2. Assembly of Particles

As the concentration of particles in solution increases, a broad peak appears at roughly 2*pi/d, where d is the distance that may be approximated to the mean interparticle distance (strictly speaking, it should be called the correlation distance). For particles behaving like a hard sphere (no interaction between), the structure factor has been analytically developed. With the hard sphere potential, scattering intensities are calculated:

Blue, red, green, magenta and black curves are for systems with particles (5nm radius) with volume fractions of 0, 0.1, 0.2, 0.3, and 0.4, respectively.

As particles get closer due to a lack of space (Imagine drying solvent from a particle solution), a broad peak suggesting amorphous structure arises in SAXS curve. Effective hydrodynamic radii of the particles are 5nm for top and 7.5nm for bottom figures, respectively.

Dotted line on both curves show q position corresponding to 2*pi/diameter (effective hydrodynamic diameter) of particles.

Structure factors from the hard sphere potential are compared to that of FCC. Lattice parameter of FCC is determined so that the particles in the lattice touch each other, meaning that the nearest neighbor distance is the diameter of the particle. Volume fraction 0.7 is almost the same with that of FCC, but peak positions and the number of peaks in the hard sphere potential structure factor are different from those in the FCC.

The hard sphere structure factor is the most commonly used one in the literature for SAXS data analysis. Although it was developed for homogeneous solution system, it has been applied to the aggregate system as well, where particles are not homogeneously distributed in a sampling volume, as long as the interpretation is done carefully considering the outcome of the brute force.

Structures with high packing would likely show certain degree of ordering unless particles are so polydisperse in size or shape. So one may need to compute the structure factor assuming a crystalline structure.

Once particles are packed into a crystalline assembly, one can use the powder diffraction theory to compute the structure factor like below:

Reference: 83. DNA-nanoparticle superlattices formed from anisotropic building blocks

Matthew R. Jones, Robert J. Macfarlane, Byeongdu Lee, Jian Zhang, Kaylie L. Young, Andrew J. Senesi & Chad A. Mirkin

Nature Materials 2010, 9, 913-917. DOI:10.1038/nmat2870

When a single type of spherical particles are composed of a unit cell, the structure factor contains only the positional information. When the particles are not spherical any more, than the structure factor will include particles' orientational information as you see from the above simulations.