Extracting short-ranged interactions from structure factors

Abstract

Inverting scattering experiments to obtain effective interparticle interactions is generally a poorly conditioned problem. L. Reatto [Phil. Mag. A 58, 37 (1986)] showed that for atomic liquids close to the triple point, inversions are hard because the structure closely resembles that of an equivalent hard-sphere fluid. Here I demonstrate that at low concentrations and for particles with short-ranged attractive potentials, S(k) also exhibits a very weak dependence on potential shape. Instead, different potentials all generate an S(k) that closely resembles that of the Baxter model with a similar second-virial coefficient. By contrast, in this energetic fluid regime, the inversion of an attractive interaction from real-space correlations such as the radial distribution function g(r) is well conditioned. Nevertheless, one may extract further information from S(k) by measuring isosbestic points, values of k where the scattering intensity I(k) or the structure factor S(k) is invariant to changes in interaction-potential well depth. These points suggest a new extended corresponding states principle for particles in solution based on the packing fraction, the second osmotic virial coefficient, and a new measure of effective potential range.

Keywords:

• short-ranged potentials
• structure factors
• Baxter model

Acknowledgements

The author is very pleased to contribute to this Special Issue of Molecular Physics dedicated to Professor Luciano Reatto in recognition of his important contributions to the field, and wishes him much success in the coming years. The author also thanks P. Bartlett, J.P.K. Doye for valuable discussions. He also thanks Remco Tuinier and Gerrit A. Vliegenthart for sharing with him their independent discovery of isosbestic points in structure factors 13.

Notes

Notes

1. More sophisticated closures 3 could be used, but for the low η regime discussed here, the PY approximation is adequate. This is demonstrated in Figure 3, where the g(r) from a molecular dynamics simulation, performed with the MOLDY code 14 on a box of 1024 atoms, is virtually indistinguishable from the PY result. Similar accuracy was found for other potentials and parameters, and confirms the findings of other authors (see e.g. 15,16 and references therein.). Another reason I use the PY closure is to facilitate comparisons with the Baxter model 11, which is only solved within PY.

2. I should add that this is not an implicit critique of 8, where other physical considerations were indeed used to infer the suggested short-ranged potential.

3. There are several similar ways to define an effective σ_eff for a LJ-n potential, see e.g. 3. A more careful analysis shows that for our examples, these don't differ much from σ, and have a marginal effect on the isosbestic points.

4. The isosbestic points typically move to a marginally higher k with increasing η, an effect also seen for the Baxter model; this can be used for a small correction to isosbestic points.