Viscosity master curves

Theories based on the uniformly effective medium have the practical advantage that they can be extended quite easily to polydisperse systems (227). Viscosity master curves can be predicted from the molecular weight distribution, for example. The only new assumption is that the entanglement time at equilibrium for a chain of molecular weight M in a polydisperse system has the form suggested by the Rouse theory (15) ... 

Fig. 3.24. Complex-viscosity master curves at 25 °C for nearly monodisperse samples of linear and three-arm-star 1,4-polybutadiene [28].

Fig. 3.26. A comparison of reduced-viscosity master curves for nearly mono-disperse linear and star polymers.

Shear stress versus parent shear rate for two capillaries of different diameter but the same L/R = 60. ABS polymer melt squares, R = 0.26 mm, 230°C barrel circles, R = 1.6 mm, 232°C barrel. The solid line represents isothermal flow estimated at high Y from the viscosity master curve (Hg-ure 2.6.1) the dashed line, adiabatic boundary condition, viscous dissipation with no heat transfer to the wall, for / = 1.6 mm. Replotted from Cox and Macosko (1974a) and Cox (1973). 

Figure 10.11 Viscosity master curves for a HDPE (solid line) and two sets of GPC fractions obtained from different HDPE whole polymers.The shift factor for shear rate includes the branching factor based on [ )], which the authors call g,a MWD factor and A, which is 2 tp.The superposition of data for the LDPE fractions is only approximate. From Mendelson etal. [75].

Figure 4 Viscosity master curve [absolute value of the complex viscosity ( ri ) vs. frequency x shift factor (oa, rad s" )] for copoly(ethylene/methacrylic acid (3.5 mol % acid, = 86 000, A = 9400) (reproduced by permission of John Wiley Sons, Inc. from T. R. Earnest, Jr., and W. J. MacKnight, J. Polym. ScL, Polym. Phys. Ed., 1978, 16, 143)...

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). 

For concentrated solutions of polystyrene in n-butylbenzene, Graessley [40] has shown that the reduced viscosity r red Cnred=(r ( y)- rls)/(rlo rls)) can be represented on a master curve if it is plotted versus the reduced shear rate (3 ((3= y/ ycnt= y-A0). For semi-dilute solutions a perfect master curve is obtained if (3 is plotted versus a slope corrected for reduced viscosity, T corp as shown in Fig. 16. 

Figure 14.10 Master curves of steady shear viscosity, r (at lower shear rates) and complex viscosity, r (at higher frequencies) for the first seven generations of PAMAM dendrimers at 40°C in the bulk state...

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. 4.3 Scaling representation of the spin-echo data at the first static structure factor peak Qmax- Different symbols correspond to different temperatures. Solid line is a KWW description (Eq. 4.8) of the master curve for 1,4-polybutadiene at Qmax=l-48 A L The scale r(T) is taken from a macroscopic viscosity measurement [130]. Inset Temperature dependence of the non-ergodicity parameter/(Q) near the lines through the points correspond to the MCT predictions (Eq. 4.37) (Reprinted with permission from [124]. Copyright 1988 The American Physical Society)...

PPG (at higher temperatures) behaves like a typical pseudoplastic non-Newtonian fluid. The activation energy of the viscosity in dependence of shear rate (284-2846 Hz) and Mn was detected using a capillary rheometer in the temperature range of 150-180°C at 3.0-5.5 kJ/mol (28,900 Da) and 12-13 kJ/mol (117,700 Da) [15]. The temperature-dependent viscosity for a PPG of 46 kDa between 70 and 170°G was also determined by DMA (torsion mode). A master curve was constructed using the time-temperature superposition principle [62] at a reference temperature of 150°G (Fig. 5) (Borchardt and Luinstra, unpublished data). A plateau for G was not observed for this molecular weight. The temperature-dependent shift factors ax were used to determine the Arrhenius activation energy of about 25 kJ/mol (Borchardt and Luinstra, unpublished data). 

Another often used representation of the viscoelastic flow behavior utilizes normal stress coefficients P/ = Ni/y. Figure 10 depicts flow curves of a family of PAA/water solutions differing in concentrations and therefore in their viscosities. Normalized by the zero-shear viscosity fiQ and by a constant shear rate /q shear stress value of to= 1 N/m they produce master curves for viscosity and the normal stress coefficient. The preparation... 

Fig. 8.10. Viscosity-shear rate master curve for concentrated polystyrene-n-butyl benzene solutions. The data were obtained for molecular weights ranging from 160000 to 2400000 concentrations from 0.255 to 0.55 gm/ml, and temperatures from 30° C to 60° C (155)...

Judged by the superposability of viscosity-shear rate data on the same master curve for a variety of polymers [polystyrene (155) (Fig. 8.10), poly(a-methyl... 

Table 8.1. Behavior of theoretical master curves for viscosity at high shear rates...

Fig. 8.15. Viscosity vs shear rate in concentrated solutions of narrow distribution polystyrene The solvent in n-butyl benzene, the concentration is 0.300 gm/ml and the temperature is 30° C. The symbols are O for M = 860000 and for M = 411000 at low shear rates (155) and at high shear rates (346). The solid line for M= 860000 is the master curve for monodisperse systems from Graessley (227). The solid line for M=411000 is the master curve from Ree-Eyring (341). Either master curve fits data for both molecular weights...

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... 

We have seen in the previous sections that viscoelastic scaling, employing the scaling factor ac, produces master viscosity curves for polymer-gas solutions that are identical to the master curve for the pure polymer. This means that the effect of dissolved gas on the rheology of polymer melts can be described entirely by the variation of ac with gas content. We have not, of course, demonstrated that all polymer-gas systems follow this scaling beha-... 

All polymer-gas systems studied here exhibit ideal viscoelastic scaling, whereby viscosity measurements taken at different gas compositions can be unified to a master curve of reduced viscosity r (c, y)/a versus reduced... 

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