Spectra, retardation/relaxation

The number N of retardation times needed depends on the required agreement between theory and experimental behaviour that is required. Instead of a description of viscoelastic behaviour with the aid of a discrete spectrum of relaxation and retardation times, also continuous relaxation or retardation time spectra can be used. In some cases these are easier to handle. 

Although the Maxwell-Wiechert model and the extended Burgers element exhibit the chief characteristics of the viscoelastic behaviour of polymers and lead to a spectrum of relaxation and retardation times, they are nevertheless of restricted value it is valid for very small deformations only. In a qualitative way the models are useful. The flow of a polymer is in general non-Newtonian and its elastic response non-Hookean. 

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... 

This does display the three elements of real behavior, an instantaneous elastic response, primary creep (retarded elastic response) and secondary creep (permanent deformation). However, the fit to real data is not good and again it is because real materials have behavior that is characterized by a spectrum of relaxation times. 

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. 

Let the number of individual units in multi-element models tend to infinity. For creep, an infinite number of Kelvin units gives an infinite number of retardation times this is called the spectrum of retardation times. The analagous development for stress relaxation leads to the spectrum of relaxation times. 

The proportionality constant between modulus and viscosity is known as the relaxation time, r,. For creep experiments, the proportionality constant is known as the retardation time. For the generalized models, there is a spectrum of relaxation or retardation times to account for the various viscoelastic processes occurring at different time scales in the material. 

The WLF equation was developed empirically to describe the change in viscoelastic properties of polymers above the glass transition.Starting from the principle of thermorheological simplicity, Le, that the spectrum of relaxation (or retardation times) changes with temperature by a simple shift along the time axis, it was found that the shift factor for the viscosity could be written as ... 

We assume that the above solution is valid in about the same time range as the self-similar relaxation time spectrum, Eq. 1-5. The retardation time spectrum is also self-similar. It is characterized by its positive exponent n which takes on the same value as in the relaxation time spectrum. 

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 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... 

The contribution of this component is shown in Figures 4.16 and 4.17. One of the important features to recognise about the retardation spectrum is that it only has an indirect relationship to both the zero shear rate viscosity and the high frequency shear modulus. Both these properties are contained in the relaxation spectra. We shall see in Section 4.5.7 that, whilst a relationship exists between H and L it is somewhat complex. 

In the same manner as the modulus can be related to the relaxation spectrum so the compliance can be related to the retardation spectrum ... 

This result is very interesting because whilst we have shown that G(0) has been excluded from the relaxation spectrum H at all finite times (Section 4.4.5), it is intrinsically related to the retardation spectrum L through Jc. Thus the retardation spectrum is a convenient description of the temporal processes of a viscoelastic solid. Conversely it has little to say about the viscous processes in a viscoelastic liquid. In the high frequency limit where co->oo the relationship becomes... 

The high frequency elastic modulus does not appear in the retardation spectrum but is an intrinsic part of the relaxation spectrum. These features are reinforced when the interrelationship between the spectra are considered. 

Exact Inversions from the Relaxation or Retardation Spectrum... 

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 ... 

This prescription could have been applied in determining directly the relaxation- and retardation-spectra, Eqs. (3.7) and (3.6), of the linear lattice, where the exact vibrational spectrum is well known. 

This chapter discusses the dynamic mechanical properties of polystyrene, styrene copolymers, rubber-modified polystyrene and rubber-modified styrene copolymers. In polystyrene, the experimental relaxation spectrum and its probable molecular origins are reviewed further the effects on the relaxations caused by polymer structure (e.g. tacticity, molecular weight, substituents and crosslinking) and additives (e.g. plasticizers, antioxidants, UV stabilizers, flame retardants and colorants) are assessed. The main relaxation behaviour of styrene copolymers is presented and some of the effects of random copolymerization on secondary mechanical relaxation processes are illustrated on styrene-co-acrylonitrile and styrene-co-methacrylic acid. Finally, in rubber-modified polystyrene and styrene copolymers, it is shown how dynamic mechanical spectroscopy can help in the characterization of rubber phase morphology through the analysis of its main relaxation loss peak. 

Once a relaxation (retardation) spectrum is obtained from a relaxation (creep compliance) viscoelastic function, any other function can be obtained. Alternatively, approximate methods have been developed to calculate viscoelastic functions from one another (10). By taking into account... 

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 proposed method of data treatment has two advantages (1) It allows assessment of the status of blend miscibility In the melt, and (11) It permits computation of any linear viscoelastic function from a single frequency scan. Once the numerical values of Equation 20 or Equation 21 parameters are established Che relaxation spectrum as well as all linear viscoelastic functions of the material are known. Since there Is a direct relation between the relaxation and Che retardation time spectra, one can compute from Hq(o)) the stress growth function, creep compliance, complex dynamic compliances, etc. 

The time ti / G is denoted as retardation time, applicable to bodies other than a Maxwell body. The relaxation curve according to Eq. (3.2) or its spectrum (the process may be complex and can have more than one relaxation time), was applied for fitting the pressure decay obtained after a sudden stop of the compressing of a monolayer at a certain surface pressure. A typical experimental result is shown in Fig. 3.2 (Tabak et al. 1977). 

A curve of the logarithm of the modulus against time and temperature is shown in Figure 13.25. This provides a particularly useful description of the behavior of a polymer and allows one to estimate, among other things, either the relaxation or retardation spectrum. 

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