A Comparative Study of the Size-Modified and Classical Poisson–Boltzmann Models with Sharp and Diffuse Interfaces

The classical Poisson–Boltzmann (PB) model treats each ion in the solvent surrounding a solute biomolecule to be a volumeless point and assumes the solute-solvent boundary to be a sharp interface. This theory could yield excessive ion crowding near the molecular surface. To prevent that, various size-modified Poisson–Boltzmann (SMPB) models have been developed, in which each ion will occupy a nonzero volume. However, the difference in potential and energy between the PB and SMPB models is quite small in the sharp interface setting. This work explores the finite ionic size effects in a diffuse interface setting. To this end, a unified SMPB framework in terms of the dimensionless potential is formulated with its limiting cases covering the standard PB model and sharp interface setting. A SMPB energy functional without the electric stress term is proposed and the analytical treatment of singular charge sources is presented in a regularization approach. A numerical comparative study is conducted by considering Kirkwood spheres and various proteins. Our studies reveal that a notable discrepancy between the PB and SMPB models only happens in the diffuse interface setting. A Stern layer-like ion concentration has been observed in the SMPB model with diffuse interface, i.e., the peak concentration occurs at a distance from the solute approximately equal to the ion radius, while this phenomenon is absent in the standard PB model and in the sharp interface setting. This study underscores the importance of incorporating diffuse interface setting in the SMPB model for representing finite sized ions in the solvent.