Storage/loss moduli

With increasing filler concentration, the complex/d)mamic viscosity as well as storage/loss moduli show a continually increasing trend. However, the viscous response dominates the elastic response with increasing filler concentration. The storage and loss moduli begin to depict more solid like behavior at higher concentrations and show independence with respect to frequency. 

Figure 10 shows the dynamical mechanical testing results at frequencies from 0.1 to 100 Hz and in the temperature range from room temperature to 200 °C. The storage, loss moduli and tan 8 at 1 Hz were plotted on Figure 11 for eomparison of different compositions. The storage modulus at 40 °C at 1 Hz is plotted as a function of bamboo composition in Figure 12. The maximum storage modulus of about 5 GPa is observed at a bamboo eomposition of 40 wt. %.

Fig. 2.67 Storage, loss modulus and loss tangent for ( ) PIPI, and (o) PIBI as a function of the temperature at 1 Hz. (From ref. [238])...

A fundamental quantity relating the basic viscoelastic functions (i.e., storage, loss modulus and compliance, shear viscosity) is the monomeric friction coefficient, which is a measure of the frictional resistence per monomer unit encountered by a moving chain segment. This co-... 

Figure 1. Storage, loss modulus and viscosity of the HDPE, measured at 210 °C...

Fig. 18. Resolution of the complex modulus G into two vectors, G the storage modulus, and G the loss modulus the phase angle is 5.

Fig. 21. Dynamic viscoelastic properties of a low density polyethylene (LDPE) at 150°C complex dynamic viscosity Tj, storage modulus G and loss modulus G" vs angular velocity, CO. To convert Pa-s to P, multiply by 10 to convert Pa to dyn/cm, multiply by 10.

Free- Vibration Methods. Free-vibration instmments subject a specimen to a displacement and allow it to vibrate freely. The oscillations are monitored for frequency and damping characteristics as they disappear. The displacement is repeated again and again as the specimen is heated or cooled. The results are used to calculate storage and loss modulus data. The torsional pendulum and torsional braid analy2er (TBA) are examples of free-vibration instmments. 

Fig. 2. Loss tangent as a measure of gelation time for a siUca sol (27) (a) loss (A) and storage (B) modulus as a function of aging time for H2O/TEOS mol ratio of 20 and HNO /TEOS mol ratio of 0.01 and (b) loss tangent as a function of aging time. To convert mPa to mm Hg, multiply by 7.50 x 10 . ...

This presentation format leads to the terminology EI = real modulus or storage modulus 2 = imaginary modulus or loss modulus. 

Figure 15 Storage modulus, (E ), loss tangent (tanS), and loss modulus, (E ), as a function of temperature for P7MB and P8MB at 3 Hz.

Experimentally DMTA is carried out on a small specimen of polymer held in a temperature-controlled chamber. The specimen is subjected to a sinusoidal mechanical loading (stress), which induces a corresponding extension (strain) in the material. The technique of DMTA essentially uses these measurements to evaluate a property known as the complex dynamic modulus, , which is resolved into two component parts, the storage modulus, E and the loss modulus, E . Mathematically these moduli are out of phase by an angle 5, the ratio of these moduli being defined as tan 5, Le. 

The value of this latter parameter is proportional to the energy dissipated as heat per cycle, and is known as the loss modulus. The former quantity, Gj, is proportional to the recoverable energy, and is called the storage modulus. The two are combined to form the complex modulus, G related by the equation... 

The B-series of silica samples were also blended with rubber and the compound formulation is shown in Table 17.6. The uncured gums were then tested according to ISO 5794-2 1998. The uncured samples were tested using a Mooney viscometer and an RPA, which measures the dynamic mechanical properties as the samples cure. Figure 17.7 shows the results of these two tests for the Mooney viscosity at 100°C, storage modulus, loss modulus, and tan 8. 

G" is loss modulus G is storage modulus S" is viscous stress S is elastic stress... 

The majority of the mechanical ectra revealed the typical behaviour of gels [9] The storage modulus only minimally increased when the frequency rised. However, the loss modulus was independent of the frequency in the limited range up to 0.1 Hz only. Within that range, the ratio between G and G was higher than 10 1. The mechanical spectra were similar to those of other HMP/sucrose gels [13], In order to conq>are the gels, the values at 0.01 Hz were used because of the independency of frequency for both moduli. 

The extent of the solid-like character, i.e. the strength of the samples, can be directly described by the storage modulus G. As both moduli rised during gelation, it seemed to be more efficient to take the ratio of G to G , describing the dominant elastic character of the viscoelastic sample, as a second characteristic quantity than to use the loss modulus G" itself. 

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. 

TANDEL is the loss tangent, GSP and GDP is the loss modulus. is the storage modulus... 

Dynamic oscillatory shear measurements of polymeric materials are generally performed by applying a time dependent strain of y(t) = y0sin(cot) and the resultant shear stress is a(t) = y0[G sin(a)t) + G"cos(cot)], with G and G" being the storage and loss modulus, respectively. 

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.

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