Frequency-dependent moduli, dynamic equations

This equation, also as equation (6.49) gives description of the frequency dependency of dynamic modulus at low frequencies (the terminal zone). Both in equation (6.49) and (9.35), the second terms present the contribution from the orientational relaxation branch, while the first ones present the contribution from the conformational relaxation due to the different mechanisms diffusive and reptational. 

The relaxation spectrum H(0) completely characterizes the viscoelastic properties of a material. H(0) can be found from the measured frequency dependence of the dynamic modulus of elasticity G (co) by means of the following integral equation ... 

The shift factor ap can be used to combine time-dependent or frequency-dependent data measured at different pressures, exactly as ap is used for different temperatures in Section A above, and with a shift factor ar,p data at different temperatures and pressures can be combined. It is necessary to take into account the pressure dependence of the limiting values of the specific viscoelastic function at high and low frequencies, of course, in an analogous manner to the use of a temperature-dependent Jg and the factor Tp/Topo in equations 19 and 20. The pressure dependence of dynamic shear measurements has been analyzed in this way by Zosel and Tokiura. A very comprehensive study of stress relaxation in simple elongation, with the results converted to the shear relaxation modulus, of several polymers was made by Fillers and Tschoegl. An example of measurements on Hypalon 40 (a chlorosulfonated polyethylene lightly filled with 4% carbon black) at pressures from 1 to 4600 bars and a constant temperature of 25°C... 

The principal regularities of viscoelastic properties of pc ymers are revealed in analysis of their temperature-frequency dependence. Such prr rties are described in the theory of reduced variables (i 05). WUliams, LandeU, and Ferry worked out a method of transformation of temperature and frequeru scales with the aid of which experim tal data, specifically the dynamic modulus, can be pat on one generalized curve, covering a very wide range of fr iuencies and temperatures (WLF-method). In a number of studies carried out up till now the licsbiUty of the WLF equation to filled systems, mainly to rubbers, has been proved (121—126). 

Studies of rheokinetics over the whole range of polyester curing is based (as for other materials) on a dynamic method, i.e., on measurements of the time dependence of the dynamic modulus at a fixed frequency, from which the time dependence of the degree of conversion (3(t). The observed dependence P(t) for polyester resins can be analyzed by an equation of the type used for other materials. Thus the following general equation was proposed for the kinetics of curing polyester and epoxy resins 69 72... 

Notwithstanding the simplifying assumptions in the dynamics of macromolecules, the sets of constitutive relations derived in Section 9.2.1 for polymer systems, are rather cumbersome. Now, it is expedient to employ additional assumptions to obtain reasonable approximations to many-mode constitutive relations. It can be seen that the constitutive equations are valid for the small mode numbers a, in fact, the first few modes determines main contribution to viscoelasticity. The very form of dependence of the dynamical modulus in Fig. 17 in Chapter 6 suggests to try to use the first modes to describe low-frequency viscoelastic behaviour. So, one can reduce the number of modes to minimum, while two cases have to be considered separately. 

Assuming (as it is reasonable) that for conditions in which the approximation ko 5> 1 is valid, the dynamic mobility also contains the (1 — Cq) dependence displayed by the static mobility (Equation (3.37)), one can expect a qualitative dependence of the dynamic mobility on the frequency of the field as shown in Figure 3.14. The first relaxation (the one at lowest frequency) in the modulus of u can be expected at the a-relaxation frequency (Equation (3.55)) as the dipole coefficient increases at such frequency, the mobility should decrease. If the frequency is increased, one finds the Maxwell-Wagner relaxation (Equation (3.54)), where the situation is reversed Re(Cg) decreases and the mobility increases. In addition, it can be shown [19,82] that at frequencies of the order of (rj/o Pp) the inertia of the particle hinders its motion, and the mobility decreases in a monotonic fashion. Depending on the particle size and the conductivity of the medium, the two latter relaxations might superimpose on each other and be impossible to distinguish. 

The reservations about determining equilibrium moduli and compliances for unloaded polymers, occasioned by slow approaches to elastic equilibrium, apply also to filled materials, but by making measurements at rather long times (or even dynamic measurements at low frequency, at elevated temperatures) the modulus or compliance can be obtained with sufficient accuracy to study its dependence on filler content and size. For noninteracting rigid fillers, the shear modulus appears to be independent of the particle size and increases with volume fraction of filler in accordance with an empirical equation of Eilers and van Dijk - ... 

Figure 10 outlines the dynamic moduli as function of the frequency o) for different values of static prestrain e q [71]. At lower frequencies (ca O) the storage modulus tends to a finite nonzero value with a nonzero derivative. This behavior cannot be described by (linear or nonUnear) standard viscoelastic constitutive equations. The data collated by [71] suggest a non-monotonic dependence of the storage modulus upon the static prestrain e q from 0 = 0-65 to G 0 = 0.75, the storage modulus S considerably decreases, but it increases again at e 0 = 0.95. A similar, but less accentuated, trend is shown by the loss modulus. Experiments collated in [72, 73] and more recently in [74] are in agreement with Lee and Kim s results. 

Below to , the storage modulus G (cu) is (1) independent of frequency (to leading order) and (2) vanishes as the square root of the distance to jamming (Equation 12.44), while the dynamic viscosity is (1) independent of frequency and (2) diverges as the square root of A(j) (Equation 12.45). The question is how do these functions depend on cu and A(]) above the crossover frequency ... 

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