Viscoelastic response functions

Since S(q ) has such a smooth temperature variation in the simulation, Fig. 47a, Fried and Binder [325] rely on dynamic criteria for locating the MST, motivated by the experimental finding that the MST shows up most clearly in the dynamic response of the blockcopolymer melt (e.g. by an abrupt change in the frequency dependence of viscoelastic response functions at Tmst [317-323]. The idea is that in the lamellar phase there will be a spontaneous symmetry breaking, S(q) depends on the direction of q for T < Tmst, since the orientation perpendicular to the lamellar is singled out, while for T > Tmst the symmetry of the disordered phase requires that S(q) is spherically symmetric. However, this... 

Another often-used viscoelastic response function, the complex modulus, is defined by... 

The two viscoelastic response functions J t) and G t) are related to each other as shown in the following ... 

This solution illustrates an important point for a linear viscoelastic material, it is possible to convert between the viscoelastic response functions, without Imowledge of the stress/strain/time differential equations to which th correspond. 

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. 

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,... 

Factorizability has also been found to apply to polymer solutions and melts in that both constant rate of shear and dynamic shear results can be analyzed in terms of the linear viscoelastic response and a strain function. The latter has been called a damping function (67,68). 

The term y(t,t ) is the shear strain at time t relative to the strain at time t. The use of a memory function has been adopted in polymer modelling. For example this approach is used by Doi and Edwards11 to describe linear responses of solution polymers which they extended to non-linear viscoelastic responses in both shear and extension. 

Figure 6.3 Plot of a simple non-linear viscoelastic response for (a) the stress relaxation as a function of the applied strain, (b) stress as a function of time at a shear strain y = 1 and (c) viscosity as a function of shear stress. (r (0) = 33Pas, rj(co) = 3 Pas, a = 1, P = 0.1, m = 0.35 and t = Is). Continued overleaf...

When a polymer is extruded through an orifice such as a capillary die, a phenomenon called die swell is often observed. In this case, as the polymer exits the cylindrical die, the diameter of the extrudate increases to a diameter larger than the diameter of the capillary die, as shown in Fig. 3.9. That is, it increases in diameter as a function of the time after the polymer exits the die. Newtonian materials or pure power law materials would not exhibit this strong of a time-dependent response. Instead they may exhibit an instantaneous small increase in diameter, but no substantial time-dependent effect will be observed. The time-dependent die swell is an example of the polymer s viscoelastic response. From a simplified viewpoint the undisturbed polymer molecules are forced to change shape as they move from the large area of the upstream piston cylinder into the capillary. For short times in the capillary, the molecules remember their previous molecular shape and structure and try to return to that structure after they exit the die. If the time is substantially longer than the relaxation time of the polymer, then the molecules assume a new configuration in the capillary and there will be less die swell. 

Figure 3.13 Polymer viscoelastic response as a function of temperature, showing the five regions glassy, glass to rubbery, rubber plateau, rubber flow, and liquid flow...

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... 

A unified approach to the glass transition, viscoelastic response and yield behavior of crosslinking systems is presented by extending our statistical mechanical theory of physical aging. We have (1) explained the transition of a WLF dependence to an Arrhenius temperature dependence of the relaxation time in the vicinity of Tg, (2) derived the empirical Nielson equation for Tg, and (3) determined the Chasset and Thirion exponent (m) as a function of cross-link density instead of as a constant reported by others. In addition, the effect of crosslinks on yield stress is analyzed and compared with other kinetic effects — physical aging and strain rate. 

PP bead foams of a range of densities were compressed using impact and creep loading in an Instron test machine. The stress-strain curves were analysed to determine the effective cell gas pressure as a function of time under load. Creep was controlled by the polymer linear viscoelastic response if the applied stress was low but, at stresses above the foam yield stress, the creep was more rapid until compressed cell gas took the majority of the load. Air was lost from the cells by diffusion through the cell faces, this creep mechanism being more rapid than in extruded foams, because of the small bead size and the open channels at the bead bonndaries. The foam permeability to air conld be related to the PP permeability and the foam density. 15 refs. 

It appears that the formal theories are not sufficiently sensitive to structure to be of much help in dealing with linear viscoelastic response Williams analysis is the most complete theory available, and yet even here a dimensional analysis is required to find a form for the pair correlation function. Moreover, molecular weight dependence in the resulting viscosity expression [Eq. (6.11)] is much too weak to represent behavior even at moderate concentrations. Williams suggests that the combination of variables in Eq. (6.11) may furnish theoretical support correlations of the form tj0 = f c rjj) at moderate concentrations (cf. Section 5). However the weakness of the predicted dependence compared to experiment and the somewhat arbitrary nature of the dimensional analysis makes the suggestion rather questionable. 

The effects of coupling of the DTO and RB units in not only one- but also three-dimensional arrays are discussed below and molecular weight trends illustrated. A fundamental connection between relaxation times and normal mode frequencies, shown to hold in all dimensions, allows the rapid derivation of the common viscoelastic and dielectric response functions from a knowledge of the appropriate lattice vibration spectra. It is found that the time and frequency dispersion behavior is much sharper when the oscillator elements are established in three-dimensional quasi-lattices as in the case of organic glasses. 

The common viscoelastic (7) and dielectric (8) response functions obtained from macroscopic phenomenological considerations are summarized in Tables 1 and 2, with the usual notation (9). The weighting coefficients, G , Jn, and en, may be obtained empirically or from molecular models. 

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. 

Rheological measurements also show that PS-PI diblock and PS-P1-PS triblock copolymers with /< 0.2 (for either block) exhibit a liquid-like viscoelastic response, even at temperatures below the ODT (Adams et al. 1994 Han et al. 1995 Sakamoto et al. 1997). Han et al. (1995) and Sakamoto et al. (1997) have observed that the ODT cannot be located for these samples based on a discontinuity in the isochronal shear moduli as a function of temperature but can be obtained from plots of logG versus logG" (Fig. 2.4(c)). 

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. 

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) ... 

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. 

Detailed analysis of the isothermal dynamic mechanical data obtained as a function of frequency on the Rheometrics apparatus lends strong support to the tentative conclusions outlined above. It is important to note that heterophase (21) polymer systems are now known to be thermo-rheologically complex (22,23,24,25), resulting in the inapplicability of traditional time-temperature superposition (26) to isothermal sets of viscoelastic data limitations on the time or frequency range of the data may lead to the appearance of successful superposition in some ranges of temperature (25), but the approximate shift factors (26) thus obtained show clearly the transfer viscoelastic response... 

Tc, the spring stretched to achieve a pitch of 0.1 mm, and the sample applied by hypodermic syringe as a 50% by weight solution in methylene chloride. The viscoelastic response of the spring supported resin was then followed in a dry nitrogen atmosphere as a function of time. About 10 to 15 mg of resin per run was requi red. 

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