Shapes of the Viscoelastic Functions

To show the effect of diluent o.n the detailed shapes of viscoelastic functions, it is convenient to employ corresponding-state plots as in Section C of Chapter 12. For the relaxation spectrum, we plot log H - log Tc/Mq against log t - log a o/kT). Of course, for a single polymer and its solutions the only variables are fo and c (which in the pure polymer becomes p). In Fig. 17-9, poly(vinyl acetate) is compared in this manner with its 50% solution in tri-m-cresyl phosphate. The values of log fo at 40°C for these two systems are 1.75 and -5.25, respectively—the diluent reduces the local friction coefficient by a factor of 10. The curves after reduction coincide at the bottom of the transition zone because this is fixed by the corresponding-state conditions, and are rather similar in shape throughout. However, the diluent causes the spectrum to rise somewhat more sharply from the theoretical slope of -5 at short times but at still shorter times it crosses the spectrum of the pure polymer, and its entrance into the glassy zone involves a broader maximum than the latter. 

In Fig. 17-10, poly( -butyl methacrylate)is similarly compared with two of the solutions in diethyl phthalate whose relaxation spectra appear in Fig. 17-2. The other two solutions with intermediate concentrations would fall very near the curves drawn. The differences in shape between the spectra of the pure polymer 

Corresponding-state plots of the relaxation spectra of poly(n-butyl methacrylate) and two solutions in diethyl phthalate, with indicated polymer concentration in weight percent.  

It has been pointed out by Aklonis and Rele that the steepness of the transition zone in polystyrene is diminished by diluent and that the magnitude of the change depends on the thermodynamic interaction of solvent with polymer, being least when the cohesive energy densities of the two components are closely matched. Molecular interpretation is uncertain since the molecular basis for shape of the relaxation spectrum at short times is poorly understood (Chapter 10, Section D). 

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A detailed comparison of theory with experiment is given for polyisobutylene in Fig. 10-18, where the components of the complex compliance are chosen for representation. The general aspects of the onset of the glassy zone are evidently semiquantitatively reproduced. However, the distinct difference in slope between theory and experiment for values of J and J less than 10 cm /dyne is apparent. A similar treatment was made by Shibayama and collaborators, who also introduced varying parameters for the springs and dashpots in the ladder model to modify the shapes of the viscoelastic functions predicted. But a more detailed picture of local molecular motions is needed to explain viscoelastic behavior near the glassy zone. 

The shapes of the viscoelastic functions in the transition zone are grossly similar for all polymers, as illustrated by the curves in Chapter 2, being qualitatively in accord with the predictions of the flexible chain theories outlined in Chapter 10. 

It is difficult to compare the shapes of the viscoelastic functions for single crystal mats and bulk crystalline polymer from the data of Figs. 16-1, 16-2, 16-S, and 16-6 without extensive recalculation under circumstances where approximation methods give poor precision because the functions vary so slowly with time or frequency cf. equation 40 of Chapter 4). Comparisons of the two types of samples will be shown for isochronal viscoelastic functions in Section B below. 

The corresponding-state plot of Fig. 17-10 implies, insofar as the curves coincide in the transition region, that all relaxation times are proportional to ib and that the magnitudes of contributions to H are all proportional to c as well as to T. This is, of course, exactly what the bead-spring chain theories predict, as in equations 18 and 20 of Chapter 9, for example (recalling that n is proportional to c). On this basis, a scheme of reduced variables can be devised to combine measurements at different concentrations as well as temperatures, without actually calculating fo and without requiring that the detailed shapes of the viscoelastic functions conform to the simple theories. 

Solution properties depend on polymer concentration and molecular weight, originally leading to the hope that one could apply reduction schemes and transform measurements of the shear moduli at different c and M to a few master curves. This hope was not met. Writing in 1980, Ferry observed It is evident... that the concentration reduction scheme for the transition zone described... above cannot be applied in the plateau zone, and indeed that no simple method for combining data at different concentrations can exist . ..the shapes of the viscoelastic functions change significantly with dilution. ) Pearson s 1987 review concluded The results... 

This chapter removes the hmitations described by Ferry and Pearson. Section 13.2 supplies an ansatz that correctly predicts the shapes of major viscoelastic functions. Comparison with experiment shows that the ansatz describes experiment well. Material-dependent viscoelastic parameters determined from actual measurements have simple concentration and molecular weight dependences, as is reasonably expected for rational physical properties, leading to a description for the c and M dependences of the viscoelastic functions, and thus to a coherent description of the variations in the shapes of the viscoelastic functions when polymer concentration and molecular weight are changed. 

In this chapter, the discussion of temperature and pressure dependence has emphasized the shifts of the viscoelastic functions on the logarithmic scale of time or frequency. The actual change in magnitude of a viscoelastic property such as G(t) or J will depend on the shape of the function whose argument is shifted, as expressed in equation 74. The shapes of these functions are discussed in the next few chapters. 

Since time-temperature shifting, as illustrated in Figs. 2-4, enjoys widespread use in linear viscoelasticity, it is important to have a consistent set of criteria for its validity. It is advisable that one have nearly exact matching of shapes of adjacent curves with over more than half-range overlap, that the shift factor have a reasonable form (e.g. WLF, Arrhenius) and possess the same value for all of the viscoelastic functions. A sharp test of how well one has time-temperature... 

In the preceding sections, we have presented the material functions derived from various constitutive equations for steady-state simple shear flow. During the past three decades, numerous research groups have reported on measurements of the steady-state shear flow properties of flexible polymer solutions and melts. There are too many papers to cite them all here. The monographs by Bird et al. (1987) and Larson (1988) have presented many experimental results for steady-state shear flow of polymer solutions and melt. In this section we present some experimental results merely to show the shape of the material functions for steady-state shear flow of linear, flexible viscoelastic molten polymers and, also, the materials functions for steady-state shear flow predicted from some of the constitutive equations presented in the preceding sections. 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

A pseudo solid-like behavior of the T2 relaxation is also observed in i) high Mn fractionated linear polydimethylsiloxanes (PDMS), ii) crosslinked PDMS networks, with a single FID and the line shape follows the Weibull function (p = 1.5)88> and iii) in uncrosslinked c/.s-polyisoprenes with Mn > 30000, when the presence of entanglements produces a transient network structure. Irradiation crosslinking of polyisoprenes having smaller Mn leads to a similar effect91 . The non-Lorentzian free-induction decay can be a consequence of a) anisotropic molecular motion or b) residual dipolar interactions in the viscoelastic state. 

In order to solve viscoelastic problems, we must select the most convenient model for the stress and then proceed to develop the finite element formulation. Doue to the excess in non-linearity and coupling of the viscoelastic momentum equations, three distinct Galerkin formulations are used for the governing equations, i.e., we use different shape functions for the viscoelastic stress, the velocity and the pressure... 

A final comment seems to be pertinent. In most cases actual measurements are not made at the frequencies of interest. However, one can estimate the corresponding property at the desired frequency by using the time (fre-quency)-temperature superposition techniques of extrapolation. When different apparatuses are used to measure dynamic mechanical properties, we note that the final comparison depends not only on the instrument but also on how the data are analyzed. This implies that shifting procedures must be carried out in a consistent manner to avoid inaccuracies in the master curves. In particular, the shape of the adjacent curves at different frequencies must match exactly, and the shift factor must be the same for all the viscoelastic functions. Kramers-Kronig relationships provide a useful tool for checking the consistency of the results obtained. 

The equilibrium modulus and the memory function m(t-t ) can be obtained from measurements in the linear viscoelastic region. Oscillatory shear data are most appropriate to determine the linear viscoelastic functions We will not go further into this matter here, since the present article is concerned with the comparison of the shape of the nonlinear tensor functionals of different materials. 

Exact matching of the shapes of adjacent curves has already been cited as one criterion for the applicability of reduced variables. Two others which should be applied to any experimental example when possible are (a) the same values of ar must superpose all the viscoelastic functions (b) the temperature dependence of flr niust have a reasonable form consistent with experience. 

C. SHAPES OF THE SPECTRA AND VISCOELASTIC FUNCTIONS IN THE TRANSITION ZONE... 

Relation of the Shape of ff to Those of Other Viscoelastic Functions... 

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