Relaxation, modulus spectrum

The linear viscoelastic behavior of liquid and solid materials in general is often defined by the relaxation time spectrum 11(1) [10], which will be abbreviated as spectrum in the following. The transient part of the relaxation modulus as used above is the Laplace transform of the relaxation time spectrum H(l)... 

The scaling of the relaxation modulus G(t) with time (Eq. 1-1) at the LST was first detected experimentally [5-7]. Subsequently, dynamic scaling based on percolation theory used the relation between diffusion coefficient and longest relaxation time of a single cluster to calculate a relaxation time spectrum for the sum of all clusters [39], This resulted in the same scaling relation for G(t) with an exponent n following Eq. 1-14. 

The relaxation modulus is often expressed with the relaxation time spectrum, Eq. 1-4 ... 

If the self-similar spectrum extends over a sufficiently wide time window, approximate solutions for the relaxation modulus G(t) and the dynamic moduli G (co), G"(co) might be explored by neglecting the end effects... 

F(i) is the underlying modulus spectrum for that system. As noted above, since the time scale of relaxation is so broad, results are best depicted on a logarithmic time scale. To do this, one needs the contribution to the modulus associated with or lying in the time interval between In T and In T 4- Jin T this incremental contribution to the modulus is designated as... 

The continuous function II( n T) [often simply given the symbol H(r) as in this chapter) is the continuous relaxation spectrum. Although called, by long-standing custom, a spectrum of relaxation times, it can be seen that H is in reality a distribution of modulus contributions, or a modulus spectrum, over the real time scale from 0 to < or over the logarithmic time scale from - to +. ... 

The shear relaxation modulus can in general be written as an integral over the relaxation time spectrum H. At the same time Equation 3 can also be used. Thus, we have... 

But Just like the Maxwell model, the Voigt model is seriously flawed. It is also a single relaxation (or retardation) time model, and we know that real materials are characterized by a spectrum of relaxation times. Furthermore, just as the Maxwell model cannot describe the retarded elastic response characteristic of creep, the Voigt model cannot model stress relaxation—-under a constant load the Voigt element doesn t relax (look at the model and think about it ) However, just as we will show that the form of the equation we obtained for the relaxation modulus from... 

For weakly entangled monodisperse and polydisperse polymer melts, J. des Cloizeavuc [26] proposed a theory based on time-dependent diffusion and double reptation. He combines reptation and Rouse modes in an expression of the relaxation modulus where a fraction of the relaxation spectrum is transferred from the Rouse to the reptation modes. Furthermore, he introduces an intermediate time Xj, proportional to M2, which can be considered as the Rouse time of an entangled polymer movii in its tube. But, in the cross-over region, the best fit of the experimental data is obtained by replaced Xj by an empirical combination of... 

One can express linear viscoelasticity using the relaxation spectrum H X), that is, using the relaxation time X. The relationship between the relaxation modulus and the spectra is ... 

The storage and loss moduli, G and G", are obtained from the relaxation spectrum in the usual way—that is, using G = Gi[co rl/(l + co zl)] G — G,[mT /(l -P The longest relaxation mode of the relaxation modulus in Eq. (3-67) is the dominant one it accounts for 96% of the zero-shear viscosity. Thus, the reptation model predicts that for a nearly monodisperse melt, the relaxation spectrum is dominated by a single relaxation time, T = Ta. This is in reasonable accord with experimental data at low and moderate frequencies (see the dashed line in Fig. 3-29). As the frequency increases, however, there... 

Following the same procedure, the components of the complex relaxation modulus can be expressed in terms of the relaxation spectrum. Thus, by substituting the difference Git) — Gg given in Eq. (9.5) into Eq. (6.3), we obtain the following expression for the storage relaxation modulus ... 

From this equation the relaxation spectrum is approximately obtained in terms of the loss relaxation modulus,... 

The friction coefficient is customarily obtained from either the relaxation or retardation spectrum, H x) or L x), respectively. At short times, i.e., on the transition from the glassy-like to the rubbery plateau, the viscoelastic processes obey Rouse dynamics, and the relaxation modulus is given by Eq. (11.45). Since H x) = —dG/d nx t, one obtains... 

The full-time dependence of the stress relaxation modulus ot randomly branched unentangled polymers is best derived from the fractal dynamics of Section 8.8 using the relaxation rate spectrum P( ) ... 

Eq = Maximum modulus in relaxation spectrum Eji = Relaxation modulus of segment with N residues En s Ed/N2, N < 300... 

The reduced storage modulus, reduced loss modulus, and relaxation time spectrum are given by... 

In this domain, the expression of the relaxation modulus is Eq. (11), which also leads to a distribution of relaxation times which is much broader than the spectrum of linear species. 

Another strong indication of the entanglement effect is the observation of a clear plateau in the linear relaxation modulus amj storage modulus spectrum when the polymer has a sufficiently high molecular... 

Fig. 11.9 Comparison of the measured storage-modulus spectrum (o and ) and that calculated from the Rouse theory (solid line) for sample A. The dashed line indicates the separation of the contributions from the G1 and GIO components. The arrow at 1/ti(1) indicates the frequency that is the reciprocal of the relaxation time of the first Rouse mode of the G1 component calculated from Eq. (7.57) with K = x 10, wheresis the arrow at l/ri(2) indicates the same for the GIO component.

In the short-time or high-frequency region of the viscoelastic response (relaxation modulus G t), viscoelastic spectrum G (w) or creep compliance J(t)) of a polymer melt with modulus values in the range from lO to 10 °dynes/cm (or compliance values in the range from to... 

Tobolsky and his coworkers made extensive efforts to characterize the stress relaxation characteristics of elastomers, notably polyisobutylene. The stress would decay over time to zero at a rate dependent on temperature and molecular weight (Fig. 2). They expressed the relaxation through a series of exponentials or a spectrum of relaxation times. Consider the shear stress decay a t) following an shear imposed strain yo- This may be used to define a shear relaxation modulus G i) through... 

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