Particle-based simulation of charge transport in discrete-charge nano-scale systems: the electrostatic problem
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NANO EXPRESS
Open Access
Particle-based simulation of charge transport in discrete-charge nano-scale systems: the electrostatic problem Claudio Berti1*, Dirk Gillespie2, Robert S Eisenberg2 and Claudio Fiegna1
Abstract The fast and accurate computation of the electric forces that drive the motion of charged particles at the nanometer scale represents a computational challenge. For this kind of system, where the discrete nature of the charges cannot be neglected, boundary element methods (BEM) represent a better approach than finite differences/finite elements methods. In this article, we compare two different BEM approaches to a canonical electrostatic problem in a three-dimensional space with inhomogeneous dielectrics, emphasizing their suitability for particle-based simulations: the iterative method proposed by Hoyles et al. and the Induced Charge Computation introduced by Boda et al. 1 Introduction The investigation of the properties of a large variety of physical systems requires accurate computation of the electrostatic interactions among discrete fixed or mobile charges. This problem has been faced in a large number of cases including the analysis of electronic and optoelectronic devices [1-6], the investigation of fluid properties and the simulation of ion transport through membrane pores [7-12]. In nano-scale physical systems, some of the properties of the interacting bodies are strongly localized and can be approximated as Dirac delta functions. For example, the properties of spatially homogeneous ionic solutions have been investigated using the so called primitive model for the ions, considering them as hard spheres with finite radii and discrete point charges placed at the center of the spheres. On the other hand, in relatively large physical systems, the charge can be described by a continuous function of the spatial coordinates representing volume (or surface) charge density. The electrostatic problem requires the solution of Poisson’s differential equation and boundary conditions. The numerical solution of Poisson’s equation requires the discretization of the partial differential equation into a system of
algebraic equations on a discretization grid of the simulation domain. In the case of nano-scale systems, the discrete nature of charges cannot be neglected. If the primitive model is adopted for the charges, the electrostatic interactions among them can be computed from Coulomb’s law. If the system of interest includes different dielectric regions (or phases) characterized by different values of the permittivity and separated by abrupt boundaries, Coulomb’s law is not enough. The charges on the boundary are not discrete. Indeed, every charge can interact significantly with every other charge through the boundary condition and a two-body treatment of electric forces (like Coulomb’s law) is incomplete, and in that sense incorrect. In the boundary element method (BEM), the polarization effects associated with the discontinuity of permittivity at boundaries is accounted for by adding to the system t
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