Temperature-frequency superposition

The dependence of the characteristic quantities (6.58) on temperature is mainly determined by the monomer friction coefficient Co, which depends on temperature, concentration, and (for small M) of molecule length (Berry and Fox 1968). The dependencies were recently discussed by Tsenoglou (2001). The monomer friction coefficient Co is a material characteristic of the system, its value is strongly determined by chemical structure of macromolecule as was shown for polybutadiene by Allal et al. (2002). 

The value of the coefficient of friction is connected with relative motion of small portions of the macromolecule, so that its temperature dependence is similar to that found for low-molecular-weight liquids, and can be written in the following form at temperatures much higher than the glass transition point 

We note that, since the parameters B and x are practically independent of temperature, the shape of the curves showing G/nT as a function of the non-dimensional frequency t u does not change as the temperature increases, so that we can make a superposition using a reduction coefficient obtained from the temperature dependence of the viscosity. 

To determine the procedure for the reduction, we shall write down the dynamic modulus at two different temperatures, one of which is a reference temperature Tref and the other is an arbitrary temperature T, 

One can consider the parameters B and x to be independent of the temperature and change the argument in the first line in such a way as to exclude the non-dimensional function. Then we write down the rule for reduction as 

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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. 

The time-temperature or frequency-temperature superposition scheme discussed in Chapter 4, Section B, is applicable to secondary relaxations as well as to the glass transition, assuming that the observed secondary relaxation peaks are well resolved. When the shift factors are obtained for these secondary relaxations, it is found that their temperature dependencies do not obey the WLF equation but follow an equation of the Arrhenius form, that is ,... 

Most response funetions of polymers obey a time-temperature or frequency-temperature superposition [43,44] A change in temperature is equivalent to a shift of the logarithmie frequency axis ... 

Partial master curves of 10 g.dL"l solutions of a,o)-alkaline earth dicarboxylato PBD in xylene at 297 K are reported in Figure 10, and result from a good frequency-temperature superposition of the experimental data.l7 Only the G" master curve of the solution of Be-based HTP is ill-defined due to the poor accuracy in the determination of the very small values of G". The shift factors support an apparent Arrhenius-type of dependence (Figure 11), from which the activation energy of the observed secondary ionic relaxation process was calculated and found to decrease as the radius of the alkaline earth cations increases (Figure 12). One also observes that the relaxation spectrum calculated by the first order approximation of Ninomiya and Ferry S is displaced along the time scale in relation with the cation size (Figure 13). The dynamic behavior of the 10 g.dL solution is obviously... 

Upon substitution of (9.82) into the moduli (9.68), we find the frequency-temperature superposition principle such that a modulus-frequency curve at any temperature T can be superimposed onto a single curve at the reference temperature 7b, if it is vertically and horizontally shifted properly. Such construction of the master curve is described by the equation... 

The separate ot-mode contribution of NR in the NR/PUR blend can be clearly observed in Figure 9.9 where the imaginary part of the electric modulus is depicted versus temperature and frequency. The temperature and frequency range was from -100 to 50 °C and 10 to 10 Hz, respectively. Generally speaking, the frequency-temperature superposition shifts the loss peak position... 

The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

Dynamic mechanical measurements for elastomers that cover wide ranges of frequency and temperature are rather scarce. Payne and Scott [12] carried out extensive measurements of /a and /x" for unvulcanized natural mbber as a function of test frequency (Figure 1.8). He showed that the experimental relations at different temperatures could be superposed to yield master curves, as shown in Figure 1.9, using the WLF frequency-temperature equivalence, Equation 1.11. The same shift factors, log Ox. were used for both experimental quantities, /x and /x". Successful superposition in both cases confirms that the dependence of the viscoelastic properties of rubber on frequency and temperature arises from changes in the rate of Brownian motion of molecular segments with temperature. 

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... 

Time-temperature superposition. Because of the relatively strong relaxations in the frequency range at room temperature (300 K), oscillation measurements were also performed at 345, 390 and 435 K in addition the D networks were measured at 265 K. 

The time-temperature superposition, implying that the functional form does not appreciably depend on temperature (see e.g. [34, 111]). For instance, mechanical or rheological data corresponding to different temperatures can usually be superimposed if their time/frequency scales are shifted properly taking a given temperature Tr as reference. 

To obtain as much information as possible on a material, an empirical technique known as time-temperature superposition (TTS) is sometimes performed. This technique is applicable to polymeric (primarily amorphous) materials and is achieved by performing frequency sweeps at temperatures that differ by a few degrees. Each frequency sweep can then be shifted using software routines to form a single curve called a master curve. The usual method involves horizontal shifting, but a vertical shift may be employed as well. This method will not... 

Fig. Z4 (a) Temperature ramp at a frequency a> = lOrads (strain amplitude A = 2%) for a nearly symmetric PEP-PEE diblock with Mn = 8.1 X 104gmol l, heating from the lamellar phase into the disordered phase. The order-disorder transition occurs at 291 1 °C, the grey band indicates the experimental uncertainty on the ODT (Rosedale and Bates 1990). (b) Dynamic elastic shear modulus as a function of reduced frequency (here aT is the time-temperature superposition shift factor) for a nearly symmetric PEP-PEE diblock with Mn = 5.0 X 1O g mol A Shift factors were determined by concurrently superimposing G and G"for w > and w > " respectively. The filled and open symbols correspond to the ordered and disordered states respectively. The temperature dependence of G (m < oi c) for 96 < T/°C 135 derives from the effects of composition fluctuations in the disordered state (Rosedale and Bates 1990). (c) G vs. G"for a PS-PI diblock with /PS = 0.83 (forming a BCC phase) (O) 110°C (A) 115°C ( ) 120°C (V) 125°C ( ) 130°C (A) 135°C ( ) 140°C ( ) 145°C. The ODT occurs at about 130°C (Han et at. 1995).

Fig. 94 a Dependence of E" as a function of frequency in the HMDA system, b Result of the high temperature superposition (from [63])... 

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