Viscoelasticity time-temperature superposition

If the modulus data of a material satisfy Equation (3.27), this material is referred to as thermorheologically simple and obeys the (viscoelastic) time-temperature superposition. 

The polymer SBR is mechanically simple, and it was found that the results for G for a given substrate could be superimposed using the Williams, Landel and Ferry (WLF) technique (see Viscoelasticity - time-temperature superposition). The master curves for each substrate were close to being parallel to each other and to a similar curve for cohesive fracture of the SBR (Fig. 1). 

For a rubber adhesive, Gent found that he could use the WLF equation to obtain a master curve of breaking stress (see Viscoelasticity - time-temperature superposition). He used a Fracture mechanics analysis to link the breaking stress to critical strain energy density. 

Viscoelasticity - time-temperature superposition D W AUBREY Shift factor, WLE equation... 

Since we are interested in this chapter in analyzing the T- and P-dependences of polymer viscoelasticity, our emphasis is on dielectric relaxation results. We focus on the means to extrapolate data measured at low strain rates and ambient pressures to higher rates and pressures. The usual practice is to invoke the time-temperature superposition principle with a similar approach for extrapolation to elevated pressures [22]. The limitations of conventional t-T superpositioning will be discussed. A newly developed thermodynamic scaling procedure, based on consideration of the intermolecular repulsive potential, is presented. Applications and limitations of this scaling procedure are described. 

Time-temperature superposition [10] increases the accessible frequency window of the linear viscoelastic experiments. It applies to stable material states where the extent of reaction is fixed ( stopped samples ). Winter and Chambon [6] and Izuka et al. [121] showed that the relaxation exponent n is independent of temperature and that the front factor (gel stiffness) shifts with temperature... 

There are two superposition principles that are important in the theory of Viscoelasticity. The first of these is the Boltzmann superposition principle, which describes the response of a material to different loading histories (22). The second is the time-temperature superposition principle or WLF (Williams, Landel, and Ferry) equation, which describes the effect of temperature on the time scale of the response. 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

Mechanical and viscoelastic behaviour of materials can be determined by different kind of instrumental techniques. Broadband viscoelastic spectroscopy (BVS) and resonant ultrasound spectroscopy (RUS) are more commonly used to test viscoelastic behavior because they can be used above and below room temperatures and are more specific to testing viscoelasticity. These two instruments employ a damping mechanism at various frequencies and time ranges with no appeal to time-temperature superposition. Using BVS and RUS to study the mechanical properties of materials is important to understanding how a material exhibiting viscoelasticity will perform. 

The peak of the dielectric loss of this process reflects its viscoelastic nature by obeying the time-temperature superposition principle, wherein the peak is shifted to higher temperatures for shorter times (higher frequencies) and vice versa. This process has been described by the Havriliak-Negami empirical formula [106, 108]... 

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... 

Detailed analysis of the isothermal dynamic mechanical data obtained as a function of frequency on the Rheometrics apparatus lends strong support to the tentative conclusions outlined above. It is important to note that heterophase (21) polymer systems are now known to be thermo-rheologically complex (22,23,24,25), resulting in the inapplicability of traditional time-temperature superposition (26) to isothermal sets of viscoelastic data limitations on the time or frequency range of the data may lead to the appearance of successful superposition in some ranges of temperature (25), but the approximate shift factors (26) thus obtained show clearly the transfer viscoelastic response... 

D lends of two incompatible polymers with different glass-transition temperatures have properties which differ from the pure components. Their viscoelastic behavior is complex and deviation from the simple time-temperature superposition is generally observed (I). The high... 

As an example of the concentration dependence of viscoelastic properties in Fig. 16.11 the shear creep compliance of poly(vinyl acetate) is plotted vs. time for solutions of poly(vinyl acetate) in diethyl phthalate with indicated volume fractions of polymer, reduced to 40 °C with the aid of the time temperature superposition principle (Oyanagi and Ferry, 1966). From this figure it becomes clear that the curves are parallel. We may conclude that the various may be shifted over the time axis to one curve, e.g. to the curve for pure polymer. In general it appears that viscoelastic properties measured at various concentrations may be reduced to one single curve at one concentration with the aid of a time-concentration superposition principle, which resembles the time-temperature superposition principle (see, e.g. Ferry, General references, 1980, Chap. 17). The Doolittle equation reads for this reduction ... 

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 cohesive fracture of conventional, non-strain crystallizing, unfilled elastomers is sensitive to rate and temperature 32.4-1.48-53) exhibiting increased values of 2J with increasing rate and decreasing temperature. The basic viscoelastic nature of the fracture of these materials is evidenced by the fact that it can be described over wide ranges of temperature and rate by time-temperature superposition as described by the WLF Equation... 

The Time-Temperature Superposition Principle. For viscoelastic materials, the time-temperature superposition principle states that time and temperature are equivalent to the extent that data at one temperature can be superimposed upon data at another temperature by shifting the curves horizontally along the log time or log frequency axis. This is illustrated in Figure 8. While the relaxation modulus is illustrated (Young s modulus determined in the relaxation mode), any modulus or compliance measure may be substituted. 

Important viscoelastic principles include the time-temperature superposition principle and its resultant WLF equation. These can be applied to understand the relationship between literature values of the glass transition temperature and actual needs. Thus, by using the growing amount of science now available in the field of damping, one can select that polymeric material which will damp most effectively. 

The inherent difficulty in the measurement of the complex dynamic moduli of viscoelastic materials is emphasized by the results of this paper. The agreement among the shifted modulus data as measured by different systems is limited by several difficulties (1) measurement inaccuracies of the instruments, (2) differences in the data reduction techniques used to apply the time-temperature superposition principle and propagation of shift curve errors and, (3) nonuniformity of the test samples. 

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