The Relaxation Spectrum

Here we continue to follow the notation recommended by the Committee on Nomenclature of the Society of Rheology. Various other symbols have been used for H, and in some cases a spectrum has been defined by an equation analogous to 19 with d logio t instead of d In r, thereby differing by a factor of 2.303. 

The characteristic zones of the viscoelastic time scale are clearly apparent in H the glassy zone to the left of the principal maximum, the trai ition zone where 

The relaxation spectrum, plotted with logarithmic scales for the eight typical polymer systems described in Chapter 2, viscoelastic liquids on left, viscoelastic solids on right, identified by numbers as described in the text, 

H drops steeply, the terminal zone where it approaches zero, and a region to the right of the transition zone in examples III, IV, and VII where H is relatively flat (the plateau) or passes through a minimum. 

In an entirely analogous manner, if the Voigt model in Fig. 1 -10 is made infinite in extent, it represents a continuous spectrum of retardation times, L, alternatively defined by the continuous analog of equation 18  

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The factor tG(t) is called the relaxation spectrum and is given the symbol H(r). 

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

An advantage of having the relaxation spectrum defined by Eq. (3.63) is that it can be adapted to expressions like this to calculate mechanical behavior other than that initially measured. 

M. S. Turner, M. E. Cates. The relaxation spectrum of polymer length distributions. J Physique 57 307-316, 1990. 

We expect that the classical framework of linear viscoelasticity also applies at the gel point. The relaxation spectrum for the critical gel is known and can be inserted into Eq. 3-3. The resulting constitutive equation will be explored in a separate section (Sect. 4). Here we are mostly concerned about the material parameters which govern the wide variety of critical gels. 

This most simple model for the relaxation time spectrum of materials near the liquid-solid transition is good for relating critical exponents (see Eq. 1-9), but it cannot be considered quantitatively correct. A detailed study of the evolution of the relaxation time spectrum from liquid to solid state is in progress [70], Preliminary results on vulcanizing polybutadienes indicate that the relaxation spectrum near the gel point is more complex than the simple spectrum presented in Eq. 3-6. In particular, the relation exponent n is not independent of the extent of reaction but decreases with increasing p. 

This equation, based on the generalized Maxwell model (e.g. jL, p. 68), indicates that G (o) can be determined from the difference between the measured modulus and its relaxational part. A prerequisite, however, is that the relaxation spectrum H(t) should be known over the entire relaxation time range from zero to infinity, which is impossible in practice. Nevertheless, the equation can still be used, because this time interval can generally be taken less wide, as will be demonstrated below. 

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 relaxation spectrum greatly influences the behaviour observed in experiments. As an example of this we can consider how the relaxation spectrum affects the storage and loss moduli. To evaluate this we need to change the kernel to that for a Maxwell model in oscillation and replace the experimental time by oscillation frequency ... 

In the limit of high frequencies the integral for the loss modulus tends to zero as the denominator in Equation 4.50 tends to infinity. The storage modulus tends to G(oo) which is just the integral under the relaxation spectrum ... 

In the limit of low frequencies the integral for the loss in the viscosity tends to zero. The storage term tends to (0) which is the integral under the relaxation spectrum after it has been multiplied by the appropriate t value at each point ... 

The range of frequencies used to calculate the moduli are typically available on many instruments. The important feature that these calculations illustrate is that as the breadth of the distributions is increased the original sigmoidal and bell shaped curves of the Maxwell model are progressively lost. A distribution of Maxwell models can produce a wide range of experimental behaviour depending upon the relaxation times and the elastic responses present in the material. The relaxation spectrum can be composed of more than one peak or could contain a simple Maxwell process represented by a spike in the distribution. This results in complex forms for all the elastic moduli. 

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. 

So suppose that we apply this property to our relaxation integral (Equation 4.47) such that the relaxation spectrum is replaced by a Dirac delta function at time rm ... 

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 ... 

Another approach we can use to describe the stress relaxation behaviour and all the linear viscoelastic responses is to calculate the relaxation spectrum H. Ideally we would like to model or measure the microstructure in the dispersion and include the role of Brownian diffusion in the loss of structural order. The intermediate scattering... 

This theory was able to account for both the molecular-weight scaling of the dynamic quantities Dg, r, and x as well as for the shape of the relaxation spectrum (see Fig. 5) apart from one important feature - the constant v in the leading exponential behaviour that multiplies the dimensionless arm molecular weight needed to be adjusted. This can be understood as follows. The prediction of the tube model for the plateau modulus from the stress Eq. (7) is... 

The shape of the relaxation spectrum predicted by Eq. (22) does indeed fit rheological data on pure star melts better than the quadratic expression calculated for stars in permanent networks [27], except at high frequencies where the assumption of activated diffusion breaks down (it may easily be verified that Ugff(s)[Pg.218]

The simplest case of comb polymer is the H-shaped structure in which two side arms of equal length are grafted onto each end of a linear cross-bar [6]. In this case the backbones may reptate, but the reptation time is proportional to the square of Mj, rather than the cube, because the drag is dominated by the dumb-bell-like frictional branch points at the chain ends [45,46]. In this case the dependence on is not a signature of Rouse motion - the relaxation spectrum itself exhibits a characteristic reptation form. The dynamic structure factor would also point to entangled rather than free motion. 

Here is the Rouse time - the longest time in the relaxation spectrum - and W is the elementary Rouse rate. The correlation function x(p,t) x p,0)) of the normal coordinates is finally obtained by ... 

Note 5 The relaxation spectrum (spectrum of relaxation times) describing stress relaxation in polymers may be considered as arising from a group of Maxwell elements in parallel. 

Attempts have been made to identify primitive motions from measurements of mechanical and dielectric relaxation (89) and to model the short time end of the relaxation spectrum (90). Methods have been developed recently for calculating the complete dynamical behavior of chains with idealized local structure (91,92). An apparent internal chain viscosity has been observed at high frequencies in dilute polymer solutions which is proportional to solvent viscosity (93) and which presumably appears when the external driving frequency is comparable to the frequency of the primitive rotations (94,95). The beginnings of an analysis of dynamics in the rotational isomeric model have been made (96). However, no general solution applicable for all frequency ranges has been found for chains with realistic local structure. 

Phenomenological blending relations for Je° have been suggested, based on the properties of rj0 and J° for narrow distribution systems and the assumption that t]0 always obeys Eq. (5.28) in blends. The relaxation spectrum for a binary system according to the linear mixing rule is (214)... 

Thus the relaxation spectrum resulting from the average coordinates equation11 of our model has the same form as that of Rouse, of Kargin and Slonimiskii, or of Bueche. In order to relate the parameters of the model to those of the Rouse theory, the time scale factor a must somehow be connected to the frictional coefficient for a single subchain of a Rouse molecule. To achieve this comparison, we may23 study the translational diffusion coefficients as computed for the two models. 

For higher modes, the ratio xjxt becomes sensitive to the correlations. As p increases, tp/t, decreases, as shown by Eq. (38). For illustration, this ratio is plotted semilogarithmically in Figure 2 as a function of pjN for a chain with 104 beads and for P = 0, 0.2,0.5, and 0.9. It is seen that in this one-dimensional model the relaxation spectrum is broadened as the energetic preference for extended conformations (P > 0) is increased. In particular, the longest and shortest relaxation times are related by... 

The shear modulus G(l) of a relaxing viscoleastic substance is a more sensitive probe of the overall distribution of relaxation times, as it does not depend so completely on either end of the relaxation spectrum. Although the present one-dimensional model cannot comprehend shear, it may be useful to study the analogous relaxation function. The relaxation function //(In x), is defined10 by... 

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