Loss modulus

The loss tangent rather than the loss modulus is plotted, also at 1 Hz. 

Chains of polybutadiene were trapped in the network formed by cooling a butadiene-styrene copolymer until phase separation occurred for the styrene, effectively crosslinking the copolymer. At 25°C the loss modulus shows a maximum which is associated with the free chains. This maximum occurst at the following frequencies for the indicated molecular weights of polybutadiene ... 

Dynamic mechanical measurements were made on PTEE samples saturated with various halocarbons (88). The peaks in loss modulus associated with the amorphous relaxation near —90°C and the crystalline relaxation near room temperature were not affected by these additives. An additional loss peak appeared near —30° C, and the modulus was reduced at all higher temperatures. The amorphous relaxation that appears as a peak in the loss compliance at 134°C is shifted to 45—70°C in the swollen samples. 

Relaxations of a-PVDF have been investigated by various methods including dielectric, dynamic mechanical, nmr, dilatometric, and piezoelectric and reviewed (3). Significant relaxation ranges are seen in the loss-modulus curve of the dynamic mechanical spectmm for a-PVDF at about 100°C (a ), 50°C (a ), —38° C (P), and —70° C (y). PVDF relaxation temperatures are rather complex because the behavior of PVDF varies with thermal or mechanical history and with the testing methodology (131). 

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.

G is called the loss modulus. It arises from the out-of-phase components of y and T and is associated with viscous energy dissipation, ie, damping. The ratio of G and G gives another measure of damping, the dissipation factor or loss tangent (often just called tan 5), which is the ratio of energy dissipated to energy stored (eq. 16). 

Plots of loss modulus or tan 5 vs temperature for polymers give peaks at energy absorbing transitions, such as the glass transition and low temperature secondary transitions (Fig. 20). Such plots are useful for characterizing polymers and products made from them. 

A viscoelastic material also possesses a complex dynamic viscosity, rj = rj - - iv( and it can be shown that r = G jiuj-, rj = G juj and rj = G ju), where CO is the angular frequency. The parameter Tj is useful for many viscoelastic fluids in that a plot of its absolute value Tj vs angular frequency in radians/s is often numerically similar to a plot of shear viscosity Tj vs shear rate. This correspondence is known as the Cox-Merz empirical relationship. The parameter Tj is called the dynamic viscosity and is related to G the loss modulus the parameter Tj does not deal with viscosity, but is a measure of elasticity. 

Fig. 20. Logarithmic decrement (related to tan 5 and loss modulus) vs temperature for a fluorocarbon dibenzoxazole (148). After drying up to 200°C, the...

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. 

Another resonant frequency instmment is the TA Instmments dynamic mechanical analy2er (DMA). A bar-like specimen is clamped between two pivoted arms and sinusoidally oscillated at its resonant frequency with an ampHtude selected by the operator. An amount of energy equal to that dissipated by the specimen is added on each cycle to maintain a constant ampHtude. The flexural modulus, E is calculated from the resonant frequency, and the makeup energy represents a damping function, which can be related to the loss modulus, E". A newer version of this instmment, the TA Instmments 983 DMA, can also make measurements at fixed frequencies as weU as creep and stress—relaxation measurements. 

The Rheometric Scientific RDA II dynamic analy2er is designed for characteri2ation of polymer melts and soHds in the form of rectangular bars. It makes computer-controUed measurements of dynamic shear viscosity, elastic modulus, loss modulus, tan 5, and linear thermal expansion coefficient over a temperature range of ambient to 600°C (—150°C optional) at frequencies 10 -500 rad/s. It is particularly useful for the characteri2ation of materials that experience considerable changes in properties because of thermal transitions or chemical reactions. 

Similar information can be obtained from analysis by dynamic mechanical thermal analysis (dmta). Dmta measures the deformation of a material in response to vibrational forces. The dynamic modulus, the loss modulus, and a mechanical damping are deterrnined from such measurements. Detailed information on the theory of dmta is given (128). 

The most important physical properties of the elastomeric polyisobutylenes are those exhibited by vulcanised compounds, especially high loss modulus and low permeabihty air. 

Rheometer. Characterizing the elastic and loss modulus over the entire temperature range of potential use and into the melt state is invaluable as a comparative and predictive tool. 

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.

The relaxation at the lowest temperature y relaxation) takes place below - I0O°C. The two polymers with shorter spacers (PDEB and PTEB) show weak relaxations overlapped with the )3 ones due to the low tanS values (0.03 and 0.04, respectively). Notwithstanding this, the y relaxation is clearly distinguished when using loss modulus plots, even in the case of PDEB, that shows the weakest maximum (see Fig. 16). For PTTB, tan6 values in the y relaxation interval are of the order of 0.05. 

It is usually considered that the y relaxation arises from crankshaft and kink movements of polymethylenic sequences, but the clear maximum of tanS and loss modulus for the three polybibenzoates here reported leads to the conclusion that the motion responsible of this relaxation also takes place when one of the methylenic... 

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... 

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