Systematic Limitations in Concentration Analysis via Anomalous Small-Angle X-ray Scattering in the Small Structure Limit

Anomalous small angle scattering measurements have been applied to diluted solutions of anionic polyacrylates decorated by specifically-interacting Pb2+ cations, revealing partial collapse of the polyacrylate into pearl-like subdomains with a size on the order of a few nanometers. From the pure-resonant scattering contribution of the Pb2+ cations, and from subsequent analysis of the resonant-invariant, the amount of Pb2+ cations condensed onto the polyanions with respect to the total amount of Pb2+ cations in the solvent was estimated. In order to scrutinize systematic limitations in the determination of the chemical concentrations of resonant scattering counterions in the collapsed phase, Monte Carlo simulations have been performed. The simulations are based on structural confinements at variable size in the range of few nanometers, which represent the collapsed subdomains in the polyanions. These confinements were gradually filled to a high degree of the volume fraction with resonant scattering counterions giving access to a resonant-invariant at a variable degree of filling. The simulations revealed in the limit of small structures a significant underestimation of the true degree of filling of the collapsed subdomains when determining chemical concentrations of Pb2+ cations from the resonant invariant.