Viscoelasticity time-temperature correspondence

So Figures 13-78 and 13-80 are our idealized representations. This correspondence is obviously interesting and important, but we will defer a discussion of the molecular origin of this until later. First we want to explore this time-temperature correspondence and the various regions of viscoelastic behavior in a little more detail. 

In thermorheological simple systems, the time-temperature correspondence principle holds. Chapter 8 gives examples of isotherms for compliance functions and relaxation moduli. The shift factors are expressed in terms of terminal viscoelastic parameters, and the temperature dependence of the shift factors is interpreted in terms of the free volume and the WLF equation. The chapter outlines methods for determining the molecular weight between entanglements, and analyzes the influence of diluents and plasticizers on the viscoelastic functions. 

In actual long term applications of polymers, however, it is well known that chemical reactions occur which actually change the viscoelastic properties of the material while it is in use. In addition, environmental factors such as exposure to solvents or even water, while not always chemically modifying a material, can have a profound influence on its viscoelastic properties in much the same way as a true chemical transformation. If predictions based exclusively on time-temperature correspondence were to be successful, the rates of all of these processes would have to vary with temperature in exactly the same m inner as does the viscoelastic spectrum. While this might be approximately true in certain special cases, it is usually not so. Thus, a more general theoretical framework is necessary to predict the properties of simultaneously chemically reacting and physically relaxing networks. 

Having examined some of the basic manifestations of the phenomenological aspects of viscoelasticity in Chapter 2, along with the efforts to model this behavior in Chapter 3, we now shift our emphasis to the results of experiments on polymers, and the interpretation of these results in terms of the accepted molecular mechanisms. We then explore the important idea of time-temperature correspondence and demonstrate the close relationship between these two variables in determining the viscoelastic responses of polymers. 

Time-temperature superposition is applicable to a wide variety of viscoelastic response tests, as are creep and stress relaxation. We illustrate the principle by considering stress relaxation test data. As a result of time-temperature correspondence, relaxation curves obtained at different temperatures can be superimposed on data at a reference temperature by horizontal shifts along the time scale. This generates a simple relaxation curve outside a time range easily accessible in laboratory experiments. This is illustrated in Figure 14.13 for polyisobutylene. Here, the reference temperature has been chosen arbitrarily to be 25°C. Data obtained at temperature above 25°C are shifted to the right, while those obtained below 25°C are shifted to the left. 

Ogbonna C I, Kakay G, Allan P S and Bevis M J (1995) The self-reinforcement of polyolefins produced by shear controlled orientation in injection molding, J Appl Polym Sci 58 2131-2135. Mano J F, Sousa R A, Reis R L, Cunha A M and Bevis M J (2001) Viscoelastic behaviour and time-temperature correspondence of HDPE with varying levels of process-induced orientation. Polymer 42 6187-6198. 

In this case, an apparent activation energy is determined, and it has higher values than secondary relaxations 100-300 kJ/mol for urethane-soybean oil networks (Cristea et al. 2013), 200-300 kJ/mol for polyurethane-epoxy interpenetrating polymer networks (Cristea et al. 2009), more than 400 kJ/mol for semicrystalline poly(ethylene terephtalate) (Cristea et al. 2010), and more than 600 kJ/mol for polyimides (Cristea et al. 2008, 2011). The glass transition temperature is the most appropriate reference temperature when applying the time-temperature correspondence in a multifrequency experiment. The procedure allows estimation of the viscoelastic behavior of a polymer in time, in certain conditions, and is based on the fact that the viscoelastic properties at a certain tanperature can be shifted along the frequency scale to obtain the variation on an extended time scale (Brostow 2007 Williams et al. 1955). The shift factor is described by the Williams-Landell-Ferry (WLF) equation ... 

The 150 °C results were obtained in dnplicate, which were in good agreement with each other. The time-temperature correspondence in the viscoelastic behaviour of elastomers may be represented with the WLF equation [17]. For the present elastomer, i.e., butadiene-acrylonitrile copolymer having 33% acrylonitrile, the temperature time shift factor, aj was previously given by [18]... 

For a fiber immersed in water, the ratio of the slopes of the stress—strain curve in these three regions is about 100 1 10. Whereas the apparent modulus of the fiber in the preyield region is both time- and water-dependent, the equiUbrium modulus (1.4 GPa) is independent of water content and corresponds to the modulus of the crystalline phase (32). The time-, temperature-, and water-dependence can be attributed to the viscoelastic properties of the matrix phase. 

In an earlier section, we have shown that the viscoelastic behavior of homogeneous block copolymers can be treated by the modified Rouse-Bueche-Zimm model. In addition, the Time-Temperature Superposition Principle has also been found to be valid for these systems. However, if the block copolymer shows microphase separation, these conclusions no longer apply. The basic tenet of the Time-Temperature Superposition Principle is valid only if all of the relaxation mechanisms are affected by temperature in the same manner. Materials obeying this Principle are said to be thermorheologically simple. In other words, relaxation times at one temperature are related to the corresponding relaxation times at a reference temperature by a constant ratio (the shift factor). For... 

Because of the time-temperature (or frequency-temperature) relation for the viscoelastic properties of polymers there is of course also a corresponding frequency dependence of the acoustic properties of polymers. In Table 14.2 frequency derivatives of sound speeds and absorptions are listed. 

For polymers, the intensity factor for fhe dielecfric quantities often (though not always) coincides with reciprocal of bj ( T/T,) for fhe viscoelastic properties. This feature has been incorporated in Equation (3.29). Corresponding to Equation (3.29), Ae and e" obeyed the time-temperature superposition ... 

There are many stress-analysis problems involving viscoelastic materials that are of a statically determinate class, i.e., the stresses in the body depend only on the applied forces and moments and not specifically on the elastic properties of the body. Such problems can be solved by invoking the correspondence principle. Then, the time and temperature dependences of the strains and flexures in the body can be obtained through the time temperature-shift properties of the viscoelastic polymer. 

The shift factor is the shift in time scale corresponding to the difference between the selected and reference temperature, and the shift factor represents the temperature dependence of the rate of the segmental motion which underlies all viscoelastic behavior the WLF equation demonstrates that all polymers, irrespective of their chemical structure, will exhibit similar viscoelastic behavior at equal temperature intervals (T-Tg) above their respective glass transition temperatures (Tg). Odian GC (2004) Principles of polymerization. John Wiley and Sons Inc., New York. Mark JE (ed) (1996) Physical properties of polymers handbook. Springer-Verlag, New York. 

Describing and predicting viscoelastic properties of polymer materials or adhesively bonded joints on the basis of analytical mathematical equations are justified only in the limits of linear viscoelasticity. Linear viscoelasticity is typically limited to strain levels below 0.5%. Furthermore, linear viscoelastic behavior is associated to the Boltzmann superposition principle, the correspondence principle, and the principle of time-temperature superposition. 

---