The Determination of Affinity Distributions: A Numerical Algorithm and its Application for Estimating the Energetic Heterogeneity of Complexing Silicas and Humic Substances

Abstract

The characterization of energetic heterogeneity has been discussed in the investigation of ion-binding and chemisorption processes. Both the calculation and the interpretation of the distribution of affinity constants are ambiguous. Methodological difficulties arise connected to the fact that electrostatic effects are difficult to separate from energetic heterogeneity because of the chemical biography of a given material. Only a close similarity between the distribution functions calculated for different ionic strengths allows the electrostatic interactions to be neglected. The numerical estimation of the distribution functions is complicated by the ill-posed nature of the problem. Some computational methods are briefly compared and methods for providing robust and unbiased estimations outlined. In contrast to differential distribution functions, the computation of integral ones may be transformed into the conventionally correct problem. On this basis, a fast and robust method for calculating integral distribution functions is proposed. In addition, this ensures numerically stable estimations of differential distribution functions. The method has been applied to a study of the energetic heterogeneity of 20 silicas chemically modified with aliphatic amines. When H⁺ ions are chemisorbed, the energetic heterogeneity observed is dependent on the surface topography and its hydration state. In addition, the binding properties of ashless fulvic and humic acids relative to H⁺, Hg²⁺ and Pb²⁺ ions have been examined. The existence of functional groups with different acidities and complexing abilities has been established.