Relaxation time spectra

There are a number of important concepts which emerge in our discussion of viscosity. Most of these will come up again in subsequent chapters as we discuss other mechanical states of polymers. The important concepts include free volume, relaxation time, spectrum of relaxation times, entanglement, the friction factor, and reptation. Special attention should be paid to these terms as they are introduced. 

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 scaling of the relaxation modulus G(t) with time (Eq. 1-1) at the LST was first detected experimentally [5-7]. Subsequently, dynamic scaling based on percolation theory used the relation between diffusion coefficient and longest relaxation time of a single cluster to calculate a relaxation time spectrum for the sum of all clusters [39], This resulted in the same scaling relation for G(t) with an exponent n following Eq. 1-14. 

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

Fig. 11. Schematic of relaxation time spectrum of the critical gel of PBD 44 (Mw = 44 000). The entanglement and glass transition is governed by the precursor s BSW-spectrum, while the CW spectrum describes the longer modes due to the crosslinking [60]. denotes the longest relaxation time of PBD44 before crosslinking...

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. 

Predictions using the observed relaxation time spectrum at the gel point are consistent with further experimental observations. Such predictions require a constitutive equation, which now is available. Insertion of the CW spectrum, Eq. 1-5, into the equation for the stress, Eq. 3-1, results in the linear viscoelastic constitutive equation of critical gels, called the critical gel equation ... 

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... 

Andrews,R.D., Tobolsky.A. V. Elastoviscous properties of pdyisobutylene. IV. Relaxation time spectrum and calculation of bulk viscosity. J. Polymer Sci 7, 221-242 (1951). 

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... 

Figure 15.8 A model of temperature-dependent relaxation time spectrum for PMN. G (r, T) is the number of polar regions having a relaxation time r, T/ is the freezing temperature, and td is the inverse Debye frequency (from [16]).

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... 

Relaxation-time spectrum Elastic constant of the cantilever... 

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

Eq. (13.80) is valid only in a rather limited time interval. If the behaviour over a longer time period must be described, a number of equations of this type can be superposed, each with a different relaxation time. Ultimately, a whole relaxation time spectrum may be developed ... 

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 ... 

Winter HH (1997) Analysis of dynamic mechanical data inversion into a relaxation time spectrum and consistency check. J Non-Newtonian Fluid Mech 68 225-39. 

From the concept of separability, the memory function of the linear viscoelasticity is required. This memory function can be related to a discrete relaxation time spectrum, available firom dynamic experiments, given by ... 

Calculate the relaxation time spectrum from the equation... 

Find the complex dynamic moduli corresponding to a box distribution of relaxation times spectrum, i.e., that... 

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 ... 

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