Polymer rheology relaxation spectrum

A few rheometers are available for measurement of equi-biaxial and planar extensional properties polymer melts [62,65,66]. The additional experimental challenges associated with these more complicated flows often preclude their use. In practice, these melt rheological properties are often first estimated from decomposing a shear flow curve into a relaxation spectrum and predicting the properties with a constitutive model appropriate for the extensional flow [54-57]. Predictions may be improved at higher strains with damping factors estimated from either a simple shear or uniaxial extensional flow. The limiting tensile strain or stress at the melt break point are not well predicted by this simple approach. 

Thus, for miscible polymer blends, the relaxation spectrum is a linear function of the relaxation spectra of the components and their weight fractions, Wj, hence one may use rheological functions to detect miscibility/immiscibility of polymer blends. An example is presented in Figure 7.14 [Utracki and Schlund, 1987]. 

A complementary use of polymer viscometry is the indirect evaluation of the MWD of a polymer from dynamic viscosity measurements [28-30]. The methods used to correlate the MWD of polymers to rheological data are based on the previous determination of the polymer relaxation spectrum from linear oscillatory shear experiments [31, 32]. MWDs obtained from viscometric data analysis can help in the determination of the MWD curve from online measurements, or in cases where this curve cannot be easily determined from size exclusion chromatography (SEC) [30, 31]. 

The second dependence in Eq. (2.32) is valid when all fractions are either entangled or not. In consequence, the relaxation spectrum of a miscible polymer blend is a linear combination of the component relaxation spectra and their weight fractions, W(. A strong deviation from linearity in plots of log Hq versus Mw/Afn and log Wmax versus log r]o indicates immiscibility [87, 88]. The principle that in miscible blends polydispersity can be calculated and used to test for system miscibility was extended to other rheological functions sensitive to polydispersity, namely, the power-law exponent (n), the cross-point coordinates (G, o) ), the free volume gradient of viscosity, the initial slope of stress growth function, and so on [3]. 

He then defined a polydispersity index of relaxation times as (r )/(r ) and pointed out that this parameter increases as the MWD becomes broader. In an entangled melt, if we limit our attention to the plateau and terminal zones, and if the relaxation spectrum function is known over the full range of times, it can be shown that this ratio of times is equal to /f. As we have seen, the product /° G indicates the breadth ofthe molecular weight distribution of a linear polymer and can be calculated directly from rheological data. For example, if the relaxation modulus in the plateau and terminal zones is represented by a single exponential ... 

SmaU, R., Carrot, C., GuiUet, J. Physically meaningful discrete relaxation spectrum Rheological behavior of monodisperse polymer melts. Macromol. Theory Simul. (1996) 5, pp. 645-661... 

Garda-Franco, C. A., Mead, D. W. Rheological and molecular characterization of linear backbone flexible polymers with the Cole-Cole model relaxation spectrum. Rheol. Acta (1999) 38, pp. 34—47... 

Dynamic mechanical techniques are important in the characterization of the rheological properties of polymers. In dynamic mechanical analysis, a small-amplitude oscillatory strain is applied to a sample, and the resulting dynamic stress is measured as a function of time. The dynamic mechanical technique allows the simultaneous measurement of both the elastic and the viscous components of the stress response. Typically, the temperature and deformation frequencies are changed in order to determine the mechanical relaxation spectrum of the system. The molecular basis of changes in the dynamic mechanical properties can be investigated using dynamic IR dichroism. 

It is well known that the linear viscoelastic properties of polymer melts and concentrated solutions are strong function of molecular structure, average molecular mass and molecular mass distribution (MWD). The relaxation time spectrum is a characteristic quantity describing the viscoelastic properties of polymer melts. Given this spectrum, it is easy to determine a series of rheological parameters. The relaxation time spectrum is not directly accessible by experiments. It is only possible to obtain the spectrum from noisy data. 

For polyacrylamide there are two rheological effects which can be explained in terms of its random coil structure. Firstly, it was discussed above that polyacrylamide is much more sensitive than xanthan to solution salinity and hardness. This is explained by the fact that the salinity causes the molecular chain to collapse, which results in a much smaller molecule and hence in a lower viscosity solution. The second effect which can be explained in terms of the polyacrylamide random coil structure is the viscoelastic behaviour of this polymer. This is shown both in the dynamic oscillatory measurements and in the flow through the stepped capillaries (Chauveteau, 1981). When simple models of random chains are constructed, such as the Rouse model (Rouse, 1953 Bird et al, 1987), the internal structure of these bead and spring models gives rise to a spectrum of relaxation times, Analysis of this situation shows that these relaxation times define response times for the molecule, as indicated in the simple Maxwell model for a viscoelastic fluid discussed above. Thus, because of the internal structure of a flexible coil molecule, one would expect to observe some viscoelastic behaviour. This phenomenon is discussed in much more detail by Bird et al (1987b), in which a range of possible molecular models are discussed and the significance of these to the constitutive relationship between stress and deformation rate and deformation history is elaborated. 

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