Thermorheologically simple

For linear thermorheologically simple materials a single temperature-dependent shift factor, aT T), can be used to predict the transient thermal response [20]. The mechanical response is history dependent and involves the use of reduced times, ( ) and (t), which can be found from the shift factor as... 

Materials to which time-temperature superposition is applicable, are sometimes termed thermorheologically simple materials. The amount of shift, log aT, which is required to bring measurements at a given temperature, T, into superposition with measurements at a reference temperature, Tr, is described, usually within a range of temperatures Tg < T < (Tg + 100), by the WLF equation (6) ... 

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. 

Since the relaxation mechanisms characteristic of the constituent blocks will be associated with separate distributions of relaxation times, the simple time-temperature (or frequency-temperature) superposition applicable to most amorphous homopolymers and random copolymers cannot apply to block copolymers, even if each block separately shows thermorheologically simple behavior. Block copolymers, in contrast to the polymethacrylates studied by Ferry and co-workers, are not singlephase systems. They form, however, felicitous models for studying materials with multiple transitions because their molecular architecture can be shaped with considerable freedom. We report here on a study of time—temperature superposition in a commercially available triblock copolymer rubber determined in tensile relaxation and creep. 

R is the gas constant and Ea the flow activation energy. The latter is a material-specific factor that is not dependent on the molar mass of the polymer and for thermorheological simple polymers also is not dependent on the shear stress. The activation energy Ea for polymer melts varies between 25 and 80 kj/mol. It can be determined from the slope of the line in the Arrhenius plot. 

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... 

Almost always the data from the apparatus above is analyzed by using the time-temperature superposition principle to form a master curve over a wide frequency range at a selected reference temperature. The basis for this procedure is that for thermorheologically simple materials the effect of a change in temperature on... 

A fundamental characteristic of the so-called thermorheologically simple systems is that consecutive isotherms have similar habits, so they overlie each other when they are shifted horizontally along the log t axis. In other words, the time-temperature correspondence principle holds. This property in creep experiments can be expressed by the relation (2,3)... 

However, for thermorheologically simple materials, that is, for those materials for which the time-temperature superposition principle holds, the mechanical properties data can be shifted parallel to the time or frequency axis. This fact suggests an additional hypothesis that can be very useful in solving some specific thermoviscoelastic problems. According to this hypothesis, the net effect of temperature in the response must be equivalent to a variation in the rates of creep or relaxation of the material. Thus for T > Tq the process occurs at a higher rate than at Tq. 

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. 

The Eqs. (2.1a) and (2.1b) apply thus actually to a rate scale and, in the frequait case of cyclic exposure, to a frequency scale co. If a thermorheologically simple system is considei ed the fr juency scale can be replaced by a temperature sale 1/T. Steps A that satisfy Eqs. (2.1a) and (2.1b) appear then in the response-functions for systmis of this nature that are measured as a function of temperatiue at pven, fixed paturbation rate. The temperature at whidi the steps occur depends, however, on the rate of external i rturbution. The temperature-dependent thawing of conformational isomers in thermorheologically complicated systems can be similariy observed in the response-functions, but the steps no longer satisfy Eqs. (2.1 a) and (2.1 b). These two equations lose, in addition, their validity with rrapect to the rate scale if, as is the case of polymers, several mutually independent, internal variables are required in order to uniquely define the conformational isomerism. In this case. Eqs. (2.1a) and (2.1b) become inequalities... 

T > To are shifted to longer times, and measurements for T < Tq aie shifted to shorter times. A well-defined reduced curve means the viscoelastic response is thermorheologically simple (Schwarzl and Staverman, 1952). It represents log Jp(t) at To over an extended time range. The time scale shift factors aj that were used in the reduction of the creep compliance curves to obtain the reduced curve constitute the temperature dependence, ar is fitted to an analytical form, which is often chosen to be the Williams-Landel-Ferry (WLF) equation (Ferry, 1980),... 

The most common means to extend the frequency scale is to invoke time-temperature superpositioning (Ferry, 1980). If all motions of a polymer contributing to a particular viscoelastic response are affected the same by temperature, then changes in temperature only alter the overall time scale such a material is thermorheologically simple. Thermorheological simplicity means conformance to the time-temperature superposition principle, whereby lower and higher strain rate data can be obtained from measurements at higher and lower temperatures, respectively. 

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. 

PVE 271.3 12.0 36.8 [127] component resolved In the dielectric relaxation spectra (not thermorheologically simple). ar,a of local segmental motion of PVE from... 

A very simple result is obtained if thermorheologically simple behavior is imposed on this mechanical model as an additional restriction. In this case, all spring constants of the model are temperature-independent while the viscosities all have a similar temperature dependence. By a change from the temperature T to the... 

In addition, the partially miscible polymer blends are thermorheologically simple, so that all considerations available for homogeneous systems can be applied in their case as well (Utracki 1990). The composition of the two phases from the vicinity of phase separation is not the same as in the case of immiscible mixtures where it is supposed that, if droplets are present in their corresponding morphology, they are composed from one of the mixture components and that their total volume is equal to that of the blend composition. This theory is not applicable to partially... 

It should be noted that the use of creep and shear compliances as material property input allows Poisson s ratio to be time-dependent. Hence, the present formulation is applicable to any thermorheologically simple isotropic viscoelastic material over any length of time. 

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