Retardation time distribution

In Section II, models were discussed that had only a single relaxation or retardation time. Actual polymers have a large number of relaxation or retardation times distributed over many decades of time. E(t) is then the sum of individual contributions, so equation (5) becomes... 

Figure 16 shows an example of a G" master curve [13]. In both cases (Figs. 15 and 16), the master curves obtained from the experimental data agreed remarkably well with the values of G and G" predicted from the stress relaxation data. If we obtained the relaxation time distribution spectrum, H(t), or the retardation time distribution spectrum, Z t), then the dynamic data can be interrelated with the static data. 

A distribution obtained by the use of equation (13) is only a first approximation to the real distribution. The corresponding distribution of retardation times is designated as L(T). It may be estimated from the slope of a compliance curve D(0 or J(t), for tensile or shear creep, respectively, plotted on a logarithmic time scale according to the equation (for shear creep)-... 

To get accurate distributions of relaxation or retardation times, the expetimcntal data should cover about 10 or 15 decades of time. It is impossible to get experimental data covering such a great range of times at one temperature from a single type of experiment, such as creep or stress relaxation-t Therefore, master curves (discussed later) have been developed that cover the required time scales by combining data at different temperatures through the use of time-temperature superposition principles. 

Distributions of relaxation or retardation times are useful and important both theoretically and practicably, because // can be calculated from /.. (and vice versa) and because from such distributions other types of viscoelastic properties can be calculated. For example, dynamic modulus data can be calculated from experimentally measured stress relaxation data via the resulting // spectrum, or H can be inverted to L, from which creep can be calculated. Alternatively, rather than going from one measured property function to the spectrum to a desired property function [e.g., Eft) — // In Schwarzl has presented a series of easy-to-use approximate equations, including estimated error limits, for converting from one property function to another (11). 

The distribution of relaxation or retardation times is much broader for cystallinc than for amorphous polymers, the Boltzmann superposition... 

As with the elastic solid we can see that as the stress is applied the strain increases up to a time t = t. Once the stress is removed we see partial recovery of the strain. Some of the strain has been dissipated in viscous flow. Laboratory measurements often show a high frequency oscillation at short times after a stress is applied or removed just as is observed with the stress relaxation experiment. We can replace a Kelvin model by a distribution of retardation times ... 

When Z is sufficiently large, it is convenient to replace the sums involved in the above expressions by integrals and define continuous distributions of relaxation and retardation times. These spectral functions are usually defined on log time scales through the relations (9) ... 

Theoretical retardation and relaxation time distributions ought to obey Eq. (3.5). 

In reality, the data on isothermal contraction for many polymers6 treated according to the free-volume theory show that quantitatively the kinetics of the process does not correspond to the simplified model of a polymer with one average relaxation time. It is therefore necessary to consider the relaxation spectra and relaxation time distribution. Kastner72 made an attempt to link this distribution with the distribution of free-volume. Covacs6 concluded in this connection that, when considering the macroscopic properties of polymers (complex moduli, volume, etc.), the free-volume concept has to be coordinated with changes in molecular mobility and the different types of molecular motion. These processes include the broad distribution of the retardation times, which may be associated with the local distribution of the holes. 

In the various formulations of the mathematical theory of linear viscoelasticity, one should differentiate clearly the measurable and non-measurable fimctions, especially when it comes to modelling apart from the material functions quoted above, one may also define non measurable viscoelastic functions which Eu-e pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and tiie memory function. These mathematical tools may prove to be useful in some situations for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the difierential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 

Figure 9.7 Double logarithmic plots of the retardation spectra as a function of the retardation time for different fractions of polystyrene of narrow molecular weight distributions (0) 4.7 x 10 (0) 9.4 x 10 ( ) 1.9 x 10 (O) 6.0 x 10 and (3) 3.8 X 10. The spectra were shifted to superpose them in the glassy-like region. (Reference temperature 100°C molecular weight, 1.9 x 10 ). (From Ref. 8.)...

Figure 9.8 Double logarithmic plot of the retardation spectrum versus the retardation time for a polystyrene fraction of molecular weight 3400 with a narrow molecular weight distribution. (From Ref. 8.)...

The relaxation and retardation functions for these models in the case of a discrete distribution of relaxation or retardation times are obviously given by... 

A similar method is used to consider compliance functions. Here, however, the distribution of retardation times L(t), defined as... 

It must be emphasized, however, that while the concept of the Deborah number provides a reasonable qualitative description of material behavior consistent with observation, no real material is characterized by a simple response time. Therefore, a more realistic description of materials involves the use of a distribution or continuous spectrum of relaxation or retardation times. We address this point in the following section. 

The restraining influence of the crystallite alters the mechanical behavior by raising the relaxation time T and changing the distribution of relaxation and retardation times in the sample. Consequently, there is an effective loss of short T, causing both the modulus and yield point to increase. The creep behavior is also curtailed and stress relaxation takes place over much longer periods. Semicrystalline polymers are also observed to maintain a relatively higher modulus over a wider temperamre range than an amorphous sample. 

Modeling of relaxation behaviour in the glass can be done using either phenomenological models or models that attempt to describe bulk behaviour using thermodynamic or molecular arguments. An example of the former is the transparent mulitparameter model of Kovacs, Aklonis, Hutchinson, and Ramos (29), now commonly referred to as the KAHR model. They used a sum of exponentials and a normalized departure from equilibrium, 8 = (v-v ,)/vco, where v is the volume at time t and Voo is the volume at equilibrium. The distribution of relaxation or retardation times is a function of 6 and can be written ... 

Here, the indices e and rj are again omitted because the strain is the same in both elements. In Equation (11-57), 700 is a constant and /ret the retardation time. As a rule, there is usually a whole spectrum of retardation times in the retardation processes also. Retardation and relaxation time distributions are similar, but they are not identical since they pertain to different models of the deformation behavior. 

Energy distributions are conveniently measured using retarding potential analysis (Chapter 12). Residence time distributions are measured by pulse techniques. ... 

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