Relaxation time shift factors

The second term in Equation (3.69) represents the terminal entanglement relaxation of PI occurring at Ig h (= p, for PI relaxing much faster than PtBS, as explained earlier). As explained for Equation (3.63), this term is approximately expressed in terms of the modulus Gp, bulk of bulk PI, the PI volume fraction ([jpi, the relaxation time shift factor Ah = Xg,h/xg, and the intensity factor Ipi given by Equation (3.65). 

The parameter is made temperature dependent according to the rheological relaxation time shift factor Uj-... 

The pattern can be obtained from the polymer temperature or concentration variations in addition to the change of G°N. The relaxation function may be too complicated a mathematical expression ever to be calculated, nonetheless, it obeys a property of invariance which allows the superposition of all normalised relaxation curves to one another by adjusting a suitable factor to the time scale of each curve. The time shift factor is found to obey the equation... 

The agreement is satisfactory but it is worth noting that the fit will be poorer if the high molecular tail is not described properly or more generahy if the relaxation time shift function X,(M) is not correct. For example, we showed [19] that failure to take into accovint the shift factor X leads to a large discrepancy between the model and the experimental data. 

Let us consider a material undergoing stress relaxation. The shift factor aj (T) describes the change of material time-scale with temperature. When temperature rises, the material time-scale shortens so that the relaxation proceeds faster, as if the material has an internal time clock —one unit of material time is equivalent to (7) units of observe time t. That is, the relaxation process of the material is calculated in a pseudo-time tlor (7). Consider a material undergoing stress relaxation. The relaxation speed is determined by an internal time-scale (or clock) within the material. As temperature rises, so does the amount of molecular motion occurring in one unit of an observation time, and the material s time scale shortens so that the relaxation proceeds faster. The shift factor aj gives a shift of material time-scale with temperature. 

When structural-rheological simplicity does not hold and, for example, fi in equation (7) changes with physical aging, exact reduction of the creep or stress relaxation cnrves cannot be accomplished. However, it is estimated by the author that a change in the KWW parameter of approximately 5-10% would result in a deviation of the logarithm of the aging time shift factor (log ate) of 0.05 over two decades in time scale. Scatter in reduced curves of this order of magnitude is often observed because of the scatter in the creep or stress relaxation data (23,39,40). Researchers have not systematically looked for such small deviations in their rednced curves. 

Application of this formulation by Prest to the data of Fig. 13-12 indicated that V = Vi but V2 s vl, at le t for 0.4. The corresponding time shift factors Xi and X2 were functions of Mw and respectively. It is of interest that, if the relaxation spectrum is to be independent of molecular weight in the transition zone where H is proportional to (Section C2 of Chapter 10), the following equation must be fulfilled ... 

The effect of pressure on linear viscoelastic properties can also be accounted for in terms of shift factors. One can define an isothermal time-shift factor flp(P) that accounts for the effect of pressure on the relaxation times at constant temperature, and it has been found that this factor follows the well-known Bams equation ... 

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. 

It was shown earlier that the variation of creep or relaxation moduli with time are as illustrated in Fig. 2.9. If we now introduce temperature as a variable then a series of such curves will be obtained as shown in Fig. 2.58. In general the relaxed and unrelaxed modulus terms are independent of temperature. The remainder of the moduli curves are essentially parallel and so this led to the thought that a shift factor, aj, could be applied to move from one curve to another. 

The range of semi-dilute network solutions is characterised by (1) polymer-polymer interactions which lead to a coil shrinkage (2) each blob acts as individual unit with both hydrodynamic and excluded volume effects and (3) for blobs in the same chain all interactions are screened out (the word blob denotes the portion of chain between two entanglements points). In this concentration range the flow characteristics and therefore also the relaxation time behaviour are not solely governed by the molar mass of the sample and its concentration, but also by the thermodynamic quality of the solvent. This leads to a shift factor, hm°d, is a function of the molar mass, concentration and solvent power. 

The glass transition involves additional phenomena which strongly affect the rheology (1) Short-time and long-time relaxation modes were found to shift with different temperature shift factors [93]. (2) The thermally introduced glass transition leads to a non-equilibrium state of the polymer [10]. Because of these, the gelation framework might be too simple to describe the transition behavior. 

Thus (he time scale / at /, divided by an is equivalent to the scale at On a log scale, log a, is thus the horizontal shift factor required for superposition. An important consequence of equation (22) is that a, or log (ii is the same for a given polymer (or solution) no matter what experiment is being employed. Thai is. creep and stress-relaxation curves are shifted by the same amount. 

Several attempts have been made to superimpose creep and stress-relaxation data obtained at different temperatures on styrcne-butadiene-styrene block polymers. Shen and Kaelble (258) found that Williams-Landel-Ferry (WLF) (27) shift factors held around each of the glass transition temperatures of the polystyrene and the poly butadiene, but at intermediate temperatures a different type of shift factor had to be used to make a master curve. However, on very similar block polymers, Lim et ai. (25 )) found that a WLF shift factor held only below 15°C in the region between the glass transitions, and at higher temperatures an Arrhenius type of shift factor held. The reason for this difference in the shift factors is not known. Master curves have been made from creep and stress-relaxation data on partially miscible graft polymers of poly(ethyl acrylate) and poly(mcthyl methacrylate) (260). WLF shift factors held approximately, but the master curves covered 20 to 25 decades of time rather than the 10 to 15 decades for normal one-phase polymers. 

The first comparison is based on the T values of gaseous, liquid and adsorbed molecules. Unfortunately, no measurements are available for butenes in the gas or liquid phase. Nevertheless a reasonable parallel can be drawn with propylene where the three different phases were investigated (35) at 295 K for the gas (1 atm) Tj of C2 is 0.095 s in the liquid state (2.6. M in CDCI3) 59.9, 58.7 and 65.2 s for Cj, C2 and C3 respectively adsorbed on NaY zeolite 0.81, 1.6 and 0.81 s. The shortest relaxation times characterize the gas phase where the spin-rotation mechanism (NOE factor n = 0) is very effective (30,35). In the liquid, dipole-dipole and spin-rotation mechanisms both play a role and the total relaxation rate is about three orders of magnitude lower than in the gas phase. The adsorbed molecules show therefore an intermediate behaviour between gas and liquid, as it was also suggested by chemical shift data. 

For transport in amorphous systems, the temperature dependence of a number of relaxation and transport processes in the vicinity of the glass transition temperature can be described by the Williams-Landel-Ferry (WLF) equation (Williams, Landel and Ferry, 1955). This relationship was originally derived by fitting observed data for a number of different liquid systems. It expresses a characteristic property, e.g. reciprocal dielectric relaxation time, magnetic resonance relaxation rate, in terms of shift factors, aj, which are the ratios of any mechanical relaxation process at temperature T, to its value at a reference temperature 7, and is defined by... 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

Fig. 1.3 Relaxation map of polyisoprene results from dielectric spectroscopy (inverse of maximum loss frequency/w// symbols), rheological shift factors (solid line) [7], and neutron scattering pair correlation ((r(Q=1.44 A )) empty square) [8] and self correlation ((t(Q=0.88 A" )) empty circle) [9],methyl group rotation (empty triangle) [10]. The shadowed area indicates the time scales corresponding to the so-called fast dynamics [11]...

To is known as the Vogel-Fulcher temperature and is located about 30 K below Tg. r is the asymptotic value of the relaxation time of the correlator 4> for T—>oo. Also the rheological shift factors a (T) mentioned above approximately follow such temperature dependences [34] ... 

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