The cell model for polyelectrolyte systems. Exact statistical mechanical relations, Monte Carlo simulations, and the Poisson–Boltzmann approximation

Abstract

Some exact statistical mechanical relations have been derived for polyelectrolyte systems within the primitive model. Using the cell model, the osmotic pressure is determined through an explicit evaluation of the derivative of the partition function. Planar, cylindrical, and spherical systems are considered and for a planar charged wall the contact value theorem [Henderson and Blum, J. Chem. Phys. 69, 5441 (1978)] is obtained as a special case. Analogous relations are derived for the cylindrical and spherical geometries. It is argued that the exact relations can be used as consistency tests for analytical approximations. It is pointed out that one merit of the Poisson–Boltzmann approximation is that the validity of the exact equations is retained. Finally, a simple method is devised for determining the osmotic pressure from Monte Carlo simulations. Results from such simulations are used to assess the accuracy of the osmotic pressure calculated using the Poisson–Boltzmann equation. For monovalent ions, the pressure is overestimated by 10%–50% in the cases studied, while with divalent counterions the error is substantially larger and a discrepancy of one order of magnitude is found.

Where the linearized Poisson–Boltzmann cell model fails: Spurious phase separation in charged colloidal suspensions

“…(70). In the next section we show that the linearized semi-grand-canonical osmotic-pressure difference β∆P DH is intrinsically thermodynamically unstable in the infinite-dilution limit and we compare with expressions previously obtained by Deserno and von Grünberg.…”

“…23 This simple functional form still remains valid at the PM (beyond mean-field) level for WS-cells of various geometries, 70 although the mean-field prediction for the equilibrium boundary densitȳ n(R) will (in general) disagree with the corresponding rigorous PM result due to the neglect of intracell microion-microion correlations and finite ionic sizes. Henceforth, to simplify the notation, we will omit the bar to denote equilibrium properties.…”

On the fluid–fluid phase separation in charged-stabilized colloidal suspensions

“…(25) in the third column. The corresponding ratio Z/Z bare is indicated in the last column with the Boltzmann distribution ρ + (r) = c s e −βqφ(r) (26) ρ − (r) = c s e +βqφ(r) (27) where φ(r) is the local electrostatic potential with respect to the reservoir. Taking the product of Eqs.…”

Macroion correlation effects in electrostatic screening and thermodynamics of highly charged colloids

“…directly from the WS cell or jellium model [12,13,14,15,16,19,20]. These models are usually solved using the non-linear PB equation including only one colloidal particle, or MC simulation [5,12,21,22]. In this work we will use non-linear PB equation with the jellium boundary conditions.…”