Abstract
Ion-mediated interactions between like-charged polyelectrolytes have been paid much attention, and the Poisson–Boltzmann (PB) theory has been shown to fail in qualitatively predicting multivalent ion-mediated like-charge attraction. However, inadequate attention has been paid to the ion-mediated interactions between oppositely charged polyelectrolytes. In this work, the potentials of mean force (PMF) between oppositely charged nanoparticles in 1:1 and 2:2 salt solutions were investigated by Monte Carlo simulations and the PB theory. Our calculations show that the PMFs between oppositely charged nanoparticles are generally attractive in 1:1 and 2:2 salt solutions and that such attractive PMFs become weaker at higher 1:1 or 2:2 salt concentrations. The comprehensive comparisons show that the PB theory can quantitatively predict the PMFs between oppositely charged nanoparticles in 1:1 salt solutions, except for the slight deviation at very high 1:1 salt concentration. However, for 2:2 salt solutions, the PB theory generally overestimates the attractive PMF between oppositely charged nanoparticles, and this overestimation becomes more pronounced for nanoparticles with higher charge density and for higher 2:2 salt concentration. Our microscopic analyses suggest that the overestimation of the PB theory on the attractive PMFs for 2:2 salt solutions is attributed to the underestimation of divalent ions bound to nanoparticles.
Highlights
Ion-mediated interactions between like-charged polyelectrolytes have been paid much attention, and the Poisson–Boltzmann (PB) theory has been shown to fail in qualitatively predicting multivalent ionmediated like-charge attraction
We investigated the potential of mean force (PMF) between oppositely charged nanoparticles in symmetrical salt solutions by Monte Carlo (MC) simulations and the PB theory
We quantitatively examine the relative deviation of the potentials of mean force (PMF) between the PB theory and the MC simulations, which is characterized by
Summary
This overestimation becomes more pronounced for nanoparticles with high charge densities and for high 2:2 salt concentrations. To understand the efficient role of divalent ions over monovalent ions in screening the Coulomb attraction between oppositely charged nanoparticles, we further calculated another kind of equivalent 1:1 salt concentration to 2:2 salt by comparing the net ion charge fraction Q(r); see Eq 1. To further characterize the relative deviation ∆∆g of the PMFs between the PB theory and MC simulations, an apparent parameter |Z|∆Q* of the difference in total net ion charges was defined as |Z|∆Q(r) at r = 14 Å, where the maximum values of |Z|∆Q(r) are located at high 1:1 and 2:2 salts; see Figs S2D–F and S4D–F in the SM. |Z|∆Q* can be served as an apparent parameter to well characterize the deviation degree of the PMFs between the PB theory and MC simulations for oppositely charged nanoparticles
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