corresponds to a critical molecular weight for extensional deformation of polymer
coils. In the bulk polymer phase, such a critical situation makes a yielding-like
behavior of the viscous fluid, raising nonlinear viscoelasticity due to a large
stretching of the chain.
The spinning of polymers corresponds to uniaxial stretching, while the film
blowing/stretching corresponds to biaxial stretching. Three typical relationships
between extensional viscosityeand extensional stresssexist in extensional flows,
as illustrated in Fig.7.11. Type A, a typical Newtonian fluid, shows the indepen-
dence ofons, which is often observed for short-chain polymers such as Nylon
6-6, POM, PMMA, etc.; Type B shows the viscosity increases with the increase of
stress when it is above a critical value,s>sc. This type ofextensional thickening
is observed for branched chains like LDPE; Type C shows that the viscosity
decreases with the increase of stress whens>sc. Such a behavior ofextensional
thinningis often observed for long linear chains like HDPE and PP. Interestingly,
the critical stressscnormally corresponds to the critical shear stress above which
the shear thinning occurs.
The molecular theory of extensional viscosity of polymer melts is again based on
the standard tube model. It considers the linear viscoelastic factors such as
reptation, tube length fluctuations, and thermal constraint release, as well as the
nonlinear viscoelastic factors such as segment orientations, elastic contraction
along the tube, and convective constraint release (Marrucci and Iannirubertok
2004 ). Thus, it predicts the extensional stress–strain curve of monodispersed linear
polymers, as illustrated in Fig.7.12. At the first stage, the extensional viscosity of
polymer melts exhibits the Newtonian-fluid behavior, following Trouton’s ratio
e¼ 3 0 ; the second state begins at the reciprocal of the tube relaxation time, when
polymer chains are oriented but not yet stable in their deformation to display the
extensional-thinning phenomenon similar to that observed in shear flow. Since
there is no transverse velocity gradient, the role of convective constraint release
is not evident. The third stage begins at the reciprocal of the Rouse relaxation time,
when polymer deformation has been stable to display thestrain-hardeningphe-
nomenon. The fourth stage restores the Newtonian-fluid behaviors due to the
Fig. 7.11Illustration of three
typical cases of extensional
viscosity of polymer melt
changing with extensional
stress
138 7 Polymer Flow