Storage modulus frequency dependence

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. 

The growth rate decreases with frequency. Figure 24 shows this schematically for the storage modulus. The frequency dependence is the same for both G and G", and it also follows a power law (Fig. 25). 

There are also some far-fetched proposals for the LST a maximum in tan S [151] or a maximum in G" [152] at LST. However, these expectations are not consistent with the observed behavior. The G" maximum seems to occur much beyond the gel point. It also has been proposed that the gel point may be reached when the storage modulus equals the loss modulus, G = G" [153,154], but this is contradicted by the observation that the G — G" crossover depends on the specific choice of frequency [154], Obviously, the gel point cannot depend on the probing frequency. Chambon and Winter [5, 6], however, showed that there is one exception for the special group of materials with a relaxation exponent value n = 0.5, the loss tangent becomes unity, tan Sc = 1, and the G — G" crossover coincides with the gel point. This shows that the crossover G = G" does not in general coincide with the LST. 

Figure 6. Dependence of storage modulus for PMMA and EPDM on temperature at a frequency of 1Hz and a strain of 0.1%.

Figure 4. Frequency dependence of the storage modulus G at 303 K. Key PDMS-B11, (comblike crosslinks) , PDMS-C1, (tetrafunctional cross-links, randomly introduced) PDMS-A2 (tetrafunctional cross-links, end-linked network).

Generally, the rheology of polymer melts depends strongly on the temperature at which the measurement is carried out. It is well known that for thermorheological simplicity, isotherms of storage modulus (G (co)), loss modulus (G"(complex viscosity (r (co)) can be superimposed by horizontal shifts along the frequency axis ... 

Fig. 9.9 Reduced frequency dependence of storage modulus, loss modulus and complex viscosity of neat PLA and various nanocomposites (PLANCs). Reprinted from [40], 2003, Elsevier Science.

Now these expressions describe the frequency dependence of the stress with respect to the strain. It is normal to represent these as two moduli which determine the component of stress in phase with the applied strain (storage modulus) and the component out of phase by 90°. The functions have some identifying features. As the frequency increases, the loss modulus at first increases from zero to G/2 and then reduces to zero giving the bell-shaped curve in Figure 4.7. The maximum in the curve and crossover point between storage and loss moduli occurs at im. 

At low frequencies the loss modulus is linear in frequency and the storage modulus is quadratic for both models. As the frequency exceeds the reciprocal of the relaxation time ii the Rouse model approaches a square root dependence on frequency. The Zimm model varies as the 2/3rd power in frequency. At high frequencies there is some experimental evidence that suggests the storage modulus reaches a plateau value. The loss modulus has a linear dependence on frequency with a slope controlled by the solvent viscosity. Hearst and Tschoegl32 have both illustrated how a parameter h can be introduced into a bead spring... 

We can consider the friction coefficient to be independent of the molecular weight. At times less than this or at a frequency greater than its reciprocal we expect the elasticity to have a frequency dependence similar to that of a Rouse chain in the high frequency limit. So for example for the storage modulus we get... 

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]...

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 15. Panel a shows the strain amplitude sweep experiment on the 2D film of Ag nanopartides. The storage modulus, C p), is higher than the loss modulus, C" (O) at low strain amplitudes. Panel b shows the frequency dependence of interfacial storage, C ( ), and loss, C" (o), moduli of the film. Reproduced from ref 33. Copyright 2007 American Chemical Society. 

Figure 4.17. Linear melt-state rheological properties as a function of oscillatory frequency (a) storage modulus, G and (b) loss modulus, G" (c) Dependence of complex viscosity on temperature for ABS nanocomposites. Reprinted with permission from ref (68).

Wu et al. (73) studied the viscoelastic properties, viz. storage modulus (GO and complex viscosity (r 0 with respect to frequency (co) of PLA-carboxylic-acid-functionalized MWCNTs nanocomposites using a rheometer (HAAKE RS600, Thermo Electron Co., USA). The dynamic frequency sweep measurements were carried out at the pre-strain level of 1%. They observed that the addition of carboxylic-acid-functionalized MWCNTs weakened the dependence of G on go, especially at higher loading levels (Figure 9.12). This indicates... 

Figure 10.12. Temperature dependence of the storage modulus E and loss modulus E" of different PEEK/SWCNT nanocomposites with 1 wt% CNT content, obtained from DMA measurements performed in the tensile mode at frequency 1 Hz and heating rate of 2°C/min. The inset is a magnification showing the increment in Tg for the nanocomposites. From ref 11.

The temperature dependences of the storage modulus G are given in Figure 12.13 for unfilled and filled EPDM. The dynamic mechanical experiments were conducted from -100 to 150°C at a frequency of 1 Hz in tensile mode using a DMA (TA 2980). Heating rate was 5°C/min. 

Fig. 1 a,b. Strain amplitude dependence of the complex dynamic modulus E E l i E" in the uniaxial compression mode for natural rubber samples filled with 50 phr carbon black of different grades a storage modulus E b loss modulus E". The N numbers denote various commercial blacks, EB denotes non-commercial experimental blacks. The different blacks vary in specific surface and structure. The strain sweeps were performed with a dynamical testing device EPLEXOR at temperature T = 25 °C, frequency f = 1 Hz, and static pre-deformation of -10 %. The x-axis is the double strain amplitude 2eo... 

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