Linear charge density distribution

For fixed values of the linear charge densities, the counterion distribution can be studied by numerically solving Eqs. (31)—(33). Such an analysis reveals three distinct phases of counterion distributions labeled I, II, and III. 

In the last few years, several workers have analyzed charge density distribution in molecular crystals with non-linear optical (NLO) properties [71-74]. The NLO response can, in principle, be explained by an anharmonic distortion of the electron density distribution due to the electric field of an applied optical pulse. The polarization P induced in a molecule is... 

Trimerized organic conductors are of special interest, because two electrons per three sites constitute the simplest situation, where both electronic transitions resulting in single- and double-site occupation take place [21]. As one considers larger n-mers, two complications arise. First, the number of equations that should be solved sharply increases. The second complication is the increase in the number of n-meric normal modes, which are coupled to an external electromagnetic field. Recently, Yartsev et al. [22] have proposed using the linear response theory for several variables to describe the optical properties of trimers with arbitrary equilibrium charge density distribution. This approach can be extended to any cluster—the size is limited only by computer facilities. 

Counterion condensation theory, however, does not provide a detailed picture of the distribution of the condensed Ions. Recent research using the Poisson-Boltzmann approach has shown that for cylindrical macroions exceeding the critical linear charge density the fraction of the counterions described by Manning theory to be condensed remain within a finite radius of the macroion even at infinite polyion dilution, whereas the remaining counterions will be infinitely dispersed in the same limit. This approach also shows that the concentration of counterions near the surface of the macroion is remarkably high, one molar or more, even at infinite dilution of the macromolecule. In this concentrated ionic milieu specific chemical effects related to the chemical identities of the counterions and the charged sites of the macroion may occur. 

LA-, and NA-heparlns were determined. Although some differences in these distributions were evident (11), it is also evident that HA-heparins exist within a similar range of sulfation as do the LA and NA heparins. Clearly, HA-heparin is not all hlgh-sulfated heparin and LA-or NA-heparins are not all low-sulfated heparins. These results show that fractionation according to antithrombin affinity probably did not occur to any extensive degree and, hence, the results in Fig 2 do Indeed demonstrate that anticoagulant activity varies with the linear charge density or factors related to it. 

This asymptotic behavior is a consequence of the different distribution of counterions on the middle and at the extremes of the chain. Therefore, as the size of the chain increases, end-effects become less important. Polyelectrolytes with a greater linear charge density achieve a larger 0p " value at a larger chain size j ateau End effects are more important in this case, due to the larger counterion condensation in these systems. The plateau values are 0 o.78 0.63 0.43... 

From a general point of view, the distribution of counterions is imposed by that of the electrostatic potential. Our purpose is to treat the thermodynamic behaviour of a cylindrical system without excess added salt the polyelectrolyte is always considered as a thin rod characterized by its linear charge density. Two approaches are investigated the first one needs the resolution of a Poisson-Boltzmann equation without... 

A thin half-ring carries a charge uniformly distributed along the ring with a linear charge density r = 10 nC/m. Determine the electric potential (p in the center of ring O. 

Ire boundary element method of Kashin is similar in spirit to the polarisable continuum model, lut the surface of the cavity is taken to be the molecular surface of the solute [Kashin and lamboodiri 1987 Kashin 1990]. This cavity surface is divided into small boimdary elements, he solute is modelled as a set of atoms with point polarisabilities. The electric field induces 1 dipole proportional to its polarisability. The electric field at an atom has contributions from lipoles on other atoms in the molecule, from polarisation charges on the boundary, and where appropriate) from the charges of electrolytes in the solution. The charge density is issumed to be constant within each boundary element but is not reduced to a single )oint as in the PCM model. A set of linear equations can be set up to describe the electrostatic nteractions within the system. The solutions to these equations give the boundary element harge distribution and the induced dipoles, from which thermodynamic quantities can be letermined. 

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