^{1,2}, M. W. Kim

^{1,2}, P. A. Pincus

^{1,2,3}and Frank L. H. Brown

^{3,4,a)}

### Abstract

A numerical algorithm based on the image charge method is introduced to calculate the electrostatic potential, energy, and forces present in systems involving multiple point charges embedded in an inhomogeneous dielectric environment composed of five parallel dielectric slabs. The methodology is implemented within Monte Carlo simulations to calculate the thermal properties of two charged dielectric plates of finite thickness immersed in water.

This research was supported by the National Science Foundation (Grant Nos. DMR-0520415, DMR-0503347, DMR-0710521, CHE-0349196, and CHE-0321368). F.L.H.B. is an Alfred P. Sloan Research Fellow and a Camille Dreyfus Teacher-Scholar. Y. S. Jho and M. W. Kim have been supported by Korean KISTEP under Grant No. I-03-064 and a grant of the Korea Health 21 R&D Project.

I. INTRODUCTION

II. COMPUTING THE ELECTROSTATIC POTENTIAL FOR A TWO SLAB GEOMETRY

III. NUMERICAL EVALUATION OF THE POTENTIAL

IV. RESULTS

V. DISCUSSION AND CONCLUSION

### Key Topics

- Dielectrics
- 41.0
- Surface charge
- 22.0
- Electrostatics
- 21.0
- Double layers
- 19.0
- Dielectric constant
- 17.0

## Figures

Schematic illustration of the systems modeled in this work. Two parallel slabs with dielectric constant are immersed in a bulk medium with dielectric constant . The slabs have thickness and their proximal faces are separated by . Static surface charges are arranged on a square lattice lying immediately below the dielectric interface between each slab and the sandwiched water region. These charges are neutralized by counterions that are freely diffusing in the central water region. Periodic boundary conditions in the plane of the slabs are assumed. The various length scales used in the figure are defined in Table I. Note that interface B coincides with the plane .

Schematic illustration of the systems modeled in this work. Two parallel slabs with dielectric constant are immersed in a bulk medium with dielectric constant . The slabs have thickness and their proximal faces are separated by . Static surface charges are arranged on a square lattice lying immediately below the dielectric interface between each slab and the sandwiched water region. These charges are neutralized by counterions that are freely diffusing in the central water region. Periodic boundary conditions in the plane of the slabs are assumed. The various length scales used in the figure are defined in Table I. Note that interface B coincides with the plane .

Counterion density at midplane between the two dielectric slabs is plotted as a function of the width of dielectric slabs . More counterions are driven to the center for narrow slabs. The dashed line represents the expected density for an ideal gas, i.e., uniform distribution of all counterions within region 3. The case corresponds to a system without any dielectric contrast between the slabs and bulk solution. The geometry of the system and physical parameters for the simulation are as follows: , , , , , and .

Counterion density at midplane between the two dielectric slabs is plotted as a function of the width of dielectric slabs . More counterions are driven to the center for narrow slabs. The dashed line represents the expected density for an ideal gas, i.e., uniform distribution of all counterions within region 3. The case corresponds to a system without any dielectric contrast between the slabs and bulk solution. The geometry of the system and physical parameters for the simulation are as follows: , , , , , and .

Pressure between the two slabs as a function of the slab width. Physical parameters are identical to those in Fig. 2. The presented error bars in this figure and subsequent figures represent standard deviations over several Monte Carlo runs; i.e., these error bars reflect finite sampling of the thermal ensemble and do not indicate any inherent errors associated with the image charge methodology, which is, for all practical purposes, exact.

Pressure between the two slabs as a function of the slab width. Physical parameters are identical to those in Fig. 2. The presented error bars in this figure and subsequent figures represent standard deviations over several Monte Carlo runs; i.e., these error bars reflect finite sampling of the thermal ensemble and do not indicate any inherent errors associated with the image charge methodology, which is, for all practical purposes, exact.

Counterion density at midplane between the two dielectric slabs as a function of the slab dielectric constant. Two different values are considered, as indicated. The remaining physical parameters are identical to those in Fig. 2, except that and .

Counterion density at midplane between the two dielectric slabs as a function of the slab dielectric constant. Two different values are considered, as indicated. The remaining physical parameters are identical to those in Fig. 2, except that and .

Pressure between two slabs as a function of the slab dielectric constant. Physical parameters are identical to those in Fig. 4.

Pressure between two slabs as a function of the slab dielectric constant. Physical parameters are identical to those in Fig. 4.

The density of counterions as a function of the position for three different dielectric constants of the slabs . Physical parameters are otherwise identical to those in Fig. 4.

The density of counterions as a function of the position for three different dielectric constants of the slabs . Physical parameters are otherwise identical to those in Fig. 4.

Counterion density at midplane between the two dielectric slabs is plotted as a function of the interslab separation . Physical parameters are same as in Fig. 2, except that .

Counterion density at midplane between the two dielectric slabs is plotted as a function of the interslab separation . Physical parameters are same as in Fig. 2, except that .

The density of counterions as a function of position for the system studied in Fig. 7, assuming the two values of indicated. The inset overlays the regions for both cases in the immediate vicinity of the interface and further compares to simple exponential decay.

The density of counterions as a function of position for the system studied in Fig. 7, assuming the two values of indicated. The inset overlays the regions for both cases in the immediate vicinity of the interface and further compares to simple exponential decay.

Pressure between two slabs as a function of the interslab separation for the system in Fig. 7.

Pressure between two slabs as a function of the interslab separation for the system in Fig. 7.

## Tables

Notations used in the paper.

Notations used in the paper.

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