Adsorption free energy of variable-charge nanoparticles to a charged surface in relation to the change of the average chemical state of the particles

Variable-charge nanoparticles such as proteins and humics can adsorb strongly to charged macroscopic surfaces such as silica and iron oxide minerals. To model the adsorption of variable-charge particles to charged surfaces, one has to be able to calculate the adsorption free energy involved. It has been shown that the change in the free energy of variable-charge particles is related to the change in their average chemical state upon adsorption, which is commonly described using surface complexation models. In this work, expressions for the free-energy change in variable-charge particles due to changes in chemical binding are derived for three ion-binding models (i.e., the Langmuir, Langmuir-Freundlich, and NICA models) and for changes due to nonspecific binding for the Donnan model. The expressions for the adsorption free energy of the variable-charge particles to a charged surface are derived on the basis of the equality of the (electro)chemical potential of the particles in the bulk solution and adsorption phase. The expressions derived are general in the sense that they account for the competition between charge-determining ions that bind chemically to the particles, and they also apply in case of the formation of chemical bonds between particle ligands and surface sites. The derived expressions can be applied in the future to model the adsorption of variable-charge nanoparticles to charged surfaces. The results obtained for the NICA-Donnan model make it possible to apply this advanced surface complexation model to describe the adsorption of humics to minerals.