Internal Rotation and Kerr Effect in Polymer Molecules

Abstract

A theory of the Kerr effect (the electro-optical birefringence) of dilute polymer solutions is developed. If interactions between polymer molecules are neglected, the molar Kerr constant of polymer molecules is expressed as Km=(2πNA/45kT){(3〈utrγu〉−Tr〈u2γ〉)(kT)−1+[3Tr〈γγ′〉−〈(Trγ)(Trγ′)〉]},where NA is the Avogadro number; k is the Boltzmann constant; T is the absolute temperature; u, γ, and γ′ are the dipole-moment vector and the optical and static polarizability tensors of a polymer molecule, respectively; utr is the transpose of u; Tr denotes the trace of a matrix; and the averages refer to those about a free polymer molecule. In addition to Km, the two quantities 〈r2〉F/〈r2〉 and 〈x2〉F/〈y2〉F are calculated, which, respectively, are considered to be measures of the degrees of deformation and orientation of a polymer molecule by an electric field of strength F. Here r is the absolute value of the end-to-end vector r = (x y z)tr, the field being assumed along the x axis; the averages with and without subscripts refer to those in the presence and in the absence of the field, respectively. Alternative expressions for Km, etc., are derived by assuming the additivity of bond dipole moments and bond polarizabilities. An application is made to the polyethylene molecule, by using geometrical models proposed by Hoeve and by Nagai and Ishikawa, and also using Denbigh's and Bunn and Daubeny's values for bond polarizabilities. It is found, for example, that the elimination of the GG′ conformation for neighboring bonds, which has been known to increase markedly 〈r2〉, increases only slightly Km. It is pointed out that the molecular-weight dependence of specific Kerr constants, found by Le Fèvre and his collaborators for a variety of polymer—solvent pairs, should be ascribed to the excluded volume. Thus need for a theory of the effect of the excluded volume on Kerr constants, and for experimental data obtained at the so-called Θ condition, is emphasized.