Horizontal shift

Curves for the viscosity data, when displayed as a function of shear rate with temperature, show the same general shape with limiting viscosities at low shear rates and limiting slopes at high shear rates. These curves can be combined in a single master curve (for each asphalt) employing vertical and horizontal shift factors (77—79). Such data relate reduced viscosity (from the vertical shift) and reduced shear rate (from the horizontal shift). 

Time-temperature superposition at the gel point does not let us distinguish between the vertical and the horizontal shift, since the spectra are given by... 

Table I The values of the horizontal shifts log aj for the network families K, D and N with respect to the reference temperature T0 = 300 K.

Generally, the rheology of polymer melts depends strongly on the temperature at which the measurement is carried out. It is well known that for thermorheological simplicity, isotherms of storage modulus (G (co)), loss modulus (G"(complex viscosity (r (co)) can be superimposed by horizontal shifts along the frequency axis ... 

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. 

The method of relating the horizontal shifts along the log time scale to temperature changes as developed by Williams, Landel, and Ferry from equation (24) is known as the WI.F method. The amount of horizontal slut of (he log time scale is givvn by log a,-. If the glass transition temperature is chosen as the reference temperature, the temperature dependence ni the shift lactoi lor most amorphous polymers is... 

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... 

There is increasing evidence, however, that a magmatic water component cannot be excluded in some volcanic systems. As more and more data have become available from volcanoes around the world, especially from those at very high latitudes, Giggenbach (1992) demonstrated that horizontal shifts are actually the excep-... 

The blending of short glass fibres results in an increase of the E-modulus to e g. its threefold over the whole temperature region. If the slope of the log E - T curve is small, such as with semi-crystalline polymers between Tg and T, a vertical shift causes a considerable horizontal shift (see MT 8.1.2.), so a strong increase of the softening points, which are of importance in applications at higher temperatures. 

Thus both curves can be made to coincide, provided that Fowler s equations (3) and (4) apply to the experimental results. Finally the shift is expressed by the position of the origin of system a in system 6. The horizontal shift is eo /(A 7 ) and gives the potential the vertical shift gives the constant M (see Fig. 31 in which the origin of system a is marked by a cross). 

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... 

With the aid of Eq. (5.47), experimental Tg vs t curves obtained at different cure temperatures may be horizontally shifted to obtain a master curve. Equation (5.47) may be written for a reference temperature, Tr ... 

FIGU RE 8.39 A flow cell constructed for non-coaxial shealth flow. The additional vertical position inlet of this flow cell allows positioning of the sample flow in the center of the channel. The horizontal control ports are used to position the sample flow with a horizontal shift. The perspective view, top view, and two cross-section views are shown [891]. Reprinted with permission from the Royal Society of Chemistry. 

There is one further important practical aspect that has to be considered when taking this approach to performing an experimental bifurcation analysis impedance measurements can only be carried out with stable stationary states it is not feasible to measure unstable stationary states, or states close to a bifurcation. However, as we discussed above, N-NDR and HN-NDR systems become unstable due to ohmic losses in the circuit, whereas they are always stable for vanishing R< >. Being aware that an ohmic series resistor causes only a horizontal shift of the impedance spectrum in the complex plane, it is apparent that it is possible to infer about the existence of bifurcations from impedance measurements at sufficiently low solution resistance (or when invoking an ZR-compensation, an option many potentiostats provide). This is illustrated with the schematic impedance spectrum shown in Fig. 12, which depicts a typical impedance spectrum of an N-NDR system. The spectrum possesses two... 

Horizontal shift distances at different points between the relaxation isotherms were measured. For PC (7), PST (8), and the 25/75 blend, the shift factors are independent of time, and complete master curves can be obtained by simple tj nf g laxation isotherms along... 

The slopes of those parallel curves, with temperature as a parameter, are equal to 2.303R/Vnc, and the relationship between the horizontal shift log bT and the temperature is... 

FIG. 13.73 Eyring plots of horizontal shift of the polycarbonate results presented in Fig. 13.71. 

If stress relaxation curves are obtained at a number of different temperatures, it is found that these curves can be superimposed by horizontal shifts to produce what is called a master curve .42 This concept of time-temperature equivalence is very important to understanding and predicting polymer behavior. As an example, a polymer at very low... 

For many purposes I feel that it is best not to reduce the horizontal shift between the two curves so as to make them absolutely coincide in order to detect any variations in the curve which may be purely transient. If the curves are very nearly superimposed upon one another variations that appear up or down at corresponding points on both curves must be due to the variation of the light coming through the photographic negatives which are permanent and not transient. 

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