Such a behavior of stress relaxation is calledtime-strain separability(Larson
1988 ). The factorh(g)is called the damping function. The empirical damping
functions that are often used in polymer melts include the exponential function
hðgÞ¼expðngÞ (7.10)and the reciprocal function
hðgÞ¼ð 1 þag^2 Þ^1 (7.11)Herenandaare both positive constants. Doi and Edwards also derived a
complicated damping function from the tube model, which is close to the empirical
reciprocal function witha¼0.2 (Larson 1988 ).
According to the interpretation of Doi-Edwards tube theory (Doi and Edwards
1986 ), the shear-thinning phenomenon of polymers under high shear rates is related
to the segment orientation in the region between shear rates separately corresponding
to the relaxation timettandtR, as well as the extensional deformation being able to
performcontour length relaxationalong the tube. However, that theory predicts a
maximum on the curve of shear stress with respect to the shear rate, and an over-
estimation of shear-thinning above 1/tt, leading to a negative shear viscosity and
instability of numerical solutions. Marrucci introduced a new mechanism of chain
motion to correct this theory, which is calledconvective constraint release(CCR)
(Marrucci 1996 ). This model suggests that the transverse velocity gradient brought by
the high shear stress facilitates disentanglement of polymer chains, and thus releases
some of chain entanglement around the tube, leading to a decay of the shear viscosity.
This shear-thinning mechanism suppresses the early deformation predicted by the
Doi-Edwards theory, and hence avoids an over-estimation. In addition, in combina-
tion with other models, it can predict the shear-thickening phenomenon when
polymer coils perform stable extensional deformation under higher shear rates.
Fig. 7.5 Illustration of the
universal curve for the shear
stress of bulk polymers over a
broad range of shear rates.
The labels are the same as in
Fig.7.4
7.2 Characteristics of Polymer Flow 133