Complex viscoelastic functions

Relationships between complex viscoelastic functions similar to those given in Tables 4.1 and 4.2 are obtained in the frequency domain. The difference is that the viscoelastic magnitudes in the frequency domain are of the complex type, that is, they have real and imaginary components. 

Linear viscoelasticity theory predicts that one component of a complex viscoelastic function can be obtained from the other one by means of the Kronig-Kramers relations (10-12). For example, the substitution of G t) — Ge given by Eq. (6.8b) into Eq. (6.3) leads to the relationship... 

There are a great number of techniques for the experimental determination of viscoelastic functions. The techniques most frequently found in the literature are devoted to measuring the relaxation modulus, the creep compliance function, and the components of the complex modulus in either shear, elongational, or flexural mode (1-4). Although the relaxation modulus and creep compliance functions are defined in the time domain, whereas the complex viscoelastic functions are given in the frequency domain, it is possible, in principle, by using Fourier transform, to pass from the time domain to the frequency domain, or vice versa, as discussed earlier. 

As mentioned above, it is very difficult, for experimental reasons, to measure the relaxation modulus or the creep compliance at times below 1 s. In this time scale region, dynamic mechanical viscoelastic functions are widely employed (5,6). However, in these methods the measured forces and displacements are not simply related to the stress and strain in the samples. Moreover, in the case of dynamic experiments, inertial effects are frequently important, and this fact must be taken into account in the theoretical methods developed to calculate complex viscoelastic functions from experimental results. 

Fig. 19. Complex viscoelastic moduli for linear chains and stars of different functionalities, without intramolecular interactions (ideal) and with a repulsive potential (EV). Reprinted with permission from [89]. Copyright (1996) American Institute of Physics... 

The relaxation modulus is the core of most of the viscoelastic descriptions and the above expression can be checked from experimental viscoelastic functions such as the complex shear modulus G (co) for instance. In addition to the molecTilar weight distribution function P(M), one has to know a few additional parameters related to the chemical species the monomeric relaxation time x,... 

For a specific food, magnitudes of G and G are influenced by frequency, temperature, and strain. For strain values within the linear range of deformation, G and G" are independent of strain. The loss tangent, is the ratio of the energy dissipated to that stored per cycle of deformation. These viscoelastic functions have been found to play important roles in the rheology of structured polysaccharides. One can also employ notation using complex variables and define a complex modulus G (o) ... 

In spite of the complex dependence of v(t) on the viscoelastic functions, the limit values of the Poisson ratio can easily be obtained. Thus the theorem of the initial and final values establishes that if a function/(t) has a limit, the following relationships hold ... 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

The shift factors are usually obtained by empirical methods that involve the horizontal translation of the isotherm representing the reduced viscoelastic functions in the time or frequency domains, in double logarithmic plots with respect to the reference isotherm. However, analysis of the components of the complex relaxation moduli in the terminal region ( 0) permits... 

The proposed method of data treatment has two advantages (1) It allows assessment of the status of blend miscibility In the melt, and (11) It permits computation of any linear viscoelastic function from a single frequency scan. Once the numerical values of Equation 20 or Equation 21 parameters are established Che relaxation spectrum as well as all linear viscoelastic functions of the material are known. Since there Is a direct relation between the relaxation and Che retardation time spectra, one can compute from Hq(o)) the stress growth function, creep compliance, complex dynamic compliances, etc. 

The dependences of complex viscoelastic modulus (E) of mixed adsorption layers, as well as its elastic (real part, and viscous (imaginary part, E,p) components as functions of MR concentration at the smallest 0.007 i s and the biggest 0.62 rad/s frequency values of the applied deformation are presented in Figs. 1 and 2 of Section 5.8. In Fig.3 of Section 5.8, the dependences of phase angle on MR concentration at the same conditions are shown. 

DMA experiments are performed under conditions of very small strain so that the material response is in the linear viscoelastic range. This means that the magnitude of stress and strain are linearly related and the deformation behavior is completely described by the complex modulus function, which is a function of time only. The theory applies both for the case of a tensile deformation or simple extension and for shear. In the latter case the comparable modulus is with components G ico) and G" co). As a first-order approximation, E = 3G. The theory is developed assuming deformation under isothermal conditions, and temperature does not appear (nor is implicit) as a variable. 

Here it is emphasized that the definition of the elastic compliance J = 1/G is not valid for the viscoelastic compliance J(t) used in eqnation 10. Rather it is the complex compliance = l/G (.co). In addition, all the linear viscoelastic functions can be related one to the other. Full discussion of these relationships can be found in Ferry (9) and Tschoegl. (10). 

Initially, for characterisation by mechanical spectroscopy, the strain dependence of, for example, the complex shear modulus (G ) is established. Typical results are shown schematically in Figure 2.8. This experiment establishes the linear viscoelastic region of the system, within which the viscoelastic functions are independent of strain. In other words, the applied strain does not perturb the sample. For entanglement networks the linear viscoelastic region extends to approximately 25% strain. 

Analysis of dynamics at the gel point and theory of viscoelasticity provide a method to determine the static scaling exponents. Indeed, scaling arguments allow to show that the viscoelastic functions, G and G , at different stages of the network formation, can be superimposed into a master curve, provided that frequency and complex modulus are renormalized by appropriate reaction time (tr) dependent factors. The theory shows that the renormalisation factors for the frequency and the complex modulus are the longest relaxation time (iz) and the steady-state creep... 

The appropriate viscoelastic functions are ordinarily the complex modulus or the complex viscosity, and the corresponding quantities extrapolated to infinite dilution are the intrinsic storage and loss moduli... 

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. 

It is in the transition zone between glasslike and rubberlike consistency that the dependence of viscoelastic functions on temperature is most spectacular, just as is the dependence on time or frequency. An example is given in Fig. 11-1 for the real part of the complex compliance of poly(/t-octyl methacrylate). Below —5°C, the experimental frequency range appears to correspond to the glassy zone the compliance is quite low, around 10 - cm dyne Pa ), and does not change... 

Providing tests are performed at low strain amplitude, small enough for the complex modulus to exhibit no strain dependency, then dynamic testing yields in principle linear viscoelastic functions. This implies that, with an unknown material, a preliminary strain sweep test is performed in order to experimentally detect the maximum strain amplitude for a linear response to be observed [i.e. G lo, f(Y)]-As illustrated in Fig. 6 with data from Dick and Pawlowsky [20], such a requirement is practically never met within the available experimental window with filled rubber materials, whose linear region tends to move back to a lower and lower strain range as the filler content increases. 

Time and frequency do not enter the above calculations. However, the solutionlike-meltlike transition suggested a structure for fixed points of the Altenberger-Dahler renormalization group. An ansatz extending the structure from a single concentration variable to a two-variable concentration-time plane indicated a possible form for the complex viscosity(14). Chapter 13 successfully compares the ansatz predictions with experiment. This two-parameter temporal scaling approach has since been applied successfully to describe viscoelastic functions of linear polymers and soft-sphere melts(15), of star polymers(16), and of hard-sphere colloids(17). 

Dispersed systems, i.e. suspensions, emulsions and foams, are ubiquitous in industry and daily life. Their mechanical properties are often tested using oscillatory rheological experiments in the linear regime as a function of temperature and frequency [29]. The complex response function is described in terms of its real part (G ) and imaginary part (G"). Physical properties like relaxation times or phase transitions of the non-perturbated samples can be evaluated. The linear rheology is characterized by the measurement of the viscoelastic moduli G and G" as a function of angular frequency at a small strain amplitude. The basics of linear rheology are described in detail in several textbooks [8, 29] and will not be repeated here. The relations between structure and linear viscoelastic properties of dispersed systems are well known [4,7, 26]. 

Both selection and design of a fibre for a given end-use must take into account a wide range of properties. However, satisfactory mechanical performance is always essential. Various aspects of the mechanical properties of polymers are discussed in detail in other chapters of this work and elsewhere." " For fibres with cylindrical symmetry, the fundamental approach requires determination of five independent viscoelastic functions," which is prohibitively time-consuming. Besides, in practice, the fibres assembled in an end-product are often deformed in a complex way. The magnitudes and directions of the applied forces vary with time in a manner which precludes an exact mathematical description. At the same time, the fibres may be exposed to a changing environment. Consequently, it is often necessary to use various end-use oriented tests for a practical evaluation of a fibre. 

The dynamic viscoelastic functions for anisotropic materials may be obtained simply be expressing the various components in equations (18) and (19) in complex form, i.e. 

Experimentally studied simultaneous IPNs were prepared from PU (based on POPG MM 2000, TDI, and trimethylolpropane (TMP)) and PUA, based on the same components (POPG MM 700) and HEMA. For various component ratios the temperature dependencies of the real and imaginary parts of the complex moduli were determined for pure networks and IPNs. The dependencies of viscoelastic functions calculated from Eqs. 129-132 are presented in Figs. 33 and 34 for individual networks (curves 1 and 2) and IPNs of various com-... 

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