Bulk polarization, equation

Methods for determining permanent dipole moments and polarizabilities can be arbitrarily divided into two groups. The first is based on measuring bulk phase electrical properties of vapors, liquids, or solutions as functions of field strength, temperature, concentration, etc. following methods proposed by Debye and elaborated by Onsager. In the older Debye approach the isotope effects on the dielectric constant and thence the bulk polarization, AP, are plotted vs. reciprocal temperature and the isotope effect on the polarizability and permanent dipole moment recovered from the intercept and slope, respectively, using Equation 12.5. 

The observed bulk polarization is given by an expression analogous to equation 19 ... 

The first and third parentheses on the right hand side of Eq. (86) represent the electrodes and electrolyte contact resistances and the second term represent the bulk resistance of the electrolyte. Substitution of Eqs. (78-86) into (85) gives the following current-voltage relation, or polarization equation, for PEMFC... 

We begin with a relatively simple model, which was suggested some years ago by Mukeqee and which provides considerable insight on solubilization of small amounts of solutes in spherical micelles. Suppose that an aqueous micellar solution has reached its solubilization limit and is in equilibrium with an excess liquid phase of a pure hydrocarbon or some other compound of low polarity. Equating the chemical potentials j,g and of the solute in the bulk organic phase and in the micelles, we have... 

At this stage the molecular and optical properties are neatly entwined. In its semiclassical macroscopic counterpart, Eq. (17) is termed a material equation because of its engagement of a bulk polarization P the microscopic and bulk polarisations are, for simple cubic systems, related through the succinct expression... 

According to Frohlich, a pure condensed dielectric consisting of polarizable molecules with a permanent dipole moment p may be formally represented by a continuum permittivity accounting for the molecular polarizability, embedded in the bulk continuum with the effective permittivity 8. The fundamental polarization equation for such a polar dielectrics is... 

The variation in absorption due to the electric field modulation (Equation 19.16) is a nonlinear optical effect. We now consider the origin of nonlinear behavior in materials. In a classical description [89-91], the electric field interacts with the charges (q) in an atom through the force (qF). which displaces the centre of the electron density away from the nucleus. This results in charge separation and thus in a field-induced dipole pi. For an assembly of atoms, the average summation over all atoms ultimately gives rise to the bulk polarization P vector of the material. P opposes the externally applied field and is given by ... 

Macedo and others (Hodge et al. [1975, 1976]) have stressed the electric modulus formalism (M = 1/c ) for dealing with conducting materials, for the reason that it emphasizes bulk properties at the expense of interfacial polarization. Equation (112) transforms to... 

Relating Macroscopic and Molecular Properties. For many or-ganic/polymeric materials, the bulk polarization is best understood through averaging of the molecular polarizabilities of the weakly-interacting constituent materials. The molecular polarizability is expressed in a series expansion, similar to equation 2 for the polarization, (28)... 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

Electrode reactions are heterogeneous since they occur at interfaces between dissimilar phases. During current flow the surface concentrations Cg j of the substances involved in the reaction change relative to the initial (bulk) concentrations Cy p Hence, the value of the equilibrium potential is defined by the Nemst equation changes, and a special type of polarization arises where the shift of electrode potential is due to a change in equilibrium potential of the electrode. The surface concentrations that are established are determined by the balance between electrode reaction rates and the supply or elimination of each substance by diffusion [Eq. (4.9)]. Hence, this type of polarization, is called diffusional concentration polarization or simply concentration polarization. (Here we must take into account that another type of concentration polarization exists which is not tied to diffusion processes see Section 13.5.)... 

Consider the case when the equilibrium concentration of substance Red, and hence its limiting CD due to diffusion from the bulk solution, is low. In this case the reactant species Red can be supplied to the reaction zone only as a result of the chemical step. When the electrochemical step is sufficiently fast and activation polarization is low, the overall behavior of the reaction will be determined precisely by the special features of the chemical step concentration polarization will be observed for the reaction at the electrode, not because of slow diffusion of the substance but because of a slow chemical step. We shall assume that the concentrations of substance A and of the reaction components are high enough so that they will remain practically unchanged when the chemical reaction proceeds. We shall assume, moreover, that reaction (13.37) follows first-order kinetics with respect to Red and A. We shall write Cg for the equilibrium (bulk) concentration of substance Red, and we shall write Cg and c for the surface concentration and the instantaneous concentration (to simplify the equations, we shall not use the subscript red ). 

This equation links the current density to surface concentration. In the case discussed (where there is no activation polarization), the Nemst equation unequivocally links the electrode s polarization to the difference between surface and bulk concentration ... 

All these equations differ from the corresponding equations for diffusion polarization, only in that the equilibrium concentration Cq appears in them instead of bulk concentration Cy. Formally, diffusion can be regarded as a first-order reaction, the fimiting diffusion flux being proportionaf to the first power of concentration. 

Many approaches have been used to correlate solvent effects. The approach used most often is based on the electrostatic theory, the theoretical development of which has been described in detail by Amis [114]. The reaction rate is correlated with some bulk parameter of the solvent, such as the dielectric constant or its various algebraic functions. The search for empirical parameters of solvent polarity and their applications in multiparameter equations has recently been intensified, and this approach is described in the book by Reich-ardt [115] and more recently in the chapter on medium effects in Connor s text on chemical kinetics [110]. 

The question whether this reaction is monoeidic or enieidic was not discussed by that author, but in view of the low polarity of bulk THF it seems likely that the principal propagating species is ion-pairs but since there is no definite evidence we will denote the rate constant by kp, but write the equations in terms of x rather than Exr. For this equilibrium polymerisation therefore... 

In the equation s is the measured dielectric constant and e0 the permittivity of the vacuum, M is the molar mass and p the molecular density, while Aa and A (po2) are the isotope effects on the polarizability and the square of the permanent dipole moment respectively. Unfortunately, because the isotope effects under discussion are small, and high precision in measurements of bulk phase polarization is difficult to achieve, this approach has fallen into disfavor and now is only rarely used. Polarizability isotope effects, Aa, are better determined by measuring the frequency dependence of the refractive index (see below), and isotope effects on permanent dipole moments with spectroscopic experiments. 

Equation (4) states that the linear deposition rate vj is a diffusion controlled boundary layer effect. The quantity Ac is the difference in foulant concentration between the film and that in the bulk flow and c is an appropriate average concentration across the diffusion layer. The last term approximately characterizes the "concentration polarization" effect for a developing concentration boundary layer in either a laminar or turbulent pipe or channel flow. Here, Vq is the permeate flux through the unfouled membrane, 6 the foulant concentration boundary layer thickness and D the diffusion coefficient. 

It is also necessary to note that the success of TSR techniques to obtain information on trapping states in the gap depends on whether or not the experiment can be performed under conditions that justify equation (1.2) to be reduced to simple expressions for the kinetic process. Usually, the kinetic theory of TSR phenomena in bulk semiconductors—such as thermoluminescence, thermally stimulated current, polarization, and depolarization— has been interpreted by simple kinetic equations that were arrived at for reasons of mathematical simplicity only and that had no justified physical basis. The hope was to determine the most important parameters of traps— namely, the activation energies, thermal release probabilities, and capture cross section— by fitting experimental cnrves to those oversimplified kinetic descriptions. The success of such an approach seems to be only marginal. This situation changed after it was reahzed that TSR experiments can indeed be performed under conditions that justify the use of simple theoretical approaches for the determination of trapping parameters ... 

The first equation is scalar, and has a wave solution with velocity Vi = -J c /p). This is the longitudinal wave of eqn (6.7). It is sometimes called an irrotational wave, because V x u = 0 and there is no rotation of the medium. The second equation is vector, and has two degenerate orthogonal solutions with velocity v = s/(cu/p)- These are the transverse or shear waves of eqn (6.6) the degenerate solutions correspond to perpendicular polarization. They are sometimes called divergence-free waves, because V u = 0 and there is no dilation of the medium. Waves in fluids may be considered as a special case with C44 = 0, so that the transverse solutions vanish, and C = B, the adiabatic bulk modulus. 

The temperature variation of ttv may be analyzed by a relationship analogous to the Clapeyron equation to yield the two-dimensional equivalent to the heat of vaporization. The numerical values obtained for this quantity more nearly resemble the bulk values for hydrocarbons than those for polar molecules. This suggests that most of the change in the surface transition involves the hydrocarbon tail of the molecule rather than the polar head. 

Concentration polarization occurs when the concentrations of reactants or products are not the same at the surface of the electrode as they are in bulk solution. For Reaction 17-1, the Nemst equation should be written... 

A simple model of a solute molecule of dipole moment fj, and molecular radius a, surrounded by a mixture of two solvents N and P of polarity functions f(DN), f Df) and bulk mole fractions N and xv yields an equation for dielectric enrichment as the local mole fractions change to j>N, yP the equilibrium condition is... 

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