Analysis of Dynamic Loss Tangent (tanδ) -Temperature Curve Relating to Molecular Motion of Polymer Chain in Amorphous Regions in Fibers and Fiber-Forming Polymeric Materials.

Part 2 : Viscoelastic Body Composed of Broad Distribution of Relaxation Time

Abstract

An attempt has been made to establish the relationships between the distribution F_τ (τ) of relaxation timeτand the shape and characteristic parameters of dynamic loss tangent (tanδ) -temperature (T) curve. For this purpose, the theoretical equation representing tanδ-T curve is derived for ten kinds of function formula for F_τ (τ) : seven functions of which are given by F_τ (τ) =k_n^'τn' (where, k_n' : constant independent ofτ, n'=-2-3) (1), and three functions are given by triangles. WLF and Arrhenius types have been assumed for temperature dependence of τ.On the basis of the theoretical equations derived here, tanδ-T curve has numerically been calculated with the aid of an electronic computer. In the case of n'≥0 in eq. (1), the peak value of tanδ-T curve, (tanδ) max, peak tamperature Tmax, and the ratio of half value width ΔT_1/2/ΔT_{1/2 (s)} (the suffix s indicates the system composing of a single relaxation time) depend on the average relaxation timeτ, F_τ (τ), the total number of Maxwell elements in a system N, the ratio of elastic modulus before and after relaxation α, and the minimum relaxation time τ_l in F_τ (τ).In the case of n'≤-0.5 in eq. (1), the shoulder is significantly observed in the lower temperature region than Tmax in tanδ-T curve. The necessary conditions for appearance of this kind of shoulder is (tanδ) max≥3.5/ (n'+3) ⁴ and/N≥ √10.The tanδ-T curve (especially, ΔT_1/2/T_{1/2 (s)} , mearsured in the temper ature range of apperance of dynamic absorption arising from microbrownian movement of polymer seg ment in amorphous region of semi-crystalline polymers (α_a), is remarkably different from that calculated by using the above mentioned functions for F_τ (τ). Contrary to this, tanδ-T curve determined experimentally in the temperature range of αa absorption of amorphous polymer is in fairly agreement with that calculated by using n'=-2-0 in F_τ (τ) =kn^'τn' .In the actual experiment, no shoulder is observed in tanδ-T curve,