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Many interacting particles in solution. II. Screening-ranged expansion of electrostatic forces

Sergii V. Siryk, Walter Rocchia

Abstract

We present a fully analytical integration of the Maxwell stress tensor and derive exact relations for interparticle forces in systems of multiple dielectric spheres immersed in a polarizable ionic solvent, within the framework of the linearized Poisson--Boltzmann theory. Building upon the screening-ranged (in ascending orders of Debye screening) expansions of the potentials developed and rigorously analyzed in the accompanying works arXiv:2512.08407, arXiv:2512.08684, arXiv:2512.09421, we construct exact screening-ranged many-body expansions for electrostatic forces in explicit analytical form. These results establish a rigorous foundation for evaluating screened electrostatic interactions in complex particle systems and provide direct analytical connections to, and systematic improvements upon, various earlier approximate or limited-case formulations available in the literature, both at zero and finite ionic strength.

Many interacting particles in solution. II. Screening-ranged expansion of electrostatic forces

Abstract

We present a fully analytical integration of the Maxwell stress tensor and derive exact relations for interparticle forces in systems of multiple dielectric spheres immersed in a polarizable ionic solvent, within the framework of the linearized Poisson--Boltzmann theory. Building upon the screening-ranged (in ascending orders of Debye screening) expansions of the potentials developed and rigorously analyzed in the accompanying works arXiv:2512.08407, arXiv:2512.08684, arXiv:2512.09421, we construct exact screening-ranged many-body expansions for electrostatic forces in explicit analytical form. These results establish a rigorous foundation for evaluating screened electrostatic interactions in complex particle systems and provide direct analytical connections to, and systematic improvements upon, various earlier approximate or limited-case formulations available in the literature, both at zero and finite ionic strength.

Paper Structure

This paper contains 17 sections, 48 equations, 4 figures.

Table of Contents

  1. Introduction
  2. General construction of exact screening-ranged expansions of potentials and energies
  3. Boundary value problem statement
  4. Representation of potentials through their spherical Fourier coefficients
  5. Exact quantification of interaction forces and their expansions in ascending orders of Debye screening factors
  6. General analytical force relations in multi-sphere systems
  7. General construction of expansions (in ascending orders of Debye screening factors) for forces
  8. Example: $N$ spheres with centrally-located point charges
  9. Numerical modeling
  10. Summary and outlook
  11. Acknowledgments
  12. Derivation of formulas \ref{['force_i']}-\ref{['aleph_beth_definitions']}
  13. Non-screened ("0-screened") components of interaction force are zero: a rigorous proof
  14. Derivation of \ref{['force_kappa_zero_BBKS_0']}
  15. Details on calculation of forces in a system of $N$ spheres with centrally-located point charges
  16. ...and 2 more sections

Figures (4)

  • Figure 1: The subfigure in the upper left corner shows the arrangement of spheres at $R=5.3$ Å. The remaining subfigures show the Cartesian components of the total electrostatic force exerted on sphere 3 depending on $R$ which varies from $5.3$ Å to $10$ Å with step $0.05$ Å (continuous lines are used to guide the eye): Line F depicts the full (true) $\mathbf F_i$ for $i=3$ (see \ref{['force_i']}), while Lines 1, 2, 3, and 4 illustrate the convergence of screening-ranged force expansion \ref{['Force_expansion_vec']} depicting $\mathbf F_i^{(1)}$, $\mathbf F_i^{(1)}+\mathbf F_i^{(2)}$, $\mathbf F_i^{(1)}+\mathbf F_i^{(2)}+\mathbf F_i^{(3)}$, and $\mathbf F_i^{(1)}+\mathbf F_i^{(2)}+\mathbf F_i^{(3)}+\mathbf F_i^{(4)}$, respectively. (Here, $\mathrm k$ is the Boltzmann constant, $\mathrm T$ is the temperature.)
  • Figure 2: Force depending on intercenter distance $R$, acting on sphere 2 in the two-sphere system. (Line legend is as in Fig. \ref{['ArgGlu_forces_sphere3']}; embedded insets show close-up views.)
  • Figure 3: Force components depending on distance $R$ varying from $36\text{~\AA}$ to $50\text{~\AA}$. (Line legends are as in Fig. \ref{['ArgGlu_forces_sphere3']}; embedded insets show close-up views.)
  • Figure 4: Dimensionless (by dividing by $\mathrm k \mathrm T/e$) potential on the plane $z=0$ and its isolines at $R=36$ Å. Arrows indicate the normalized (unit) directions of the forces acting on the particles.

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4