Quantitative Assessment of the Accuracy of the Poisson–Boltzmann Cell Model for Salty Suspensions

Abstract

The cell model is a ubiquitous, fast, and relatively easily implemented model used to estimate the osmotic pressure of a colloidal dispersion. It has been shown to yield accurate approximations of the pressure in dispersions with a low salt content. It is generally accepted that it performs well when long-ranged interactions are involved and the structure of the dispersion is solidlike. The aim of the present work is to determine quantitatively the error committed by assuming the pressure computed with the cell model is the real osmotic pressure of a dispersion. To this end, cell model pressures are compared to a correct estimation of the actual pressures obtained from Poisson-Boltzmann Brownian dynamics simulations including many-body electrostatics and the thermal motion of the colloids. The comparison is performed for various colloidal sizes and charges, salt contents, and volume fractions. It is demonstrated that the accuracy of the cell model predictions is a function of only the average intercolloid distance scaled by Debye's length κd̅ and the normalized colloidal charge. The cell model is accurate for κd̅ < 1 and not reliable for κd̅ > 5 independently of the colloidal charge. In the 1 < κd̅ < 5 range, covering a wide set of experimental conditions, the colloidal surface charge has a large influence on the error associated with the cell approximation. The results presented in this article should provide a useful reference to determine a priori if the cell model can be expected to predict accurately an equation of state for a given set of physicochemical parameters.

Ultrafiltration of charge-stabilized dispersions at low salinity

“…According to a multi-colloid PB Brownian dynamics simulation study by Hallez et al [60], the identification of the suspension osmotic pressure Π with the microion osmotic pressure calculated using the PB cell model is quite accurate for κ res d 1, but not reliable for κ res d 5, with a larger error introduced for intermediate values of κ res d that is strongly depending on (l B /a)|Z bare |. Here, d = a φ −1/3 − 1 is roughly twice the average intercolloid distance.…”

Osmotic pressure of permeable ionic microgels: Poisson-Boltzmann theory and exact statistical mechanical relations in the cell model

“…The shape of the macroion dictates the appropriate geometry of the cell (planar, cylindrical, spherical), the symmetry of which eases the computational solution of the nonlinear PB equation. Predictions for microion densities and osmotic pressure prove to be most accurate for systems with relatively low salt concentration [9,26].…”

Jellium and cell model for titratable colloids with continuous size distribution

“…for the cell model and 28) for the renormalized jellium model. The above boundary conditions, although necessary to solve the cell model and the renormalized jellium model do not prejudge (define) either the nature of the colloidal charge or the response of the latter to colloid density or to a change in the nature and concentration of the electrolyte solution.…”

“…This should not come as a surprise since the PCM is known to neglect the entropic and contact contributions to the osmotic pressure 28,58 . As discussed by Hallez et al 28 , it is found Figure 9.…”

“…From a simulation point of view, representative and realistic models of charged polydisperse colloidal suspensions are indeed tractable neither at the primitive model level of approximation, where the degrees of freedom of the solvent molecules are averaged out through a dielectric continuum, nor at the level of the mean field approximation [26][27][28] , where the many ions are further replaced by a mean electrostatic potential obtained from a) Electronic mail: guillaume.bareigts@u-bourgogne.fr b) Electronic mail: christophe.labbez@u-bourgogne.fr solving the Poisson-Boltzmann equation. An amenable and attractive approach, first introduced by Beresford-Smith 29 , consists, instead, in whittling the colloidal system down to the colloidal particles only, i.e.…”