Birefringence polarizability

Cao J, Berne BJ (1993) Theory of polarizable liquid crystals optical birefringence. J Chem Phys 99(3) 2213-2220... 

Birefringence is one of the simplest methods for the characterization of molecular orientation in polymers. The polarizability of a structural unit is usually not equivalent in all directions, leading to three independent refractive indices along its principal axes. In an isotropic sample, a single averaged macroscopic refractive index is observed whereas birefringence or trirefringence is observed... 

The theory of strain birefringence is elaborated in terms of the RIS model as applied to vinyl polymer chains. Additivity of the polarizability tensors for constituent groups is assumed. Stress-birefringence coefficients are calculated for PP and for PS. Statistical weight parameters which affect the Incidences of various rotational states are varied over ranges consistent with other evidence. The effects of these variations are explored in detail for isotactic and syndlotact/c chains. 

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)]. 

The birefringence An of a nematic phase depends on the anisotropic polarizabilities a, a L and the degree of order S (Eq. (3), p. 114). The polarizabilities ae and of a nematic phase parallel and perpendicular to the director respectively obey the following relations 53)... 

The anisotropy of polarizability can be positive (eg, polycarbonate) as well as negative (eg, polystyrene). This offers the possibility of minimizing birefringence by copolymerization or blending of suitable polymers with the right mixture ratio, eg, blends of poly(phenylene ether) (PPE) and polystyrene (PS). The magnitude of birefringence of axial-symmetrically oriented polymers vs their molecule orientation has been described (182). 

The third-rank tensors Zafir od a1 aft describe non linear response of the electron cloud to first order in E. These quantities are sometimes referred to as hypermagnetizabilities, which are related to Cotton-Motton effect (4-6), (CME- the birefringence of light in gases in a constant magnetic field) and as shielding polarizabilities (7-14). 

From these expressions, the birefringence from the scattering of oriented, anisotropic particles can be calculated. The dichroism can also be calculated, but in this limit, it is only present if the polarizability has an imaginary component. In other words, only absorbing particles are predicted to be dichroic. 

The polarizability tensor, a, introduced in section 4.1.2, is a measure of the facility of the electron distribution to distortion by an imposed electric field. The structure of the electron distribution will generally be anisotropic, giving rise to intrinsic birefringence. This optical anisotropy reflects the average electron distribution whereas vibrational and rotational modes of the molecules making up a sample will cause the polarizability to fluctuate in time. These modes are discrete, and considering a particular vibrational frequency, vk, the oscillating polarizability can be modeled as... 

The Lorentz-Lorenz equation can be used directly to model the birefringence of a solution of rigid rod molecules subject to an orienting, external field. Figure 7.2 shows a representative molecule, which is modeled as having a uniaxial polarizability of the form... 

The Lorentz-Lorenz equation can be used to express the components of the refractive index tensor in terms of the polarizability tensor. Recognizing that the birefringence normalized by the mean refractive index is normally very small, ( A/i / 1), it is assumed that Aa /a 1, where the mean polarizability is a = (al + 2oc2)/3 and the polarizability anisotropy is Aa = a1-a2. It is expected that the macroscopic refractive... 

The approach just sketched in terms of effective properties has also been applied to other vibrational spectroscopies, such as Raman [9], IR linear dichroism [10], VCD [11] and VROA [12], as well as to (hyper)polarizabilities [47 19] and birefringences of systems in a condensed phase (see refs. [50,51] and the contribution by Rizzo in this... 

This approach is based on the introduction of molecular effective polarizabilities, i.e. molecular properties which have been modified by the combination of the two different environment effects represented in terms of cavity and reaction fields. In terms of these properties the outcome of quantum mechanical calculations can be directly compared with the outcome of the experimental measurements of the various NLO processes. The explicit expressions reported here refer to the first-order refractometric measurements and to the third-order EFISH processes, but the PCM methodology maps all the other NLO processes such as the electro-optical Kerr effect (OKE), intensity-dependent refractive index (IDRI), and others. More recently, the approach has been extended to the case of linear birefringences such as the Cotton-Mouton [21] and the Kerr effects [22] (see also the contribution to this book specifically devoted to birefringences). 

Birefringences are mostly observed in condensed phases, especially pure liquids or solutions, since the strong enhancement of the effects allows for reduced dimensions (much shorter optical paths) of the experimental apparatus. Nowadays measurements of linear birefringences can be carried out on liquid samples with desktop-size instruments. Such measurements may yield information on the molecular properties, molecular multipoles and their polarizabilities. In some instances, for example KE, CME and BE, measurements (in particular of their temperature dependence) have been carried out simultaneously on some systems. From the combination of data, information on electric dipole polarizabilities, dipole and quadrupole moments, magnetizabilities and higher order properties were then obtained. 

Rizzo reviews in a unitary framework computational methods for the study of linear birefringence in condensed phase. In particular, he focuses on the PCM formulation of the Kerr birefringence, due to an external electric field yields, on the Cotton-Mouton effect, due to a magnetic field, and on the Buckingham effect due to an electric-field-gradient. A parallel analysis is presented for natural optical activity by Pecul Ruud. They present a brief summary of the theory of optical activity and a review of theoretical studies of solvent effects on these properties, which to a large extent has been done using various polarizable dielectric continuum models. 

A decrease of An or Ak upon addition of salt is quite common for flexible polyelectrolytes (see e.g. [56, 57]). It is generally interpreted as being a consequence of the conformational change brought about by the rise of ionic strength. When the coil size is reduced, the optical and electrical polarizability of the polyions is diminished. This leads to the observed drop of electric birefringence. Coiling of the polyion can also lead to an increase of counted-... 

The simplified schematic in Figure 2a shows the essential features of the effect. Optically anisotropic molecules in the solution are preferentially oriented by the applied field E(t), resulting in a difference of refractive indices for components of polarized light parallel and perpendicular to the bias field which is measured as a birefringence. The basic theoretical problem is to evaluate this effect in terms of anisotropies of polarizability Aa. referred to molecular axes which produce a time dependent effect when the molecules are preferentially oriented by the field. For no anisotropy in absence of the field, the effect must be an evgn function of field strength, and at low fields proportional to E. A remarkable feature of the effect is that for molecules with permanent dipole moments the response af-... 

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