallites,68 represent a rarer situation in which the underlying symmetry of the crystal structure is not preserved in the particle shape. This is also apparent in the powder and simulated X-ray patterns from the distinctive (200) reflection at 29.1° in 2θ, which shows both a sharp reflection from the nanorod long axis and a broader reflection from the short axes. The sharp (200) reflection unambiguously identifies the (100) as the direction of oriented attachment in PbSe nanorods.