Spectra relaxation

Fig. XIV-2. Dielectric relaxation spectrum of a water-in-oil emulsion containing water in triglyceride with a salt concentration of 5 wt % at a temperamre of 25°C. The squares are experimental points and the lines are fits to Eq. XIV-4. (From Ref. 9.)...

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. 

While a plastic usually exhibits not one but many relaxation times, each relaxation time is affected by the temperature in exactly the same manner as another. That is the whole relaxation spectrum shifts in unison along the logarithmic no longer applicable in these materials, because the crystalline morphology changes with the temperature. 

In other words, the following description of the orientational relaxation spectrum is valid at such pressures when the Q-branch has already collapsed and starts to broaden. 

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. 

Measurement of the equilibrium properties near the LST is difficult because long relaxation times make it impossible to reach equilibrium flow conditions without disruption of the network structure. The fact that some of those properties diverge (e.g. zero-shear viscosity or equilibrium compliance) or equal zero (equilibrium modulus) complicates their determination even more. More promising are time-cure superposition techniques [15] which determine the exponents from the entire relaxation spectrum and not only from the diverging longest mode. 

A self-similar relaxation spectrum with a negative exponent (-n) has the property that tan S is independent of frequency. This is convenient for detecting the instant of gelation. However, it is not evident that the claim can be reversed. There might be other functions which result in a constant tan S. This will be... 

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 continuous function II( n T) [often simply given the symbol H(r) as in this chapter) is the continuous relaxation spectrum. Although called, by long-standing custom, a spectrum of relaxation times, it can be seen that H is in reality a distribution of modulus contributions, or a modulus spectrum, over the real time scale from 0 to < or over the logarithmic time scale from - to +. ... 

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 ... 

Figure 4.11 A log normal relaxation spectrum centred on a modal time of Is...

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... 

Figure 5.19 A plot of a typical relaxation spectrum calculated from Equation (5.59)...

Substitution of Eq. (10) irto Eq. (ll)gives the well-known Doi-Edwards relaxation spectrum ... 

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