The Viscoelastic Response

Elastic behavior in polymers is rubberlike and thus not Hookean, and viscous behavior is shear thinning and thus not Newtonian. Further, real polymers display characteristics that are a combination of such elastic and viscous responses. This is best illustrated by the Maxwell model [75, 78], which considers the viscoelastic 

De 1 and De 1 correspond to predominantly elastic and viscous responses respectively. De determines the level of shear stress that can relax during the melt processing, and hence the residual chain orientation, morphology, and properties in the solid-state article fabricated by melt processing. 

The Maxwell model [Eq. (36)] can also be expressed in the integral form of Eq. (38) [75], That is, the stress can be looked upon as the sum of every incremental strain y(t )At multiplied by the corresponding exponentially decreasing modulus 

---

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

The relaxation and creep experiments that were described in the preceding sections are known as transient experiments. They begin, run their course, and end. A different experimental approach, called a dynamic experiment, involves stresses and strains that vary periodically. Our concern will be with sinusoidal oscillations of frequency v in cycles per second (Hz) or co in radians per second. Remember that there are 2ir radians in a full cycle, so co = 2nv. The reciprocal of CO gives the period of the oscillation and defines the time scale of the experiment. In connection with the relaxation and creep experiments, we observed that the maximum viscoelastic effect was observed when the time scale of the experiment is close to r. At a fixed temperature and for a specific sample, r or the spectrum of r values is fixed. If it does not correspond to the time scale of a transient experiment, we will lose a considerable amount of information about the viscoelastic response of the system. In a dynamic experiment it may... 

Adsorption of rubber over the nanosilica particles alters the viscoelastic responses. Analysis of dynamic mechanical properties therefore provides a direct clue of the mbber-silica interaction. Figure 3.22 shows the variation in storage modulus (log scale) and tan 8 against temperature for ACM-silica, ENR-silica, and in situ acrylic copolymer and terpolymer-silica hybrid nanocomposites. 

The reptation model for polymer diffusion would predict that the thickness of the gel phase reflects the dynamics of disentanglement. The important factors here are chain length, solvent quality and temperature since they affect the dimensions of the polymer coils in the gel phase. The precursor phase, on the other hand, depends upon solvency and temperature only through the osmotic force it can generate in the system and the viscoelastic response of the system in the region of the front. These factors should be independent of the PMMA molecular weight. 

We have developed the idea that we can describe linear viscoelastic materials by a sum of Maxwell models. These models are the most appropriate for describing the response of a body to an applied strain. The same ideas apply to a sum of Kelvin models, which are more appropriately applied to stress controlled experiments. A combination of these models enables us to predict the results of different experiments. If we were able to predict the form of the model from the chemical constituents of the system we could predict all the viscoelastic responses in shear. We know that when a strain is applied to a viscoelastic material the molecules and particles that form the system gradual diffuse to relax the applied strain. For example, consider a solution of polymer... 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

The viscoelastic response of the UHMWPE pseudo-gels depends also on the solution concentration. As shown in Figure 10, the G increased with concentration at frequency 50 sec"l over the temperature range from -20 C to 100 C. This is probably due to the higher entanglement density and/or higher crystallinity in the higher concentration solution. 

The time of gelation changes significantly with sol-gel chemistry. One method of measuring determines the viscoelastic response of the gel as a function of shear rate. 

The relation between friction and viscosity goes beyond the Stokes relation. The Navier-Stokes hydrodynamics has been generalized by Zwanzig and Bixon [23] to include the viscoelastic response of the medium. This generalization provides an elegant expression for the frequency-dependent friction which depends among other things on the frequency-dependent bulk and shear viscosities and sound velocity. 

Figure 4 depicts the imaginary part of the frequency-dependent viscosity which clearly demonstrates the bimodality of the viscoelastic response. In the same figure the prediction from the Maxwell s relation have also been plotted. In the latter the relaxation time xs is calculated by the well-known... 

The viscoelastic response of a series of CMIMx copolymers has been determined as a function of temperature [79]. The loss modulus, El, at 1 Hz, of a series of copolymers with molar content of maleimide units varying from 5 to 25% is plotted in Fig. 127 for comparison the curve for pure PMMA is also drawn. 

But it is difficult to apply zero shear predictions to measurements that have been performed at low, but nonzero, shear rates. Neither s nor u can be decisively determined at a fixed frequency near the gel point the only way to truly obtain s and u exponents from experiments is through the use of a theory capable of predicting the entire frequency dependence of the viscoelastic response. 

In these systems which are very simple from chemical point of view the viscoelastic responses are very complexes but what is the main contribution is the splitting of the dipolar relaxations from those of conductive origin in the dielectric spectra The striking consequence of this transformation is that M (T) spectra exhibit two families of peacks, attributed to conductive and dipolar relaxations which can be analyzed following different procedures [40]... 

Another interesting family of saturated cyclic poly(methacrylate)s are those containing norbornyl groups which show a very interesting behavior from viscoelastic point of view. Scheme 2.6 show these structures in which as the previous cases an spacer group have been inserted in order to get confidence about the effect of small structural changes on the viscoelastic responses. 

The viscoelastic response of polymer melts, that is, Eq. 3.1-19 or 3.1-20, become nonlinear beyond a level of strain y0, specific to their macromolecular structure and the temperature used. Beyond this strain limit of linear viscoelastic response, if, if, and rj become functions of the applied strain. In other words, although the applied deformations are cyclic, large amplitudes take the macromolecular, coiled, and entangled structure far away from equilibrium. In the linear viscoelastic range, on the other hand, the frequency (and temperature) dependence of if, rf, and rj is indicative of the specific macromolecular structure, responding to only small perturbations away from equilibrium. Thus, these dynamic rheological properties, as well as the commonly used dynamic moduli... 

The main feature about molten high polymers (molecular weights higher than about 104) concerns the broadness of the relaxation spectrum that characterises the viscoelastic response of these systems. This broad two-dispersion spectrum may spread over a range of relaxation times going from about 10 9 up to several seconds [4]. It is well illustrated from the modulus of relaxation observed after applying a sudden stress to the polymer the resulting sudden deformation of the sample is then kept constant and the applied stress is released in order to avoid the flow of the polymer. For example, the release of the constraint oxy(t) is expressed as a function of the shear modulus of relaxation Gxy(t) ... 

Beside the consideration of the up-cycles in the stretching direction, the model can also describe the down-cycles in the backwards direction. This is depicted in Fig. 47a,b for the case of the S-SBR sample filled with 60 phr N 220. Figure 47a shows an adaptation of the stress-strain curves in the stretching direction with the log-normal cluster size distribution Eq. (55). The depicted down-cycles are simulations obtained by Eq. (49) with the fit parameters from the up-cycles. The difference between up- and down-cycles quantifies the dissipated energy per cycle due to the cyclic breakdown and re-aggregation of filler clusters. The obtained microscopic material parameters for the viscoelastic response of the samples in the quasi-static limit are summarized in Table 4. 

Isothermal measurements of the dynamic mechanical behavior as a function of frequency were carried out on the five materials listed in Table I. Numerous isotherms were obtained in order to describe the behavior in the rubbery plateau and in the terminal zone of the viscoelastic response curves. An example of such data is shown in Figure 6 where the storage shear modulus for copolymer 2148 (1/2) is plotted against frequency at 10 different temperatures. 

Lightly cross-linked elastomers follow a simple pattern of ultimate behaviour. Smith (1958) has shown that the ultimate properties of this class of polymers follow a time-temperature equivalence principle just as the viscoelastic response to small non-destructive stresses does. 

In this section, the composite system with the properties given by Eq. (58) will be used. Since glassy polymers are not in thermodynamic equilibrium, the change in the nonequilibrium glassy state and its relaxation define the viscoelastic response. The relaxation modulus is given by Eq. (40). 

According to Partridge [163], toughening is efficient when, by comparison to the neat homopolymer tested under the same conditions, the impact resistance is multiplied by a factor of 10, without losing more than 25% of stiffness. The upper temperature limit for the use of rubber-modified blends is controlled by the matrix melt temperature, Tm, their lower limit by the glass transition temperature, Tg, of the particles. As soon as the viscoelastic response of the latter is too slow to accommodate an external loading, the polymer assumes a glassy state and breaks in a brittle way. 

---