Complex dynamic viscosity

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. 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.

Fig. 10. Complex dynamic viscosity as function of temperature for three different aliphatic hyperbranched polyesters based on bismethylol propionic acid and having different end-group structure - (O) propionate end-groups, ( ) benzoate end-groups, ( ) hydroxyl end-groups [118]...

An integrally monlded composite article comprised of (I) a nonfoam layer formed from a thermoplastic elastomer powder composition (A) and (II) a foam layer formed from a foamable composition comprised of (i) (B) a thermoplastic synthetic resin powder, and (C) a heat decomposable foaming agent and (D) a liquid coating agent, wherein the thermoplastic elastomer powder (A) is comprised of a composition of an ethylene-alpha-olefm copolymer mbber and a polyolefin resin or thermoplastic elastomer powder comprised of a partially crosslinked composition of an ethylene-alpha-olefm copolymer mbber and a polyolefin resin, the thermoplastic elastomer powder having a complex dynamic viscosity at 250 deg C and a freqnency of 1 radian/sec of not more than 1.5x1,000,000 poise and having a Newtonian viscosity index n, calculated by a specific formula. 

A series of different hydroxyfunctional hyperbranched polyesters (H1-H6) with increasing ratio TMP bis-MPA was studied. The tests were made on samples quenched from melt. As discussed previously, the molar masses for these polymers are difficult to determine and the results ate therefore presented as a function of the ratio bis-MPA TMP, which can be directly related to the theoretical molar mass. The complex dynamic viscosity (r ) of hyperbranched polyesters show an increase in viscosity with size which levels out at a certain value (Figure 11). The corresponding linear polymers would exhibit a linear relationship q versus log molar mass and hence have a higher melt viscosity. The hydroxyfunctional polyesters exhibit a Newtonian behavior within a medium shear range (10 -10 rad s ). 

Fi re 11. Complex dynamic viscosity (q ) for hydroxyfunctional hyperbranched polyesters as a function of the ratio bis-MPA TMP, polyesters, H1-H6. 

The principle rheological properties which reflect the polymer process dynamics are the loss modulus (C), storage modulus (G"), dynamic complex viscosity (n ), and tan delta parameters. In simplified form the loss modulus describes the viscous or fluid component of viscosity. That is, how easily the molecules can move past each other. The storage modulus describes the elastic or network entanglement structure of the polymers. It is, therefore, sensitive to cross linking, reaction formation and the elastomeric modifiers. The complex dynamic viscosity is the combined effect of both moduli discussed. It, therefore. 

We note that the dynamic viscosity and the dynamic rigidity are components of the complex dynamic viscosity, rf-. 

Figure 11.5 Complex dynamic viscosity as a function of temperature for a main-chain polyether consisting of a methyl stilbene mesogen and a mixture of seven-and nine-carbon aliphatic spacers. The polymer has a molecular weight of 36,000. The diamonds and squares are for temperature ramp rates of 0.1 °C and 2.0°C/min, respectively the open and closed symbols are for heating and cooling, respectively. The dashed line marks the isotropic-nematic transition. (From Gillmor et al. 1994, with permission from the Journal of Rheology.)...

Figure 21.1 shows the log plots of rubber process analyzer (RPA) complex dynamic viscosity (r) ) vs shear rate (from changes in frequency, in radians/ second). The results characterize the rubber as a pseudoplastic material in... 

Figure 21.1 Complex dynamic viscosity as a function of shear rate (100 °C, 1 Hz) for the mica-filled NR composites.

Complex/dynamic viscosity Creep compliance Relaxation modulus Static/dynamic force Temperature... 

There is a surprising but useful relationship between the steady shear viscosity t](y) and the amplitude of complex dynamic viscosity (or simply the complex... 

Figure 34. (a) The magnitude of the complex dynamic viscosity, j / as a function of strain amplitude, coyo, and (b) the magnitude of the complex... 

The real or in-phase component n] of the complex dynamic viscosity is related to the loss modulus by the equation n] = G"/(ji (equation 32 of Chapter 1). It is mostly useful for viscoelastic liquids because at low frequencies it approaches the steady-flow viscosity tjq. It is plotted with logarithmic scales in Fig. 2-5. 

FIG. 2-5. Real part of the complex dynamic viscosity, plotted logarithmically for the eight systems identified as in Fig. 2-1. 

Viscosity-shear rate studies also provide clues as to compatibility in block and graft copolymer systems. It has thus been empirically established that there is a match for a given linear amorphous polymer of the shear rate dependence of apparent viscosity with the frequency dependence of the complex dynamic viscosity [42]. Thus, the differences between these two may be a measure of the incompatibility and domain structure in the melts of graft and block copolymers [41]. 

An example ofthe mechanical spectra (dependences of G, G" and T] on CO, the SI physical unit is the Pascal-second, Pa.s, equivalent to N.s/m or kg/m s) of a polysaccharide gel is shown in Figure 7.37. The gel has a much higher value of the storage modulus G than the value of the loss modulus G and the gel therefore essentially behaves as a sohd. Both modules are independent of strain rate (co) and the gel is therefore highly elastic. The complex dynamic viscosity (t] ) decreases with increasing strain rate (co) values as the gel becomes more fluid, which means that it is thixotropic. 

Oscillatory shear experiments using, for example, cone-and-plate devices constitute the third main group of viscometric techniques. These techniques enable the complex dynamic viscosity rj ) to be measured as a function of the angular velocity (cu). The fundamental equations are presented in section 6.2 (eqs (6.22H6.27)). Another arrangement is two rotating parallel excentric discs by which the melt is subjected to periodic sinusoidal deformation. 

Complex dynamic viscosity rj, and its components—viscous (t) ) and elastic ( "), shear relaxation modulus G and its components—storage modulus G and loss modulus G". 

M average molecular weight complex dynamic viscosity... 

Dynamic rheological properties are dynamic viscosity and dynamic modulus. They depend on frequency, shear rate, time, and temperature. The parameters generally used are complex dynamic viscosity (r ) and its viscous (11 ) and elastic (ri"") components, and shear relaxation modulus (G ) and its storage modulus (G ) and loss modulus (G ) components (Figures 5.8). G for LDPE has the highest value, and G for LLDPE and mPE are very similar. The study of tan 6 = ti /ti" is of great interest for processing. 

The viscoelastic characteristics are usually obtained from the results of dynamic experiments in which knowledge of the law of the change in the molecular orientation with variable loads is even more important for the correct interpretation of the results than in the case of steady-state shear flow. The absence of a homogeneous orientation in each cycle can be the cause of the almost constantly observed difference in the values of the complex dynamic viscosity Tj and the related characteristic in the established flow mode in conditions of the comparison of y = co (co is the angular frequency). Generally speaking, despite the qualitative correlation between the dynamic and steady-state characteristics of LC polymers, quantitative coincidence between them is usually... 

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