The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering

Abstract

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 favoured 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.

Figure 1. Schematics of the wide-angle scattering experiment.

In Born’s approximation, the far-field scattering intensity reads I(q)∝|∫ρ(r)e^iqrd³r|², where ρ(r) is the scattering density of the particle centered at r=0. After decomposing r into components parallel (r_||) and perpendicular (r_⊥) to the projection vector n_p=k_in+q/2, which is by definition perpendicular to q, the intensity can be recast as , representing the Fourier transform of the projected density ρ(r_⊥)=∫ρ(r)dr_|| on a plane with normal vector n_p. (a) For small scattering angles, the approximation n_p||k_in is valid, which inhibits access to any structural information along this direction. (b) The variation of n_p with q for large scattering angles provides access to the 3D properties of the particle. (c) In the experiment, single-shot diffraction patterns of silver particles intersecting the FEL photon beam are captured by the 2D detector.

Figure 2. Comparison of measured and theoretical scattering images.

(a–d) Selected experimental scattering patterns of single Ag particles and MSFT simulation results for matched geometries (as indicated). False-colour images show the scattering intensity (logarithmic scale) as function of the transverse components of the scattering vector. The dark spot in the centre of the experimental data originates from a hole in the detector for direct beam transmission. Cluster shapes are drawn as seen from the direction of the incident beam. The size is given by the radius r of the polyhedra’s circumscribed sphere. (e–h) Same cluster shapes as in a–d imaged at different orientations with respect to the incident beam. Trunc, truncated.

Figure 3. Comparison of different approximation levels.

False-colour images show the simulated scattering intensity (logarithmic scale) of a truncated octahedron (cf. Fig. 2b) as function of the transverse components of the scattering vector within different approximations. (a) Small-angle approximation corresponding to an effective scattering density projected onto a plane. (b) Born’s approximation taking into account the full 3D geometry but no absorption and refraction. (c) Same as b but including a simplified absorption model. (d) Full FDTD simulations using the optical properties of bulk-silver.

Figure 4. Optimization of model parameters.

(a) Experimental and (b) simulated scattering patterns for a single Ag particle with the optimized shape of a truncated octahedron as depicted in c using the FDTD method (false-colour on logarithmic scale). Parameter optimization for truncation and radius was performed by minimization of the mean-squared deviation (MSD) of experimental data from theory as exemplarily shown in d for the particle size. The optimal radius is r=95 nm with an uncertainty of ±8%, estimated from the curvature of the normalized MSD around the minimum. (e) R-factor as a function of rotation angle for the truncated octahedron depicted in c. For these calculations, the model shape is rotated away from the optimal orientation in c around an axis parallel to the upper edge of the hexagonal front facet.