# A robust and accurate formulation of molecular and colloidal electrostatics

## TL;DR

This paper introduces a boundary integral reformulation for molecular and colloidal electrostatics that eliminates singularities, enabling more accurate and stable numerical computations of potentials, fields, and forces near boundaries and contact points.

## Contribution

It presents a boundary regularized integral equation formulation that removes singularities from the conventional BIM, improving accuracy and computational stability in electrostatics modeling.

## Key findings

- Accurate evaluation of surface integrals without special numerical procedures.
- Enhanced precision in calculating electric fields and forces at boundaries.
- Reliable handling of close boundary configurations without numerical instability.

## Abstract

This paper presents a re-formulation of the boundary integral method (BIM) for the Debye-Huckel model of molecular and colloidal electrostatics that removes the mathematical singularities that have been accepted as an intrinsic part of the conventional BIM. The essence of the present boundary regularized integral equation formulation consists of subtracting a known solution from the conventional BIM in such a way as to cancel out the singularities associated with the Green's function. This approach better reflects the non-singular physical behavior of the systems on boundaries with the benefits of (i) the surface integrals can be evaluated accurately using quadrature without any need to devise special numerical integration procedures, (ii) being able to use quadratic or spline function surface elements to represent the surface more accurately and the variation of the functions within each element is represented to a consistent level of precision by appropriate interpolation functions, (iii) being able to calculate electric fields, even at boundaries, accurately and directly from the potential without having to solve hypersingular integral equations and this imparts high precision in calculating the Maxwell stress tensor and consequently, intermolecular or colloidal forces, (iv) a reliable way to handle geometric configurations in which different parts of the boundary can be very close together without being affected by numerical instabilities, therefore potentials, fields and forces between surfaces can be found accurately at surface separations down to near contact, and (v) having the simplicity of a formulation that does not require complex algorithms to handle singularities will result in significant savings in coding effort and in the reduction of opportunities for coding errors. These advantages are illustrated using examples drawn from molecular and colloidal electrostatics.