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

Part 5 : Dynamic Loss Tangent (tan δ) -Temperature Curve of Complex Systems

Abstract

In order to evaluate the tan δ value of the complex system consisting of m components with different relaxation time, a new approximation equation has been derived. For the parallel model in which dynamic models representing the viscoelasticity of each component are arranged in parallel, the tan δ value of a whole system can be approximated by the tan δ_e. defined as :
tanδ_e=^mΣ_i=1 V _(i) ·tanδ _(i)
For the series model, in which dynamic models are arranged in the series manner, by the tan δ_e, defined as :
tanδ_e=^mΣ_i=1 J _(i) ·tanδ _(i)
(i) stands for the value of i-th component. V (i) and J (i) are modulus fraction and compliance fraction of i-th component respectively and are defined by;
V _(i) =E _{(i) 1}υ _(i) /^mΣ_i=1 E _{(i) 1}υ _(i) (1)
J _(i) = (υ _(i) /E _{(i) 1}) /^mΣ_i=1 (υ _(i) /E _{(i) 1}) (2)
where, E _{(i) 1} is the modulus after relaxation and V _(i) is the volume fraction of i-th component. If the peak value of tan δ of the component, (tan δ) _max is same with each other and (tan δ) _max≤0.3, the relative error originated by using the tan δ_e for representing tan δ value of the parallel model is within 5%. This error decreases monotonically with an increase in the distribution width of the relaxation time of its component. This strongly supports the validity of the approximation equationof tan δ_??_tan δ_e in parallel model proposed in our previous paper (this journal, 30, T85 (1977)) for establishing the analyzing procedure of the tan δ-temperature curve.