Cole Complex viscosity

Montfort et al. [138] used Cole-Cole complex viscosity plots to analyze data for binary polystyrene blends. They superposed data at several temperatures by plotting rff versus rf and obtained curves showing contributions from the two components of the blend. Figure 5.30 shows plots of data for three blends. Blend (a) contains 5% polystyrene with M = 4 10 in a matrix of a polystyrene with M= 3.5 10. Labaig etal. [141] used Eq. 5.65 to fit data for branched polyethylenes. 

For diluted chains, a Cole-Cole plot of the complex viscosity (Fig. 19) exhibits a relaxation domain well-separated from the matrix allowing one to measure the same average relaxation times as above. However, the weight average time x, has to be corrected by the matrix contribution. Watanabe [20] cast it into the form ... 

Figure 19 The terminal relaxation domain of diluted polymethylmethacrylate (a) and diluted polyisoprene (b) can easily be distinguished from the matrix one (dashed lines) in a Cole-Cole representation of complex viscosities [19]-...

Figure 6. A Cole-Cole representation of the imaginary part of the complex viscosity (TI") as a function of its real part (TI ) for pure EMA-Zn-20 at 150 and 18O°C.

The rheological behavior of these materials is still far from being fully understood but relationships between their rheology and the degree of exfoliation of the nanoparticles have been reported [73]. An increase in the steady shear flow viscosity with the clay content has been reported for most systems [62, 74], while in some cases, viscosity decreases with low clay loading [46, 75]. Another important characteristic of exfoliated nanocomposites is the loss of the complex viscosity Newtonian plateau in oscillatory shear flow [76-80]. Transient experiments have also been used to study the rheological response of polymer nanocomposites. The degree of exfoliation is associated with the amplitude of stress overshoots in start-up experiment [81]. Two main modes of relaxation have been observed in the stress relaxation (step shear) test, namely, a fast mode associated with the polymer matrix and a slow mode associated with the polymer-clay network [60]. The presence of a clay-polymer network has also been evidenced by Cole-Cole plots [82]. 

Another type of Cole-Cole representation of rheological data is a plot of the imaginary versus the real parts of the complex viscosity. Such a plot for a monodisperse polystyrene is shown in Fig. 5.29 [139]. Note that the viscosities are strong functions of temperature, and data taken at various temperatures therefore do not superpose. However, if time-temperature superposition is obeyed, a temperature-independent plot can be obtained by use of reduced complex viscosity components as shown below. 

Marin etal. [140] found that their data at a given temperature fell on a circular arc, and extrapolating the arc to the horizontal axis gives the limiting value of the real part of the complex viscosity at zero frequency, which is equal to the zero-shear viscosity. The circular arc could be fitted by the Cole-(2ole function (Eq. 5.60) in the form of Eq. 5.65. 

The real and imaginary components of the complex viscosity, as modeled by the Cole-Cole function (Eq. 5.60) can be derived from Eq. 5.65. 

Numerically calculated Cole-Cole plots as shown in Figure 9.2 could be successfully compared to experimental data, and this allows us to calculate the parameter and, hence, the ratio %reakl ep- From measurements at low angular frequencies, it is possible to determine the zero shear viscosity Tio, which usually coincides with the magnitude of the complex viscous resistance. The plateau modulus Gq will be obtained by extrapolation of the Cole-Cole data to the horizontal axis. The terminal relaxation time is then obtained from x = ri(/Go-... 

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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