Kuhn–Grün analysis of polarized Rayleigh and Raman scattering experiments to deduce segmental orientation in polymeric systems

Abstract

The intensity of Rayleigh and Raman scattered light from molecular structural units is proportional to the quadratic polarizability tensor and the derived polarizability tensor, respectively. The orientation of the polymer skeletal backbone is directly related to the orientation of the scattering structural units comprising it. The mathematical structure of the quadratic scattering tensors for a single Kuhn bond are deduced in terms of the unit vector along a Kuhn bond from symmetry considerations alone (Boehler 1987). Subsequent application of the Kuhn–Grun conditional probability analysis (Kuhn and Grun, Kolloid Z 101:248–271, 1942), which uses a freely jointed chain model, yields a general expression for the quadratic Raman and polarizability tensors for a single chain segment with five independent terms. Each term is multiplied by a spectroscopic parameter that is a complex function of the intrinsic spectroscopic tensors and the orientation distribution of monomers within an elementary Kuhn bond. A small stretch analysis of the Kuhn–Grun representation of the quadratic polarizability reveals that independent fourth moments of the segmental orientation distribution function can only be determined experimentally when the deformation or stretch of the flexible polymer is large and finite, thus severely restricting a primary advantage of the Raman and Rayleigh scattering methods. A general segmental additivity theorem is rigorously proven which demonstrates that polarized scattering experiments physically reflect the average orientation and stretch of flexible polymer skeletal backbone segments, or sub-segments, independent of chain architecture or molecular weight. Constitutive equations are fundamentally constructed to determine Kuhn bond orientation and are intrinsically related to the Kuhn–Grun analysis. The decoupling approximation, which is always invoked in Doi–Edwards type models of entangled polymeric liquids, is examined in light of the Kuhn–Grun analysis.