The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering
The diversity of nanoparticle shapes generated by condensation from gaseous matter reflects the fundamental competition between thermodynamic equilibration and the persistence of metastable configurations during growth.
In the kinetically limited regime, intermediate geometries that are favored only in early formation stages can be imprinted in the finally observed ensemble of differently structured specimens. Here we demonstrate that single-shot wide-angle scattering of femtosecond soft X-ray free-electron laser pulses allows three-dimensional characterization of the resulting metastable nanoparticle structures. For individual free silver particles, which can be considered frozen in space for the duration of photon exposure, both shape and orientation are uncovered from measured scattering images. We identify regular shapes, including species with fivefold symmetry and surprisingly large aspect ratio up to particle radii of the order of 100 nm. Our approach includes scattering effects beyond Born’s approximation and is remarkably efficient—opening up new routes in ultrafast nanophysics and free-electron laser science.
Naturally grown particles exhibit a vast variety of architectures ranging from simple, almost spherical shapes (for example, fog droplets), to highly symmetric polyhedral (for example, clusters or certain viruses) and fascinating, complex geometries such as snowflakes or pollen grains. Studying growth processes of nanostructures addresses the key fundamental question of how geometric structure and stability are determined by the occurrence of thermodynamically metastable shapes during particle formation. An illustrative case is the condensation of metal nanoparticles. Albeit optimal equilibrated shapes can be predicted from energetic considerations such as the Wulff construction1, manifold morphologies are frequently reported2, 3, 4 that are far away from the ideal equilibrium geometries and often beyond theoretical predictability. Conventional microscopy methods have enabled high-resolution imaging of individual particles grown or deposited on surfaces and revealed insight into their geometric properties3, 5. Single free, unsupported particles, however, elude experimental access via microscopy since they cannot be immobilized without an interacting substrate. The current knowledge on the morphology of free metal particles is therefore based on orientation- and ensemble-averaged approaches such as electron diffraction6, 7, 8, drift measurements9 and photoelectron spectrocopy10.
An unambiguous experimental morphology characterization of free particles has to face a number of challenges. First, the co-existence of diverse shapes requires circumvention of ensemble averaging by the study of individual particles. Second, to reliably resolve the particle shape irrespective of its orientation, three-dimensional (3D) imaging methods are required. Third, free particle growth is a statistical process such that a specific combination of shape, orientation and size cannot be repeatedly prepared, excluding tomographic techniques11, 12 that rely on multiple measurements of the same object or of equivalent replicas. Considerable efforts have been devoted to obtaining 3D structure information of nanosystems via diffractive imaging with intense femtosecond pulses from X-ray free-electron lasers (X-FELs)13, 14, 15, 16, 17, 18, 19, 20, 21. In-flight characterization of single nanoparticles by X-ray small-angle scattering has been successfully demonstrated, revealing the effective two-dimensional (2D) projection of the electron density14, 16, 20. The 3D reconstruction based on single-shot small-angle X-ray scattering data can only be achieved by exploiting additional symmetry information, as was shown for the case of deposited particles18. The reconstruction of wide-angle X-FEL scattering has been advocated as an enabling technology for the complete single-shot 3D structure determination of individual nanosystems15. However, the short X-FEL wavelengths utilized in previous single-shot in-flight particle imaging studies precluded the detection of sufficient wide-angle signal to apply this technology.
Here we show that this limitation can be overcome by employing soft X-FEL pulses. Based on the resulting single-shot wide-angle scattering images, we demonstrate the identification of symmetry, morphology and orientation of individual gas-phase Ag particles by means of a simple and efficient procedure based on fast simulations. Our approach is complementary to the full reconstruction via phase retrieval methods14, 15, 16, 18, 19, 21, 22, 23 as it allows extracting the relevant 3D structure information from a single scattering pattern of an individual particle without computational inversion of the scattering process, provided that the particles can adequately be described by a parametric geometry model. The reported results provide evidence for metastable shapes of unsupported Ag particles in a so far inaccessible size regime.
Wide-angle scattering experiment
The key experimental requirement for the 3D characterization is the ability to resolve the scattering signal up to large angles. For an illustrative motivation of this requirement, it is convenient to assume validity of the first Born approximation. In the limit of small-angle scattering, the scattered far-field can essentially be described by a 2D Fourier transform of the object’s shape projected onto a plane (characterized by normal vector np||kin) perpendicular to the incident beam direction, see Fig. 1a. This follows from the fact that the transfer momentum q is small in magnitude (|q|≪|kin|) and therefore essentially perpendicular to the incident wave vector kin. The resulting intensity distribution (i) reflects only effective 2D information on the object’s density distribution and (ii) is point-symmetric with respect to q=0, which impedes unique identification of the target orientation.
These limitations can be overcome by recording scattering under large angles (|q|≈|kin|) because the scattering pattern then reflects dependent projections of the density (on planes with normal vectors ), see Fig. 1b. In a sense, wide-angle scattering enables single-shot tomography as the direction of the projection plane varies with scattering angle within a single image.
However, due to the drastic decrease of the scattering intensity I(q) with increasing |q| (Porod’s law), significant signal from the particle shape can in practice be detected only up to a critical angle. Considering near spherical shapes, this angle is roughly proportional to the wavelength and limited to a few degrees when using hard X-ray radiation in the keV range (see Methods). Here we remedy this limitation by employing 90 eV soft X-ray laser pulses to access the wide-angle regime required for the 3D characterization of individual particles.
In the experiments (see scheme in Fig. 1c), silver particles were prepared in a cluster machine24 equipped with a magnetron sputtering source and directed into the focus of the 100 fs soft X-ray pulses provided by the free-electron laser (FEL) facility FLASH (see Methods for details). Note that for the employed wavelength of 13.5 nm, the detailed atomic structure is averaged out in the scattering images, simplifying the shape analysis tremendously. Scattering images were captured by a 2D detector with 78° acceptance angle17, 25. A total of 25,000 scattering images with significant intensity have been recorded.
Scattering patterns and particle morphologies
The measured single-particle scattering images (Fig. 2, left column) show highly symmetric patterns with twofold (a), threefold (b), fivefold (c), and sixfold (d) symmetry. The patterns consist of one or more closed ring-like features near the centre followed by discontinuous higher order rings that form streak-like features accompanied by a faint fine structure. Scattering patterns with odd number of mirror axes (Fig. 2b,c), that is, with broken point symmetry, immediately demonstrate that the wide-angle scattering data contains true 3D structure information. Because of substantial absorption of the soft X-ray radiation inside the particles (penetration depth ≈12.5 nm for bulk silver), the morphology identification method must account for scattering effects beyond the Born approximation, which excludes application of conventional iterative reconstruction techniques. We use a simple and efficient multislice Fourier transform (MSFT) algorithm that includes an effective treatment of absorption to calculate scattering images from 3D trial shapes based on a large systematic set of polyhedra (see Methods).
Excellent qualitative agreement between measured scattering images and MSFT results (Fig. 2) is achieved by adjusting size and orientation of the trial model shapes (see Methods for the detailed procedure). In most cases, even fine details are well reproduced, such as the spots in between x-shaped main features in Fig. 2a. Among the morphologies that match the experimental data are decahedra (a), truncated octahedra (b), icosahedra (c) and surprisingly flat hexagonal particles, which correspond to truncated twinned tetrahedra (d). For each of the above shapes, particles with different size and orientation are identified in the data set (see examples in Fig. 2e–h), confirming the repeated occurrence of the identified geometries. Although some of the compact shapes (such as those in Fig. 2b,c) deviate only weakly from a sphere, the scattering images taken for different orientations are extremely diverse (compare Fig. 2b,c to Fig. 2f,g respectively; see Supplementary Movie 1 for a simulation of all high-symmetry orientations of a truncated octahedron). The strong directional and shape sensitivity (even for nearly spherical shapes) demonstrates the 3D capabilities required for the unique identification of particle morphologies. Striking evidence for the transition from small- to wide-angle scattering regimes can be found within a single image for large clusters if it shows broken point symmetry (see Fig. 2b,g). In these cases, the symmetry changes from even, close to the image centre, to odd in the outer regions.
Benchmark of the MSFT method
To corroborate the reliability of the MSFT method and to elucidate the role of absorption, we have compared theory results for different levels of approximation for the case of the truncated octahedron (cf. Fig. 2b). The resulting data within the small-angle approximation (Fig. 3a), implemented via a 2D Fourier transform of the projected scattering density, predicts point symmetric scattering images (the power spectrum of a real-valued function is symmetric) and fails to resemble the experiment. This point symmetry is lifted in the 3D simulation (Fig. 3b) obtained within Born’s approximation. Comparison to MSFT including effective absorption (Fig. 3c) shows that absorption induces a broadening of scattering features, a relative intensity increase in higher diffraction orders and a reduction of the scattering angle of the first-order intensity maximum. Differences between the MSFT result and that of the full treatment of multiple scattering within the finite-difference time-domain (FDTD) framework (Fig. 3c versus Fig. 3d) are more subtle and in most cases irrelevant for shape identification, justifying the effective absorption treatment in MSFT. Compared with the 3D simulations in Fig. 3b–d, the 2D scattering image in Fig. 3a shows similar features close to the centre but deviates significantly at large scattering angles, illustrating the transition from small-angle to wide-angle scattering within a single image.
The MSFT method enables efficient shape identification from a finite set of trial shapes and yields a rough estimate of the parameters of the respective geometry model. A more accurate description of the scattering process via FDTD simulations offers further refinement of the free parameters by direct minimization of the mean-squared deviation of calculated and experimental scattering patterns (see Methods for technical details). This is exemplarily illustrated in Fig. 4 for the truncated octahedron from Fig. 2b, yielding even better agreement between experiment and simulation (R-factor ≈0.18, see Fig. 4e). The resulting shape (Fig. 4c) was obtained after optimizing the degree of truncation and the radius of the particle, leading to a slightly reduced value of the predicted radius of r=95 nm (Fig. 4d) and a relative truncation of 0.39, as compared with the MSFT estimate of r=100 nm obtained under the assumption of an ideal Archimedean shape with a truncation of 1/3. This scheme can be extended to other parameters such as orientation (see Fig. 4e) or shape asymmetries to obtain precise information on the geometry of the individual nanoparticle.