The Rouse–Mooney model for coherent quasielastic neutron scatterings of single chains well entangled in polymer melts

Abstract

The dynamic structure factor (DSF) for single (labeled) chains well entangled in polymer melts has been developed based on the Rouse-Mooney picture; the DSF functions derived from the Langevin equations of the model in both discrete and continuous forms are given. It is shown that for all practical purposes, it is sufficient to use the continuous form to analyze experimental results in the "safe" q region (q being the magnitude of the scattering wave vector q) where the Rouse-segment-based theories are applicable. The DSF form reduces to the same limiting form as that of the free Rouse chain as q(2)a(2) or q(2)R(2)-->infinity (a and R being the entanglement distance and the root mean square end-to-end distance, respectively), confirming what has been expected physically. The natural reduction to the limiting form allows the full range of DSF curves to be displayed in terms of the reduced Rouse variable q(2)(Z(d)t)(0.5) in a unified way. The displayed full range represents a framework or "map," with respect to which effects occurring in different regions of the DSF may be located and studied in a consistent manner. One effect is the significant or noticeable deviations of the theoretical DSF curves from the limiting curve in the region approximately 4>q(2)(Z(d)t)(0.5)> approximately 0.1 (a time region where t<tau(1) (e)) to the faster side as qa is in the range 1-5. This is supported by the comparison of the experimental results of an entangled poly(vinylethylene) sample with the theoretical curves. The DSF functional forms predict plateaus with heights depending on the value of q-q-split plateaus-as can be experimentally observed in the time region greater than the relaxation time tau(1) (e) of the lowest Rouse-Mooney mode, when qa falls between approximately 1 and approximately 7. High sensitivity of the distribution of the q-split plateaus to a enables its value to be extracted from matching the calculated with the experimental results. The thus obtained a value for a well-entangled poly(ethylene-co-butene) polymer is in close agreement with the rheological result. It is shown that the fixed-end boundary conditions in the Rouse-Mooney model are responsible for the correct prediction of the distribution of the q-split plateaus.