^{1,a)}, Andrey Vlasov

^{2}, Lucian Anton

^{3,b)}and Andrew J. Masters

^{3}

### Abstract

We investigate the screening properties of Gaussian charge models of electrolyte solutions by analysing the asymptotic behaviour of the pair correlation functions. We use a combination of Monte Carlo simulations with the hyper-netted chain integral equation closure, and the random phase approximation, to establish the conditions under which a screening length is well defined and the extent to which it matches the expected Debye length. For practical applications, for example, in dissipative particle dynamics, we are able to summarise our results in succinct rules-of-thumb which can be used for mesoscale modeling of electrolyte solutions. We thereby establish a solid foundation for future work, such as the systematic incorporation of specific ion effects.

We would like to acknowledge the contribution of Dr. Ming Li, who noted the structure of the RPA solution for ions and neutral spheres in a related context. A.V. acknowledges partial support for trips to Manchester from grant GOSK #14.513.12.0003 (Ministry of Science and Education of Russia).

I. INTRODUCTION

II. MODEL

III. TOOLS

A. Pair correlation functions and screening

B. Integral equation theory

C. Monte Carlo methods

IV. RESULTS

A. RPA solution for the URPM

B. RPA solution for the solvated case

C. Comparison between HNC and MC

D. Screening properties from HNC

E. Worked example

F. The choice of σ

V. DISCUSSION

### Key Topics

- Electrolytes
- 26.0
- Correlation functions
- 21.0
- Solvents
- 21.0
- Monte Carlo methods
- 19.0
- Electrostatics
- 7.0

## Figures

Pair correlation functions for the URPM at two state points on opposite sides of the Kirkwood line (see text for details), plotted as |r h ±±| versus r to illustrate the asymptotic behaviour. Lines are HNC, data points with error bars are MC. Lengths are expressed in units of σ.

Pair correlation functions for the URPM at two state points on opposite sides of the Kirkwood line (see text for details), plotted as |r h ±±| versus r to illustrate the asymptotic behaviour. Lines are HNC, data points with error bars are MC. Lengths are expressed in units of σ.

URPM thermodynamics along two isotherms showing −p ex (solid HNC lines with square MC data points) and −⟨U⟩/3V (dashed HNC lines with diamond MC data points) as a function of density. The Debye-Hückel limiting law is shown as a dotted line in the two cases. Lengths and densities are expressed in units of σ, and thermodynamic quantities in units of k B T/σ3.

URPM thermodynamics along two isotherms showing −p ex (solid HNC lines with square MC data points) and −⟨U⟩/3V (dashed HNC lines with diamond MC data points) as a function of density. The Debye-Hückel limiting law is shown as a dotted line in the two cases. Lengths and densities are expressed in units of σ, and thermodynamic quantities in units of k B T/σ3.

Pair correlation functions for a solvated model at the indicated state point. Lines are HNC, data points with error bars are MC. From top to bottom, the curves are: g +−; g 0 + = g 0 − ≈ g 00; and g ++ = g −−. The difference between g 0 ± and g 00 is tiny and not resolved in this plot. Lengths and densities are expressed in units of r c .

Pair correlation functions for a solvated model at the indicated state point. Lines are HNC, data points with error bars are MC. From top to bottom, the curves are: g +−; g 0 + = g 0 − ≈ g 00; and g ++ = g −−. The difference between g 0 ± and g 00 is tiny and not resolved in this plot. Lengths and densities are expressed in units of r c .

Total correlation function h ++(r) from HNC for the URPM at l B = 1 and ρ z = 0.01(1)5 (top to bottom). Curves have been displaced for clarity. The Kirkwood transition from pure exponential decay (ρ z ≲ 0.03) to damped oscillatory (ρ z ≳ 0.03) is clearly seen. Lengths and densities are expressed in units of σ.

Total correlation function h ++(r) from HNC for the URPM at l B = 1 and ρ z = 0.01(1)5 (top to bottom). Curves have been displaced for clarity. The Kirkwood transition from pure exponential decay (ρ z ≲ 0.03) to damped oscillatory (ρ z ≳ 0.03) is clearly seen. Lengths and densities are expressed in units of σ.

The Kirkwood line for the URPM. The solid line with circles is from HNC. The dashed line is the RPA, from Eq. (16) . If solvent particles and short range repulsions are added, this map is practically unchanged.

The Kirkwood line for the URPM. The solid line with circles is from HNC. The dashed line is the RPA, from Eq. (16) . If solvent particles and short range repulsions are added, this map is practically unchanged.

The screening length for the URPM, comparing the value extracted by fitting the asymptotic tails of h αβ in HNC, to the RPA value from Eq. (15) . Lengths and densities are expressed in units of σ.

The screening length for the URPM, comparing the value extracted by fitting the asymptotic tails of h αβ in HNC, to the RPA value from Eq. (15) . Lengths and densities are expressed in units of σ.

Comparison between a fully solvated model (solid line, black) and the URPM equivalent (dashed line, blue). Both are calculated using HNC. The dotted line indicates λRPA from Eq. (15) . Lengths and densities are expressed in units of r c .

Comparison between a fully solvated model (solid line, black) and the URPM equivalent (dashed line, blue). Both are calculated using HNC. The dotted line indicates λRPA from Eq. (15) . Lengths and densities are expressed in units of r c .

HNC results for a 1:2 electrolyte. Note that h +− changes sign at r ≈ 0.4 r c , giving the appearance of a double peak (the primary peak in g +− is at r ≈ 0.8 r c ). Lengths and densities are expressed in units of r c .

HNC results for a 1:2 electrolyte. Note that h +− changes sign at r ≈ 0.4 r c , giving the appearance of a double peak (the primary peak in g +− is at r ≈ 0.8 r c ). Lengths and densities are expressed in units of r c .

Ratio between RPA screening length and Debye length for a 1:1 electrolyte, as a function of concentration, for three choices of σ. The lower (upper) horizontal axis shows the concentration in physical (simulation) units. Each curve terminates when the model system crosses the Kirkwood line. The dashed line is at λRPA/λD = 0.95.

Ratio between RPA screening length and Debye length for a 1:1 electrolyte, as a function of concentration, for three choices of σ. The lower (upper) horizontal axis shows the concentration in physical (simulation) units. Each curve terminates when the model system crosses the Kirkwood line. The dashed line is at λRPA/λD = 0.95.

## Tables

Sample calculation for a 0.1 M 1:1 electrolyte.

Sample calculation for a 0.1 M 1:1 electrolyte.

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