Discrete relaxation modulus

Figure 3-10. Discrete distribution of relaxation times and associated partial modulus values. The continuous line represents the stress relaxation modulus based on this distribution.

This is an ill-posed problem and small pertmbations in (measmed) G ( >) or G"(o)) can produce large pertmbations in H(X). In addition to H(X), various techniques have been described to determine the discrete relaxation spectrum, in terms of a set of modulus-relaxation time-pairs, using the generalised Maxwell model [Ferry, 1980], However, infinitely many parameter sets may be derived, all of which are adequate for the pmpose of representing experimental data. 

The behavior predicted by Eqs. (39) for values of Ei and E2 appropriate to the glass transition in an amorphous polymer (compare Figure 14.1) is shown in Figure 14.6. The model accounts for the qualitative features of experimentally observed transitions, namely a step-like drop in modulus as to decreases below r , and a characteristic peak in tan <5. However, more complex models involving many relaxation times, that is, a discrete or continuous relaxation time spectrum, are necessary if more quantitative agreement with experiment is to be obtained (for an example of a discrete relaxation time spectrum derived from a molecular model, see Section 14.3.3) [10,11[. 

Another approach to linear viscoelasticity that has been widely used in the past is the relaxation spectrum H(X). Using it provides a continuous function of relaxation time X rather than a discrete set. The relation between the relaxation modulus and the spectra is... 

Often, enough discrete parameters to provide reasonable response models can be extracted from experimental creep or stress-relaxation data using Tobolsky s Procedure X [4]. This procedure will be illustrated for stress-relaxation data in the form of the relaxation modulus Gr(t). According to Equation 15.20Z),... 

However, even when such a model is not being used, it is often useful to describe the relaxation modulus by use of Eq. 4.16 where the constants are inferred from experimental data by an empirical procedure. The resulting constants, T , G , are said to constitute a discrete relaxation spectrum. While these empirical parameters have no physical significance, in the limit of large N, they should, in principle, approach the underlying function G(f), which is a material property. Methods of determining the constants for a discrete spectrum from experimental data are described in Section 4.4. 

A discrete or line spectrum is often used to describe the relaxation modulus of molten polymers. It lends itself to the conversion of one response functional into another and can be inferred from data in such a way that it describes those data with a precision limited only by that of the data themselves. It is important to note that the parameters of a discrete spectrum obtained by fitting data are not related to those of a molecular theory. For example, one cannot obtain the Rouse time from a discrete spectrum obtained by fitting data. 

In practice, the conversion of an experimental data set, G, G, o, into a relaxation modulus function is usually carried out by representing G(t) in terms of a generalized Maxwell model. Thus, the oscillatory shear data are transformed into a set [Gj, Tj], i.e., a discrete relaxation spectrum. This transformation is based on the discrete form of Eqs. 4.40a, b ... 

Making use of equations presented earlier in this section, one can show that if the discrete spectrum is an accurate representation of the entire relaxation modulus function, the moments defined above can be used to calculate combinations of linear properties as follows ... 

As we have seen, the preferred experiment for establishing the linear viscoelastic behavior of polymers is small-amplitude oscillatory shear, which yields the storage and loss moduli rather than the relaxation modulus. In Section 4.4 we described a number of techniques for inferring either a continuous or discrete relaxation spectrum, given G (o). However it is important to note that in the transformation to a discrete spectrum, some information is always lost, and this can affect the reliability of subsequent calculations that make use of this spectrum. 

Further Applicatioiis. Detailed evaluations and numerical emulations of the theory are found in Vleeshouwers thesis. (31,38) One examine concerns the frequency and temperature dependence of the complex bulk modulus. Another is the simulation of cooling the melt over a transition region into the glassy state. To proceed the continuous temperature profile is approximated by a series of discrete steps depending on the cooling rate. The relaxing free volume and volume following each T-jump are then computed. 

The modulus recovery experiments allowed measuring the terminal relaxation time of reptation motion of bulk and surface immobilized chains, supporting the hypothesis that theie is no interphase per se when nano-scale is considered. In order to bridge the gap between the continuum interphase on the microscale and the discrete molecular structure of the matrix consisting of freely reptating chains in the bulk and retarded reptatiug chains in contact with the inclusions, higher order elasticity combined with a suitable molecular dynamics model could be utilized [151-155]. 

Another piece of information that may be added is the plateau modulus. If contributions from high-frequency Rouse and glassy modes of relaxation can somehow be eliminated from the data, this should be equal to the sum of the discrete moduli. 

---