Smeared charges

Because of the limitation intrinsic to the adoption of an explicit parametrised density model, many crystallographers have been dreaming of disposing of such models altogether. The thermally-smeared charge density in the crystal can of course be obtained without an explicit density model, by Fourier summation of the (phased) structure factor amplitudes, but the resulting map is affected by the experimental noise, and by all series-termination artefacts that are intrinsic to Fourier synthesis of an incomplete, finite-resolution set of coefficients. 

However, these studies predicted larger polyion sizes and smaller electrostatic potentials than experimentally measured [85]. These discrepancies primarily arise from the assumption of smeared charge distributions in the polyion domain, whereas the potential around the polyion actually varies strongly with distance from the polyion. 

The surface potential of a bilayer is a result of having charged lipids in this bilayer (for reviews see References 63 and 66). In fluid phase bilayers, rapid translational diffusion of the lipids allows the surface charge associated with the lipids to be considered a smeared charge and the electrostatic potential at the surface of the bilayer, J o, is well described by the Gouy-Chapman equation ... 

Figure 8.17, Minimum energy position of divalent cations on discrete charge sites (—) of tetrahedrally charged clays (a) and smeared charge sites (—) of octahedrally charged clays (b). This diagram is idealized in that charge positions on clays are not likely to be evenly spaced.

Uni-univalent and One Divalent Electrolyte Case. The electrical potential of a smeared charged membrane has been solved by Abraham-Shrauner for 2-1-1 electrolytes (one divalent cation, one monovalent cation, and one monovalent anion). The electrical potential (l/(x) at a distance X from the membrane surface is expressed by the following equation ... 

Figure 22. Comparison between spatially averaged discrete charge (continuous line) and smeared charge (dashed line) potentials, where (A) = 35 mM, (B) = 1 mM and...

Smeared Charge Multipole Model for Electrostatics and Its Parameterization Protocol 129... 

Going Beyond Point Charges The Smeared Charge with Multipole Model... 

In the smeared charge multipole model, each atom i is represented by a smeared charge which consists of a nuclear... 

The interaction between a smeared charge and a point dipole is... 

The parameterization scheme for the smeared charge with multipole model is illustrated in Fig. 4.4 the parameterization starts with geometry optimizations and electronic structural calculations... 

Figure 4.4 Parameterization scheme of the smeared charge with multipole electrostatic model.

Similarly, the energy of the complex consists of two terms. The first term is the energy of the sphere and of the uniformly smeared charge of the wrapped polyelectrolyte this term favors electrically neutral complexes. The energy corrections from the... 

As noted above, we find that this kinetics formulation is applicable to cases where the salt concentration is low, and deposition extensive, so that the "smeared charge approximation holds. Assuming that depositing DNA-SWCNTs form a homogenous layer on the surface implies that the surface charge will increase with deposited density Ps (in p.m of nanotubes per p.m of surface). As surface charge increases, surface potential relative to bulk solution will also increase. Within the linearized Debye-Hiickel approximation, the surface potential is related to surface charge as ... 

An alternative way of describing the electrostatics is to introduce smeared charges in place of point charges. A model using smeared charges with a point polarizability representation has been developed. The pair functions predicted by this model are in accord with experimental results both for ambient conditions and for elevated temperatures and reduced densities. 

For the sake of discussion, let us consider the smeared charge distribution so that Eq. (6.76) can be written as... 

In Eq. (6.86), the Q s are the partition functions of individual components in the presence of a field. Explicitly, for the polyelectrolyte chain with smeared charge distribution along the backbone, single chain partition function is given by... 

The Thole model is based on a modified dipole-dipole interaction, which can be reformulated in terms of the interaction of smeared charge densities. This eliminates the divergence of the head-to-taU dipole-dipole interaction at small interseparations (Angstrom scale) [103-105]. Smearing out the charge distribution mimics the nature of the quantum mechanical wavefunction, which effectively guards against this unphysical polarization catastrophe. 

Thermodynamic properties of polyelectrolyte solutions are mostly determined by ionic distribution around the polyion skeleton. To explain thermodynamic properties, therefore, the rod-like model may be effective. It is assumed that the polyion is a rod of infinite length and has smeared charges distributed uniformly over the surfaces of the rod. That is, a real polyion chain consists of a series of discrete charges and each charge is surrounded by its own ionic atmosphere. If the radius of ionic atmosphere is... 

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