Electrostatic free energy density

The free energy density terms introduced so far are all used in the description of the smectic phases made by rod-like molecules, the electrostatic term (6) being characteristic for the ferroelectric liquid crystals made of chiral rod-like molecules. To describe phases made by bent-core molecules one has to add symmetry allowed terms which include the divergence of the polar director (polarization splay) and coupling of the polar director to the nematic director and the smectic layer normal ... 

In this section we describe some examples of functionals proposed to compute the electrostatic potential (Ik which is used in Equation (1.72) to solve for the electrostatic interaction energy between the charge density p0 and the dielectric medium. This class contains functionals which are not energy functionals, in the sense that their minimization does not lead to the electrostatic free energy, Equation (1.72). However, at the end of the variational process they provide an electrostatic potential (or a polarization) which satisfies Equation (1.73) and thus it can be used to compute the electrostatic energy. 

Assume that the polyion is a charged cylinder of contour length L and radius R. On evaluating the surface charge density ct of the cylinder by a self-consistent methodology, the electrostatic free energy pgeiec was expressible as [44]... 

The electrostatic free energy includes the usual macroscopic free energy density ED (with the displacement field D = sqE + P) and also the interaction of the dipoles with the additional field Ep due to the nonuniform polarization [15] ... 

This problem is best approached by considering the electrostatic free energy, F, oi protein molecule due to its net charge and its interaction with an ion atmosphere. For a spherical ion of uniform charge density, according to the Debye-Hiickel theory (Cohn and Edsall, 1943, p. 473)... 

Clearly, Eq. (13) concerns the electrostatic interactions only, so that a suitably chosen hard-core contribution, e.g. of Camahan-Starling type [25] must be added to the free energy densities. Differentiation with respect to the densities of the species finally yields the chemical potential and the activity coefficients required for evaluating the mass action law determining the concentrations of free ions and ion pairs. 

As presented above, the degree of substitution of ionic groups or charge density is an important factor for ion-binding ability of these derivatives. An electrostatic free energy per ionic group, at the degree of ionization, a, may be estimated... 

A many-body perturbation theory (MBPT) approach has been combined with the polarizable continuum model (PCM) of the electrostatic solvation. The first approximation called by authors the perturbation theory at energy level (PTE) consists of the solution of the PCM problem at the Hartree-Fock level to find the solvent reaction potential and the wavefunction for the calculation of the MBPT correction to the energy. In the second approximation, called the perturbation theory at the density matrix level only (PTD), the calculation of the reaction potential and electrostatic free energy is based on the MBPT corrected wavefunction for the isolated molecule. At the next approximation (perturbation theory at the energy and density matrix level, PTED), both the energy and the wave function are solvent reaction field and MBPT corrected. The self-consistent reaction field model has been also applied within the complete active space self-consistent field (CAS SCF) theory and the eomplete aetive space second-order perturbation theory. ... 

For a lamellar microstructure, the wave vectors of the Fourier components are antiparallel, and the electrostatic contribution to the free energy density is... 

Forces on Disclination Lines. Disclination lines have associated with them an extended distortion field, as illustrated in Fig. 32. They interact indirectly through this distortion field with other defects. To lowest order, the free energy density, including an electrostatic term, is [14]... 

The first term in Eq. [133] is also what one would find from classical electrostatics for the energy of a plane with charge density s in the presence of the Debye-Hiickel potential due to a second plane with charge density s+ at a distance 2R. The electrostatic free energy is now fovmd from the interaction potential and Eq. [107]... 

The final class of methods that we shall consider for calculating the electrostatic compone of the solvation free energy are based upon the Poisson or the Poisson-Boltzmann equatior Ihese methods have been particularly useful for investigating the electrostatic properties biological macromolecules such as proteins and DNA. The solute is treated as a body of co stant low dielectric (usually between 2 and 4), and the solvent is modelled as a continuum high dielectric. The Poisson equation relates the variation in the potential (f> within a mediu of uniform dielectric constant e to the charge density p ... 

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