^{1,a)}, Jean-Pierre Hansen

^{2,3}and Gerhard Kahl

^{1}

### Abstract

We investigate the phase separation of the “ultrasoft restricted primitive model” (URPM), a coarse-grained representation of oppositely charged, interpenetrating polyelectrolytes, within a mean-field description based on the “chemical picture.” The latter distinguishes between free ions and dimers of oppositely charged ions (Bjerrum pairs) which are in chemical equilibrium governed by a law of mass action. Interactions between ions, and between ions and dimers are treated within linearized Poisson-Boltzmann theory, at four levels of approximation corresponding to increasingly refined descriptions of the interactions. The URPM is found to phase separate into a dilute phase of dimers, and a concentrated phase of mostly free (unpaired) ions below a critical temperature *T* _{ c }. The phase diagram differs, however, considerably from the predictions of recent simulations; *T* _{ c } is about three times higher, and the critical density is much lower than the corresponding simulation data [D. Coslovich, J. P. Hansen, and G. Kahl, Soft Matter7, 1690 (2011)10.1039/c0sm01090a]. Possible reasons for this unexpected failure of mean-fieldtheory are discussed. The Kirkwood line, separating the regimes of monotonically decaying and damped oscillatory decay of the charge-charge correlation function at large distances is determined within the random phase approximation.

We thankfully acknowledge Daniele Coslovich for many helpful discussions and for providing the MD simulation data. This work has been supported by the Marie Curie ITN-COMPLOIDS (Grant Agreement No. 234810), by the Austrian Science Foundation (FWF) under Project No. P19890-N16 and by the Studienstiftung des Deutschen Volkes.

I. INTRODUCTION

II. THE MODEL

III. THE “CHEMICAL PICTURE”

IV. MEAN-FIELDTHEORY: FOUR LEVELS OF APPROXIMATION

V. RESULTS

VI. THE KIRKWOOD LINE

VII. DISCUSSION AND CONCLUSIONS

### Key Topics

- Mean field theory
- 65.0
- Free energy
- 17.0
- Phase diagrams
- 17.0
- Correlation functions
- 8.0
- Molecular dynamics
- 8.0

## Figures

vs. *x* for *T* = 0.02, 0.04, 0.06, and 0.08 (from top to bottom), and *n* _{1} = 0.0035 (panel a) and *n* _{2} = 0.01 (panel b). The arrows along the *x*-axis indicate the position of the cut-off distance *X*(*T*, *n*) for *T* = 0.02.

vs. *x* for *T* = 0.02, 0.04, 0.06, and 0.08 (from top to bottom), and *n* _{1} = 0.0035 (panel a) and *n* _{2} = 0.01 (panel b). The arrows along the *x*-axis indicate the position of the cut-off distance *X*(*T*, *n*) for *T* = 0.02.

Fraction α of free ions vs. temperature *T* along the isochore *n* = 0.0035 (panel a) and *n* = 0.01 (panel b) calculated within approximation levels B, C, and D.

Fraction α of free ions vs. temperature *T* along the isochore *n* = 0.0035 (panel a) and *n* = 0.01 (panel b) calculated within approximation levels B, C, and D.

Fraction α of free ions vs. total ion density *n* along the isotherms *T* = 0.02 (panel a) and *T* = 0.04 (panel b).

Fraction α of free ions vs. total ion density *n* along the isotherms *T* = 0.02 (panel a) and *T* = 0.04 (panel b).

Phase diagrams of the URPM in the (*n*, *T*) plane from approximation levels B, C, and D, and from MD simulations.^{15,16}

Phase diagrams of the URPM in the (*n*, *T*) plane from approximation levels B, C, and D, and from MD simulations.^{15,16}

Positions of purely imaginary poles (open symbols), and of complex poles (filled symbols) in the (*q* _{1}, *q* _{2}) plane (upper right quadrant) for a low value of and a large value of .

Positions of purely imaginary poles (open symbols), and of complex poles (filled symbols) in the (*q* _{1}, *q* _{2}) plane (upper right quadrant) for a low value of and a large value of .

Full *h* _{+−}(*x*) (panel a), and its asymptotic limit (45) (panel b) as a function of *x* for two cases: one state in the monotonic regime (low ), and one state in the damped oscillatory regime (high ). Note that for low the logarithmic plot ln[*xh* _{+−}(*x*)] is a straight line for large *x*, indicating pure exponential decay at longest range, whereas for high exponentially damped oscillations persist for all distances *x*.

Full *h* _{+−}(*x*) (panel a), and its asymptotic limit (45) (panel b) as a function of *x* for two cases: one state in the monotonic regime (low ), and one state in the damped oscillatory regime (high ). Note that for low the logarithmic plot ln[*xh* _{+−}(*x*)] is a straight line for large *x*, indicating pure exponential decay at longest range, whereas for high exponentially damped oscillations persist for all distances *x*.

Kirkwood line in the (*n*, *T*) plane, relative to the approximate phase diagram of the URPM.

Kirkwood line in the (*n*, *T*) plane, relative to the approximate phase diagram of the URPM.

Integrand χ(*x*) of the integral ξ^{3} vs. *x*. Solid lines show the integrand for the exact potential *v* _{+−}, while the dashed lines show the results for χ(*x*) based on the quadratic expansion (11) of *v* _{+−}(*x*). Curves are for *T* = 0.02, 0.04, and 0.08 from top to bottom.

Integrand χ(*x*) of the integral ξ^{3} vs. *x*. Solid lines show the integrand for the exact potential *v* _{+−}, while the dashed lines show the results for χ(*x*) based on the quadratic expansion (11) of *v* _{+−}(*x*). Curves are for *T* = 0.02, 0.04, and 0.08 from top to bottom.

Positions of the maximum *X* _{1}(*T*) and the minimum *X* _{2}(*T*) of the integrand χ(*x*) vs. temperature *T*, and the cut-off distances *X*(*T*), *X*(*T*, *n* _{1} = 0.0035), and *X*(*T*, *n* _{1} = 0.01) vs. *T* (see text).

Positions of the maximum *X* _{1}(*T*) and the minimum *X* _{2}(*T*) of the integrand χ(*x*) vs. temperature *T*, and the cut-off distances *X*(*T*), *X*(*T*, *n* _{1} = 0.0035), and *X*(*T*, *n* _{1} = 0.01) vs. *T* (see text).

Ratio vs. *T* for cut-off distances *X*(*T*), *X*(*T*, *n* _{1} = 0.0035), and *X*(*T*, *n* _{1} = 0.01).

Ratio vs. *T* for cut-off distances *X*(*T*), *X*(*T*, *n* _{1} = 0.0035), and *X*(*T*, *n* _{1} = 0.01).

Ratio ζ_{0}/ζ of the reduced electric polarizabilities vs. *T* for cut-off distances *X*(*T*), *X*(*T*, *n* _{1} = 0.0035), and *X*(*T*, *n* _{1} = 0.01).

Ratio ζ_{0}/ζ of the reduced electric polarizabilities vs. *T* for cut-off distances *X*(*T*), *X*(*T*, *n* _{1} = 0.0035), and *X*(*T*, *n* _{1} = 0.01).

Cut-off distances *X*(*T* = 0.02, *n* _{1}) and *X*(*T* = 0.06, *n* _{1}) vs. *n* _{1}.

Cut-off distances *X*(*T* = 0.02, *n* _{1}) and *X*(*T* = 0.06, *n* _{1}) vs. *n* _{1}.

## Tables

Critical temperature *T* _{ c }, density *n* _{ c }, pressure *P* _{ c }, and osmotic coefficient *Z* _{ c }, estimated from approximations B, C, and D, and from MD simulations.^{15,16}

Critical temperature *T* _{ c }, density *n* _{ c }, pressure *P* _{ c }, and osmotic coefficient *Z* _{ c }, estimated from approximations B, C, and D, and from MD simulations.^{15,16}

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