Excess polarizability

The excess polarizability a is related to the (optical) dielectric constants o and of the solvent and of the solution, respectively, in the following manner ... 

The particles of the preceding treatment are here replaced by small elements of volume of the solution. The excess polarizability of one of these volume elements due to the deviation of its concentration from the average may be written... 

Equation (14.5) establishes the linear relationship between the RI sensitivity and the resonant mode s frequency shift for a given number of captured molecules at the OFRR surface. Also, note that the magnitude of the shift is directly proportional to the surface density and the excess polarizability of the bound molecules. Although (14.5) is developed for the OFRR, it is generally applicable to other types of ring resonator sensors, as is shown by Zhu, et al.27... 

As has been pointed out (63), this is a rather artificial model and, moreover, its application is quite unnecessary. In fact, (a> can be calculated from the refractive index increment (dnjdc), as has extensively been done in the field of light scattering. This procedure is applicable also to the form birefringence effect of coil molecules, as the mean excess polarizability of a coil molecule as a whole is not influenced by the form effect. It is still built up additively of the mean excess polarizabilities of the random links. This reasoning is justified by the low density of links within a coil. In fact, if the coil is replaced by an equivalent ellipsoid consisting of an isotropic material of a refractive index not very much different from that of the solvent, its mean excess polarizability is equal to that of a sphere of equal volume [cf. also Bullough (145)]. 

Temperature and pressure fluctuations in the solvent medium create density fluctuations with nonzero excess polarizability. However, the contributions of these fluctuations to the scattered intensity can generally be neglected compared to those of molecular or colloidal solutes. As a result, the volume integral in Eq. (11) is dominated by contributions from solute particles. Suppose that the y th solute particle has volume Vj and center-of-mass located at position rj(t). Then the contribution of this particle to the integral in Eq. (11) can be expressed as an integral... 

The excess polarizability cannot be determined experimentally, but it can be shown to be given by... 

In general, the polarizability is a tensor whose invariants, trace and anisotropy, give rise to polarized and fully depolarized light scattering, respectively. Collision-induced light scattering is caused by the excess polarizability of a collisional pair (or a larger complex of atoms or molecules) that arises from the intermolecular interactions. In Section I.l, we are concerned with the definition, measurement, and computation of interaction-induced polarizabilities and their invariants. 

As we already discussed in Section 2.5.3 for excess polarizabilities of molecules dissolved in a solvent, the London dispersion interactions between molecules in a solvent medium may be very different from those of isolated molecules in free space. The intrinsic permanent dipole moment, p, and polarizability of an isolated gas molecule, a, may be different in the liquid state or when dissolved in a medium, and this can only be determined by experiment. 

As we have already discussed in Section 2.5.3 for excess polarizabilities of molecules dissolved in a solvent, and in Section 2.6.4 for van der Waals interactions in a medium, when two molecules 1 and 2 are dissolved in a medium 3, the van der Waals forces between them are reduced because of the dielectric screening of the medium. This reduction is particularly important for liquids with high dielectric constants. The attraction force is decreased by a factor of the medium s er for Keesom and Debye interactions and by a factor of e] for London dispersion interactions. This strong reduction in the attractive pair potential means that the contributions of molecules further apart tend to be relatively minor, and each interaction is dominated only by contributions from its nearest neighbors. 

Bi is an internal field function (equal to 1 for long rods), and ai and a2 are the excess polarizabilities of the particle. The orientation distribution function, / (0), becomes... 

All the parameters in Equation (9-39) except the polarizability a can be measured directly, a is the excess polarizability, that is, the difference between the polarizability of the solute and that of the displaced solvent in dilute solutions. The polarizability of gases is related to the dielectric constant e via e — 1 = AnaiN K), where N is the number of molecules in the volume F. 

In these, and n are the unit column vectors in the polarization directions of scattered and incident light, respectively, r(a, s ) the vector between contour points a and a, r(a) the excess polarizability tensor per unit length at contour point a relative to a laboratory-fixed coordinate system, and the superscript T the tianspose of the column vector. 

Theoretical considerations showed that at moderate concentrations the dispersion forces dominate over the electrostatic ones and that dy/dcj should be proportional to where Bj, in turn, is proportional to the static excess polarizability of the ion in water, R -Rw (Bostrbm et al. 2001). For multivalent anions (not considered in (Bostrom et al. 2001)) Bj/zi needs to be taken into account to obtain the excess polarizability, in terms of the molar refraction Ri/zi -Rw, where Bw = 3.67 cm mol and some Rj values (taken from (Marcus 1998)) are shown in Table 2.1. The correlation is far from perfect, as seen in Fig. 4.1, but the general trend is confirmed. 

The next step of interpretation of Eq. (3) is the development of an expression for the excess polarizability, a, caused by the dissolution of the polymer in the solvent. This is done by remembering that the polarizability at the frequency of light is related to the squares of the refractive index, n (as well as the permittivity, the dielectric constant, e). The connection between a and n is written at the top of Fig. A.3.4. Its background can be found in any physical chemistry text. Assuming the square of the refractive index changes linearly with concentration, one can draw the sketch in the figure. The refractive index n refers to the pure solvent, while n is the refractive index of the solution of concentration Cj. 

The polarizabilities and the medium permittivity are, however, real quantities. Its exact form depends on the molecular details of the three substances. For the case of isolated (gas phase) molecules, e iiv) = 1, and if there is only one electronic absorption frequency, the ionization frequency Vj, the excess polarizabilities are equal to the total ones, which are shown to be... 

The theoretical description of the scattering function is facilitated by dividing the chain into m identical subxmits characterized by an excess polarizability density. All the subunits are identical in composition, so it is not necessary to weight the result by the local polarizability density. The scattering function can then be expressed as ... 

Figure 247. Spatial variations of the local solvent density and the polymer concentration lead to fluctuation in the excess polarizability a x- Th plot in b shows along the white line in a.

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