The relaxation time spectrum

The determination of the relaxation spectrum of a visco-elastic fluid from various dynamic shear measmements has been discussed by many workers (e.g. see, Orbey and Dealy, 1991 Baumgaertel and Winter, 1989 Sulhvan et al, 1994) and, in the case of a visco-elastic fluid, the problem of determining the relaxation spectrum from oscillatory shear measmements involves the inversion of the following pair of integral equations  

This is an ill-posed problem and small pertmbations in (measmed) G ( ) or G (o)) can produce large pertmbations in H(X). In addition to H(X), various techniques have been described to determine the discrete relaxation spectrum, in terms of a set of modulus-relaxation time-pairs, using the generalised Maxwell model [Ferry, 1980], However, infinitely many parameter sets may be derived, all of which are adequate for the pmpose of representing experimental data. 

Recently, the issue of sampling localisation in determining the relaxation spectrum has been considered [Davies and Anderssen, 1997]. It is usually assumed that G and G measured over the frequency range a Wmax yield information about the relaxation spectrum over the range of relaxation 

Simple moving-average formulae which can be applied to oscillatory shear data to recover estimates of the relaxation spectrum have been reported [Davies and Anderssen, 1998]. These formulae represent an improvement over previous commercial software in that they take into accoimt the limits imposed by sampling localization and yield accurate spectra very rapidly on a PC. 

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The linear viscoelastic behavior of liquid and solid materials in general is often defined by the relaxation time spectrum 11(1) [10], which will be abbreviated as spectrum in the following. The transient part of the relaxation modulus as used above is the Laplace transform of the relaxation time spectrum H(l)... 

The relaxation modulus is often expressed with the relaxation time spectrum, Eq. 1-4 ... 

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. 

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. 

The shear relaxation modulus can in general be written as an integral over the relaxation time spectrum H. At the same time Equation 3 can also be used. Thus, we have... 

As a liquid is cooled at a finite rate, the relaxation time spectrum will shift to longer times and a temperature region will eventually be reached where the sample is no longer in volume equilibrium. If the sample continues to be cooled at this rate it will become a glass. A glass is a nonequilibrium, mechanically unstable amorphous solid. If the sample is held at a fixed temperature near Tg the volume will relax towards its equilibrium value. In this section we will restrict our attention to equilibrium liquids at temperatures near... 

Baumgaertel M, De Rosa ME, Machado J, Masse M, Winter HH (1992) The relaxation time spectrum of nearly monodisperse polybutadiene melts. Rheol Acta 31(l) 75-82 Baumgarter A, Muthukumar M (1996) Polymers in disordered media. Adv Chem Phys 94 625-708. Eds I. Prigogine and S.A. Rice, Wiley, New York Berry GC, Fox TG (1968) The viscosity of polymers and their concentrated solutions. Adv Polym Sci 5 261—357... 

Often/(f) is approximated by the summation C, exp(—f/i,) in which the combination of T values is called the relaxation time spectrum. 

By looking at Eqs. (27) and (28), Eq. (32) confirms the customary way of relating P of the stretched exponential function, Eq. (19), to the relaxation time spectrum. The glassy state relaxation is dominated by the part of the spectrum having longer relaxation times. The fractal dynamics of holes are diffusive, and the diffusivity depends strongly on the tenuous structure in fractal lattices, v is the exponent in the power-law relationship between local diffusivity and diffusion length ... 

Calculate the relaxation time spectrum from the equation... 

In a general case, the reconstruction of the form of the time dependence of function (3/2) [cos Q t) — 1/3], i.e. of the mean square cosine of the rotation angle of the oscillator, and a direct determination of the shape of the relaxation time spectrum (all and t/) from the experimental curve Y(T/ri) are complicated mathematical problems (see Sect. 5). They may be solved rigorously (under the condition that... 

Distribution of relaxation times, H(t). In this discussion, using plots of 3/J(t), we shall discuss the distribution of characteristic response times in terms of relaxation times. The relaxation-time spectrum, H(t), can be determined as a first approximation (18) by the following relationship ... 

There are formal mathematical relationships between the complex and stress-relaxation moduli, but a practical approach has often been to use approximate relationships to derive the relaxation-time spectrum from either type of measurement. Similar approximations give retardationtime spectra from complex or creep compliances. Further details can be found in the books referred to as (1) in section 7.7. 

Some of the manifestations of viscoelasticity are delayed relaxation of stress after cessation of flow phase shift between stress and strain rate in oscillatory shear flow shear thinning (decrease of viscosity) at shear rates exceeding the reciprocal of the longest relaxation time and normal stress differences in shear flow, whose magnitudes are related to the relaxation time spectrum. A very convenient observation for experimentalists is that there is a close similarity between the shear viscosity and first normal stress difference as functions of shear rate and the corresponding parameters, complex viscosity and storage modulus, as functions of frequency in a small amplitude oscillatory shear. 

This expresses itself in slow power law dsmamics for both the linear relaxation modulus G(t) and the relaxation time spectrum H(X) (12-14)... 

The relaxation time spectrum can be expressed to a similar degree of approximation in terms of the real and imaginary parts of the complex modulus ... 

Figure 4.19 The Alfrey approximation for the relaxation time spectrum H ty. (a) from the stress relaxation modulus G(t) (b) from the real and imaginary parts G and G2, respectively, of the complex modulus G(co)...

However, the modes just defined do not have a very precise meaning. According to classical mechanics, the notion of modes requires the relaxation time spectrum of a chain (with given N) to consist of a discrete series of peaks. Actually, owing to the non-linear coupling among modes that is introduced by the above effects (a) and (b), the spectrum is a continuum, and one cannot expect sharp peaks the notion of modes is probably meaningless. 

It is well known that the linear viscoelastic properties of polymer melts and concentrated solutions are strong function of molecular structure, average molecular mass and molecular mass distribution (MWD). The relaxation time spectrum is a characteristic quantity describing the viscoelastic properties of polymer melts. Given this spectrum, it is easy to determine a series of rheological parameters. The relaxation time spectrum is not directly accessible by experiments. It is only possible to obtain the spectrum from noisy data. 

Experiments can be carried out either in the frequency domain or in the time domain. In the first case, one obtains both storage, G (co), and loss, G"(co), moduli, while the second method gives the relaxation modulus, G(t). In any case, the linear viscoelasticity theory predicts that G (co), G"(o)) or G(t) can be described using a unique function the relaxation time spectrum, H(A) [1]. ff(A) is related to G(t) by... 

Assuming the Maxwellian hypothesis, the relaxation time spectrum can be expressed in a discrete form, leading to... 

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