Glass master curve

Figure 6.19 Loss and storage moduli as a function of frequency for 2Si02-Na20 glass (master curve see also Section 6.4.S.4). (Reprinted from Mills, 1974 with permission from Elsevier.)...

Note that subtracting an amount log a from the coordinate values along the abscissa is equivalent to dividing each of the t s by the appropriate a-p value. This means that times are represented by the reduced variable t/a in which t is expressed as a multiple or fraction of a-p which is called the shift factor. The temperature at which the master curve is constructed is an arbitrary choice, although the glass transition temperature is widely used. When some value other than Tg is used as a reference temperature, we shall designate it by the symbol To. 

FIGURE 26.2 Master curve of the friction coefficient of an acrylate-butadiene rubber (ABR) gum compound on smooth clean dry glass, referred to room temperature. (From Grosch, K.A., Proc. Roy. Soc., A 274, 21, 1963.)... 

The shape of the maser curve not only depends on the rubber compound, but also on the surface on which it slides. On dry, clean polished glass the friction master curve for gum rubbers rises from very small values at low log ajv to a maximum which may reach friction coefficients of more than 3 and falls at high log ajv to values which are normally associated with hard materials, i.e., 0.3 shown for an ABR gum compound in Figure 26.2. If the position of the maximum on the log a-fV axis for different gum rubbers is compared with that of their maximum log E frequency curves, a constant length A = 6 X 10 m results which is of molecular dimension, indicating that this is an adhesion process [10]. 

FIGURE 26.8 The friction master curves of the acrylate-hutadiene rubber (ABR) gum mbber on (a) dry glass, (b) dry clean silicon carbide 180, (c) dry silicon carbide dusted with MgO powder, (d) Alumina 180 wetted with distilled water, and (e) wetted with water +5% detergent. 

FIGURE 26.20 The log a v speed function of the previous chart is combined with the friction master curves for a natural rubber (NR) and a styrene-butadiene rubber (SBR) gum compound on glass showing the limited range of friction values (and their position on the log a-iv axis for different testing conditions) which are obtained when the sliding speed is increased. 

FIGURE 26.35 Master curve of the side force coefficient of two tread compounds with different glass transition temperatures on wet Alumina 180 A 3,4 IR Tg = —21°C and OESBR Tg = —46°C. Two spot measurements at two different water temperatures show that the ranking if the two compounds reverses. (From Grosch, K.A., Kautschuk, Gummi, Kunststojfe, 6 m, 432, 1996.)... 

Fig. 12. Dynamic moduli master curves of PBD 44 precursor (p = 0) and PBD 44 critical gel [60]. The entanglement and glass transition regime is hardly affected by the crosslinking. Open symbols correspond to G, filled ones to G"...

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 free-volume concept was applied most widely in the theory of viscoelastic properties of polymers developed by Williams, Landel and Ferry (WLF theory), presented in detail in12. According to WLF theory, the changes in liquid viscosity with frequency and temperature from glass temperature T% to T may be plotted on a single master curve by using the reduction factor... 

Stress relaxation master curve. For the poly-a-methylstyrene stress relaxation data in Fig. 1.33 [8], create a master creep curve at Tg (204°C). Identify the glassy, rubbery, viscous and viscoelastic regions of the master curve. Identify each region with a spring-dashpot diagram. Develop a plot of the shift factor, log (ax) versus T, used to create your master curve log (ot) is the horizontal distance that the curve at temperature T was slid to coincide with the master curve. What is the relaxation time of the polymer at the glass transition temperature ... 

At the melting point (Tm) and at the glass transition point (Tg) its value is nearly zero in the intermediate region a maximum (vmax) is observed at a temperature Tk. Gandica and Magill (1972) have derived a master curve, valid for all "normal" polymers, in which the ratio v/vmax is plotted vs. a dimensionless crystallisation temperature ... 

Figure 28. The excess wing exponent y for ten different glass formers (cf. Fig. 27) rescaled by " g = y(Tg) so that all the datasets coincide on a master curve. The solid line represents Eq. (40). For more details see Ref. [275],...

Figure 32. The a-relaxation times for the glass formers studied in the present work (cf. Fig. 27). In addition data of diglycidyl ether of bisphenol A (DGEBA) and phenyl glycidyl ether (PGE) are included time constants as obtained from DS data sets of m-TCP and 2-picoline were combined with xrl from conductivity and light scattering measurements, respectively, (a) Relaxation times as a function of T Ts. The systems differ by the slope of Ta at Tg. (b) By plotting xr, as a function of the rescaled temperature z = m(T/Tg — 1) the effect of an individual fragility is removed and a master curve is obtained for systems with similar To. Solid line represents Eq. (41) with Kf) — 17. (c) Upper part master curve for xa according to Eq. (42). Deviations of the data from Eq. (42) (solid line) indicate break-down of the VFT equation. Lower part The ratio lg(ra/rvft) shows deviations from a VFT behavior most clearly. Dashed vertical lines indicate shortest and fastest tx, respectively, observed. All the figures taken from Ref. [275].

The left-hand panel of Fig. 11-17 contains sketches of typical stress relaxation curves for an amorphous polymer at a fixed initial strain and a series of temperatures. Such data can be obtained much more conveniently than those in the experiment summarized in Fig. 11-8, where the modulus was measured at a given time and a series of temperatures. It is found that the stress relaxation curves can be caused to coincide by shifting them along the time axis. This is shown in the right-hand panel of Fig. 11-17 where all the curves except that for temperature Tg have been shifted horizontally to form a continuous master curve at temperature T%. The glass transition temperature is shown here to be Tj at a time of 10" min. The polymer behaves in a glassy manner at this temperature when a strain is imposed within 10 min or less. 

It is common practice now to use the glass transition temperature measured by a very slow rate method as the reference temperature for master curve construction. Tlien the shift factor for most amorphous polymers is given fairly well by... 

The use of this time-temperature equivalence allows one to obtain "master curves" at a reference temperature, which enlarges considerably the experimental window. For glass-forming materials such as polystyrene, polymethylmetacrylate, polycarbonate, polymerists describe the shift factor aj in terms of the WLF equation ... 

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