Approximations Leading to a Unified Exponential/Power-Law Approach to Small-Angle Scattering

TLDR

A mass fractal such as a polymer coil has two structural levels—overall radius of gyration and substructural persistence length—each described by a Guinier and an associated power‑law regime. The paper presents a new approach to small‑angle scattering analysis for complex systems with multiple related structural levels and provides the arguments that lead to this unified method. The authors derive a function that models Guinier exponential and power‑law regimes, accounting for low‑q and high‑q limits, using only the parameters employed in local fits. The unified approach distinguishes buried Guinier regimes, applies to diverse systems, has been successfully fitted to data with multiple regimes, and its usefulness is shown by comparisons with model calculations for polymer coils, polydisperse spheres, ellipsoids, rods, and discs.

Abstract

A new approach to the analysis of small-angle scattering is presented that describes scattering from complex systems that contain multiple levels of related structural features. For example, a mass fractal such as a polymer coil contains two structural levels, the overall radius of gyration and the substructural persistence length. One structural level is described by a Guinier and an associated power-law regime. A function is derived that models both the Guinier exponential and structurally limited power-law regimes without introducing new parameters beyond those used in local fits. Account is made for both a low-q and a high-q limit to power-law scattering regimes. The unified approach can distinguish Guinier regimes buried between two power-law regimes. It is applicable to a wide variety of systems. Fits to data containing multiple power-law and exponential regimes using this approach have previously been reported. Here, arguments leading to the unified approach are given. The usefulness of this approach is demonstrated through comparison with model calculations using the Debye equation for polymer coils (mass fractal), equations for polydisperse spheres (Porod scattering) and randomly oriented ellipsoids of revolution with diffuse interfaces, as well as randomly oriented rod and disc-shaped particles.