Modulus, imaginary reduced

Figure 7.03. Real and imaginary parts of the electric modulus vs. reduced frequency coTf) for 0.4 Ca(N03)2 0.6KNOj glass (After Howell ct. al.. 1974)...

Figure 4b. Imaginary reduced modulus plotted with the Maxwellian relaxation time-frequency product, oiTjJao for PPO (A), 59° scattering angle ...

Rheological properties of filled polymers can be characterised by the same parameters as any fluid medium, including shear viscosity and its interdependence with applied shear stress and shear rate elongational viscosity under conditions of uniaxial extension and real and imaginary components of a complex dynamic modulus which depend on applied frequency [1]. The presence of fillers in viscoelastic polymers is generally considered to reduce melt elasticity and hence influence dependent phenomena such as die swell [2]. 

Fig. 2.2. Real part [G ]R and imaginary part [G"]R of reduced complex modulus of Zimm s theory as evaluated by Tschoegl (59)...

Fig. 2.4. Real part [G ] and imaginary part [G"] of reduced complex modulus plotted against reduced angular frequency toR for 4 armed star polymer for Zimm-Kilb theory (49). Thick lines are calculated from eigenvalues given in Zimm-Kilb paper solid lines for h- oo and dashed lines for h- 0. Thin lines are calculated from eigenvalues of difference equation with Nb = 100 solid lines for h = 0.25 and dashed lines for h = 0.30 (73)...

Fig. 3.2. Real part G M/cR T (filled circles) and imaginary part (G" -mrj M/cR T (open circles) of reduced complex modulus plotted against reduced angular frequency ojqstispM/cR T for polyisobutylene (M = 1.76x 106) in benzene at 24° C at three concentrations as indicated. Lines are drawn following Zimm theory with...

Fig. 3.5. Real part [O ] and imaginary part [G"] of intrinsic complex modulus plotted against reduced angular frequency o > [ ] tjs for polystyrene solutions in two good solvents, Aroclor 1232 and a-chloronaphthalene, at 25° C. Curves are drawn following Tschoegl theory with e = 0.13 and h — 40 (3)...

Fig. 3.8. Real part [G ]R and imaginary part [G"]R of reduced intrinsic complex modulus plotted against reduced angular frequency wz01 for nine-armed polystyrene in two 0-solvents open circles, Decalin at 15° C and filled circles, DOP at 21° C. Curves are drawn following Zimm-Kilb theory with / = 9, Nb = 100 and...

Fig. 4.1. Real part G R and imaginary part GR of reduced complex. modulus plotted against reduced angular frequency wR for stiff chain model of Hearst-Harris with (contour length)/(persistenee length) 0.002 (111)...

The effect of varying T and hence tjs, was extensively studied for polystyrene and poly-a-methyl styrene solutions in Aroclor 129,130). The results revealed that (G — m ls)/e T is a unique function of co t] — t]s)/QT for each solution ova wide ranges of r s and to, where q is the density of the solution. Examples of the result of the reduction are given in Fig. 4.6 and 4.7. Here the real part G p and the imaginary part G" — cor]s)p of a reduced complex modulus (G — iojrjs) q0 TJq T) are plotted against coar for polystyrene solutions in Aroclor where the... 

Fig. 4.6. Real part G p and imaginary part (G" — mrjs)p of complex modulus reduced to 25° C for polystyrene of low molecular weight in chlorinated diphenyl. Subscript p indicates multiplication of data with q0T0/qT. Curves are drawn following Peterlin theory with parameters indicated 129)...

The <0 is the Brillouin shift in radians per second. The reduced real and imaginary modulus data are shown in Figures 4a and 4b. The plateau in (M — K)Joo extends to wt/sJoo 10 . In the region of solid like behavior reduction fails below 220 K. However, the juxtaposition of the low- and high-angle reduced data and the extent of the plateau region over some three decades in (ot sJqo lends support to the reduction procedure used. The solid curve is a single-relaxation model with a lumped relaxation time, T, of T = 0.06 tm- The dashed curve is based on a two-relaxation-time model with ti = 0.09 tm and to = 0.002 tm. 

Resonance Techniques. Resonance methods (183-185) are used for measurements below about 100 kHz. The complex modulus (usually Young s or shear) is determined over a limited frequency range at a number of fixed temperatures, usually over one to three decades of frequency and over the useful temperature range of the material. From the measured data, reduced frequency plots at constant temperature are generated by the application of the time-temperature principle (3). In these plots, the real and imaginary parts of the complex modulus are plotted over many decades of frequencies, typically, as many as six or more decades of frequency than were actually measured. 

Dynamic Modulus. The complex shear compliance is being obtained in a Periy-Fitzgerald transducer apparatus and preliminary results are illustrated here.f9) Figure 10 shows the real part of the compliance, reduced to 25 0 Figure 11, the imaginary part. 

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