Percus–Yevick structure factors made simple

Measuring the structure factor, S(q), of a dispersion of particles by Small-Angle X-ray Scattering provides a unique method to investigate the spatial arrangement of colloidal particles. However, it is impossible to find the exact location of the particles from S(q) because some information is inherently lacking in the SAXS signal. The two standard ways to analyse an experimental structure factor are then to compare it either to structure factors computed from simulated systems, or to analytical structure factors calculated from approximated systems. For liquids of monodisperse hard spheres, the latter method provides analytical structure factors through the Ornstein-Zernike equation used with the Percus-Yevick closure equation. The structure factors obtained in this way were not adequate for the more common dispersions of polydisperse particles. However, Vrij, Bloom and Stell were able to demonstrate that the same mathematical framework could be extended to yield accurate approximations for the experimental structure factor. Still, this solution has remained underused because of its mathematical complexity. In the present work, we derive and report the complete Percus-Yevick solution for general polydisperse hard-spheres systems in a concise form that is straightforward to use. The form of the solution is made simple enough to give ready solutions of several important particle-radius distributions (Schulz, truncated normal and inverse Gaussian). We also discuss in detail the case of the power-law radius distribution, relevant in the case of systems made of an Apollonian packing of spheres, as recently discovered experimentally in high internal-phase-ratio emulsions.