Interaction Potentials in Liquids: the Inverse Problem

This work advances a new method for extracting effective pair-potentials from static structure factors (accessible through radiation scattering measurements) of spherically symmetric dense fluids and supramolecular fluids such as colloids and micellar solutions. Liquid-state theories, including integral-equation and perturbation theories, have traditionally concerned themselves with the prediction of structural and thermodynamic properties of fluids from assumed or known interparticle potentials. This is known as the forward problem. In contrast, the question of whether or not it is possible to calculate effective pair-potentials from experimental structure factors has so far received insufficient attention. This problem, known as the inverse problem, is the focus of this dissertation. First, we show that simple single-step inversions based on closures to the Ornstein-Zernike integral equation (which are, to this day, used in studying liquid metals) are inadequate for extracting reliable pair-potentials. We also illustrate, using specific examples, why the integral -equation methods fail. Second, we demonstrate the robustness and accuracy of a recently introduced predictor-corrector inversion method based on perturbation theory for one-component systems. Both experimental and computationally generated structure factors for a wide variety of strongly interacting systems (liquids of rare gases, liquid metals as well as a model colloidal dispersion) are used for this purpose. An error analysis is also presented to identify the influence of experimental errors in the structure factors on the extracted potentials. Third, we extend the above method to two-component systems and apply it to theoretical partial structure factor data for a model binary Lennard-Jones mixture to demonstrate that the original potentials can be recovered with remarkable accuracy. This method is the first formal solution of the inverse problem for binary mixtures, converges rapidly and is not numerically intensive. Our results demonstrate that structure factors retain many of the details concerning microscopic interaction forces and that these details can be recovered with remarkable accuracy in the case of both one- and two-component systems using suitable methods, if sufficiently accurate experimental data over a wide range of scattering angles, including the small-angle region, are available.