Vertical shift

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. 

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). 

From this relatively simple test, therefore, it is possible to obtain complete flow data on the material as shown in Fig. 5.3. Note that shear rates similar to those experienced in processing equipment can be achieved. Variations in melt temperature and hypostatic pressure also have an effect on the shear and tensile viscosities of the melt. An increase in temperature causes a decrease in viscosity and an increase in hydrostatic pressure causes an increase in viscosity. Topically, for low density polyethlyene an increase in temperature of 40°C causes a vertical shift of the viscosity curve by a factor of about 3. Since the plastic will be subjected to a temperature rise when it is forced through the die, it is usually worthwhile to check (by means of Equation 5.64) whether or not this is signiflcant. Fig. 5.2 shows the effect of temperature on the viscosity of polypropylene. 

The residuals relative to the smoothed trace (to ymean if o smoothing has been done) are plotted a vertical shift and an expansion factor can be chosen. 

Fig. 3 a UV-Vis DRS spectra of dehydrated TS-1 catalyst reporting the typical 208 nm (48000cm i) LMCT hand, see Fig. 2h also reported are the four excitation laser lines used in this Raman study near-lR (dotted), visible (full), near-UV (dashed) and far-UV (dot-dashed), b Raman spectra of dehydrated TS-1 obtained with four different lasers emitting at 7 = 1064,422,325, and 244 nm (dotted, full, dashed, and dot-dashed lines, respectively). Raman spectra have been vertically shifted for clarity. Although the intensity of each spectrum depends upon different factors, the evolution of the 7(1125)//(960) ratio by changing the laser source is remarkable. The inset reports the Raman spectrum collected with the 244 nm laser in its full scale, in order to appreciate the intensity of the 1125 cm enhanced mode. Adapted from [48] with permission. Copyright (2003) by The Owner Societies 2003... 

Fig. 30. Absorption spectrum of a Q-CdS sol (upper part) and of six fractions (lower part). Spectra are vertically shifted...

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

Fig. 68 Comparison of temperature-dependent intensity of first-order Bragg peak for bare matrix copolymer (A) containing 0.5 wt% nanocomposites with plate-like (V), spherical (o) and rod-like ( ) geometry. Data are vertically shifted for clarity. Inset dependence of ODT temperature on dimensionality of fillers (spherical 0, rod-like 1, plate-like 2). Vertical bars width of phase transition region. Pure block copolymer is denoted matrix . From [215]. Copyright 2003 American Chemical Society...

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... 

Before finding the Laplace-transformed probability density wj(s, zo) of FPT for the potential, depicted in Fig. A 1(b), let us obtain the Laplace-transformed probability density wx s, zo) of transition time for the system whose potential is depicted in Fig. Al(c). This potential is transformed from the original profile [Fig. Al(a)] by the vertical shift of the right-hand part of the profile by step p which is arbitrary in value and sign. So far as in this case the derivative dpoints except z = 0, we can use again linear-independent solutions U(z) and V(z), and the potential jump that equals p at the point z = 0 may be taken into account by the new joint condition at z = 0. The probability current at this point is continuous as before, but the probability density W(z, t) has now the step, so the second condition of (9.4) is the same, but instead of the first one we should write Y (0) + v1 (0) = YiiOje f1. It gives new values of arbitrary constants C and C2 and a new value of the probability current at the point z = 0. Now the Laplace transformation of the probability current is... 

Fig. 8 Temperature dependence of din f>/d(T 1), i.e., slope of the Arrhenius plot as a function of temperature for (a) (EDT-TTFBr2)FeBr4 at various pressures - the data for 0, 5.8 and 10.1 kbar are vertically shifted up by 60, 40 and 20 K, respectively, for clarity (b) (EDO-TTFBr2)2GaCl4 and (EDO-TTFBr2)2FeCl4 at 11 kbar. TMl and TN are the metal-insulator transition temperature and the Neel temperature, respectively, hi (b) the metal-insulator transition is observed as two separate peaks...

Fig. 6.13 Highly repeatability separation and detection of DNT from nitrotoluene and DNT mixtures. SPME extractions are performed at room temperature, (a) Nitrotoluene extraction time 3 s, DNT extraction time 20 s. (b) Nitrotoluene extraction time 1 s, DNT extraction time 30 s. DNT peak emerges at approximately 40 s. Curves are vertically shifted for clarity...

Fig. 6. Raman spectra of sample 1 (Ti-free silicalite), and samples 3, and 5 (TS-1 with Ti02 wt% being 2 and 3, respectively), (a) Spectra collected with a A = 1064 nm (9398 cm-1) excitation, (b) Spectra collected with a A = 224 nm (40,984 cm x) excitation. Inset UV-DRS spectrum of sample 5. Vertical line indicates the position of the excitation wavelength A used for collecting the sample reported in part (b). Vertical dotted lines are placed at 960 cm 1. Spectra of both parts have been vertically shifted for clarity [Reprinted from Ricchiardi et al (41) with permission. Copyright (2001) American Chemical Society].

Fig. 9. (a) Infrared spectra of outgassed thin pellets of Ti-free silicalite (curve 1) and TS-1 with increasing Ti content x (curves 2-5). Spectra were normalized by means of the overtone bands between 1500 and 2000 cm-1 (not shown) and vertically shifted for clarity. The thick horizontal line represents the fwhm of the 960 cm-1 band for sample 2. By assuming that this band has a constant fwhm for any x, the absorbance W obtained is plotted as the ordinate in panel b, where the band has the same fwhm as in curve 2 (horizontal thin lines), (b) Intensity W of the 960 cm-1 infrared band (normalized absorbance units) as a function of x (full squares) and corresponding Raman counts (open squares) [Reprinted from Ricchiardi et al. (41) with permission. Copyright (2001) American Chemical Society]. 

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. 

Fig. 4 Effect of nanoclay loading on neat SEBS a Lorentz -corrected SAXS profiles (vertically shifted for better clarity) showing effect of nanoclay arrows indicate peak positions, b Lengths corresponding to first- and second- order scattering vector positions along with the 2D SAXS patterns for each sample of clay-loaded nanocomposites...

The vertical shift has arisen from the application of an absolute potential difference of d to a hypothetical interface, initially with zero potential difference across it, i.e., zhj) = 0. But the argument is valid for any change of potential across the interface. Thus, if the double-layer potential is initially Atye (i.e., the interface is at equilibrium) and then the potential is change to zf< ), the Morse curve for the initial state is shifted vertically through an energy F(Aty — Atye), or Ft],... 

Fig. 9.13. As a consequence of the vertical shift of one linear curve, the critical activation energy is altered.

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). 

DC-Correction is applied to compensate for a DC-offset of the FID, i.e. a vertical shift of the FID with respect to the zero-line, which occurs in quadrature detection mode if the two channels are not matched to each other. The effect is most pronounced for very weak samples and manifests itself, after Fourier transformation, as a spike in the centre of the spectrum at the center or carrier frequency. 

In these cases, the standard free energy of adsorption can be obtained from the equilibrium condition and is a simple exponential function of the potential which does not depend significantly on the charge distribution at the interface for an uncharged adsorbate. The chemisorption thus corresponds to a vertical shift in the free energy curves as depicted in Fig. 12 and affects the energy of activation [76]. 

The effect of the chemisorption of electrode reaction intermediates was first considered by Butler for the hydrogen evolution reaction [33]. Considering the adsorption in quasi-equilibrium or the steady-state approach, the effect of adsorption of an intermediate is a vertical shift in the corresponding free energy—reaction coordinate curve as depicted in Fig. 12. 

The main catalytic influence of the nature of the electrode material is through the adsorption of intermediates of complex electrode reactions. Hortiuti and Polanyi [58] suggested that the activation energy of an electrode reaction should be related to the heat of adsorption of adsorbed intermediates by a relationship of the form of the Br0nsted rule in homogeneous solutions. This corresponds to a vertical shift of the potential energy curves by an amount j3Aif°ds with (5 a symmetry factor as discussed in Sect. 6.4 and depicted in Fig. 12. 

The fact that the polyreaction of diacetylenes is topochemically controlled is especially well documented by the polymerization behavior of the sulfolipid (22)23 . (22) forms two condensed phases when spread on an acidic subphase at elevated temperatures (Fig. 10). UV initiated polymerization can only be carried out at low surface pressures in the first condensed phase, where the molecules are less densely packed. Apparently, in the second phase at surface pressures from 20 to 50 mN/m the packing of the diyne groups is either too tight to permit a topochemical polymerization or a vertical shift of the molecules at the gas/water interface causes a transition from head packing to chain packing (Fig. 10), thus preventing the formation of polymer. 

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... 

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