Retardation and Relaxation Time Spectra

Having established the properties of the single-time relaxation process, we have now also a means to represent a more complex behavior. This can be accomplished by applying the superposition principle, which must always hold in systems controlled by linear equations. Considering shear properties again, we write for a dynamic compliance J (uj) with general shape a sum of Debye-processes with relaxation times ti and relaxation strengths AJi 

Often it is more appropriate to employ a representation in integral form 

The characteristic function in this integral which specifies the relaxation properties of the system is Lj(logr), being called the retardation time spectrum of the shear compliance J  

The identical function can be used in order to describe the result of a creep experiment on the system. One has just to substitute the dynamic compliance of the Debye-process by the associated elementary creep function, as given by Eq. (5.61). This leads to 

The immediate reaction with amplitude Jy here is represented by the heavy-side function 0(t), which is unity for t 0 and zero for t 0. 

Alternative to the shear compliances, J t) and one can also use for the description of the properties under shear the shear moduli, G t) and G u ). As we shall find, this can drastically change the values of the relaxation times. Let us first consider a single Debye process, now in combination with a superposed perfectly elastic part and calculate the associated dynamic modulus. We have 

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The data are further analyzed mathematically, hi particular, it is of interest to establish retardation and relaxation time spectra that fit the measured data using Voigt or Maxwell models. Adding the temperature dependence of the data leads to the interesting observation that time and temperature effects are often coupled by the time-temperature superposition principle. Effects caused by an increase in temperature can also be produced by an increase in time scale of the experiment. The ratio of modulus to temperature, when plotted versus the logarithm of time for different temperatures,... 

Mechanical models, retardation and relaxation time spectra... 

One can imagine that, of necessity, there must be many different molecular configurations contributing to the viscoelastic behavior of any one sample. For this reason the models in Fig. 6.20 are drawn as combinations of many elements i. These combinations of the elements of the model are linked to combinations of retardation and relaxation times (spectra). Naturally, there may also be combinations of both models needed for the description. For most polymeric materials, viscoelastic behavior can be found for sufficiently small amplitudes of deformation. Rigid macromolecules (see Fig. 1.7) show relatively little deviation from elasticity, so that the major application of DMA is to flexible, linear macromolecules. 

Chapter 9 examines transient and nontransient viscoelastic functions in terms of the retardation times (comphance functions) and relaxation times (relaxation moduli) and compares retardation and relaxation times. Methods to determine retardation and relaxation spectra from compliance functions and relaxation spectra, respectively, are presented. 

The main goal in DMA, and at the same time the greatest difficulty, is to relate the macroscopic responses (as expressed, for example, by retardation or relaxation time spectra) to microscopic bond deformation and molecular conformation changes that originate from the readjustment of the molecular arrangement via segmental diffusion. 

The functions FfA) and /(A) characterize the retardation and the relaxation spectra, respectively, and thus represent the distribution of retardation and relaxation time values in a specific material. Further details are given in (Haddad 1995 de With 2006). 

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 this chapter we describe the common forms of viscoelastic behaviour and discuss the phenomena in terms of the deformation characteristics of elastic solids and viscous fluids. The discussion is confined to linear viscoelasticity, for which the Boltzmann superposition principle enables the response to multistep loading processes to be determined from simpler creep and relaxation experiments. Phenomenological mechanical models are considered and used to derive retardation and relaxation spectra, which describe the time-scale of the response to an applied deformation. Finally we show that in alternating strain experiments the presence of the viscous component leads to a phase difference between stress and strain. 

Relaxation Time Spectra and Retardation Time Spectra... 

This is different for the star core. Figure 57 provides a comparison of the spectra at two Q-values with those from an equivalent full star (sample 3). Over short periods of time, both sets of spectra nearly coincide. However, over longer periods of time, the relaxation of the star core is strongly retarded and seems to reach a plateau level. This effect may be explained by the occurrence of interarm entanglements as recently proposed by scaling arguments [135]. 

Fig. 1. Retardation time spectra corresponding to the relaxation box (A), slope- /2 wedge (B), and slope-3/2 wedge (C). Schematics are all drawn on same scale...

On the other hand knowledge of these functions and of the spectra of relaxation (or retardation) times derived from them, is very helpful for obtaining insight into the molecular mechanisms by which they are originated. Analysis of the time dependency of mechanical properties thus provides a powerful tool to investigate the relations between structure and properties. 

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. 

Baumgartel M and Winter HH (1989) Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheol Acta 28 511-9. 

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. 

In practice the relationship between creep and stress-relaxation data is usually considered in terms of relaxation- or retardation-time spectra (see section 7.2.5) or by the use of various approximate methods. 

Relaxation and retardation time spectra can be calculated exactly from stress relaxation, creep and dynamic mechanical measurements using Fourier or Laplace... 

The reader may use the Alfrey approximation (see Section 4.3.2) to derive relaxation and retardation time spectra from the data of Figure 6.1. These spectra can be approximated by a wedge and box distribution [3], shown by the dotted lines in Figure 6.2. 

FIG. 9-7. Line spectra (crosses) for relaxation and retardation as predicted by the Rouse theory, with equivalent continuous spectra at short times (solid lines) ti is longest or terminal relaxation time. 

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