Counterions distribution around spheres

Theoretical determinations of the counterion distribution around a PE were done by Solms by approximating them as tangentially bonded spheres which interact via the coulomb potential [64]. In his work, the Osmotic pressure and thermodynamic properties were determined via analytical integrals [64]. 

In this chapter, we first discuss the case of completely salt-free suspensions of spheres and cylinders. Then, we consider the Poisson-Boltzmann equation for the potential distribution around a spherical colloidal particle in a medium containing its counterions and a small amount of added salts [8]. We also deals with a soft particle in a salt-free medium [9]. 

Monte Carlo calculations on simplified model systems representing the DNA, counterions, and water solvent have been carried out by several research groups. Le Bret and Zimm o reported two such calculations. The first used an impenetrable cylinder embedded with a linear array of charges to represent DNA backbone and the other used a double-helical charge array on an impenetrable cylinder. The mobile ions were treated as hard spheres, and the ionic interaction between the ion and the model DNA were modulated by the solvent, which was treated as a dielectric continuum with a dielectric constant of 80. Ion distributions around the cylinders were calculated and compared, but there were no significant differences between the two models, possibly because... 

Figure 1.11 gives the scaled potential distribution y(r) around a positively charged spherical particle of radius a with yo = 2 in a symmetrical electrolyte solution of valence z for several values of xa. Solid lines are the exact solutions to Eq. (1.110) and dashed lines are the Debye-Hiickel linearized results (Eq. (1.72)). Note that Eq. (1.122) is in excellent agreement with the exact results. Figure 1.12 shows the plot of the equipotential lines around a sphere with jo = 2 at ka = 1 calculated from Eq. (1.121). Figures 1.13 and 1.14, respectively, are the density plots of counterions (anions) (n (r) = exp(+y(r))) and coions (cations) ( (r) = MCxp(—y(r))) around the sphere calculated from Eq. (1.121). 

Initially, counterions were distributed randomly around the nanocapsule, which was filled and surrounded by water. In the internal cavity of the POM 172 water molecules were placed 72 waters fulfilling the molybdenum coordination sphere of the pentagonal Mo(Mo)s units and a structureless 100 H2O cluster. Following a standard protocol [24], a large number of configurations were collected through the MD trajectory and analyzed. Then, the radial (RDF) and spatial (SDF) distribution functions of the centers of the capsule-oxygen water atoms were computed. 

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