Relaxation spectrum function

Tobolsky and his coworkers found that the H t) relaxation spectrum function for polyisobutylene may be represented by the combination of a wedge and box, i.e., as... 

However, because of the concentration of relaxation information at very short times, it is generally preferable to work with a logarithmic time scale. This leads to a relaxation spectrum function, H( t), which is a time-weighted spectrum function defined as F T, so that the relaxation modulus is given by ... 

He then defined a polydispersity index of relaxation times as (r )/(r ) and pointed out that this parameter increases as the MWD becomes broader. In an entangled melt, if we limit our attention to the plateau and terminal zones, and if the relaxation spectrum function is known over the full range of times, it can be shown that this ratio of times is equal to /f. As we have seen, the product /° G indicates the breadth ofthe molecular weight distribution of a linear polymer and can be calculated directly from rheological data. For example, if the relaxation modulus in the plateau and terminal zones is represented by a single exponential ... 

Figure 5.3 Relaxation spectrum function inferred from the storage and loss moduli shown in Fig. 5.4 using the technique of Roths etal. [29] (logarithmic scales).

Equation 5.12 can be written in terms of the relaxation spectrum function H(t), which was defined by Eq. 4.19. 

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. 

A self-similar relaxation spectrum with a negative exponent (-n) has the property that tan S is independent of frequency. This is convenient for detecting the instant of gelation. However, it is not evident that the claim can be reversed. There might be other functions which result in a constant tan S. This will be... 

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 relaxation spectrum H is independent of the experimental time t and is a fundamental description of the system. The exponential function depends upon both the experimental time and the relaxation time. Such a function in the context of this integral is called the kernel. In order to describe different experiments in terms of a relaxation spectrum H or retardation spectrum L it is the kernel that changes. The integral can be formed in time or frequency depending upon the experiment being modelled. The inclusion of elastic properties at all frequencies and times can be achieved by including an additional process in the relaxation... 

So suppose that we apply this property to our relaxation integral (Equation 4.47) such that the relaxation spectrum is replaced by a Dirac delta function at time rm ... 

The Dirac delta function clearly provides one form of spectra which has an analytical transform to the viscoelastic experimental regimes discussed so far. An often overlooked function was developed by Tobolsky6 and Smith.7 They noted that particular forms of the relaxation or retardation spectra have exact analytical transforms. These functions give well defined spectra and provide good fits to experimental data. The relaxation spectrum is defined by the function ... 

Here is the Rouse time - the longest time in the relaxation spectrum - and W is the elementary Rouse rate. The correlation function x(p,t) x p,0)) of the normal coordinates is finally obtained by ... 

For higher modes, the ratio xjxt becomes sensitive to the correlations. As p increases, tp/t, decreases, as shown by Eq. (38). For illustration, this ratio is plotted semilogarithmically in Figure 2 as a function of pjN for a chain with 104 beads and for P = 0, 0.2,0.5, and 0.9. It is seen that in this one-dimensional model the relaxation spectrum is broadened as the energetic preference for extended conformations (P > 0) is increased. In particular, the longest and shortest relaxation times are related by... 

The shear modulus G(l) of a relaxing viscoleastic substance is a more sensitive probe of the overall distribution of relaxation times, as it does not depend so completely on either end of the relaxation spectrum. Although the present one-dimensional model cannot comprehend shear, it may be useful to study the analogous relaxation function. The relaxation function //(In x), is defined10 by... 

Fig. 3. Effect of bond correlations on the relaxation spectrum of a one-dimensional chain of 10,000 links. The slope d log H/d log-z of the relaxation function is plotted as a function of reduced relaxation time t/t,. Open circles indicate points at which p/N = 0.25 filled circles, pjN = 0.5.

In the low-frequency range (with x spectral function L(z) depends weakly on frequency x. Then Eq. (32) comes to the Debye-relaxation spectrum given by Eq. (33). Its main characteristics, such as the dielectric-loss maximum Xd and its frequency xD, are given by Eq. (34). A connection between these quantities and the model parameters becomes clear in an example of a very small collision frequency y. In this case, relations (34) come to... 

For example, Figs. 2.43 and 2.44 present the measured [55] viscosity and first normal stress difference data, respectively, for three blow molding grade high density polyethylenes along with a fit obtained from the Papanastasiou-Scriven-Macosko [59] form of the K-BKZ equation. A memory function with a relaxation spectrum of 8 relaxation times was used. 

Equations 3.4-3 and 3.4-4 form the molecular theory origins of the Lodge rubberlike liquid constitutive Eq. 3.3-15 (23). For large strains, characteristic of processing flows, the nonlinear relaxation spectrum is used in the memory function, which is the product of the linear spectrum and the damping function h(y), obtained from the stress relaxation melt behavior after a series of strains applied in stepwise fashion (53)... 

In the example given, the constitutive equation used is a multimode Phan Tien Tanner (PTT). It requires the evaluation of both linear and nonlinear material-response parameters. The linear parameters are a sufficient number of the discrete relaxation spectrum 2, and r]i pairs, which are evaluated from small-strain dynamic experiments. The values of the two nonlinear material-response parameters are evaluated as follows. Three semiarbitrary initial values of the two nonlinear parameters are chosen and the principal normal stress difference field is calculated for each of them using the equation of motion and the multimode PTT. They are compared at each field point (i, j) to the experimentally obtained normal stress difference and used in the following cost function F... 

The molecular relaxation process has been studied by the autocorrelation function of normal modes for a linear polymer chain [177]. The relaxation spectrum can be analyzed by the Kohlrausch-Williams-Watts function [177,178] ... 

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) ... 

As a result of these very general considerations, one expects the dielectric response function, as expressed by the complex permittivity, k (oj), or the attenuation function, a(oi), of ordinary molecular fluids to be characterized, from zero frequency to the extreme far-infrared region, by a relaxation spectrum. To first order, k (co) may be represented by a sum of terms for individual relaxation processes k, each given by a term of the form ... 

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