The Local Electric Field

In the case of polar crystals such as the P-phase of PVDF, the elastic and dielectric properties are strongly coupled. This coupling, described formally in 

(6)-(9), necessitates a self-consistent treatment of the thermal, elastic, and dielectric properties. 

In the case where the independent variables are stress Oj, temperature T, and applied electric field strain and polarization Pf become the dependent quantities. 

Equations (6) and (7) express these relationships. are the elastic compliance constants OC are the linear thermal expansion coefficients 4 and d jj,are the direct and converse piezoelectric strain coefficients, respectively Pk are the pyroelectric coefficients and X are the dielectric susceptibility constants. The superscript a on Pk, Pk, and %ki indicates that these quantities are defined under the conditions of constant stress. If is taken to be the independent variable, then O and are the dependent quantities  

Q are the elastic stiffness constants, are the thermal stress coefficients, and gkj and are the direct and converse piezoelectic stress coefficients, respectively. The superscript , on Pk, p k, and Xki indicates that these quantities are now defined under the conditions of constant strain. 

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Because protein samples are actually ampholytes, when samples are loaded onto the gel and a current is appHed, the compounds migrate through the gel until they come to their isoelectric point where they reach a steady state. This technique measures an intrinsic physicochemical parameter of the protein, the pi, and therefore does not depend on the mode of sample appHcation. The highest sample load of any electrophoretic technique may be used, however, sample load affects the final position of a component band if the load is extremely high, ie, high enough to titrate the gradient ampholytes or distort the local electric field. 

Attempts have also been made to separate non-specific effects of the local electrical field from hydrogen-bonding effects for a small group of ionic liquids through the use of the k scale of dipolarity/polarizability, the a scale of hydrogen bond donor acidity, and the (i scale of hydrogen bond basicity (see Table 3.5-1) [13, 16]. 

From comparison of the optical properties of particles deposited on the same substrate and differing by their organization (Figs. 7 and 8) it can be concluded that the appearance of the resonance peak at 3.8 eV is due to the self-organization of the particles in a hexagonal network. This can be interpreted in terms of mutual dipolar interactions between particles. The local electric field results from dipolar interactions induced by particles at a given distance from each other. Near the nanocrystals, the field consists of the ap-... 

The fluid model is a description of the RF discharge in terms of averaged quantities [268, 269]. Balance equations for particle, momentum, and/or energy density are solved consistently with the Poisson equation for the electric field. Fluxes described by drift and diffusion terms may replace the momentum balance. In most cases, for the electrons both the particle density and the energy are incorporated, whereas for the ions only the densities are calculated. If the balance equation for the averaged electron energy is incorporated, the electron transport coefficients and the ionization, attachment, and excitation rates can be handled as functions of the electron temperature instead of the local electric field. 

The influence of structured electrodes or multipoint electrodes, which enhance the local electric field [20,21], as well as the effect of discharge polarity [20-22] and gap length [21,23,24], was investigated. 

A major advantage of fluorescence as a sensing property stems from the sensitivity to the precise local environment of the intensity, i.e., quantum yield (excited state lifetime (xf), and peak wavelength (Xmax). In particular, it is the local electric field strength and direction that determine whether the fluorescence will be red or blue shifted and whether an electron acceptor will or will not quench the fluorescence. An equivalent statement, but more practical, is that these quantities depend primarily on the change in average electrostatic potential (volts) experienced by the electrons during an electronic transition (See Appendix for a brief tutorial on electric fields and potentials as pertains to electrochromism). The reason this is more practical is that even at the molecular scale, the instantaneous electric... 

Fig. 5 Absorption and fluorescence emission spectra of the 3-hydroxychromone dye F4N1 in the absence (black) and presence (red) of a local electric field, which promotes the excitation charge transfer leading from the ground state to the N state. In the presence of the local electric field, the energy of the N state is reduced, causing a red shift of the N emission peak and an increase in its intensity relative to the T emission peak. The change in relative intensities of the N and T peaks reflects a shift in the excited state tautomeric equilibrium toward the N state...

As suggested before, the role of the interphasial double layer is insignificant in many transport processes that are involved with the supply of components from the bulk of the medium towards the biosurface. The thickness of the electric double layer is so small compared with that of the diffusion layer 8 that the very local deformation of the concentration profiles does not really alter the flux. Hence, in most analyses of diffusive mass transport one does not find any electric double layer terms. For the kinetics of the interphasial processes, this is completely different. Rate constants for chemical reactions or permeation steps are usually heavily dependent on the local conditions. Like in electrochemical processes, two elements are of great importance the local electric field which affects rates of transfer of charged species (the actual potential comes into play in the case of redox reactions), and the local activities... 

The second cause of broadening of electronic spectra is the fluctuations in the structure of the solvation shell surrounding the fluorophore. The distribution of solute-solvent configurations and the consequent variation in the local electric field leads to a statistical distribution of the energies of the electronic transitions. This phenomenon is called inhomogeneous broadening (for a review see Nemkovich et al., 1991). 

The excitation spectrum proves even more useful near the surface. Since anisotropic molecules at the surface of a liquid tend to orient relative to the surface tangent, one might expect the excitation spectrum to be sensitive to such orientation. For example, suppose we take the extreme case in which molecules at the surface are oriented with their transition moments perpendicular to the surface tangent. Then the only field component which can excite these molecules is the radial field at the surface. When one recalls that only the N type vector field has radial components, one expects that a calculation of the excitation spectrum of such a molecular layer will yield half as many resonant features as shown in Figure 8.4. Indeed this is the case. Figure 8.7 shows the calculated surface average of the square modulus of the radial component of the local electric field, < E er 2>J, where sr is the radial unit vector. 

Field emission is the emission of electrons from a solid under an intense electric field, usually at ambient temperatures. It occurs by the quantum mechanical tunneling of electrons through a potential barrier (Fig. 13.1). This leads to an exponential dependence of emission current density J on the local electric field, as given by the Fowler Nordheim equation,... 

In general, (Eioc) 5 E, since the local electric field is averaged over the atomic sites and not over the spaces between these sites. In metals, where valence electrons are free (nonlocalized electrons), the assumption (Eioc) = E is reasonable, but for bound valence electrons (dielectrics and semiconductors) this relation needs to be known. However, for our purpose of a qualitative description of optical properties, we will still retain this assumption. 

Free diffusion of molecules in solution is characteristically a haphazard process with net directionality determined only by solute gradients and diffusion coefficients. Within cellular and extracellular spaces, however, diffusion can be strongly influenced by noncovalent interactions of solvent and solute molecules with membranes as well as the cellular and extracellular matrix. Channels and orifices can also alter the movement of solute and solvent molecules. These interactions can greatly alter the magnitude of the diffusion coefficient for a molecule from its isotropic value D in water to apparent diffusion coefficient D (which often can be directionally resolved into D, Dy, and D ). The parameter A, known as the tortuosity, equals DID y. In principle, A has X, y, and z components that need not be equal if there is any anisotropy in the local electrical fields or porosity of the matrix. 

In order to understand the diffuse layer in detail, we need to go back to the fnndamental eqnations of electrostatics due to J.C. Maxwell. The equation of interest relates the local electric field E(r) at the position vector r to the net local electric charge density p(r) ... 

Orientation polarization can occur in materials composed of molecules that have permanent electric dipole moments. The permanent dipoles tend to become aligned with the apphed electric field, but entropy and thermal effects tend to counter this alignment. Thus, orientation polarization is highly temperature-dependent, unlike the forms of induced polarization which are nearly temperature-independent. In electric fields of moderate intensity, the orientation polarization is proportional to the local electric field, as for the other forms of polarization... 

The supporting medium (aqueous or organic solvents membrane-mimetic compartments) also has a profound influence on the optical and electro-optical properties of nanosized semiconductor particles. This dielectric confinement (or local field effect) originates, primarily, in the difference between the refractive indices of semiconductor particles and the surrounding medium [573, 604], In general, the refractive index of the medium is lower than that of the semiconductor particle, which enhances the local electric field adjacent to the semiconductor particle surface as compared with the incident field intensity. Dielectric confinement of semiconductor particles also manifests in altered optical and electro-optical behavior. 

Equation (2) expresses the model of the crystal polarization used in the molecular modeling of PVDF reported in this chapter, where is the dipole of each repeat unit of the single chain in vacuum, Ap is the change in dipole moment of the repeat unit of the chain in going from the vacuum environment to the environment of the packed crystal and (cos tp) is the attenuation of the dipole moment of the repeat unit along the fe-axis due to thermally stimulated oscillations about thec-axis. Ap is directly related to the local electric field (Eioc, V/m) through the repeat unit polarizability (ot, m ) ... 

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