Charged dielectric spheres interacting in electrolytic solution: A linearized Poisson–Boltzmann equation model

Abstract

We present an analytical theory of electrostatic interactions of two spherical dielectric particles of arbitrary radii and dielectric constants, immersed into a polarizable ionic solvent (assuming that the linearized Poisson–Boltzmann framework holds) and bearing arbitrary charge distributions expanded in multipolar terms. The presented development entails a novel two-center re-expansion analytical theory that expands upon and improves the existing ones, bypassing the conventional expansions in modified Bessel functions. On this basis, we develop a specific matrix formalism that facilitates the construction of asymptotic expansions in ascending order of Debye screening terms of potential coefficients, which are then employed to find exact closed-form expressions for the total electrostatic energy. In particular, this work allows us to explicitly and precisely quantify the k-screened terms of the potential coefficients and mutual interaction energy. Specific cases of monopolar and dipolar distributions are described in particular detail. Comprehensive numerical examples and tests of series convergence and the relative balance of leading and higher-order terms of the mutual interaction energy are presented depending on the inter-particle distance and particles’ radii. The results of this work find application in soft matter modeling and, in particular, in computational biophysics and colloid science, where the availability of increasingly larger experimental structures at the atomic-level resolution makes numerical treatment challenging and calls for more efficient expressions and an increased range of validity.

Arbitrary-Shape Dielectric Particles Interacting in the Linearized Poisson–Boltzmann Framework: An Analytical Treatment

“…This, however, does not diminish the importance of studying the behavior of the LPBE also in the case of highly charged objects: their electrostatics may still be correctly described at sufficiently long distances (as compared to the Debye length) by the usual DH approximation provided that the sources of the electric field are properly renormalized. − (See also the recent ref for additional comments concerning the ranges of applicability of the DH theory.) This once again emphasizes the importance of a thorough study of the DH approximations, both theoretically and numerically, and justifies the constant stream of works related to the LPBE (see recent refs , , and − , and references therein).…”

“…During biomolecular recognition and binding, electrostatics provides specificity by helping to tune the delicate balance between the desolvation of the binding interfaces and the descreened charged and polar interactions . At larger distances, the long-range nature of electrostatics allows the interacting partners to spend enough time in close proximity so as to increase the probability of finding the best mutual conformation for binding …”

“…Recently, Filippov and Derbenev and co-workers have published several works ,, exploring the electrostatic force between two charged polarizable spheres immersed in an electrolytic solution or in equilibrium plasma. Other relevant studies important to mention here are refs , , , , and − . Let us note that the existing literature largely focuses on treating the case of a system of two spherical particles with special symmetries also in the charge (especially, azimuthal symmetry).…”

“…In the same work, it was further demonstrated that even in the (simplest) case of two centrally located point charges the ( k ≥ 2)-screened terms may significantly exceed the conventional singly screened ( k = 1) DLVO term. ,,,, This imbalance can only increase when higher-order multipoles are present or when particles have a large dissimilarity (in size, charge, etc.). This emphasizes the importance of developing a rigorous (exact at least at the DH level) electrostatic analytical theory for arbitrary-shape polarizable dielectric particles at arbitrary interparticle separations and without any assumption on charge or system symmetries and particle sizes (as often found also in the recent literaturesee the detailed overview in ref , section III). It is worth noting that, apart from the extensively discussed case of spherical particles, recently exact (analytical) results were also obtained for interaction between a charged dielectric sphere and a planar surface and between two dielectric spheroids in the Poisson limit, i.e., at zero ionic strength.…”

Exact polarization energy for clusters of contacting dielectrics

“…However, for particles suspended in ionic solutions, more efforts are needed to incorporate the effects of ionic screening and the packing of polar solvents in the confined gap region. 28,29 In addition, the role of polarization in the stability of particulate aggregates remains to be quantified experimentally. To put our results into tests, it is advantageous to examine the aggregates formed from particles with high permittivity, such as BaTiO 3 , suspended in low permittivity liquids, such as apolar organic liquids.…”