Effective charge versus bare charge: an analytical estimate for colloids in the infinite dilution limit

Abstract

We propose an analytical approximation for the dependence of the effective charge on the bare charge for spherical and cylindrical macro-ions as a function of the size of the colloid and salt content, for the situation of a unique colloid immersed in a sea of electrolyte (where the definition of an effective charge is non ambiguous). Our approach is based on the Poisson-Boltzmann (PB) meanfield theory. Mathematically speaking, our estimate is asymptotically exact in the limit κa ≫ 1, where a is the radius of the colloid and κ the inverse screening length. In practice, a careful comparison with effective charges parameters obtained by numerically solving the full non-linear PB theory proves that it is good down to κa ∼ 1. This is precisely the limit appropriate to treat colloidal suspensions. A particular emphasis is put on the range of parameters suitable to describe both single and double strand DNA molecules under physiological conditions.

Does Electrical Conductivity of Linear Polyelectrolytes in Aqueous Solutions Follow the Dynamic Scaling Laws? A Critical Review and a Summary of the Key Relations

“…During the last decades, a number of studies have focused on the phenomenon of counterion condensation, both experimental [31,32] and theoretical, and by simulations [33,34].…”

Screening of spherical colloids beyond mean field: A local density functional approach

“…Still, one often talks about effective charges, which mimic the stronger condensation of nonlinear theory within a linearized treatment [39][40][41][42][43][44]. That, however, is clearly not a physical but rather a formal renormalization, necessitated by the simplified linear treatment, and is thus a different story.…”

Like-charge attraction in confinement: myth or truth?

“…Applying the correction procedure as described above, eventually leads to perfect agreement with a screened Coulomb potential with Z * ≈ 18700 ± 1500 and κ -1 ≈ 55 ± 5nm (Fig.5c). Finally we calculated the bare charge Z bare obtained from Z * and κ -1 using an analytical expression based on the PoisssonBoltzmann mean-field theory 29 and obtain Z bare = 19210 and 19275 for the low and high ionic concentrations, respectively. The almost perfect agreement is an independent consistency check for the corrected U(r) and demonstrates that our correction leads to reasonable parameters.…”