Macroions, electrostatic potential

The theory accounts for the fact that some fraction of counterions escapes from the interior of the macromolecular coil. The macromolecular coil on the whole possesses by the electrostatic charge and interacts via the screened electrostatic potential with low-molecular ions escaped from the interior of the macroion. 

The localization of counterions in the intramolecular volume is a common feature manifested in dilute salt-free solutions of branched polyions of different topologies, including dendritic (star-burst), randomly (hyper)branched PEs, PE molecular brushes, etc. The physical reason for this phenomena is the same as outlined for PE stars a strongly charged, branched, macroion creates a high electrostatic potential, which attracts counterions and retains them in the intramolecular volume, in spite of a significant loss in the translational entropy. The effect is most pronounced in a dilute solution, where the concentration of counterions in the bulk is extremely low. 

The spherical cell model was used early to examine the distribution of the small ions near a charged macroion and the thermodynamics of such systems. In the past, the model was used to investigate, e.g., the electrostatic potential and the small ion distribution [67- 74], the self-diffusion of coim-terions [75-78], the micelle formation of charged surfactants [79], and full phase diagrams of charged surfactant-water systems [80]. Many of those investigations were based on the PB equation, providing an approximate description of the model system. 

The simulation has been carried out for a poly(2-acrylamido-2-methyl-propanesulfonic acid) (PAMPS) gel that is a fully ionized polyelectrolyte having sulfonic groups as macroions and as counterions. The radius of the polymer chain of PAMPS is r, = 6.08 X 10 m. The other constants used are b = 2.55 X 10 m, = 78 (e = c Cq, where q is the dielectric constant in a vacuum), and T = 300 K. The simulation was carried out for a cubic box with a side length of r (1/2 of a periodic unit cell) and then copied to the whole periodic unit according to the symmetry. The simulation box was divided into 10 x 10 x 10 lattice cells. To make the surface of poly ions on lattice points, r = lOr, was used. Equation 1 was expressed by Poisson s difference equations and the electrostatic potential on lattice points was calculated by Liebmann s method. A relaxation parameter of 0.5 has been used to accelerate the calculation. An iterative difference value of Ae = 10 kT was used to determine the accuracy of the calculation. Symmetric conditions have been used for walls of the simulation box i.e., the normal derivatives of the potential on the walls are zero. 

The Poisson-Boltzmann approach to desaiption of the electrostatic potential near macroions fails when there are strong correlations between counterions. This takes place either in the system with multivalent counterions or in the case of large values of the Bjenum length 1b (see for details References 14 and 84-87). 

Figure 3a shows separately the electrostatic (DLVO-like) and excluded volume contributions to the effective interaction potential between two macroions, W (r). The electrostatic contribution is all the way repulsive and decays mono-... 

A) The physical reason for the weakening of the macroion structure as, e.g., the macroion radius is reduced (System FV System II), is a nonlinear response. Table 1 shows the macroion-counterion potential energy at contact amoimts to /mi = 3.6 and 28.5, respectively, and Fig. 6 shows a superlinear increase of the coimterion accumtdation. Although the direct macroion-macroion repulsion also increases as 71i is increased, the effect of the screening grows faster. In fact, the macroion structure decreases as the electrostatic coupling / ll is raised when Tn > 5/Zr [22]. 

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