Polymer Physics

The scenario above was called the thermorheological simplicity, which was first
conveyed by Ferry in 1950 under the protocol of linear viscoelasticity (Ferry 1950 ).
However, in the short-time (or high-frequency) region, the viscoelastic response of
molten polymers is much related to the energetic interactions between chain units;
while in the long-time (or low-frequency) region, the response is much related to
the entropy gain of chain conformation. Therefore, the temperature dependence of
the relaxation time in the short-time region is generally stronger than that in the
long-time region, making the shifting curves deviated from the time-temperature
superposition principle. Such a scenario is also called the thermorheological com-
plexity (Plazek et al. 1995 ), which allows the identification of different relaxation
mechanisms, especially in a picture of spatial dynamic heterogeneity for glass-
forming polymers (Ediger 2000 ; Lin 2011 ).

6.2.4 Dynamic Mechanical Analysis

Since the mechanical response of materials is related to the time or frequency of the
imposing stress, one can measure the hierarchical characteristic relaxation times of
the materials via continuous scan of imposing frequency. Often, the solid materials
are characterized by the dynamic mechanical spectroscopy or dynamic viscoelastic
spectroscopy, while the liquid materials are characterized by the rheometer. Nowa-
days, advanced instruments can measure the continuous change from liquid to solid.
In a typical case of dynamic mechanical analysis, a small stress oscillates
periodically in a sinusoidal mode with amplitudesand frequencyo, and the
small strainefollows the modulation with a certain phase lagd. The sinusoidal
stress is the imposed stimulation, and in a complex form,

s^ ¼sexpðiotÞ (6.29)

The sinusoidal strain is the detected response with a phase lag, as

e^ ¼eexpðiotidÞ (6.30)

Accordingly, the complex modulus can be obtained as

E^ ¼

s^

e^

¼

s

e

expðidÞ¼Eðcosdþi sindÞ¼E^0 þiE^00 (6.31)

whereE’keeps pace with the stimulation, as thestorage modulus, andE"misses the
steps with the stimulation, as theloss modulus. Their ratio is defined as theloss
factor, which is the tangent of the phase lagd,

E^00

E^0

¼tand (6.32)

6.2 Relaxation of Polymer Deformation 105