^{1}, David S. Dean

^{2}, Thomas C. Hammant

^{3}, Ronald R. Horgan

^{3}and Rudolf Podgornik

^{4}

### Abstract

The one-dimensional Coulomb lattice fluid in a capacitor configuration is studied. The model is formally exactly soluble via a transfer operator method within a field theoretic representation of the model. The only interactions present in the model are the one-dimensional Coulomb interaction between cations and anions and the steric interaction imposed by restricting the maximal occupancy at any lattice site to one particle. Despite the simplicity of the model, a wide range of intriguing physical phenomena arise, some of which are strongly reminiscent of those seen in experiments and numerical simulations of three-dimensional ionic liquid based capacitors. Notably, we find regimes where over-screening and density oscillations are seen near the capacitor plates. The capacitance is also shown to exhibit strong oscillations as a function of applied voltage. It is also shown that the corresponding mean-fieldtheory misses most of these effects. The analytical results are confirmed by extensive numerical simulations.

R.P. acknowledges support from the Leverhulme Trust, the ARRS through the program P1-0055 and the research project J1-0908 as well as the University of Toulouse for a one month position of Professeur invité.

I. INTRODUCTION

II. ONE-DIMENSIONAL LATTICE COULOMB FLUID MODEL

A. Without external charges

B. With charges on the boundaries or constant applied voltage

III. MEAN-FIELD APPROXIMATION

IV. EXACT COMPUTATION

A. Other formulation and limit of highly charged boundaries

V. NUMERICAL COMPUTATION

A. Free energy for charged boundaries

B. Average potential drop for fixed surfacecharges

C. Average charge for fixed potential difference

D. Capacitance

E. Charge density

VI. NUMERICAL SIMULATIONS

VII. CONCLUSION

### Key Topics

- Mean field theory
- 69.0
- Capacitance
- 33.0
- Surface charge
- 30.0
- Capacitors
- 19.0
- Double layers
- 13.0

##### H01G4/00

## Figures

Lattice Coulomb fluid model. Dark balls stand for cations and light balls for anions. Boundary charges are at sites −1 and *M*.

Lattice Coulomb fluid model. Dark balls stand for cations and light balls for anions. Boundary charges are at sites −1 and *M*.

Dimensionless grand potential as a function of the system size for γ = 1 and μ = 100.

Dimensionless grand potential as a function of the system size for γ = 1 and μ = 100.

Dimensionless free enthalpy as a function of the system size for γ = 1 and μ = 100.

Dimensionless free enthalpy as a function of the system size for γ = 1 and μ = 100.

Average potential drop as a function of the imposed charge for γ = 1 and μ = 1 and system size 10^{4}.

Average potential drop as a function of the imposed charge for γ = 1 and μ = 1 and system size 10^{4}.

Left: Average surface charge as a function of the imposed voltage, for γ = 1 and μ = 1 with system size 10^{4}. Right: Average surface charge as a function of the imposed voltage, for γ = 1 and μ = 0.5 with system size 10^{4}.

Left: Average surface charge as a function of the imposed voltage, for γ = 1 and μ = 1 with system size 10^{4}. Right: Average surface charge as a function of the imposed voltage, for γ = 1 and μ = 0.5 with system size 10^{4}.

Difference of the bulk pressure as a function of the parameters γ and μ. Contour line where λ_{0}(*Q* = 0) = λ_{0}(*Q* = 0.5) (blue solid line) compared with the approximate transition line (65) (red dashed line).

Difference of the bulk pressure as a function of the parameters γ and μ. Contour line where λ_{0}(*Q* = 0) = λ_{0}(*Q* = 0.5) (blue solid line) compared with the approximate transition line (65) (red dashed line).

Left: Capacitance as a function of the voltage drop for μ = 1: appearance of the peaks as γ increases for a bell shaped capacitance. Right: Capacitance as a function of the voltage drop for μ = 0.03: appearance of the peaks as γ increases for a camel shaped capacitance. The system size is 10^{4}.

Left: Capacitance as a function of the voltage drop for μ = 1: appearance of the peaks as γ increases for a bell shaped capacitance. Right: Capacitance as a function of the voltage drop for μ = 0.03: appearance of the peaks as γ increases for a camel shaped capacitance. The system size is 10^{4}.

Left: Capacitance as a function of the voltage drop for μ = 1 and γ = 1. Right: Capacitance as a function of the voltage drop for μ = 0.03 and γ = 0.3. The system size is 10^{4}.

Left: Capacitance as a function of the voltage drop for μ = 1 and γ = 1. Right: Capacitance as a function of the voltage drop for μ = 0.03 and γ = 0.3. The system size is 10^{4}.

Capacitance as a function of the voltage drop for γ = 1 and different values of the fugacity. The system size is 10^{4}.

Capacitance as a function of the voltage drop for γ = 1 and different values of the fugacity. The system size is 10^{4}.

PZC capacitance as a function of μ for different values of γ, for an infinite system.

PZC capacitance as a function of μ for different values of γ, for an infinite system.

Left: Charge density close to the left electrode (located at *x* = −1) as a function of the position for γ = 1, μ = 1, and *Q* = 1. Right: Charge density close to the left electrode (located at *x* = −1) as a function of the position for γ = 1, μ = 1, and *Q* = 4.25. The system size is 80.

Left: Charge density close to the left electrode (located at *x* = −1) as a function of the position for γ = 1, μ = 1, and *Q* = 1. Right: Charge density close to the left electrode (located at *x* = −1) as a function of the position for γ = 1, μ = 1, and *Q* = 4.25. The system size is 80.

Left: Exact result for the mean charge density close to the left electrode (located at *x* = −1) as a function of the position for γ = 1, μ = 1, and *Q* = 0.5, for different values of the fugacity. Right: Exact result for the mean charge density as a function of the position for γ = 1, μ = 1, and *Q* = 0.5, for different values of the fugacity. The system size is 80.

Left: Exact result for the mean charge density close to the left electrode (located at *x* = −1) as a function of the position for γ = 1, μ = 1, and *Q* = 0.5, for different values of the fugacity. Right: Exact result for the mean charge density as a function of the position for γ = 1, μ = 1, and *Q* = 0.5, for different values of the fugacity. The system size is 80.

Exact result for the mean charge density as a function of the position for γ = 1 and μ = 100, for different values of the boundary charge. The system size is 80.

Exact result for the mean charge density as a function of the position for γ = 1 and μ = 100, for different values of the boundary charge. The system size is 80.

Exact result for the mean charge density as a function of the position for μ = 1000 and *Q* = 0.5, for an odd number of sites (*M* = 81).

Exact result for the mean charge density as a function of the position for μ = 1000 and *Q* = 0.5, for an odd number of sites (*M* = 81).

Mean charge density ρ_{0} of the first layer as a function of the surface charge *Q* for γ = 1 and μ ∈ {3, 10, 100} (the two big arrows indicate the change when μ increases). The system size is 500.

Mean charge density ρ_{0} of the first layer as a function of the surface charge *Q* for γ = 1 and μ ∈ {3, 10, 100} (the two big arrows indicate the change when μ increases). The system size is 500.

Left: Theory and simulation for mean charge ⟨*Q*⟩ versus Δ*v* for γ = 1, μ = 1 on a lattice with *M* = 128. Right: Theory and simulation for mean voltage difference ⟨Δ*v*⟩ versus *Q* for γ = 1, μ = 1 on a lattice with *M* = 1024.

Left: Theory and simulation for mean charge ⟨*Q*⟩ versus Δ*v* for γ = 1, μ = 1 on a lattice with *M* = 128. Right: Theory and simulation for mean voltage difference ⟨Δ*v*⟩ versus *Q* for γ = 1, μ = 1 on a lattice with *M* = 1024.

Theory and simulation in the fixed *Q* ensemble for the mean charge density ρ versus distance *x* from the left-hand plate located at *x* = −1 for γ = 1, μ = 10, *Q* = −0.5 on a lattice with *M* = 1024. Note that there is mild over-screening at *x* = 0.

Theory and simulation in the fixed *Q* ensemble for the mean charge density ρ versus distance *x* from the left-hand plate located at *x* = −1 for γ = 1, μ = 10, *Q* = −0.5 on a lattice with *M* = 1024. Note that there is mild over-screening at *x* = 0.

Left: Theory and simulation for capacitance *c* _{Δv } versus Δ*v* in the fixed Δ*v* ensemble for γ = 1, μ = 1 on lattice with *M* = 128. Right: Theory and simulation for capacitance *c* _{Δv } versus Δ*v* in the fixed Δ*v* ensemble for γ = 0.3, μ = 0.03 on lattice with *M* = 128.

Left: Theory and simulation for capacitance *c* _{Δv } versus Δ*v* in the fixed Δ*v* ensemble for γ = 1, μ = 1 on lattice with *M* = 128. Right: Theory and simulation for capacitance *c* _{Δv } versus Δ*v* in the fixed Δ*v* ensemble for γ = 0.3, μ = 0.03 on lattice with *M* = 128.

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