^{1}, Alexandre P. dos Santos

^{1}and Yan Levin

^{1}

### Abstract

The thermodynamic properties of highly charged colloidalsuspensions in contact with a salt reservoir are investigated in the framework of the renormalized Jellium model (RJM). It is found that the equation of state is very sensitive to the particular thermodynamic route used to obtain it. Specifically, the osmotic pressure calculated within the RJM using the contact value theorem can be very different from the pressure calculated using the Kirkwood-Buff fluctuation relations. On the other hand, Monte Carlo simulations show that both the effective pair potentials and the correlation functions are accurately predicted by the RJM. It is suggested that the lack of self-consistency in the thermodynamics of the RJM is a result of neglected electrostaticcorrelations between the counterions and coions.

This work was partially supported by the CNPq, FAPERGS, INCT-FCx, and by the US-AFOSR under the Grant No. FA9550-09-1-0283.

I. INTRODUCTION

II. THEORETICAL BACKGROUND

A. The renormalized Jellium model

B. The Donnan equilibrium

C. The Kirkwood-Buff relation

III. MONTE CARLO SIMULATIONS

IV. RESULTS

A. Colloid-colloid correlations

B. Ion-ion correlations

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Colloidal systems
- 93.0
- Suspensions
- 33.0
- Mean field theory
- 14.0
- Electrostatics
- 13.0
- Thermodynamic properties
- 10.0

##### B01J13/00

## Figures

Effective pair potentials calculated using the MC simulations (squares), bare DLVO pair potential (dashed line), and DLVO with RJM effective parameters (solid line) for *Z* _{ bare } = 20, *a* = 10 Å, and λ_{ B } = 7.2 Å.

Effective pair potentials calculated using the MC simulations (squares), bare DLVO pair potential (dashed line), and DLVO with RJM effective parameters (solid line) for *Z* _{ bare } = 20, *a* = 10 Å, and λ_{ B } = 7.2 Å.

Colloid-colloid pair correlation functions obtained using the MC simulations (Ref. 52) and the RJM-OZ approach, for (a) ρ_{ s } = 24.9 mM and (b) ρ_{ s } = 249 mM. In both cases, the bare charge is *Z* _{ bare }λ_{ B }/*a* = 21.6, and the volume fraction is η = 0.0084.

Colloid-colloid pair correlation functions obtained using the MC simulations (Ref. 52) and the RJM-OZ approach, for (a) ρ_{ s } = 24.9 mM and (b) ρ_{ s } = 249 mM. In both cases, the bare charge is *Z* _{ bare }λ_{ B }/*a* = 21.6, and the volume fraction is η = 0.0084.

Reduced osmotic compressibility as a function of the reservoir salt concentration ρ_{ s } for colloidal particles of radius *a* = 30 Å and bare charge *Z* = 1000. The colloidal volume fractions are: (a) η = 10^{−5}, (b) η = 10^{−4}, (c) η = 10^{−3}, and (d) η = 10^{−2}. We see a dramatic discrepancy between the predictions of the JEOS (solid lines) and the Kirkwood-Buff fluctuation theory (dashed lines), especially at intermediate salt concentrations and high volume fractions.

Reduced osmotic compressibility as a function of the reservoir salt concentration ρ_{ s } for colloidal particles of radius *a* = 30 Å and bare charge *Z* = 1000. The colloidal volume fractions are: (a) η = 10^{−5}, (b) η = 10^{−4}, (c) η = 10^{−3}, and (d) η = 10^{−2}. We see a dramatic discrepancy between the predictions of the JEOS (solid lines) and the Kirkwood-Buff fluctuation theory (dashed lines), especially at intermediate salt concentrations and high volume fractions.

Comparison between the osmotic pressure calculated using the JEOS, (solid line) and the explicit integration of Eq. (9) (dashed line), with the experimental results reported in Ref. 53. The reservoir salt concentration is ρ_{ s } = 8μM, while the Bjerrum length is λ_{ B } = 2.38 nm, the colloidal radius is *a* = 21.9 nm, and colloidal charges are: (a) *Z* = 34 and (b) *Z* = 40.

Comparison between the osmotic pressure calculated using the JEOS, (solid line) and the explicit integration of Eq. (9) (dashed line), with the experimental results reported in Ref. 53. The reservoir salt concentration is ρ_{ s } = 8μM, while the Bjerrum length is λ_{ B } = 2.38 nm, the colloidal radius is *a* = 21.9 nm, and colloidal charges are: (a) *Z* = 34 and (b) *Z* = 40.

Comparison between the osmotic compressibilities calculated using the JEOS Eq. (6) (solid lines); the JEOS with explicit ionic correlations Eqs. (6) and (17) (point lines); and the KB fluctuation theory, Eq. (9) (dashed lines). The radius of colloidal particles is *a* = 10 Å, the bare colloidal charge is *Z* = 1000, and the Bjerrum length is λ_{ b } = 7.2 Å. The volume fractions are: (a) η = 0.01 and (b) η = 0.05.

Comparison between the osmotic compressibilities calculated using the JEOS Eq. (6) (solid lines); the JEOS with explicit ionic correlations Eqs. (6) and (17) (point lines); and the KB fluctuation theory, Eq. (9) (dashed lines). The radius of colloidal particles is *a* = 10 Å, the bare colloidal charge is *Z* = 1000, and the Bjerrum length is λ_{ b } = 7.2 Å. The volume fractions are: (a) η = 0.01 and (b) η = 0.05.

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