^{1,a)}, Simon Petris

^{1}and Derek Y. C. Chan

^{1}

### Abstract

The force between two parallel charged flat surfaces, with discrete surfacecharges, has been calculated with Monte Carlo simulations for different values of the electrostatic coupling. For low electrostatic coupling (small counterion valence, small surface charge, high dielectric constant, and high temperature) the total force is dominated by the entropic contribution and can be described by mean field theory, independent of the character of the surfacecharges. For moderate electrostatic coupling, counterion correlation effects lead to a smaller repulsion than predicted by mean field theory. This correlation effect is strengthened by discrete surfacecharges and the repulsive force is further reduced. For large electrostatic coupling the total force for smeared out surfacecharges is known to be attractive due to counterion correlations. If discrete surfacecharges are considered the attractive force is weakened and can even be turned into a repulsive force. This is due to the counterions being strongly correlated to the discrete surfacecharges forming effective, oppositely directed, dipoles on the two walls.

This work was supported by the Australian Research Council, the Particulate Fluids Processing Centre at The University of Melbourne, the Victorian Partnership for Advanced Computing, and the Wenner–Gren Foundations.

I. INTRODUCTION

II. THE MODEL AND NUMERICAL METHOD

A. The model for two charged walls with counterions

B. The Monte Carlo method

C. Calculating the pressure

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Surface charge
- 41.0
- Electrostatics
- 38.0
- Double layers
- 35.0
- Mean field theory
- 22.0
- Charge coupled devices
- 20.0

## Figures

In the model used, negative wall charges are located a distance behind the walls. In the plane, the wall charges are distributed on a square lattice with lattice constant . The walls are infinite in the plane and separated a distance in the direction.

In the model used, negative wall charges are located a distance behind the walls. In the plane, the wall charges are distributed on a square lattice with lattice constant . The walls are infinite in the plane and separated a distance in the direction.

The pressure as a function of the distance between two charged walls with monovalent counterions . The lines are the results from MC simulations with a low degree of charge discretization while the symbols are results from corresponding simulations with a high degree of charge discretization . The different coupling strengths are (dashed line and diamonds), (dotted line and squares) and (solid line and circles). The standard deviations are smaller than the symbol sizes.

The pressure as a function of the distance between two charged walls with monovalent counterions . The lines are the results from MC simulations with a low degree of charge discretization while the symbols are results from corresponding simulations with a high degree of charge discretization . The different coupling strengths are (dashed line and diamonds), (dotted line and squares) and (solid line and circles). The standard deviations are smaller than the symbol sizes.

The electrostatic correlation and entropic components of the total pressure for different coupling strengths (a) , (b) , and (c) . Results corresponding to a high degree of charge discretization are shown in symbols and those that correspond to a low degree of charge discretization are shown in lines. The electrostatic correlation components are given in squares and dashed lines, the entropic components are given in diamonds and dotted lines and the total pressures are given in circles and solid lines. The standard deviations are smaller than the symbol sizes (even for the insets).

The electrostatic correlation and entropic components of the total pressure for different coupling strengths (a) , (b) , and (c) . Results corresponding to a high degree of charge discretization are shown in symbols and those that correspond to a low degree of charge discretization are shown in lines. The electrostatic correlation components are given in squares and dashed lines, the entropic components are given in diamonds and dotted lines and the total pressures are given in circles and solid lines. The standard deviations are smaller than the symbol sizes (even for the insets).

The pressure between two charged walls as a function of the electrostatic coupling for , , , and . (a) The lines are the results from MC simulations where the discrete wall charges are allowed to move while the symbols are results from the corresponding simulations with fixed discrete wall charges. The degrees of charge discretization are (dashed line and squares) and (solid line and circles). The inset shows corresponding results for small . (b) The components of the pressure for the two cases with moving wall ions, (lines without symbols) and (lines with symbols). The solid lines shows the total pressure, the dotted lines the electrostatic correlation pressure, and the dashed lines the entropic pressure. The standard deviations are smaller than the symbol sizes.

The pressure between two charged walls as a function of the electrostatic coupling for , , , and . (a) The lines are the results from MC simulations where the discrete wall charges are allowed to move while the symbols are results from the corresponding simulations with fixed discrete wall charges. The degrees of charge discretization are (dashed line and squares) and (solid line and circles). The inset shows corresponding results for small . (b) The components of the pressure for the two cases with moving wall ions, (lines without symbols) and (lines with symbols). The solid lines shows the total pressure, the dotted lines the electrostatic correlation pressure, and the dashed lines the entropic pressure. The standard deviations are smaller than the symbol sizes.

Distribution functions for the counterions. (a) The distribution of counterions in the direction, perpendicular to the walls, for (solid line), (dotted line), and (dashed line). and . The left inset provides details close to a wall and the right inset provides details around the midplane. (b) The distribution of counterions in the plane, parallel to the walls, for (solid line), (dashed line), and (dotted line). and . Also shown is the distribution for , , and (circles). Only part of the wall in the minimal image box is shown.

Distribution functions for the counterions. (a) The distribution of counterions in the direction, perpendicular to the walls, for (solid line), (dotted line), and (dashed line). and . The left inset provides details close to a wall and the right inset provides details around the midplane. (b) The distribution of counterions in the plane, parallel to the walls, for (solid line), (dashed line), and (dotted line). and . Also shown is the distribution for , , and (circles). Only part of the wall in the minimal image box is shown.

The pressure between two charged walls as a function of the wall-wall distance for different values of . The dashed lines are the results from MC simulations where while the solid lines with symbols are results from the corresponding simulations with . The following values have been used, (circles), 2:1 (squares), 1:2 (diamonds), and 2:2 (triangles). The dashed lines are not individually labeled since they are similar. The simulations are performed for .

The pressure between two charged walls as a function of the wall-wall distance for different values of . The dashed lines are the results from MC simulations where while the solid lines with symbols are results from the corresponding simulations with . The following values have been used, (circles), 2:1 (squares), 1:2 (diamonds), and 2:2 (triangles). The dashed lines are not individually labeled since they are similar. The simulations are performed for .

A schematic overview of the effect of going from a low degree of charge discretization (or smeared out wall charges) to a high degree of charge discretization.

A schematic overview of the effect of going from a low degree of charge discretization (or smeared out wall charges) to a high degree of charge discretization.

## Tables

The entropic component , electrostatic component , and the total pressure for , , and at different degrees of charge discretization and electrostatic coupling . Also shown is the standard deviation for the total pressure and the change in pressure when going from a low degree of charge discretization to a high degree of charge discretization, .

The entropic component , electrostatic component , and the total pressure for , , and at different degrees of charge discretization and electrostatic coupling . Also shown is the standard deviation for the total pressure and the change in pressure when going from a low degree of charge discretization to a high degree of charge discretization, .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content