# Efficient Algorithms for Electrostatic Interactions Including Dielectric Contrasts

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Planar Interfaces: Image Charges

**Figure 1.**Schematic summation scheme for Image Charge MMM2D (ICMMM2D): in order to take into account dielectric boundaries, image charges are introduced outside the dielectric boundaries, to the left and right of the original box. The dielectric contrasts, ${\Delta}_{l}$ and ${\Delta}_{r}$, are computed from the dielectric jump at the left and right boundaries, respectively. Depending on the dielectric contrasts, charges will either be repelled by the surface (as in this sketch) or attracted to it. Note that in usual computer simulations, the system is periodically replicated in the dimensions parallel to the interfaces, as indicated in the figure.

**Figure 2.**(

**a**) Sketch of our simulation setup, a 3:1 electrolyte, e.g., AlCl${}_{3}$, between two walls with a dielectric constant different from that of water. The size difference between the ion types is neglected in this simulation. The slab is periodically replicated parallel to the walls, vertical in the sketch. (

**b**) Density distribution of anions and cations of the trivalent electrolyte, near the dielectric interface. The dielectric interfaces is placed at $x=0$, and a repulsive potential maintains a minimum distance of 0.5 nanometers for all ions. A good dielectric ${\epsilon}_{C}=800$, representing conducting material, strongly attracts cations, while anions are less attracted. A bad dielectric ${\epsilon}_{C}=2$, representing a typical biological membrane, repels cations. The univalent anions are much less repelled.

#### 2.1. Example: Electrolyte between Dielectric Walls

## 3. Arbitrarily-Shaped Interfaces: Induced Charges

#### 3.1. Example: Ion Distribution in a Pore

**Figure 3.**(

**a**) The induced charge calculation (ICC*) example system: positive and negative ions are displayed as red and blue spheres, the ICC* discretization points by grey spheres; (

**b**) ion density of both species in the pore measured close to the center of the pore.

## 4. Metallic Interfaces: Corrections

**Figure 4.**(

**a**) Illustration of the dielectric boundary problem of single charge q outside of a grounded conducting sphere. The problem can be solved by assuming an image charge, ${q}^{\prime}$, inside the sphere, leading to zero potential Φ on its surface. If the sphere is assumed to be conducting, but isolated, the excess charge, ${q}^{\prime}$, has to be canceled by adding a second charge, ${q}^{\u2033}$, in the center of the sphere, which leads to a constant surface potential, Φ. (adapted from [25]). (

**b**) A more complex geometry with an upper and lower electrode (yellow). The electrodes are treated with the ICC* algorithm. If a Coulomb solver with periodic boundary conditions (BCs) in the vertical direction is applied, the potential difference between both electrodes is automatically zero. This is because the periodicity yields zero potential difference between an electrode and its periodic image, and the ICC* algorithm ensures that the two electrodes connected over periodic BCs are on equal potential.

#### 4.1. Verification: Field and Potential in Metallic BCs

**Figure 5.**(

**a**) Sketch of the model system that was used to probe the influence of grounded and isolated metallic boundary conditions. The two possible setups are depicted by adding a switch to electrically connect the two plates. (

**b**) The resulting electrostatic field, E, perpendicular to the plates and the electrostatic potential in the slab.

## 5. Smooth Variations: Local Method

**Figure 6.**Maxwell Equations Molecular Dynamics (MEMD) interpolation of the charges onto the lattice. (

**a**) The electric current, $\mathit{j}$, permittivity ε and electric field $\mathit{D}$ are interpolated to the adjacent lattice links. The magnetic field component, $\mathit{B}$, is placed on the lattice plaquettes via a finite-differences curl $(\nabla \times )$ operator. (

**b**) The numerical error of the algorithm is dependent on the lattice spacing. The error can be reduced by applying a coarser mesh, coming from the right in this graph, and, thereby, increasing the field propagation speed in the system. However, at large lattice spacings a, the interpolation error at small distances dominates and diverges. In a densely-populated system, like the examples seen here, a minimal relative error of ${10}^{-3}$ is achieved at mesh sizes comparable to the minimum distance of two charges, denoted here by σ. For reference, the errors compared to a high precision P3M force calculation are included for three example systems.

#### 5.1. Example: Colloid with Dielectric Jump and Continuous Dielectric Constant

**Figure 7.**(

**a**) A charged colloid (charge $Z=60e$, radius $R=12.75\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$) is suspended in a salt solution (concentration $c=50\phantom{\rule{0.166667em}{0ex}}\mathrm{mmol}/\text{L}$). The dielectric constant is modeled as an abrupt jump to the bulk water permittivity with the MEMD and ICCP${}^{3}$M algorithms (gray points, green curve) or a linear radial increase within two ion diameters from the colloid surface between the two regimes (red). (

**b**) The resulting radial charge density profiles, which exhibit a drastic difference between the dielectric jump in model 1 and a smooth interpolation in model 2.

## 6. Conclusion

## Acknowledgments

## Conflicts of Interest

## References

- Ewald, P.P. Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys.
**1921**, 369, 253–287. [Google Scholar] [CrossRef] - Heyes, D.M. Electrostatic potentials and fields in infinite point charge lattices. J. Chem. Phys.
**1981**, 74, 1924–1929. [Google Scholar] [CrossRef] - De Leeuw, S.W.; Perram, J.W.; Smith, E.R. Simulation of electrostatic systems in periodic boundary conditions. I. Lattice sums and dielectric constants. Proc. R. Soc. Lond. A
**1980**, 373, 27–56. [Google Scholar] [CrossRef] - De Leeuw, S.W.; Perram, J.W.; Smith, E.R. Simulation of electrostatic systems in periodic boundary conditions. II. Equivalence of boundary conditions. Proc. R. Soc. Lond. A
**1980**, 373, 57–66. [Google Scholar] [CrossRef] - Hockney, R.W.; Eastwood, J.W. Computer Simulation Using Particles; IOP: London, UK, 1988. [Google Scholar]
- Darden, T.; York, D.; Pedersen, L. Particle Mesh Ewald: An N log(N) method for Ewald sums in large systems. J. Chem. Phys.
**1993**, 98, 10089–10092. [Google Scholar] [CrossRef] - Essmann, U.; Perera, L.; Berkowitz, M.L.; Darden, T.; Lee, H.; Pedersen, L. A smooth Particle Mesh Ewald method. J. Chem. Phys.
**1995**, 103, 8577–8593. [Google Scholar] [CrossRef] - Deserno, M.; Holm, C. How to mesh up Ewald sums. I. A theoretical and numerical comparison of various particle mesh routines. J. Chem. Phys.
**1998**, 109, 7678–7693. [Google Scholar] [CrossRef] - Widmann, A.H.; Adolf, D.B. A comparison of Ewald summation techniques for planar surfaces. Comput. Phys. Commun.
**1997**, 107, 167–186. [Google Scholar] [CrossRef] - Yeh, I.C.; Berkowitz, M.L. Ewald summation for systems with slab geometry. J. Chem. Phys.
**1999**, 111, 3155–3162. [Google Scholar] [CrossRef] - Arnold, A.; de Joannis, J.; Holm, C. Electrostatics in periodic slab geometries I. J. Chem. Phys.
**2002**, 117, 2496–2502. [Google Scholar] [CrossRef] - De Joannis, J.; Arnold, A.; Holm, C. Electrostatics in periodic slab geometries II. J. Chem. Phys.
**2002**, 117, 2503–2512. [Google Scholar] [CrossRef] - Levrel, L.; Maggs, A.C. Boundary conditions in local electrostatics algorithms. J. Chem. Phys.
**2008**, 128, 214103. [Google Scholar] [CrossRef] [PubMed] - Thompson, D.; Rottler, J. Local monte carlo for electrostatics in anisotropic and nonperiodic geometries. J. Chem. Phys.
**2008**, 128, 214102. [Google Scholar] [CrossRef] [PubMed] - Smith, E.R. Electrostatic potentials for thin layers. Mol. Phys.
**1988**, 65, 1089–1104. [Google Scholar] [CrossRef] - Tyagi, S.; Arnold, A.; Holm, C. ICMMM2D: An accurate method to include planar dielectric interfaces via image charge summation. J. Chem. Phys.
**2007**, 127, 154723. [Google Scholar] [CrossRef] [PubMed] - Tyagi, S.; Arnold, A.; Holm, C. Electrostatic layer correction with image charges: A linear scaling method to treat slab 2D + h systems with dielectric interfaces. J. Chem. Phys.
**2008**, 129, 204102. [Google Scholar] [CrossRef] [PubMed] - Tyagi, C.; Süzen, M.; Sega, M.; Barbosa, M.; Kantorovich, S.; Holm, C. An iterative, fast, linear-scaling method for computing induced charges on arbitrary dielectric boundaries. J. Chem. Phys.
**2010**, 132, 154112. [Google Scholar] [CrossRef] [PubMed] - Kesselheim, S.; Sega, M.; Holm, C. Applying ICC* to DNA translocation. Effect of dielectric boundaries. Comput. Phys. Commun.
**2011**, 182, 33–35. [Google Scholar] [CrossRef] - Maggs, A.C.; Rosseto, V. Local simulation algorithms for coulombic interactions. Phys. Rev. Lett.
**2002**, 88, 196402. [Google Scholar] [CrossRef] [PubMed] - Pasichnyk, I.; Dünweg, B. Coulomb interactions via local dynamics: A molecular-dynamics algorithm. J. Phys. Condens. Matter
**2004**, 16, 3999–4020. [Google Scholar] [CrossRef] - Arnold, A.; Lenz, O.; Kesselheim, S.; Weeber, R.; Fahrenberger, F.; Roehm, D.; Košovan, P.; Holm, C. ESPResSo 3.1—Molecular Dynamics Software for Coarse-Grained Models. In Meshfree Methods for Partial Differential Equations VI; Griebel, M., Schweitzer, M.A., Eds.; Springer: Berlin, Germany, 2013; Volume 89, Lecture Notes in Computational Science and Engineering; pp. 1–23. [Google Scholar]
- Limbach, H.J.; Arnold, A.; Mann, B.A.; Holm, C. ESPResSo—An extensible simulation package for research on soft matter systems. Comput. Phys. Commun.
**2006**, 174, 704–727. [Google Scholar] [CrossRef] - Arnold, A.; Holm, C. Efficient Methods to Compute Long Range Interactions for Soft Matter Systems. In Advanced Computer Simulation Approaches for Soft Matter Sciences II; Holm, C., Kremer, K., Eds.; Springer: Berlin, Germany, 2005; Volume II, Advances in Polymer Sciences; pp. 59–109. [Google Scholar]
- Jackson, J.D. Classical Electrodynamics, 3rd ed.; Wiley: New York, NY, USA, 1999. [Google Scholar]
- Arnold, A.; Holm, C. MMM2D: A fast and accurate summation method for electrostatic interactions in 2D slab geometries. Comput. Phys. Commun.
**2002**, 148, 327–348. [Google Scholar] [CrossRef] - Arnold, A.; Holm, C. A novel method for calculating electrostatic interactions in 2D periodic slab geometries. Chem. Phys. Lett.
**2002**, 354, 324–330. [Google Scholar] [CrossRef] - Ballenegger, V.; Arnold, A.; Cerda, J.J. Simulations of non-neutral slab systems with long-range electrostatic interactions in two-dimensional periodic boundary conditions. J. Chem. Phys.
**2009**, 131, 094107. [Google Scholar] [CrossRef] [PubMed] - Katsikadelis, J.T. Boundary Elements: Theory and Applications: Theory and Applications; Elsevier: Oxford, UK, 2002. [Google Scholar]
- Kesselheim, S.; Sega, M.; Holm, C. Effects of dielectric mismatch and chain flexibility on the translocation barriers of charged macromolecules through solid state nanopores. Soft Matter
**2012**, 8, 9480–9486. [Google Scholar] [CrossRef] - Dekker, N.H.; Smeets, R.M.M.; Keyser, U.F.; Krapf, D.; Wu, M.Y.; Dekker, C. Salt dependence of ion transport and DNA translocation through solid-state nanopores. Nano Lett.
**2006**, 6, 89–95. [Google Scholar] - Dekker, C. Solid-state nanopores. Nat. Nanotechnol.
**2007**, 2, 209–215. [Google Scholar] [CrossRef] [PubMed] - Siwy, Z.; Kosinska, I.D.; Fluinski, A.; Martin, C.R. Asymmetric diffusion through synthetic nanopores. Phys. Rev. Lett.
**2005**, 94, 048102. [Google Scholar] [CrossRef] [PubMed] - Keyser, U.; van der Does, J.; Dekker, C.; Dekker, N. Optical tweezers for force measurements on DNA in nanopores. Rev. Sci. Instrum.
**2006**, 77, 105105. [Google Scholar] [CrossRef] - Wanunu, M.; Morrison, W.; Rabin, Y.; Grosberg, A.; Meller, A. Electrostatic focusing of unlabelled DNA into nanoscale pores using a salt gradient. Nat. Nanotechnol.
**2009**, 5, 160–165. [Google Scholar] [CrossRef] [PubMed] - Bastian, P.; Blatt, M.; Dedner, A.; Engwer, C.; Klöfkorn, R.; Kuttanikkad, S.; Ohlberger, M.; Sander, O. The Distributed and Unified Numerics Environment (DUNE). In Proceedings of the 19th Symposium on Simulation Technique, Hannover, Germany, 12–14 September 2006.
- Bastian, P.; Heimann, F.; Marnach, S. Generic implementation of finite element methods in the Distributed and Unified Numerics Environment (DUNE). Kybernetika
**2010**, 2, 294–315. [Google Scholar] - Deserno, M.; Holm, C. How to mesh up Ewald sums. II. An accurate error estimate for the Particle-Particle-Particle-Mesh algorithm. J. Chem. Phys.
**1998**, 109, 7694. [Google Scholar] [CrossRef] - Merlet, C.; Salanne, M.; Rotenberg, B. New coarse-grained models of imidazolium ionic liquids for bulk and interfacial molecular simulations. J. Phys. Chem. C
**2012**, 116, 7687–7693. [Google Scholar] [CrossRef] - Kondrat, S.; Kornyshev, A. Superionic state in double-layer capacitors with nanoporous electrodes. J. Phys. Condens. Matter
**2011**, 23, 022201. [Google Scholar] [CrossRef] [PubMed] - Feng, G.; Zhang, J.; Qiao, R. Microstructure and capacitance of the electrical double layers at the interface of ionic liquids and planar electrodes. J. Phys. Chem. C
**2009**, 113, 4549–4559. [Google Scholar] [CrossRef] - Maggs, A.C. Auxilary field Monte Carlo for charged particles. J. Chem. Phys.
**2004**, 120, 3108–3118. [Google Scholar] [CrossRef] [PubMed] - Rottler, J.; Maggs, A.C. Local molecular dynamics with Coulombic interactions. Phys. Rev. Lett.
**2004**, 93, 170201. [Google Scholar] [CrossRef] [PubMed] - Fahrenberger, F.; Holm, C. Computing Coulomb Interaction in Inhomogeneous Dielectric Media via a Local Electrostatics Lattice Algorithm. 2013. Available online: http://arxiv.org/abs/1309.7859 (accessed on 14 October 2013).
- Pasichnyk, I.; Everaers, R.; Maggs, A.C. Simulating van der Waals interactions in water/hydrocarbon-based complex fluids. J. Phys. Chem. B
**2008**, 112, 1761–1764. [Google Scholar] [CrossRef] [PubMed] - Bonthuis, D.J.; Gekle, S.; Netz, R.R. Dielectric profile of interfacial water and its effect on double-layer capacitance. Phys. Rev. Lett.
**2011**, 107, 166102. [Google Scholar] [CrossRef] [PubMed] - Drift, W.P.J.T.V.D.; Keizer, A.D.; Overbeek, J.T.G. Electrophoretic mobility of a cylinder with high surface charge density. J. Colloid Interface Sci.
**1979**, 71, 67–78. [Google Scholar] [CrossRef] - Bonthuis, D.J.; Gekle, S.; Netz, R.R. Profile of the static permittivity tensor of water at interfaces: Consequences for capacitance, hydration interaction and ion adsorption. Langmuir
**2012**, 28, 7679–7694. [Google Scholar] [CrossRef] [PubMed] - Arnold, A.; Bolten, M.; Dachsel, H.; Fahrenberger, F.; Gähler, F.; Halver, R.; Heber, F.; Hofmann, M.; Holm, C.; Iseringhausen, J.; et al. Comparison of scalable fast methods for long-range interactions. Phys. Rev. E
**2013**. submitted. [Google Scholar] [CrossRef] [PubMed]

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Arnold, A.; Breitsprecher, K.; Fahrenberger, F.; Kesselheim, S.; Lenz, O.; Holm, C.
Efficient Algorithms for Electrostatic Interactions Including Dielectric Contrasts. *Entropy* **2013**, *15*, 4569-4588.
https://doi.org/10.3390/e15114569

**AMA Style**

Arnold A, Breitsprecher K, Fahrenberger F, Kesselheim S, Lenz O, Holm C.
Efficient Algorithms for Electrostatic Interactions Including Dielectric Contrasts. *Entropy*. 2013; 15(11):4569-4588.
https://doi.org/10.3390/e15114569

**Chicago/Turabian Style**

Arnold, Axel, Konrad Breitsprecher, Florian Fahrenberger, Stefan Kesselheim, Olaf Lenz, and Christian Holm.
2013. "Efficient Algorithms for Electrostatic Interactions Including Dielectric Contrasts" *Entropy* 15, no. 11: 4569-4588.
https://doi.org/10.3390/e15114569