Amorphous polymers time-temperature superposition

The WLF equation can be widely applied, and demonstrates the equivalence of time and temperature, the so-called time-temperature superposition principle, on the mechanical relaxations of an amorphous polymer. The equation holds up to about 100° above the glass transition temperature, but after that begins to break down. 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

The time-temperature superposition principle has practical applications. Stress relaxation experiments are practical on a time scale of 10 to 10 seconds (10 to 10 hours), but stress relaxation data over much larger time periods, including fractions of a second for impacts and decades for creep, are necessary. Temperature is easily varied in stress relaxation experiments and, when used to shift experimental data over shorter time intervals, can provide a master curve over relatively large time intervals, as shown in Figure 5.65. The master curves for several crystalline and amorphous polymers are shown in Figure 5.66. 

Many amorphous homopolymers and random copolymers show thermorheologically simple behavior within the usual experimental accuracy. Plazek (23,24), however, found that the steady-state viscosity and steady-state compliance of polystyrene cannot be described by the same WLF equation. The effect of temperature on entanglement couplings can also result in thermorheologically complex behavior. This has been shown on certain polymethacrylate polymers and their solutions (22, 23, 26, 31). The time-temperature superposition of thermorheologically simple materials is clearly not applicable to polymers with multiple transitions. The classical study in this area is that by Ferry and co-workers (5, 8) on polymethacrylates with relatively long side chains. In these the complex compliance is the sum of two contributions with different sets of relaxation mechanisms the compliance of the chain backbone and that of the side chains, respectively. 

For semi-crystalline polymers with melting points of more than 100 °C above the glass transition temperature and for amorphous polymers far above the glass transition temperature Tg (at around T = Tg + 190°C), the shift factors obtained from time-temperature superposition can be plotted in the form of an Arrhenius plot for thermally activated processes ... 

Above Tg the stress relaxation and the creep behaviour of amorphous polymers obey the "time-temperature superposition (or equivalence) principle". 

We expect that the modification creates the free volume (Vf) in wood substance from the similarity of the effect of and n on viscoelasticity. The discussion for wood, however, is impossible on the basis of a concept of the free volume, although the flexibility of molecular motion for synthetic amorphous polymers is discussed. Unfortunately, we can not directly know the created free volume because the time-temperature superposition principle is not valid for wood [19]. The principle is related to WLF equation by which the free volume is calculated. The free volume, however, relates to volumetric swelling as follows. 

The WLF equation applies to amorphous polymers in the temperature range of Tg to about Tg + lOO C. In this equation J is the reference temperature, these days taken to be the T, while and C2 are constants, initially thought to be universal (with Cx = 17.44 and C2 = 51.6), but now known to vary somewhat from polymer to polymer. These experimental observations bring up a number of interesting questions. What is the molecular basis of the time-temperature superposition principle What is the significance of the log scale and what does the superposition principle tell us about the temperature dependence of relaxation behavior And what about the temperature dependence of a7 at temperatures well below 2 ... 

Fortunately for linear amorphous polymers, modulus is a function of time and temperature only (not of load history). Modulus-time and modulus-temperature curves for these polymers have identieal shapes they show the same regions of viscoelastic behavior, and in each region the modulus values vary only within an order of magnitude. Thus, it is reasonable to assume from such similarity in behavior that time and temperature have an equivalent effect on modulus. Such indeed has been found to be the case. Viscoelastic properties of linear amorphous polymers show time-temperature equivalence. This constitutes the basis for the time-temperature superposition principle. The equivalence of time and temperature permits the extrapolation of short-term test data to several decades of time by carrying out experiments at different temperatures. 

The aforesaid extrapolations make use of a time-temperature superposition principle which is based on the fact that time and temperature have essentially equivalent effects on the modulus values of amorphous polymers. Figure 3.19 shows modulus data taken at several temperatures for poly(methyl methacrylate) [8]. Because of the equivalent effect of time and temperature, data at different... 

Fig. 6.12 Illustration of (a) the storage modulus, the loss modulus and the loss factor as a function of frequency across the glass transition temperature of amorphous polymers (b) the loss factor as a function of temperature according to the time-temperature superposition principle. Below the a peak for glass transition, there are secondary relaxation peaks...

Low Molecular Weight Amorphous Polymers. The most spectacular observations of breakdown in time-temperature superposition occur in low molecular weight polymers. The effect was first seen in PS by creep and recovery measurement (105). The recoverable compliance Jrit) for a 3400 Da molecular weight PS imdergoes a dramatic change in shape of the recoverable compliance... 

The rheological behavior of polymers depends also on the time scale (or frequency) with which stresses are applied. The well-known silicone polymer silly putty shows a wide range of behavior at room temperature. Left on the table top, a ball of silly putty will slowly flow and spread. Pulled slowly, it will extend like chewing gum. Dropped to the floor, the ball will bounce like rubber. Struck sharply with a hammer, it will shatter like glass. This exemplifies the generalization that, as the time scale is varied isothermally, a given amorphous polymer will exhibit the same range of properties that it would normally show with variation in temperature. This observation is expressed quantitatively in the Time-Temperature Superposition Principle. 

Since fibers consist primarily of oriented crystallites, it is unfair to classify them as heterophase. However, the generalizations of time-temperature superposition that work so well with amorphous polymers do not apply to fibers. Fibers do exhibit viscoelasticity qualitatively like the amorphous polymers. It comes as a surprise to some that J. C. Maxwell, who is best known for his work in electricity and magnetism, should have contributed to the mathematics of viscoelasticity. The story goes that while using a silk thread as the restoring element in a charge-measuring device. Maxwell noticed that the material was not perfectly elastic and exhibited time-dependent effects. He noticed that the material was not perfectly elastic and showed time effects. The model that bears his name was propounded to correlate the real behavior of a fiber. 

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