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 measured signal. The two standard ways to analyse an experimental S(q) are then to compare it either with structure factors computed from simulated systems or with analytical ones calculated from approximated systems. However, such approaches may prove inadequate for dispersions of variously polydisperse particles. While Vrij, Bloom and Stell established a mean-field approach that could yield fairly accurate approximations for experimental S(q), this solution has remained underused because of its mathematical complexity. In the present work, the complete Percus–Yevick solution for general polydisperse hard-sphere systems is derived in a concise form that is straightforward to use. The form of the solution has been simplified enough to provide experimentalists with ready solutions of several commonly encountered particle-radius distributions in real systems (Schulz, truncated normal and inverse Gaussian). The approach is also illustrated with a case study of the exponential radius distribution. Finally, the application of the proposed solution to the power-law radius distribution is discussed in detail by comparing the calculations with experimentally measured S(q) for an Apollonian packing of spherical droplets recently reported in high-internal-phase-ratio emulsions.