Analytical solutions of the Poisson-Boltzmann equation within an interstitial electrical double layer in various geometries

Hybrid method integrating adsorption and chemical precipitation of heavy metal ions on polymeric fiber surfaces for highly efficient water purification

A lot of research has been focused on increasing the specific surface area of adsorbents over a long period of time to remove heavy metal ions from wastewater using the adsorbent. However, porous adsorbents with high specific surface area have demonstrated drawbacks in water purification processes, such as high pressure drop and limitations in the adsorption capacity of heavy metal ions. In recent years, a mechanism-based convergence method involving adsorption/chemical precipitation has emerged as a promising strategy to surmount the constraints associated with porous adsorbents. The mechanism involves amine groups on chelating fibers dissociating OH- ions from water molecules, thereby raising the pH near the fibers. This elevated pH promotes the crystallization of heavy metal ions on the fiber surfaces. The removal of heavy metal ions proceeds through a sequence of adsorption and chemical precipitation processes. An adsorbent based on chelating fibers, integrating adsorption technology with chemical precipitation, demonstrates superior performance in removing significant quantities of heavy metal ions (ca. 1000-2000 mg/g for Cd2+, Cu2+ and Pb2+) when compared to developed porous adsorbents (ca. 50-760 mg/g for same ions). This review paper introduces advanced polymer fibers endowed with the capability to integrate hybrid technology, delves into the mechanism of hybrid technology, and examines its application in process technology for the effective removal of heavy metal ions. The versatility of these advanced fibers extends far beyond the removal of heavy metal ions in water treatment, making them poised to garner significant attention from researchers across diverse fields due to their broad range of potential applications. After further processes involving the removal of templates from chelating polymeric fibers used as supports and the reduction of precipitated heavy metal oxide crystals, the resulting heavy metal crystals can exhibit thin walls and well-interconnected porous structures, suitable for catalytic applications.

Counterion Release from Macroion Assemblies of Planar Geometry

Macroions such as nanoparticles, polyelectrolytes, ionic gels, and amphiphiles can form condensed, often self-assembled, phases that are embedded in a solvent region. The condensed phase contains not only the partially or fully immobile charges of their macroions but also corresponding counterions that are mobile and thus free to migrate out of their confinement into the solvent region where they benefit from high translational entropy. Based on the nonlinear Poisson-Boltzmann model for monovalent ions, we quantify the corresponding fraction of released counterions for a planar slab geometry of the macroion phase. Slab thickness, extension of the solvent phase, fixed background charge density provided by the macroions, and dielectric constants inside slab and solvent combine into three dimensionless parameters that the fraction of released counterions depends on. We calculate that fraction and analyze the limits of a thin macroion phase, a large solvent phase, and linearized theory, where simple analytic results become available. Of particular interest is the presence of a single-planar interface that separates a bulk macroion phase from an extended solvent region. We calculate the apparent surface charge density that emerges due to the released counterions. Our model yields a comprehensive description of counterion partitioning between a planar macroion phase and a solvent region on the level of mean-field electrostatics in the absence of added salt ions.

Surface charging parameters of charged particles in symmetrical electrolyte solutions

Surface electric charge of dispersed particles is an essential determinant of physicochemical properties, coagulation and flocculation processes, and stability of colloidal solutions. Size-dependence of surface potential, charge density, and total surface charge of suspended charged particles has recently received attention in the literature. Despite the clear significance of understanding such dependence, very few studies have been devoted to this problem, with contradictory results of the relationship type. Currently, there is no analytical formula to represent explicit relationships between surface charging parameters and particle size. This research work is directed at development of accurate physics-based formulas for quantification of curvature-dependence of surface potential, surface charge density, and total surface charge for cylindrical and spherical charged particles immersed in a symmetrical electrolyte solution. First, a non-dimensional approach is adopted to simplify the problems, overcoming the difficulty of dealing with multiple influential variables. Then, to reduce the degrees of freedom of the problems under consideration, Gauss's law is combined with the condition of electro-neutrality in an electrical double layer (EDL). Next, the resulting complex integral equations are solved to construct characteristic curves and to express the dimensionless surface charging parameters explicitly as a function of the dimensionless particle radius. The new theoretical expressions are founded on approximate analytical and numerical solutions of the nonlinear Poisson-Boltzmann (PB) equation in cylindrical and spherical geometries. Afterwards, the solutions of the non-dimensionalized problems are dimensionalized to derive accurate explicit closed-form expressions, describing how surface charging parameters are related to the radius of a charged particle, properties of the solution, and thermodynamic conditions. These analytical formulas enable researchers to properly determine surface potential, surface charge density, total surface charge, and radius of dispersed particles by characterizing only one of them. Finally, the validity of the commonly-held hypothesis that surface charge density is independent of particle size is examined at the end of this study.