Complex compliance/viscosity

This result, that the low frequency limit of the in phase component of the viscosity equates to the viscosity of the dashpot, means that for a single Maxwell model it is possible to replace rj by rj(0). Thus far we have concentrated on the description of experimental responses to the application of a strain. Similar constructions can be developed for the application of a stress. For example the application of an oscillating stress to a sample gives rise to an oscillating strain. We can define a complex compliance J which is the ratio of the strain to the stress. We will explore the relationship between different experiments and the resulting models in Section 4.6. 

Many amorphous homopolymers and random copolymers show thermorheologically simple behavior within the usual experimental accuracy. Plazek (23,24), however, found that the steady-state viscosity and steady-state compliance of polystyrene cannot be described by the same WLF equation. The effect of temperature on entanglement couplings can also result in thermorheologically complex behavior. This has been shown on certain polymethacrylate polymers and their solutions (22, 23, 26, 31). The time-temperature superposition of thermorheologically simple materials is clearly not applicable to polymers with multiple transitions. The classical study in this area is that by Ferry and co-workers (5, 8) on polymethacrylates with relatively long side chains. In these the complex compliance is the sum of two contributions with different sets of relaxation mechanisms the compliance of the chain backbone and that of the side chains, respectively. 

Thus either G (to) or G"(co) as a function of to gives the information equivalent to that included in G(t) as a function of t. The complex modulus is experimentally a more convenient quantity to describe the linear viscoelasticity of low-viscosity fluids than the relaxation modulus (1). The complex modulus is related to the complex viscosity / (co) and the complex compliance J (to) ... 

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

Polymers owe much of their attractiveness to their ease of processing. In many important teclmiques, such as injection moulding, fibre spinning and film fonnation, polymers are processed in the melt, so that their flow behaviour is of paramount importance. Because of the viscoelastic properties of polymers, their flow behaviour is much more complex than that of Newtonian liquids for which the viscosity is the only essential parameter. In polymer melts, the recoverable shear compliance, which relates to the elastic forces, is used in addition to the viscosity in the description of flow [48]. 

This equation suggests a procedure to obtain the real component of the complex viscosity from the compliance functions at zero frequency. By taking Eq. (8.29) into account, Eq. (8.27) can be rewritten as... 

The complex modulus will be employed throughout this paper except that the dynamic viscosity >j (to) will be used in some cases. In order to describe the low frequency behavior of a material, the zero-shear viscosity rj and the steady shear compliance Je° are used. They are defined... 

There exists a clear relationship between t and 1/w G (l/w) G(t). Data in a range of w = 10 " to 10 is equivalent to that over an extended period of time (thus, avoiding long-term measurements). This is very essential in creep or long-term studies. In addition to G, other complex parameters may be defined, like compliance or viscosity. 

The rheology of blends of linear and branched PLA architectures has also been comprehensively investigated [42, 44]. For linear architectures, the Cox-Merz rule relating complex viscosity to shear viscosity is valid for a large range of shear rates and frequencies. The branched architecture deviates from the Cox-Merz equality and blends show intermediate behavior. Both the zero shear viscosity and the elasticity (as measured by the recoverable shear compliance) increase with increasing branched content. For the linear chain, the compliance is independent of temperature, but this behavior is apparently lost for the branched and blended materials. These authors use the Carreau-Ya-suda model. Equation 10.29, to describe the viscosity shear rate dependence of both linear and branched PLAs and their blends ... 

Master curves (7b = 25 °C) for the complex viscosity of two nearly monodisperse 1,4-polybutadiene melts [28] are shown in Fig. 3.24. One is linear rjo = 4.8 X 10 Pa s, J° = 2.1 X 10 Pa ), the other a three-arm star rjo = 2.8 x 10 Pa s, J° - 1. 4 X 10 Pa ). Their zero-shear viscosities are similar, but their recoverable compliances differ by a factor of seven and the shapes of their curves are obviously different, too. Figures 3.25(a) and (b) compare those results with steady-shear-viscosity data for nearly monodisperse polymers, showing master curves at 183 °C for five linear polystyrene samples [29] (48 500 < M < 242 000) in Fig. 3.25(a), and master curves at 106 °C for seven polybutadiene stars [30] (45 000 < M < 184000) in Fig. 3.25(b). Values of t]o were available for all samples, so knowledge of rj y)/r o was always available. Values of J° were not generally available, so Tq for the shear-rate reduction was estimated from the onset of shear-rate dependence. Agreement with the Cox-Merz rule is evident even in this rather severe test of using different samples and even different species. The... 

Figure 3. Dipole centered in a sphere (represented by the arrow) surrounded by an environment with properties that depend on the model. In the case of the Debye model, the environment has a viscosity tj independent of time. In the DiMarzio-Bishop model the viscosity is a complex time-dependent viscosity = lit). In the Havriliak-Havriliak model the cavity is not spherical and the environment is taken to be represented by a complex tensile compliance D(

Doi and Edwards derived theoretical expressions for various quantities including the relaxation spectrum the stress relaxation modulus G t) and the in-phase and out-of-phase t ran forms G (co) and G (ca) the plateau modulus G the steady state compliance JgO the self-diffusion coefficient Dq the complex viscosity and the zero-frequency viscosity qo. For... 

Marin and Graessley [137] used Cole-Cole plots, together with the original Cole-Cole function (Eq. 5.60) to interpret data for several polystyrenes prepared by anionic polymerization. They plotted the imaginary versus the real components of the complex retardational compliance, (fiS), defined as] (o)- l/(i (U t q). They found that for the sample with a molecular weight of about 37,000, which is near the critical molecular weight for viscosity, M, a plot of " co) versus J co) took the form of a circular arc and could thus be fitted to Eq. 5.62, by analogy with Eq. 5.60. 

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