Vertical shift factor

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. 

Table II The values of the vertical shift factors bf for the four D networks with respect to TQ = 300 K.

Master curves can often be made for crystalline as well as for amorphous polymers (33-38). The horizontal shift factor, however, will generally not correspond to a WLF shift factor. In addition, a vertical shift factor is generally required which, has a strong dependence on temperature (36-38). At least part of the vertical shift factor results from the change in... 

Tobol sky and co-workers who also modified it to account for proportionality of modulus to absolute temperature (3). This has the effect of creating a slight vertical shift in the data. Ferry further modified the time-temperature superposition to account for changes in density at different temperatures which has the effect of creating an additional vertical shift factor (4). The effect of the temperature-density ratio on modulus is frequently ignored, however, since it is commonly nearly unity. 

From the vertical shift factor of the master curve, we are able to describe the mass dependence of the zero-shear viscosity in the iso-free volume state which is directly connected to the radius of gyration of the chains. In the molten state, it is generally assumed that the chains exhibit a Gaussian conformation and therefore the viscosity should be proportional to the molecular weight. 

Figure 25 the vertical shift factor of the master curve of Figure 23 gives the structure factor A as a function of molecular weight. 

Lopes da Silva et al. (1994) found that the fiequeney-temperature superposition, analogous to time-temperature superposition in transient rheologieal experiments, was applieable to a 1 % locustbean (LB) gum dispersion so that master eurves at To = 25°C were obtained for G and G" (Figure 3-38). In eontiast, smooth master eurves could not be obtained for G and G" values of 3.5% high-methoxyl pectin dispersions either separately or for both simultaneously (Figure 3-39). The discrepancies were higher for a 3.5% low-methoxyl pectin dispersion. It was concluded that the time-temperature superposition technique was not applicable to the pectin dispersions due to their aggregated structure. For the studied samples, the vertical shift factor for the moduli (Topo/ T P) had a small effect on the master curve (Lopes da Silva et al., 1994). 

Therefore, Or = 0.0012. This means that the longest relaxation time r of the polyisoprene at 100°C is 0.0012 times its value at 25°C. The viscosity tj changes roughly in proportion to the relaxation time if the small vertical shift factor is neglected. Thus, ... 

Figure 5.7 Frequency dependences of the storage (Q) and loss (-h) moduli for poly(dimethylsilox-ane) (PDMS) samples whose reactions were quenched at the times indicated (see Fig. 5-6). The data are time-temperature-shifted to the reference temperature T gf of 34°C, and they are shifted additionally by an amount A on the logarithmic axis to keep the curves from overlapping. The vertical shift factors bf are given by p T sf)T d/ p T)T), where p is the density. (From Winter and Chambon 1986, with permission from the Journal of Rheology.)...

As demonstrated before, the shifting involves three shift factors, one horizontal, usually expressed as aj, = b rip(T)/rip(Tp), where b = p T /pT is the hrst vertical shift factor that originates in the thermal expansion of the system (p is density). The subscript o indicates reference conditions, dehned by the selected reference temperature T, usually taken in the middle of the explored T-range. For homopolymer melts, as well as for amorphous resins, the two shift factors, aj, and b.j, are sufficient. However, for semi-crystalline polymers, a second vertical factor, v., has been found necessary — it accounts for variation of the crystallinity content during frequency scans at different temperatures [Ninomiya and Ferry, 1967 Dumoulin, 1988]. 

The variation of density of rock resulting from temperature variation is very small. Therefore it can be considered that the intrinsic variation of creep compliance itself is also small. Consequently the effect of vertical shift factor in eq. (17) can be neglected and only the horizontal shift factor plays a dominant role. 

Horizontal shift factors (a ) were determined empirically. No vertical shift factor (To/T) was Incorporated. The empirical shift factors were used with Arrhenlus-type [log = (. AH/2.303R)/ (1/T-l/To)] and HLF-type [log at - Ci (T-To)/(C2 + T-To)] equations. A plot of T-To/log at versus T-To yielded HLF constants and C< 2 resultant values corresponding to the glassy zone... 

Ferry suggested on the basis of a molecular theory of viscoelasticity that there should be a small vertical shift factor ro/0o/(T/0),where p is the density at T and is the density at the reference temperature T. These shifts have been made in fig. 7.14, but they are usually small and are often neglected.)... 

The horizontal shift factor reflects the temperature dependence of relaxation time, and the vertical shift factor reflects the... 

Vertical shifts are seen as a temperature dqiradait zero time shear modulus Gq the vertical shift factor is hr = GtP /G,fJ). The modulus G is calculated ftxnn the torque 1 assuming that die stress and strain ate both linear ftmctitMis of the t linder radius, as in Equation (1). 

Figure 3. Tonperature shift ftu tors, log as a function of temperature. Table II. Vertical shift factors for time - temperature and time - strain superposition...

Table IV. Vertical shift factor rate as a function of temperature and strain...

SBR elastomer with known crosslinking densities was studied in dynamic shear and tensile creep and data collected from -30 to 70 °C used to construct TTS master curves. In addition to a temperature shift factor a vertical shift factor was required from 10 to 30 °C to account for changes in density. Linear viscoelastic properties were observed in accordance with the classical theory of rubber elasticity. Standard vertical shift factors were required in a comparative TTS test with uncrosslinked polybutadiene and poly(ethylene-cu-propylene-co-diene monomer) (EPDM). ... 

The principle of TTS lies in the equivalency of time (frequency) and temperature. Due to various limitations, one cannot carry out experiments at conditions such as at very low frequencies and very high temperatures or vice versa. TTS is used to obtain data at different conditions to save experimental time. The viscoelastic data of one temperature can be related to the higher or lower temperature using a shift factor a ) to the right side, or to the left side of the time axis using a reference temperature (T f). A fully overlapped curve can be obtained for any reference temperature this is called a master curve . It is also widely accepted that a minor vertical shift factor may also be applied to more accurately model master curves. 

The horizontal and vertical shift factors for reduction of two aging creep curves are defined as... 

Interpretation of the data from the interrupted creep experiment is based on the assumption that once the material reaches steady-state creep, a change in temperature will not disturb this condition if Js is independent of temperature. Work by Leaderman on polyisobutylene above on Tg (49), as well as Plazek s work on l,3,5-tri-a -naphthylbenzene (48), indicates this to be the case. Using the interrupted creep test, the evolution of both Js and r] for a polystyrene of 3400 molecular weight have been measured (48). This work confirms that for materials with temperature-dependent Js, Js will decrease during physical aging. The interrupted creep experiment may also make it possible to experimentally verify the relationship between the vertical shift factors and Js-... 

Ferry and co-workers [7], on the basis of the molecular theory of viscoelasticity, proposed that superposition should incorporate a small vertical shift factor Fopo/TP, where p is the density at the experimental temperature T and po relates to the reference temperature Tq. Further corrections have been suggested by McCrum and Morris [8] to deal with the changes in unrelaxed and relaxed compliances with temperature. 

The average WLF parameters for PLA are Ci =3.24K and C2 = 164.9 K the later correspond to a Vogel temperature of 288.25 K. Vertical shift factors given by poTolpT are first applied to the measured values of G and G" using the known temperature dependence of density (Equation 10.13) before determining the shift factors [11]. This WLF temperature dependence is built into the accompanying software. 

This equation predicts that the vertical shift factor is proportional to and that G is located on a straight line against C/C in the double logarithmic plot. Actually, the shift factor au is proportional to M ° experimentally. But the master curve was not a straight line. Near the gel point, Eq. (5) can be written in a more appropriate way which reads as... 

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