The Relaxation and Retardation Spectra

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. 

Methods have been described in preceding sections to relate and r to viscoelastic functions in limit conditions, for example, when o) -> 0. In this section, the procedures to evaluate these parameters from the relaxation and retardation spectra are analyzed. 

Other approximations of higher order can be used in the evaluation of the relaxation and retardation spectra. Let us define the function... 

In 1990, Honerkamp and Weese published a seminal paper on the use of Tikhonov s regularization for the determination of material functions. The developed method of data treatment was found particularly useful for the computation of the relaxation and retardation spectra [Elster et al, 1991 Honerkamp and Weese, 1993]. It has also been used to compute the sphere-size distribution of the dispersed phase in binary blends [Gleinser et al, 1994a], as well as the ratio of the dispersed drop diameter divided by the interfacial tension coefficient, d/Vj [Gleinser et al, 1994b]. 

Since each of the preceding functions can be calculated from any other, it is an arbitrary matter which is chosen to depict the behavior of a system and to correlate with theoretical formulations on a molecular basis. In fact, two other derived functions are sometimes used for the latter purpose—the relaxation and retardation spectra, H and L, which will be defined in Chapter 3. Actually, different aspects of the viscoelastic behavior, and the molecular phenomena which underlie them, have different degrees of prominence in the various functions enumerated above, so it is worthwhile to examine the form of several of these functions even when all are calculated from the same experimental data. A qualitative survey of their appearance will be presented in Chapter 2. 

EXACT INTERRELATIONS AMONG VISCOELASTIC FUNCTIONS CH. 3 B. THE RELAXATION AND RETARDATION SPECTRA... 

To calculate J from J", for example, it is necessary, for each point desired (o)i), to evaluate the integral of equation 49 graphically or numerically. The integration has features similar to those of equations 21 and 22 which connect the relaxation and retardation spectra.22 There are analogous interrelations between G and... 

There are other more complicated experimental situations where viscoelastic behavior can also be predicted in terms of the relaxation and retardation spectra or other functions. These include deformations at constant rate of strain and constant rate of stress increase, stress relaxation after cessation of steady-state flow, and creep recovery or elastic recoil, all of which were mentioned in Chapter 1, as well as nonsinusoidal periodic deformations. In referring to stress a, strain y, and rate of strain 7, the subscript 21 will be omitted here although it is understood that the discussion applies to shear unless otherwise specified. 

M at which the slope changes is a characteristic value, Me, which according to the Bueche theory is related to the average molecular weight spacing between entanglement points, Me, in a rather complicated manner approximately, Me s 2 Me- Because the change in slope is of course not really a discontinuity, there has been some skepticism of a real qualitative difference between the r ons M < Me and M > Me- However, in the relaxation and retardation spectra there is a clear difference only for M > Me do two maxima appear. 

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 ... 

Material Response Time — The Deborah Number Relaxation and Retardation Spectra... 

The continuous relaxation and retardation spectra calculated from the Rouse theory are... 

Elster, C Honerkamp, J., and Weese, J. (1991) Using regularization methods for the determination of relaxation and retardation spectra. Rhed. Acta, 30 (2), 161-174. 

In the frequency region where G and G" are proportional to co / the continuous relaxation and retardation spectra in equations 23 and 24 of Chapter 9 hold and can be formulated in terms of i/o by use of equation 5. The frequency dependence of the storage and loss modulus can be expressed either in terms of rjo ... 

Persistence of Relaxation and Retardation Spectra into the Glassy Zone... 

Although the data of Fig. 15-6 have not been used to calculate relaxation and retardation spectra, it is clear from the very gradual frequency dependence of the moduli and tan 6 that the spectra are relatively flat over a wide range of frequency scale, as already illustrated as curves V in Figs. 3-3 and 3-4 for this same polymer in the glassy state. The latter were derived from shear creep data of Iwayanagi, confirmed by comparison with similar data of Lethersich. 2... 

The analysis makes use of relaxation and retardation spectra of elastomers. The estimation of the magnitude of friction has not yet been attempted. However, a major step towards that goal has been shown. In particular the wetting theory has been found to be useful in describing the strength of a countersurface in adhesion. 

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. 

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 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. 

These equations provide a formal connection between the creep and stress relaxation functions. However, this approach is of greatest interest from a purely theoretical standpoint. In practice, the problem of interchangeability of creep and stress relaxation data is usually dealt with via relaxation or retardation spectra, and by approximate methods. 

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. 57. Relaxation spectra of the fully labelled star ( , + ) and the star core (, ) at two different Q-values. The solid lines represent the result of a fit for the Zimm dynamic structure factor to the initial relaxation of the fully labelled star. The dashed lines are visual aids showing the retardation of the relaxation for the star core. (Reprinted with permission from [154]. Copyright 1990 American Chemical Society, Washington)...

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. 

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...

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. 

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. 

In the previous sections we have seen that the compliance and relaxation viscoelastic functions can be expressed in terms of the retardation and relaxation spectra, respectively. However, the spectra cannot be determined beforehand they can only be calculated from viscoelastic functions. For example, N s) and N s) in Eqs. (9.5) and (9.11) can be obtained by using the expressions... 

It should be pointed out that m is positive, and its value lies in the range 0 < m < 1. Following analogous procedures, the retardation and relaxation spectra can be obtained from dynamic relaxation and dynamic compliance functions, respectively. The pertinent equations can be found in Ref. 1. 

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