3-D x-ray diffraction imaging with nanoscale resolution

A feature of emerging nanotechnology has been the development of enhanced capability for the production of a broad and diverse range of materials in the form of nanoscale (<100 nm) particulate, and for the dispersion of such nanoparticulate in a variety of matrices to create novel composite nanostructures. The emergence of this new class of materials brings with it an imperative for improved techniques of structural characterisation. The novel diffraction imaging technique presented here utilises a crystal analyser interposed between the object and the detector to effectively deconvolute coherent components of the complex diffraction pattern from an aperiodic array of scattering objects and selectively project the resulting Fraunhofer diffraction pattern in a reciprocal space defined by the position and orientation of the analysing crystal. It exploits the 3-D geometrical construction of the Ewald sphere to record multiple 2-D distributions of coherent diffracted intensity at different locations on the Ewald sphere (Fig.1). The Fraunhofer diffraction data is collected from a very large assembly of dispersed nanoparticles, and the recordable diffracted intensity is thus magnified by a factor approximately equivalent to the number of particles. The angular intensity distribution from a given particle does not depend on the beam size or the sample-detector distance and the diffracted intensity measured in reciprocal space is thus simply the sum of the individual diffracted intensities from all particles exposed to the incident beam. In the case when all the particles are identical, the reconstructed image will represent an individual particle. However, when the dispersed particles have a spread in size, the reconstruction will show a “representative of the modal”, that is the most frequently occurring particle.

To obtain 3-D information about an object, it is necessary to exploit the full three dimensions of the Ewald sphere. In contrast to conventional real-space imaging experiments, the resulting 2-D map of the diffracted intensity from the sample represents a projection of the 3-D diffraction image on the Ewald sphere (Fig. 1). Rotation of the sample about the principal optical axis sees a change in the 3-D intensity profile projected onto the surface of the Ewald sphere and, as a result, a change in the projected 2-D distribution of coherent diffracted intensity recorded from the sample. To demonstrate the potential of this novel approach, we have selected a range of nanocomposite structures containing intermetallic (Fig. 2a) and ceramic nanoparticles (Fig. 2b), and carbon nanotubes, with characteristic dimensions ranging from of the order of ten to a few hundreds of nanometres.

Figure 3 presents experimentally-recorded 2D maps of the diffracted intensities for two azimuthal orientations of the Al-Cu alloy sample: defined as 0 degrees and 45 degrees. To reconstruct the shape of the diffracting q' nanoparticles embedded in the Al matrix, we used a modified Gerchberg-Saxton formalism. The modal size of the nano-particles was estimated from the width of the peaks in the 1D diffraction scans performed prior to the 2-D data collection. Reconstruction of the TiO₂ nanoparticles and carbon nanotubes was performed using the same approach.

The result of the reconstruction for the Al-Cu sample is presented in Fig. 4(a). The two orthogonal variants of the particle(s) reconstructed represent a representative of the modal particle in two orthogonal planes within the given aperture of the incident beam. Reconstructions for the dispersed TiO₂ nanoparticles and carbon nanotubes are presented in Figs 4(b) and (c) respectively. The reconstructed form and dimensions of the representative of the modal TiO2 nanoparticle are consistent with those of typical particles observed by TEM, Fig. 2(b). The difficulties experienced with conventional TEM imaging of embedded crystalline particles sparsely distributed in a polymer matrix serve to highlight qualitatively the sensitivity of the present technique to such dispersions of low number density and/or volume fraction. Equally, the example of Fig. 4(c) illustrates qualitatively the capability of the technique for distinguishing crystalline carbon nanotubes dispersed in a largely amorphous molecular hydrocarbon. It should be realized, however, that the variations in the nanotube diameter correspond to single pixels in the reconstructed image, and thus cannot be claimed to be real features at present. The form and scale of the rendered image are consistent with those of the multi-wall nanotubes typically used in composite nanostructures, where individual nanotubes vary from <10nm to 10-20 nm in diameter, with aspect ratios (length: diameter) in the range 2:1 to 10:1. For both of these examples, the resolution in the reconstructed images was of the order of 1 nm.

The technique described here represents an important addition to the emerging suite of tools available for 3-D x-ray characterisation of nanostructures. The principal advantage of the method is that it allows for non-destructive and in-situ analysis over large (several mm³) volumes of material, and determination of the representative of the modal size and shape of a dispersed array of nanoscale (<100nm) scattering objects with potentially sub-10nm spatial resolution. Moreover, it is, in principle, insensitive to the coherency of the radiation employed and can thus be implemented in a laboratory environment. While it does not permit direct imaging of individual objects in an array, the technique does have potential for rapidly determining in situ temporal changes in the modal dimensions and form of a scattering array as a function of changes in such external variables as temperature, pressure and applied stress. That it is able to do so in volumes of material truly representative of bulk behaviour ensures that the technique is highly complementary to alternative approaches which are commonly destructive of the material in specimen preparation and often restricted to highly localised and potentially unrepresentative volumes. A typical area of the material studied in a TEM experiment covers tens to hundreds of nanometers. A single experiment employing the present technique eclipses this total volume by at least one order of magnitude, up to a few cubic millimetres, providing the ideal input for modern computational materials science, which seeks to model material behaviour in terms of structure averaged over representative volumes.

FIGURES.

Figure 1. Schematic representation of the experimental configuration (not to scale). The crossed feature intersecting the Ewald sphere represents a 3-D diffraction image of a representative of the modal nanoparticle.

Figure 2. TEM micrographs of (a) an Al-4Cu sample aged 72h at 250°C; and (b) TiO₂ nanoparticles embedded in a thin (100nm) section of polymer matrix.

Figure 3. Projections of x-ray diffraction intensity (logarithmic scale) recorded experimentally from the dispersed Al₂Cu nanoparticles at two locations in reciprocal space separated by an azimuthal sample rotation of 45°. The ‘ghost’ intensity detected in the lower right corner of the intensity profiles is attributable to higher order diffraction/refraction phenomena from the particle distribution.

Figure 4. 3-D renderings of: (a) two orthogonal variants of plate-shaped Al₂Cu nanoparticles embedded in an Al matrix. Surface faceting observable in the reconstructed image is of a form and scale consistent with growth defects associated with particle coarsening, but it remains to be established whether they can be interpreted to have such physical significance (b) a representative of the modal TiO₂ nanoparticle dispersed in polymer matrix. (c) a representative of the modal carbon nanotube dispersed in polymer matrix.