Dielectric media

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

The third term in Equation (11.52) is the correction factor corresponding to the work done creating the charge distribution of the solute within the cavity in the dielectric medium. the gas-phase wavefimction. 

To calculate AGgi c, we must take account of the work done in creating the charge distribi w ithin the cavity in the dielectric medium. This is equal to one-half of the electrostatic i action energy between the solute charge distribution and the polarised dielectric, amd S ... 

For successful interruption of the arc plasma it is essential that the energy emitted by the arc plasma is at least equal to the capacitive energy received by the dielectric medium. At the instant of arc extinction, this phenomenon is termed energy balancing, when... 

Until the 1970s the chemical used as the impregnating and dielectric medium for capacitor units was PCB (polychlorinated biphenyl) liquid. It was found to be toxic and unsafe for humans as well as contamination of the environment. For this reason, it is no longer used. The latest trend is to use a non-PCB, non-toxic, phenyl xylyl ethane (PXE-oil), which is a synthetic dielectric liquid of extremely low loss for insulation and impregnation of the capacitor elements or to use mixed polypropylene or allpolypropylene (PP) liquids as the dielectric. A non-oil dielectric, such as epoxy resin, is also used. 

Since the interface behaves like a capacitor, Helmholtz described it as two rigid charged planes of opposite sign [2]. For a more quantitative description Gouy and Chapman introduced a model for the electrolyte at a microscopic level [2]. In the Gouy-Chapman approach the interfacial properties are related to ionic distributions at the interface, the solvent is a dielectric medium of dielectric constant e filling the solution half-space up to the perfect charged plane—the wall. The ionic solution is considered as formed... 

The simplest shape for the cavity is a sphere or possibly an ellipsoid. This has the advantage that the electrostatic interaction between M and the dielectric medium may be calculated analytically. More realistic models employ moleculai shaped cavities, generated for example by interlocking spheres located on each nuclei. Taking the atomic radius as a suitable factor (typical value is 1.2) times a van der Waals radius defines a van der Waals surface. Such a surface may have small pockets where no solvent molecules can enter, and a more appropriate descriptor may be defined as the surface traced out by a spherical particle of a given radius rolling on the van der Waals surface. This is denoted the Solvent Accessible Surface (SAS) and illustrated in Figm e 16.7. 

The dielectric medium is normally taken to have a constant value of e, but may for some purposes also be taken to depend for example on the distance from M. For dynamical phenomena it can also be allowed to be frequency dependent i.e. the response of the solvent is different for a fast reaction, such as an electronic transition, and a slow reaction, such as a molecular reorientation. 

The simplest way to treat the solvent molecules of an electrolyte explicitly is to represent them as hard spheres, whereas the electrostatic contribution of the solvent is expressed implicitly by a uniform dielectric medium in which charged hard-sphere ions interact. A schematic representation is shown in Figure 2(a) for the case of an idealized situation in which the cations, anions, and solvent have the same diameters. This is the solvent primitive model (SPM), first named by Davis and coworkers [15,16] but appearing earlier in other studies [17]. As shown in Figure 2(b), the interaction potential of a pair of particles (ions or solvent molecule), i and j, in the SPM are ... 

Wood and Blundy (2001) developed an electrostatic model to describe this process. In essence this is a continuum approach, analogous to the lattice strain model, wherein the crystal lattice is viewed as an isotropic dielectric medium. For a series of ions with the optimum ionic radius at site M, (A(m))> partitioning is then controlled by the charge on the substituent (Z ) relative to the optimum charge at the site of interest, (Fig. 10) ... 

Mixed-solvent solutions of various cosolvent-water proportions are titrated and psKa (the apparent pKa) is measured in each mixture. The aqueous pKa is deduced by extrapolation of the psKa values to zero cosolvent. This technique was first used by Mizutani in 1925 [181-183]. Many examples may be cited of pKa estimated by extrapolation in mixtures of methanol [119,161,162,191,192,196,200], ethanol [184,188-190,193], propanol [209], DMSO [212,215], dimethylformamide [222], acetone [221], and dioxane [216]. Plots of psKa versus weight percent organic solvent, Rw = 0 — 60 wt%, at times show either a hockey-stick or a bow shape [119]. For Rw > 60 wt%, S-shaped curves are sometimes observed. (Generally, psKa values from titrations with Rw > 60 wt% are not suitable for extrapolation to zero cosolvent because KC1 and other ion pairing interferes significantly in the reduced dielectric medium [223].)... 

Section 3.3.4 pointed out that cosolvents alter aqueous ionization constants as the dielectric constant of the mixture decreases, acids appear to have higher pKa values and bases appear (to a lesser extent than acids) to have lower values. A lower dielectric constant implies that the force between charged species increases, according to Coulomb s law. The equilibrium reaction in Eq. (3.1) is shifted to the left in a decreased dielectric medium, which is the same as saying that pKa increases. Numerous studies indicate that the dielectric constant in the region of the polar head groups of phospholipids is 32, the same as the value of methanol. [381,446-453] Table 5.2 summarizes many of the results. 

To illustrate the use of the vector operators described in the previous section, consider the equations of Maxwell. In a vacuum they provide the basic description of an electromagnetic field in terms of the vector quantifies the electric field and 9C the magnetic field The definition of the field in a dielectric medium requires the introduction of two additional quantities, the electric displacement SH and the magnetic induction. The macroscopic electromagnetic properties of the medium are then determined by Maxwell s equations, viz. 

The interface is, from a general point of view, an inhomogeneous dielectric medium. The effects of a dielectric permittivity, which need not be local and which varies in space, on the distribution of charged particles (ions of the electrolyte), were analyzed and discussed briefly by Vorotyntsev.78 Simple models for the system include, in addition to the image-force interaction, a potential representing interaction of ions with the metal electrons. 

Effect of off-diagonal dynamic disorder (off-DDD). The interaction of the electron with the fluctuations of the polarization and local vibrations near the other center leads to new terms VeP - V P, Vev - Vev and VeAp - VAPd, VA - VAd in the perturbation operators V°d and Vfd [see Eqs. (14)]. A part of these interactions corresponding to the equilibrium values of the polarization P0l and Po/ results in the renormalization of the electron interactions with ions A and B, due to their partial screening by the dielectric medium. However, at arbitrary values of the polarization P, there is another part of these interactions which is due to the fluctuating electric fields. This part of the interaction depends on the nuclear coordinates and may exceed the renormalized interactions of the electron with the donor and the acceptor. The interaction of the electron with these fluctuations plays an important role in processes involving solvated, trapped, and weakly bound electrons. 

Equation (89) shows that the allowance for the variation of the charge of the adsorbed atom in the activation-deactivation process in the Anderson model leads to the appearance of a new parameter 2EJ U in the theory. If U — 2Er, the dependence of amn on AFnm becomes very weak as compared to that for the basic model [see Eq. (79)]. In the first papers on chemisorption theory, a U value of 13eV was usually accepted for the process of hydrogen adsorption on tungsten. However, a more refined theory gave values of 6 eV.57 For the adsorption of hydrogen from solution we may expect even smaller values for this quantity due to screening by the dielectric medium. 

Without explicit analytical expressions for the free energy derivative (as could be obtained for the case of a Bom ion in a dielectric medium), the integral has to be evaluated numerically by simulation. 

Continuum electrostatics approximations in which the solvent is represented as a featureless dielectric medium are an increasingly popular approach for the electrostatic... 

This value is considerably larger than those reported earlier for aqueous metal sols (43) colloidal gold - 58 mv, 32 mv platinum 44 mv, 30 mv lead - 18 mv. However, such large values might be expected for a low dielectric medium such as acetone. 

Some Basics. The field theory of electrostatics expresses experimentally observable action-at-a-distance phenomena between electrical charges in terms of the vector electric field E (r, t), which is a function of position r and time t. Accordingly, the electric field is often interpreted as force per unit charge. Thus, the force exerted on a test charge q, by this electric field is qtE. The electric field due to a point charge q in a dielectric medium placed at the origin r = 0 of a spherical coordinate system is... 

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