^{1,a)}, R. A. Curtis

^{1}and L. Lue

^{1,b)}

### Abstract

The interaction between two dielectric plates immersed in an electrolyte solution is examined by using a variational perturbation approximation for the grand partition function. This approach differs from previous treatments in that the screening length between the plates is treated as a variational parameter. A key finding is that adjacent to each plate is a layer of ion depletion with thickness given by about one-half of a Bjerrum length. Consequently, for plate-plate separations less than the Bjerrum length, nearly all the electrolyte is excluded from between the plates, and the interaction is given by the sum of a van der Waals interaction and an attractive osmotic depletion force. In contrast to the predictions of previous theories, the interaction between the plates at short range increases with increasing electrolyte concentration and may provide an important contribution to the salt-induced attraction, commonly referred to as salting out. Because the range of the osmotic depletion force is roughly equal to the Bjerrum length, it increases with the square of the valency of the electrolyte. At larger plate-plate separations, the van der Waals interaction is screened as electrolyte enters the space between the plates, leading to an exponential decay of the interactions, as has been previously observed. However, this interaction is slightly stronger than that previously predicted, due to ion depletion from the surface of the interface, also this effect increases with increasing electrolyte concentration.

M.M.H. acknowledges support from an EC Marie Curie Fellowship No. (MEST-CT-2004-503750).

I. INTRODUCTION

II. THEORY

III. SINGLE-PLATE GEOMETRY

A. Uniform screening

B. Incorporation of a depletion layer

C. Comparison with simulation

IV. TWO-PLATE GEOMETRY

V. CONCLUSIONS

### Key Topics

- Electrolytes
- 83.0
- Dielectrics
- 38.0
- Van der Waals forces
- 22.0
- Electrostatics
- 18.0
- Dielectric constant
- 16.0

## Figures

The density as a function of position for and (i) (solid line) and (ii) (dotted line) The thin lines are for , and the thick lines give the result when and are optimized.

The density as a function of position for and (i) (solid line) and (ii) (dotted line) The thin lines are for , and the thick lines give the result when and are optimized.

The surface tension as predicted by the variational theory: (i) (solid line), (ii) (dashed line), and (iii) (dotted line). The thin lines give the result when , and the thick lines are from the full variational theory.

The surface tension as predicted by the variational theory: (i) (solid line), (ii) (dashed line), and (iii) (dotted line). The thin lines give the result when , and the thick lines are from the full variational theory.

The variation of the (a) thickness and (b) inverse screening length of the depletion layer with the reduced electrolyte concentration for (i) (solid line), (ii) (dashed line), and (iii) (dotted line). The inset is for the case where .

The variation of the (a) thickness and (b) inverse screening length of the depletion layer with the reduced electrolyte concentration for (i) (solid line), (ii) (dashed line), and (iii) (dotted line). The inset is for the case where .

Comparison between this work (lines) and simulation data by Henderson *et al.* (Ref. 34) (symbols). Note that and . (i) , , and (solid line and squares), (ii) , , and (dotted line and circles), and (iii) , , and (dashed line and diamonds).

Comparison between this work (lines) and simulation data by Henderson *et al.* (Ref. 34) (symbols). Note that and . (i) , , and (solid line and squares), (ii) , , and (dotted line and circles), and (iii) , , and (dashed line and diamonds).

The density in the middle of the plates as a function of plate separation for (i) , (solid line), (ii) , (dotted line), (iii) , (dashed line), and (iv) , (dashed-dotted line). The thin lines are for , and the thick lines give the result when and are optimized.

The density in the middle of the plates as a function of plate separation for (i) , (solid line), (ii) , (dotted line), (iii) , (dashed line), and (iv) , (dashed-dotted line). The thin lines are for , and the thick lines give the result when and are optimized.

The variation of the (a) thickness and the (b) inverse screening length of the depletion layer as a function of plate separation and (i) (solid line) and (ii) (dotted line). The thin lines are for , and the thick lines give the result when .

The variation of the (a) thickness and the (b) inverse screening length of the depletion layer as a function of plate separation and (i) (solid line) and (ii) (dotted line). The thin lines are for , and the thick lines give the result when .

The pressure as a function of plate separation for: (i) , (solid line), (ii) , (dotted line), and (iii) , (dashed line). The thin lines are for , and the thick lines give the result when and are optimized.

The pressure as a function of plate separation for: (i) , (solid line), (ii) , (dotted line), and (iii) , (dashed line). The thin lines are for , and the thick lines give the result when and are optimized.

The difference between the pressure and the screened van der Waals pressure as a function of plate separation for: (i) (solid line), (ii) (dashed line), and (iii) (dotted line). The inset gives the corresponding screened van der Waals pressure.

The difference between the pressure and the screened van der Waals pressure as a function of plate separation for: (i) (solid line), (ii) (dashed line), and (iii) (dotted line). The inset gives the corresponding screened van der Waals pressure.

The variation of the pressure as a function of bulk screening length at constant plate-plate separation for . (i) (solid line) and (ii) (dotted line). The thin lines are for , and the thick lines give the result when and are optimized.

The variation of the pressure as a function of bulk screening length at constant plate-plate separation for . (i) (solid line) and (ii) (dotted line). The thin lines are for , and the thick lines give the result when and are optimized.

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