^{1}, David S. Dean

^{2}and Rudolf Podgornik

^{3}

### Abstract

We investigate the statistical mechanics of an inhomogeneous Coulomb fluid composed of charged particles with static polarizability. We derive the weak- and the strong-coupling approximations and evaluate the partition function in a planar dielectric slab geometry with charged boundaries. We investigate the density profiles and the disjoining pressure for both approximations. Comparison to the case of non-polarizable counterions shows that polarizability brings important differences in the counterion density distribution as well as the counterion mediated electrostatic interactions between charged dielectric interfaces.

V.D. and D.S.D. would like to thank R. R. Horgan for a discussion about the divergences appearing in the fluctuations about the mean-field. D.S.D. acknowledges support from the Institut Universitaire de France. R.P. acknowledges support of The Leverhulme Trust and of ARRS through research program P1-0055 and research project J1-0908.

I. INTRODUCTION

II. MODEL

A. Field-action

B. Dimensionless field-action

C. Density and electroneutrality

D. Grand potential, free energy, and pressure

III. WEAK COUPLING APPROXIMATION

A. Mean-field

B. Pressure in the plane-parallel geometry

C. Second order fluctuations correction

D. Pressure

E. Density

IV. STRONG COUPLING

A. Formulation

B. Density

C. Pressure

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Mean field theory
- 62.0
- Polarizability
- 49.0
- Dielectrics
- 21.0
- Free energy
- 14.0
- Electrostatics
- 12.0

##### B01J13/00

## Figures

Polarizable counter-ions of excess polarizability α between two charged plates. The solvent in between has a permittivity ε_{w}, while the two semi-infinite regions 0 > *z* > *L* have permittivity ε_{ext}. The two surfaces at *z* = 0, *L* bear a surface charge of surface charge density σ. *R* is the radius of the ions, or more precisely of their cavity.

Polarizable counter-ions of excess polarizability α between two charged plates. The solvent in between has a permittivity ε_{w}, while the two semi-infinite regions 0 > *z* > *L* have permittivity ε_{ext}. The two surfaces at *z* = 0, *L* bear a surface charge of surface charge density σ. *R* is the radius of the ions, or more precisely of their cavity.

Pressure *P* ^{(1)}(*L*) from Eq. (34) as a function of the plate separation *L* for ε_{ext} = 0.05, Ξ = 1, and *R* = 1. The dashed lines are the mean field result, the solid lines include the fluctuations. (Left) For α = −0.1 and large plate separation, the difference is barely distinguishable. (Right) For small plate separation *L* and various values of the polarizability α. The effect of fluctuations can be quite important for large counterion polarizabilities.

Pressure *P* ^{(1)}(*L*) from Eq. (34) as a function of the plate separation *L* for ε_{ext} = 0.05, Ξ = 1, and *R* = 1. The dashed lines are the mean field result, the solid lines include the fluctuations. (Left) For α = −0.1 and large plate separation, the difference is barely distinguishable. (Right) For small plate separation *L* and various values of the polarizability α. The effect of fluctuations can be quite important for large counterion polarizabilities.

Counterions density profile within the slab—dependence on the coupling constant Ξ. (Left) Weak coupling density close to the left electrode taking into account the fluctuations around the mean field as a function of the position within the slab (), for *R* = 0.3, α = −0.3, ε_{ext} = 0.05, and Ξ = 0.3 (dashed line) or Ξ = 1 (solid line). The mean-field density itself is presented by the dotted line. (Right) Strong coupling density as a function of the position for *R* = 2, α = −0.01, ε_{ext} = 0.05, and Ξ = 10 (solid line) or Ξ = 50 (dashed line).

Counterions density profile within the slab—dependence on the coupling constant Ξ. (Left) Weak coupling density close to the left electrode taking into account the fluctuations around the mean field as a function of the position within the slab (), for *R* = 0.3, α = −0.3, ε_{ext} = 0.05, and Ξ = 0.3 (dashed line) or Ξ = 1 (solid line). The mean-field density itself is presented by the dotted line. (Right) Strong coupling density as a function of the position for *R* = 2, α = −0.01, ε_{ext} = 0.05, and Ξ = 10 (solid line) or Ξ = 50 (dashed line).

Counterions density profile within the slab—dependence on the polarizability α. (Left) Weak coupling density close to the left electrode taking into account the fluctuations around the mean field as a function of the position within the slab (), for *R* = 0.3, ε_{ext} = 0.05, Ξ = 0.3, and α = −0.3 (solid line), α = −0.1 (dashed line), or α = −10^{−6} (dotted line). (Inset) Deviation from the mean-field density. (Right) Strong coupling density as a function of the position within the dielectric slab for *R* = 1, ε_{ext} = 0.05, Ξ = 10, and α = −0.05 (solid line), α = −0.01 (dashed line), or α = −10^{−6} (dotted line).

Counterions density profile within the slab—dependence on the polarizability α. (Left) Weak coupling density close to the left electrode taking into account the fluctuations around the mean field as a function of the position within the slab (), for *R* = 0.3, ε_{ext} = 0.05, Ξ = 0.3, and α = −0.3 (solid line), α = −0.1 (dashed line), or α = −10^{−6} (dotted line). (Inset) Deviation from the mean-field density. (Right) Strong coupling density as a function of the position within the dielectric slab for *R* = 1, ε_{ext} = 0.05, Ξ = 10, and α = −0.05 (solid line), α = −0.01 (dashed line), or α = −10^{−6} (dotted line).

Strong coupling pressure as a function of the plate separation *L*. (Left) *R* = 2, ε_{ext} = 0.05, and Ξ = 10 for different values of α. (Right) α = −0.01, ε_{ext} = 0.05, and Ξ = 10 for different values of *R*.

Strong coupling pressure as a function of the plate separation *L*. (Left) *R* = 2, ε_{ext} = 0.05, and Ξ = 10 for different values of α. (Right) α = −0.01, ε_{ext} = 0.05, and Ξ = 10 for different values of *R*.

Effect of the outer permittivity on the counterion density distribution. (Left) Weak coupling density with α = −0.3, *R* = 0.3, Ξ = 0.3, and ε_{ext} = 0.05 (solid line) or ε_{ext} = 1 (dashed line). (Right) Strong coupling density with α = −0.01, *R* = 2, Ξ = 10, and ε_{ext} = 0.05 (solid line) or ε_{ext} = 1 (dashed line).

Effect of the outer permittivity on the counterion density distribution. (Left) Weak coupling density with α = −0.3, *R* = 0.3, Ξ = 0.3, and ε_{ext} = 0.05 (solid line) or ε_{ext} = 1 (dashed line). (Right) Strong coupling density with α = −0.01, *R* = 2, Ξ = 10, and ε_{ext} = 0.05 (solid line) or ε_{ext} = 1 (dashed line).

Effect of the outer permittivity on the interaction pressure. (Left) Weak coupling pressure with α = −0.1, *R* = 1, Ξ = 0.3, and ε_{ext} = 0.05 (solid line) or ε_{ext} = 1 (dashed line). (Right) Strong coupling pressure with α = −0.01, *R* = 2, Ξ = 10, and ε_{ext} = 0.05 (solid line) or ε_{ext} = 1 (dashed line).

Effect of the outer permittivity on the interaction pressure. (Left) Weak coupling pressure with α = −0.1, *R* = 1, Ξ = 0.3, and ε_{ext} = 0.05 (solid line) or ε_{ext} = 1 (dashed line). (Right) Strong coupling pressure with α = −0.01, *R* = 2, Ξ = 10, and ε_{ext} = 0.05 (solid line) or ε_{ext} = 1 (dashed line).

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