Frequency-dependent moduli, dynamic

Fig. 5. Typical forms of frequency dependences of dynamic modulus. The content of the filler increases upon the transition from durve 1 to 2 and to 3. The discussion of regions I-VI, displayed on the curves, see text...

Fig. 12. The frequency dependence of dynamic storage modulus G at 200 °C for calcium carbonate filled polypropylenes (mean particle size 0.15 pm).Filler loading wt%, (o) 0 (6) 10 (o) 20 (9) 30 [47]...

Fig. 2.17 Frequency dependence of dynamic shear moduli computed using a model for the linear viscoelasticity of a cubic phase based on slip planes, introduced by Jones and McLeish (1995). Dashed line G, solid line G".The bulk modulus is chosen to be G — 105 (arb. units). The calculation is for a slip plane density AT1 - 10 5 and a viscosity ratio rh = = 1, where rjs is the slip plane viscosity and t] is the bulk viscosity. The strain...

This equation, also as equation (6.49) gives description of the frequency dependency of dynamic modulus at low frequencies (the terminal zone). Both in equation (6.49) and (9.35), the second terms present the contribution from the orientational relaxation branch, while the first ones present the contribution from the conformational relaxation due to the different mechanisms diffusive and reptational. 

Fig. 15. Frequency dependence of dynamic modulus (G ) and viscosity (t ) for the lacquer.

During dynamic measurements frequency dependences of the components of a complex modulus G or dynamic viscosity T (r = G"/es) are determined. Due to the existence of a well-known analogy between the functions r(y) or G"(co) as well as between G and normal stresses at shear flow a, seemingly, we may expect that dynamic measurements in principle will give the same information as measurements of the flow curve [1],... 

Second. The interpretation of the results of dynamic measurements is far from being obvious. Figure 5 gives a schematic representation of some typical results of measuring frequency dependence of the modulus G (m) for compositions with different concen-... 

The peculiarities of dynamic properties of filled polymers were described above in connection with the discussion of the method of determining a yield stress according to frequency dependence of elastic modulus (Fig. 5). Measurements of dynamic properties of highly filled polymer melts hardly have a great independent importance at present, first of all due to a strong amplitude dependence of the modulus, which was observed by everybody who carried out such measurements [3, 5]. 

Moreover, if for pure polymer melts the correlation of the behavior of the functions ri (co) andrify) under the condition of comparing as y takes place, as a general rule, but for filled polymers such correlation vanishes. Therefore the results of measuring frequency dependences of a dynamic modulus or dynamic viscosity should not be compared with the behavior of the material during steady flow. 

In the case of dynamic mechanical relaxation the Zimm model leads to a specific frequency ( ) dependence of the storage [G ( )] and loss [G"(cd)] part of the intrinsic shear modulus [G ( )] [1]. The smallest relaxation rate l/xz [see Eq. (80)], which determines the position of the log G (oi) and log G"(o>) curves on the logarithmic -scale relates to 2Z(Q), if R3/xz is compared with Q(Q)/Q3. The experimental results from dilute PDMS and PS solutions under -conditions [113,114] fit perfectly to the theoretically predicted line shape of the components of the modulus. In addition l/xz is in complete agreement with the theoretical prediction based on the pre-averaged Oseen tensor. 

Fig. 3.22 Frequency-dependent dynamic modulus G"(co) from a PE chain of M =800 kg/mol at 509 K. The solid line gives the reptation prediction of G co)-cor . The peak here may not be confused with the a-relaxation of the glass dynamics. It immediately follows from the Fourier transform of strongly depends on molecular weight. The glass relaxation...

Here pg and p f are the mass densities of the gel and the solvent, respectively, K is a bulk modulus, c0 is the speed of sound, and i s is the solvent shear viscosity. The solvent bulk viscosity has been neglected. The terms proportional to / arise from an elastic coupling in the free energy between the density deviation of gel and that of solvent The p in Eq. (6.1) coincides with the shear modulus of gels treated so far. We neglect the frequency-dependence of the elastic moduli. It can be important in dynamic light scattering, however, as will be discussed in the next section. 

Dilute polyelectrolyte solutions, such as solutions of tobacco mosaic virus (TMV) in water and other solvents, are known to exhibit interesting dynamic properties, such as a plateau in viscosity against concentration curve at very low concentration [196]. It also shows a shear thinning at a shear strain rate which is inverse of the relaxation time obtained from the Cole-Cole plot of frequency dependence of the shear modulus, G(co). 

The relaxation spectrum H(0) completely characterizes the viscoelastic properties of a material. H(0) can be found from the measured frequency dependence of the dynamic modulus of elasticity G (co) by means of the following integral equation ... 

The evolution of the dynamic viscosity rp (co, x) or of the dynamic shear complex modulus G (co.x) as a function of conversion, x, can be followed by dynamic mechanical measurements using oscillatory shear deformation between two parallel plates at constant angular frequency, co = 2irf (f = frequency in Hz). In addition, the frequency sweep at certain time intervals during a slow reaction (x constant) allows determination of the frequency dependence of elastic quantities at the particular conversion. During such experiments, storage G (co), and loss G"(co) shear moduli and their ratio, the loss factor tan8(co), are obtained ... 

Figure 3a-b illustrate the LCB effect on the melt rheological properties. The response of the rheological behaviour to the copolymerisation ability and vinyl end group selectivity of the siloxy-substituted metallocenes has been investigated from their dynamic modulus curves. The frequency dependency of the dynamic modulus of the polyethenes produced with catalysts 2 is demonstrated in Fig. 3a. For comparison dynamic modulus for a linear polyethene, prepared by the catalyst -BuCp2ZrCl2, is shown in Fig. 3b. 

Thus, one may conclude that, in the region of comparatively low frequencies, the schematic representation of the macromolecule by a subchain, taking into account intramolecular friction, the volume effects, and the hydrodynamic interaction, make it possible to explain the dependence of the viscoelastic behaviour of dilute polymer solutions on the molecular weight, temperature, and frequency. At low frequencies, the description becomes universal. In order to describe the frequency dependence of the dynamic modulus at higher frequencies, internal relaxation process has to be considered as was shown in Section 6.2.4. 

One can see that the frequency dependence of the dynamic modulus is determined by two parameters B and x... 

Before we discuss the frequency dependencies of the dynamic modulus, which are shown in Fig. 16 for typical values of parameters, we shall find expressions for the characteristic quantities at B 1. The latter assumption allows us to... 

In materials which are highly attenuating, the particle velocity and particle displacement are out of phase, so the elastic modulus and density of the material are complex and dynamic ( .e. frequency dependent). For many materials, the attenuation coefficient is fairly small ( .e. a co/c), so the particle velocity and displacement are in phase and Eq. 9.4 can be replaced with ... 

---