Macroscopic fields

Next we consider the net field at the molecule. This turns out to be the sum of two effects the macroscopic field given by Eq. (10.12) plus a local field that is associated with the charge on the surface of the cavity surrounding the molecule of interest. The latter may be shown to equal (l/3)(aj j/eo). Hence the net field at the molecule is... 

Using calculated values of the macroscopic field and Eq. (13), we computed the total overpotential as a function of charge for direct comparison with the macroscopic capacitance experiment. One set of results is shown in Fig. 13. Actually, Fig. 13 shows a favorable case. Reported experiments are not all in agreement. We refer to Ref. 36 for further details. 

Here, E n = 0 on Sp (Neumann type boundary condition), where n is the unit outward normal from the pore region, and T> is compact. E can be interpreted as the microscopic electric field induced in the pore space when a unit macroscopic field e is applied, assuming insulating solid phase and uniform conductivity in the pore fluid. Its pore volume average is directly related to the tortuosity ax ... 

Evaluating the performance of a gas-solid transport system usually requires a means of macroscopic field description of the distribution of basic flow properties such as pressure, mass fluxes, concentrations, velocities, and temperatures of phases in the system. To conduct such an evaluation, the Eulerian continuum or multifluid approach is usually the best choice among the available approaches. 

In condensed matter, the density and therefore the electrostatic interaction between the microscopic dipoles is quite high. Hence, the local field Eioc at the position of a particular dipole is given by the superposition of the applied macroscopic field Eq and the sum... 

These effective properties represent the main result of the theoretical formulation of NLO phenomena for solvated systems [2-6] as they describe the response of the solute in terms of the macroscopic field in the surrounding medium and thus they may be directly related to the macroscopic properties determined from experiments. 

The response of a medium to a macroscopic field E(t) generated by the superposition of a static and an optical component (E(f) = E° + E cos(< h)) is represented by the dielectric polarization vector (dipole moment per unit of volume) P(t) ... 

In Equation (2.183) new surface charges, qex, have been introduced these charges can be described as the response of the solvent to the external field (static or oscillating) when the volume representing the molecular cavity has been created in the bulk of the solvent. We note that the effects of qex in the limit of a spherical cavity coincide with that of the cavity field factors historically introduced to take into account the changes induced by the solvent molecules on the average macroscopic field at each local position inside the medium more details on this equivalence will be given in Section 2.7.4. 

The OWB equations obtained in this semiclassical scheme analyse the effective polarizabilities in term of solvent effects on the polarizabilities of the isolated molecules. Three main effects arise due to (a) a contribution from the static reaction field which results in a solute polarizability, different from that of the isolated molecules, (b) a coupling between the induced dipole moments and the dielectric medium, represented by the reaction field factors FR n, (c) the boundary of the cavity which modifies the cavity field with respect the macroscopic field in the medium (the Maxwell field) and this effect is represented by the cavity field factors /c,n. 

Accounting for the effect of the host material on the interactions between the dipoles involves the refractive index, the relative orientation of the charges, and the local or internal field. The local or internal field problem is associated with the fact that molecules in a host medium occupy a particular volume or a cavity . This cavity description has been used to formalize the description of interactions between dipoles. The region occupied by the molecule results in an additional correction so the field acting on the molecule will be an effective local field rather than the mean macroscopic field. The field acting on the molecule may be an applied electromagnetic field (such as in absorption), the effect of another dipole or a combination of the two. 

An individual atom or ion in a dielectric is not subjected directly to an applied field but to a local field which has a very different value. Insight into this rather complex matter can be gained from the following analysis of an ellipsoidal solid located in an applied external field Ea, as shown in Fig. 2.28. The ellipsoid is chosen since it allows the depolarizing field Edp arising from the polarization charges on the external surfaces of the ellipsoid to be calculated exactly. The internal macroscopic field Em is the resultant of Ea and Edp, i.e. Ea — Edp. 

An electric field Eloc acting on a molecule is termed a local field since it may differ substantially from the macroscopic field outside the medium (because of the influence of neighboring molecules). The field will, in general, distort the electron density distribution p(r) in a molecule. Such... 

The molecular field acting at the location of a water molecule is composed of a macroscopic field, E = — (dip/ dr), a local field (the Lorentz field El) that occurs because each molecule is surrounded by a dielectric medium (constituted of all the other water molecules), El = P/3eo, plus an additional field, Ep, due to the neighboring dipoles.18,27 Assumingthat the additional fieldis generated only by the neighboring molecules from the same and adjacent icelike water layers (see Figure 2), at each of the sites (I or II) of layer j the field is provided by13... 

In addition to the field generated by the adjacent dipoles, there is a macroscopic fieldE due to the presence of charges and of the average polarization P of the medium. In the Lorentz treatment of polarization, for a constant macroscopic field in a linear and homogeneous medium of dielectric constant e (hence satisfying P = e0(e — 1)E), the local field E n(fl at a site of a selected dipole is related to the macroscopic field E via19... 

The macroscopic field at the interface is related to the electric potential, via the expression E = —chp(z)/dz, while the field generated by the water dipoles is17... 

One of the models follows the Lorentz—Debye theory, as summarized by Jackson20 and Frenkel.21 In a constant macroscopic field Eo, the macroscopic polarization P is... 

The effective Eeld E, is the sum between the macroscopic field Eo and the internal field due to the other molecules of the medium. The latter can be calculated by adding the individual contributions of the other molecules close to the selected one, Elncal, and subtracting the contribution from the same molecules treated in an average continuum approximation described by the polarization P ... 

At microscopic scale, a water molecule is subjected to a local electric field, which is the sum between a macroscopic field generated by sources from outside the medium, and a field generated by the other water molecules surrounding it [26]. For a uniformly polarized medium, the latter corresponds to the field generated in a spherical cavity in a medium, due to the polarization of the whole medium (the Lorenz field ZiLorenz = [26]). Therefore, the local field acting on a water... 

The traditional double-layer theory combines the Poisson equation with the assumption that the polarization is proportional to the macroscopic electric field, and uses Boltzmann distributions for the concentrations of the ions. The potential of mean force, which should be used in the Boltzmann distribution, is approximated by the mean value of the electrical potential. The macroscopic field E and the polarization P are related via the Poisson equation... 

In traditional Lorentz—Debye theory, the local field that acts on a water dipole is given by E + P/3so, where E is the macroscopic field and P is the polarization [5], In the present case, the local field /ilocal should also include the field Ep generated by the neighboring dipoles of a water molecule. 

In the traditional double layer theory, the average dipole moment of a water molecule is proportional to the macroscopic field (the derivative of the electrical potential)... 

The electrostatic free energy (Eq. (5a)) accounts for the interaction of the charges and dipoles with the macroscopic field in the conventional theory. However, in the polarization model, there is an additional contribution to the energy, because the dipoles interact with the field that generates their correlation. As it will be shown in the next section, Eq. (2) can be rewritten as [32] ... 

Let us first review the basics of the Lorentz theory for polarization. If one assumes that a constant macroscopic field is applied to a homogeneous medium of dielectric constant s, the polarization through the medium will be uniform. However, the polarization of a molecule is not proportional to the macroscopic electric field (created by sources external to the dielectric), but to the local electric field, which contains also the field generated by all the other molecules of the dielectric. To account for the latter, one can separate the medium in a spherical cavity (in which the central molecule and its molecular neighbors reside, see Fig. 1 A) and the rest of the medium, which... 

For extracting microscopic properties from experiment the so-called local field factors are needed [131], and, vice versa, also to obtain macroscopic values from computed (microscopic) results. The concept of relating the macroscopic field and the actual (or local) field experienced by a molecule goes back to Lorentz [113,132],... 

In condensed media consisting of molecules, the intermolecular forces such as permanent and induced dipole interactions are generally small compared to intramolecular chemical binding forces. Therefore, the molecular identities and properties are conserved to a certain extent. They nevertheless differ significantly from those of an isolated molecule in the gas phase. Therefore, both in linear and non-linear optics the question arises of how to relate molecular to macroscopic properties. More specifically, how do the individual permanent and induced dipole moments of the molecules translate into the macroscopic polarization of the medium The main problem is to determine the local electric field acting on a molecule in a medium which differs from the average macroscopic field E (Maxwell field) in this medium. 

We have shown above how the reaction field model can be used to estimate solute-solvent interactions in the absence of external fields. Now we introduce effective polarizabilities that connect the Fourier components of the induced dipole moment (33) with the macroscopic fields in the medium. In the linear case, the Fourier component / induced by an external optical field can be represented by the product of the macroscopic field amplitude " and an effective first-order polarizability a(-tu w) using (93). 

Hie polarizability a(-o) a)) is involved in several linear optical experiments including refractive index measurements. Equation (93) shows that the solute molecule experiences a local field which is larger than the macroscopic field by the cavity field factor/ " and by the reaction field factor For typical media the magnitude of the product is of the order of 1.3-1.4. In the case of... 

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