Ion distribution near a charged wall
The electrostatic interactions between charged objects in solution and their ion atmosphere play an important role in biological processes and colloidal stability. The folding of proteins and their biological activity, the compaction of genetic materials, the adsorption of ions onto lipid membranes, or the selfassembly of biomolecules are amongst the numerous examples in biology where a theoretical description of the electrostatic interactions can lead to an improved understanding of the molecular functions in cells and to a better efficacy of drugs for biomedical applications.
The PoissonBoltzmann theory has been used for nearly one century to compute the distribution of ions surrounding charged surfaces. This theory considers pointlike ions interacting via their mean field in a continuum dielectric medium. More comprehensive models based on integral equation theories or Monte Carlo simulations have shown several discrepancies with the classical PoissonBoltzmann approach due to correlation effects arising from the size of ions and from the spatial fluctuations of the electrostatic
potential. These models have pointed out the importance of different sizes in the ionion interaction for monovalent systems. The composition of the electrical double layer can be altered compared to the classical theory because the increase of the effective excluded volume of ions results in a decrease of the system bulk entropy that favors the tendency of ions to be adsorbed; it can thus lead to a surface overcharging where an apparent charge is adsorbed onto a likecharged wall. Integral equation theories and particle simulations, however, lack the simple physical picture provided by a PoissonBoltzmann type of approach.
We have therefore devised a generalized PoissonFermi formalism applicable to polydisperse systems of multiple ions. The distributions of ions are hence described in terms of bulk volume fractions of each ion species rather
than just their bulk concentrations. An arbitrary number of excluded volumes can be taken into account. Size correlation effects, which have not been studied before in the framework of such a simple theory, are exemplified next to a charged wall: underscreening, saturated layer of
mixed ions and ion stratification (Figure 1) are reported. At last, we have proposed a selfconsistent way to compute the change of the effective dielectric constant as the ion volume fractions vary across the solution.
Figure 1. Spatial repartitions of ions with different valences next to a charged wall. The solution is made of three electrolytes: a 3:1 at 10 µM, a 2:1 at 10 mM, and a 1:1 at 300 mM. The wall has a charge density of σ=0.4 C.m^{2}. All the counterions have an excluded volume of 1 nm^{3} and the coions 0.15 nm^{3}. (A) Ion concentrations given by the generalized PoissonFermi equation. The inset is a schematic of the ion stratification next to the wall. (B) Ion concentrations given by the PoissonBoltzmann theory. The numbers next to the curves on (A) and (B) indicate the ion valence. c_{i} is the ion concentrations and κx the distance from the wall normalized to the Debye length. [Phys. Rev. E 78 (2008) 061506]
