Time-temperature equivalence, viscosity

It is well established that between Tg and about Tg + 50 K, the relaxation kinetics obeys the WLF law (Williams et al., 1955). If Pr is a property depending on the macromolecular mobility (relaxation modulus, complex modulus, viscosity, diffusion rate, etc.), the time-temperature equivalence principle may be formulated as... 

The linear viscoelastic properties of all samples were characterized by dynamic shear measurements in the parallel-plate geometry. Experimental details have been previously published [9]. Using time-temperature equivalence, master curves for the storage and loss moduli were obtained. Fig. 1 shows the master curves at 140°C for the relaxation spectra and Table 3 gives the values of zero-shear viscosities, steady-state compliances and weight-average relaxation times at the same temperature. 

The time-temperature equivalence implies that the viscosity (or relaxation times) of polymers may be written as the product of two functions ... 

The time-temperature equivalence principle makes it possible to predict the viscoelastic properties of an amorphous polymer at one temperature from measurements made at other temperatures. The major effect of a temperature increase is to increase the rates of the various modes of retarded conformational elastic response, that is, to reduce the retarding viscosity values in the spring-dashpot model. This appears as a shift of the creep function along the log t scale to shorter times. A secondary effect of increasing temperature is to increase the elastic moduli slightly because an equilibrium conformational modulus tends to be proportional to the absolute temperature (13). 

Unstable branches on the P(Q) curve and the appearance of hysteresis loops can occur for various reasons usually connected with an increase in viscosity. Thus, a non-monotonic P(Q) curve was first encountered in an analysis of the flow of a hot inert (non-reactive) liquid in a cold tube when the viscosity of the liquid was strongly dependent on temperature.190 The intense dissipative heat output may have been the reason for the instability in the flow of an inert liquid.191 In both cases, the reason for the nonmonotonic in P(Q) dependence was the strong dependence of viscosity on temperature, which is equivalent here to time dependence for viscosity. Detailed investigations of the hysteresis transitions shown in Fig. 4.24 proved that they have a wave character 192 in this case, the transition occurs at a constant flow rate. 

When a Newtonian liquid, such as a hydrocarbon mixture, is subjected to a shearing stress, a velocity gradient develops within the fluid. Viscosity (or dynamic viscosity) is defined as the shear stress per unit area at any point within the fluid divided by the velocity gradient at that point. Consequently, the viscosity is a dynamic property nevertheless, for Newtonian liquids it is a state property, that is, it depends only on state properties such as temperature and pressure or density. The dimensions of viscosity are force x time/length or equivalently mass/length x time. Occasionally kinematic viscosity, which is the ratio of dynamic viscosity to fluid density, is used instead of dynamic viscosity. The dimensions of kinematic viscosity are length /time. 

Ohm s law, V=J R (voltage equals current times resistance), electricity has the same form as equation 9.1-14 which may be written as equation 9.1-15, where AP is the pressure differential, Q is the flow rate and resistance is given by equation 9.1-16, where t] is the viscosity of the fluid. Table 9.1-2 shows that the viscosity of liquids is highly temperature-dependent. Gases are much less temperature dependent because of the greater separation between molecules. If there are multiple discharge paths the equivalent resistance is the same as electrical resistors in... 

The solidity of gel electrolytes results from chain entanglements. At high temperatures they flow like liquids, but on cooling they show a small increase in the shear modulus at temperatures well above T. This is the liquid-to-rubber transition. The values of shear modulus and viscosity for rubbery solids are considerably lower than those for glass forming liquids at an equivalent structural relaxation time. The local or microscopic viscosity relaxation time of the rubbery material, which is reflected in the 7], obeys a VTF equation with a pre-exponential factor equivalent to that for small-molecule liquids. Above the liquid-to-rubber transition, the VTF equation is also obeyed but the pre-exponential term for viscosity is much larger than is typical for small-molecule liquids and is dependent on the polymer molecular weight. 

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 polyamidoamines are very high-viscosity liquids, some having viscosities over 50,000 mPa/sec. Typical amine equivalent weights are 100-150. The polyamidoamines react with bisphenol A epoxy resins at room temperature although the adhesives usually require several hours to reach sufficient molecular weight to carry a load. Cure times can be shortened to a few minutes at about 150°C. 

The nature of the counter-ion X and the temperature at which the polymerization is carried out are important. For example, in a study of THF polymerizations at 30°C initiated by equivalent amounts of triethyl-oxonium salts with different counter-ions, Dreyfuss and Dreyfuss [82] observed differences in both conversion and rates of viscosity change with time of polymerization. In the cases of BF4 and SbCl, the final viscosities and conversions that were attained were lower than when SbFg or PFg counter-ions were used. The viscosity of BF4 polymers remained constant after constant conversion was reached, so termination can be inferred. The viscosity of SbClj polymers continued to decrease even after constant conversion was attained. With the SbClg counter-ion, both termination and transfer occurred. In a comparison of rates of THF polymerization at 0°C initiated by Et3 0 BF4 and Et3 0 AlCl4, Saegusa and Matsumoto [83] confirmed the termination inferred by Dreyfuss and Dreyfuss [82] and even earlier by Vofsi and Tobolsky [86] in a study with only the BF4 counter-ion at 0°C. Saegusa and Matsumoto applied the [P ] method [53] for determining active centres described in Section 3.1.1. The termination reaction was less obvious at 0°C than at 30°C nevertheless, it was clearly evident. Further they found that termination... 

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