An Optimization Regularization Method for Determination of Partial Structure Factors from Small-Angle Scattering Experiments

A new method is introduced for extracting partial structure factors from scattering data. For a bimodal mixture of colloidal particles, a minimum of three contrast-variation scattering experiments are necessary to determine the three partial structure factors. This determination of partial structure factors is a classically ill-posed matrix inversion. In addition to the random measurement error, there is random concentration error present in the set of scattering experiments, and composition parameters must be optimally chosen to minimize their effect on the inversion. The standard methods of least squares and regularization have been applied to the inversion, but are unable to provide physically significant solutions. Partial structure factors derived from the optimization regularization technique developed here satisfy the smoothness and physical constraints that partial structure factors must obey. This technique is effective in extracting partial structure factors from data with hard-sphere and sticky-hard-sphere interactions that contain concentration errors and wavelength-smearing effects.