Loss modulus frequency dependence

Figure 7. Storage modulus (Pa) and loss modulus (Pa) dependence on frequency (Hz) of 0.32 volume fraction dispersion of 220-nm MMA-MAA (96 04) latex, containing 0.648 X 10 meq/m surface acid at pH 9 and 0.138 wt % free surfactant in the aqueous phase. Thickener HEUR 270, 0.5 wt % in the aqueous phase. Key O, storage modulus A, loss modulus.

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. 

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. 

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... 

The thus obtained high-density Mn-Zn ferrite was investigated in detail from the view of physical and mechanical properties, that is, the relationships between the composition of metals (a,) ) and <5 the magnetic properties such as temperature and frequency dependence of initial permeability, magnetic hysteresis loss and disaccommodation and the mechanical properties such as modulus of elasticity, hardness, strength, and workability. Figures 3.13(a) and (b) show the optical micrographs of the samples prepared by the processes depicted in Fig. 3.12(a) and (b), respectively. The density of the sample shown in Fig. 3.13(a) reached up to 99.8 per cent of the theoretical value, whereas the sample shown in Fig. 3.13(b) which was prepared without a densification process, has many voids. 

In addition to knowing the temperature shift factors, it is also necessary to know the actual value of ( t ) at some temperature. Dielectric relaxation studies often have the advantage that a frequency of maximum loss can be determined for both the primary and secondary process at the same temperature because e" can be measured over at least 10 decades. For PEMA there is not enough dielectric relaxation strength associated with the a process and the fi process has a maximum too near in frequency to accurately resolve both processes. Only a very broad peak is observed near Tg. Studies of the frequency dependence of the shear modulus in the rubbery state could be carried out, but there... 

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. 

Sonic absorption on the other hand is - for linear polymers - a typical constitutive property, dependent of temperature and frequency, for which no additivity techniques are available. For cross-linked polymers the integrated loss modulus-temperature function (the "loss area") in the glass-rubber transition zone shows additive properties. 

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).

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.

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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