Temperature-dependent shift factor

PPG (at higher temperatures) behaves like a typical pseudoplastic non-Newtonian fluid. The activation energy of the viscosity in dependence of shear rate (284-2846 Hz) and Mn was detected using a capillary rheometer in the temperature range of 150-180°C at 3.0-5.5 kJ/mol (28,900 Da) and 12-13 kJ/mol (117,700 Da) [15]. The temperature-dependent viscosity for a PPG of 46 kDa between 70 and 170°G was also determined by DMA (torsion mode). A master curve was constructed using the time-temperature superposition principle [62] at a reference temperature of 150°G (Fig. 5) (Borchardt and Luinstra, unpublished data). A plateau for G was not observed for this molecular weight. The temperature-dependent shift factors ax were used to determine the Arrhenius activation energy of about 25 kJ/mol (Borchardt and Luinstra, unpublished data). 

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... 

Curves of stress (divided by absolute temperature) versus log time-to-break at various temperatures can be made to coincide by introducing the temperature-dependent shift factor flT. Application of the same shift factor causes the curves of the elongation at the break br versus the logarithm of time-to-break at various temperatures to coincide. A direct consequence is that all tensile strengths (divided by absolute temperature), when plotted against elongation at break, fall on a common failure envelope, independent of the temperature of testing. Fig. 13.84 shows the behaviour of Viton B elastomer. 

Conventionally storage modulus versus frequency and temperature results are presented by extrapolated isotherms called master curves. These are plotted by shifting frequency data with a temperature dependent shift factor, a. Most published results for are based on manual graphic methods. Computer work first aimed at systemizing and automating determinations of shift factor was initiated by a desire to efficiently process the hundreds of data points from each FTMA run. 

Figure 2.15 Construction of the time-temperature superposition and derivation of temperature dependent shift factor.

Fig. 2.27. Recoverable-compliance, Ji(t), data of PPMS 5000 at temperatures -32.2°C ( ), -35.0°C U),-38.6°C (t),-40.0°C ( ), -41.1 °C (x), -42.6 °C n, -44.5 °C ( ), -45.2 °C (V), -46.9 °C (A), and -50 °C (o). The data taken at different temperatures have been shifted horizontally along the log t axis by a temperature-dependent shift factor log ut in order to superpose the curves at the short-time end with the data for -35.0 °C. The inset shows the retardation spectrum, L, as a function of the reduced retardation time X with reference temperature To = -35.0 °C, which was obtained numerically from J lf) data.

When excited by thermal agitation, both creep and recovery quicken with temperature. In many cases, thermal effects are focused on time alone and can be accounted for by means of a temperature dependent shift factor a iT), thereby D(t) D( ), where = (T) = tlajiT) and analogously for E(t). This states that the time-dependent response at all temperatures is essentially the same. 

Numerous studies have been reported on the methods that enable one to obtain temperature independent log T) versus log OjY plots, usually referred to as reduced (or master) plots, where is called a temperature-dependent shift factor. We will show different ways of obtaining Uj. The availability of reduced plots for any given polymer will enable one to estimate the shear viscosity of the polymer at any desired shear rate and temperature. There are two different ways of obtaining reduced plots for shear viscosity, depending on whether a polymer is semicrystalline or amorphous. 

The scaling the functional shape hardly depends on temperature. Curves corresponding to different temperatures superimpose in a single master curve when they are represented against a reduced time variable that includes a T-dependent shift factor. 

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).

Viscoelasticity n. (1) A mechanical property involving a combination of elastic and viscous behavior conforming to neither just simple elastic nor simple viscous behavior also, strongly dependent on temperature. See shift factor and WFL equation. (2) The property of a polymer that characterizes it as neither an ideal solid nor a viscous liquid, but seeming to have the character of both. In addition, to having some of the... 

In analyzing the viscosity data at various temperatures and shear rates, which were obtained via slit rheometry for molten PS, PMMA, polypropylene (PP), and LDPE with solubilized CO2 at varying concentrations, Royer et al. (2000, 2001) used the following expressions, analogues of Eq. (13.13), for the concentration-dependent shift factor at temperatures below T + 100... 

In the case of a solution, if all the relaxation mechanisms are assumed to have the same dependence on concentration, implying the existence of a concentration shift factor, Uq, then the combined temperature-concentration shift factor is simply UjUq (59). 

The absolute measurement of areas is not usually usefiil, because tlie sensitivity of the spectrometer depends on factors such as temperature, pulse length, amplifier settings and the exact tuning of the coil used to detect resonance. Peak intensities are also less usefiil, because linewidths vary, and because the resonance from a given chemical type of atom will often be split into a pattern called a multiplet. However, the relative overall areas of the peaks or multiplets still obey the simple rule given above, if appropriate conditions are met. Most samples have several chemically distinct types of (for example) hydrogen atoms within the molecules under study, so that a simple inspection of the number of peaks/multiplets and of their relative areas can help to identify the molecules, even in cases where no usefid infonnation is available from shifts or couplings. 

From the data in Fig. 4.8b, estimate the shift factors required to displace the data at 0 = 0.5 (consider only this point) so that all runs superimpose on the experiment conducted at 128 C at 0 = 0.5. Either a ruler or proportional dividers can be used to measure displacements. Criticize or defend the following proposition Whether a buffered aqueous solution of H2O2 and 1. containing small amounts of S2O3 and starch, appears blue or colorless depends on both the time and the temperature. This standard general chemistry experiment could be used to demonstrate the equivalency of time and temperature. The pertinent reactions for the iodine clock are... 

For most purposes only the Stokes-shifted Raman spectmm, which results from molecules in the ground electronic and vibrational states being excited, is measured and reported. Anti-Stokes spectra arise from molecules in vibrational excited states returning to the ground state. The relative intensities of the Stokes and anti-Stokes bands are proportional to the relative populations of the ground and excited vibrational states. These proportions are temperature-dependent and foUow a Boltzmann distribution. At room temperature, the anti-Stokes Stokes intensity ratio decreases by a factor of 10 with each 480 cm from the exciting frequency. Because of the weakness of the anti-Stokes spectmm (except at low frequency shift), the most important use of this spectmm is for optical temperature measurement (qv) using the Boltzmann distribution function. 

Master curves can also be constmcted for crystalline polymers, but the shift factor is usually not the same as the one calculated from the WLF equation. An additional vertical shift factor is usually required. This factor is a function of temperature, partly because the modulus changes as the degree of crystaHiuity changes with temperature. Because crystaHiuity is dependent on aging and thermal history, vertical factors and crystalline polymer master curves tend to have poor reproducibiUty. 

The measurement of correlation times in molten salts and ionic liquids has recently been reviewed [11] (for more recent references refer to Carper et al. [12]). We have measured the spin-lattice relaxation rates l/Tj and nuclear Overhauser factors p in temperature ranges in and outside the extreme narrowing region for the neat ionic liquid [BMIM][PFg], in order to observe the temperature dependence of the spectral density. Subsequently, the models for the description of the reorientation-al dynamics introduced in the theoretical section (Section 4.5.3) were fitted to the experimental relaxation data. The nuclei of the aliphatic chains can be assumed to relax only through the dipolar mechanism. This is in contrast to the aromatic nuclei, which can also relax to some extent through the chemical-shift anisotropy mechanism. The latter mechanism has to be taken into account to fit the models to the experimental relaxation data (cf [1] or [3] for more details). Preliminary results are shown in Figures 4.5-1 and 4.5-2, together with the curves for the fitted functions. 

---