STATISTICAL FIELD THEORY MODELS TO STUDY ADSORPTION OF POLYELECTROLYTES ONTO CHARGED SURFACE

The study of polyelectrolyte adsorption on oppositely charged surfaces has been a subject of research over time due to its technological and biological applications. Some examples of what has motivated the study of the principles that trigger adsorption include their potential influence on cellular metabolism, stabilization of colloidal suspensions, and formation of aggregates between polyelectrolyte chains and proteins. The transition from the adsorbed to the desorbed state occurs abruptly with variations of pH or ionic strength. A scaling relationship is observed for the critical surface charge density given by \sigma~ \kappa^\beta, where κ is the inverse Debye length and β is the exponent that depends on the geometry of the surface and the ion concentration regime. The goal of this work is to study the adsorption of various homogeneously charged polyelectrolytes on a charged spherical surface using functional methods in field theory. These methods replace the degrees of freedom associated with a particle system with degrees of freedom related to real scalar fields, which simplifies the problem since such a process decouples the interactions among particles, replacing them by independent interactions between each particle and the field variables. Through statistical field theory, we derive coupled partial differential equations that describe the system: the Edwards equation, associated with the diffusion process of chains in the solution under the influence of the external fields; the non-homogeneous modified Helmholtz equation whose source given by the density of monomers distribution coupled with the linear charge density of the chain; and the equation associating the excluded volume interaction with the monomers’ density. Using the finite difference method in one dimension, we found the salt concentration and macromolecular charge density conditions in which the adsorption is stable, resulting in the scale relation obtained experimentally. In three dimensions, we find the integral form of the solutions, since the finite difference method becomes unstable near the transition point, which will be used to obtain the monomer distribution and scaling relations.