Continuous distribution of relaxation times

R. M. Kroeker, R. M. Henkelman 1986, (Analysis of biological NMR relaxation data with continuous distributions of relaxation-times),/. Mag. Reson. 69, 218. 

The methods described above give continuous distributions of relaxation times. However, the molecular theories of Viscoelasticity of polymers as... 

In the most general sense, non-Debye dielectric behavior can be described in terms of a continuous distribution of relaxation times, G(x) [11]. 

The next issue to concern us will be anomalous relaxation in which the smearing out of a relaxation spectrum (i.e., the deviation of complex susceptibility from its Debye form) is associated with the concept of a relaxation time distribution. As is well known, this concept implies an assembly of dipoles with a continuous distribution of relaxation times of Eq. (379). 

Thus, the crossover from a strictly exponential to an anomalous relaxation pattern can be associated with the change of a continuous distribution of relaxation times (a = 1) into a fractal-like one (0 < a = df < 1). 

Figure 3-11. Continuous distribution of relaxation times expressed as H r) and the corresponding tensile stress relaxation modulus E(t).

If n is large, the summation in the equation may be approximated by the integral of a continuous distribution of relaxation times E(r). 

As can be seen, the Maxwell-Weichert model possesses many relaxation times. For real materials we postulate the existence of a continuous spectrum of relaxation times (A,). A spectrum-skewed toward lower times would be characteristic of a viscoelastic fluid, whereas a spectrum skewed toward longer times would be characteristic of a viscoelastic solid. For a real system containing crosslinks the spectrum would be skewed heavily toward very long or infinite relaxation times. In generalizing, A may thus he allowed to range from zero to infinity. The concept that a continuous distribution of relaxation times should be required to represent the behavior of real systems would seem to follow naturally from the fact that real polymeric systems also exhibit distrihutions in conformational size, molecular weight, and distance between crosslinks. 

Equation [7.2.8] defines the functions G (t) for continuous distribution of relaxation times. For a discrete spectrum, containing nj relaxation times, the distribution function takes the form... 

The range of relaxation times allowed in the fitting was usually between 0.5 ps and 1 s with a density of 12 points per decade. Relaxation rates are obtained from the moments of the peaks in the relaxation time distribution or, if the peaks overlap, from the peak maximum position. With a broad distribution of relaxation times, these inversion methods yield multiple peaks in the "unsmoothed" analysis. The "smoothing" parameter (P) was selected as 0.5 in all cases, after it was established that the number of peaks did not increase with further increase in smoothing. As a further check, an analysis was made on a simulated correlation function consisting of a broad continuous distribution of relaxation times with noise added equal to the residuals from the analysis of the experimental correlation curve. REPES recovers the original distribution except when a very low smoothing parameter (P 0) is used. 

It is generally assumed that the description of polymer properties requires a continuous distribution of relaxation times. Numerous forms of the distribution function have been assumed, often for mathematical simplicity or on the basis of physical intuition. It has been found that a fractional power law distribution of relaxation times of the form t leads to hysteresis absorption with aX = m7il2 (Np), when m < 1 (3). The disadvantage of this model is that attempts to justify this distribution of relaxation times on a molecular basis quantitatively have not been successful. Mathematically, almost any experimental result can be expressed in terms of a distribution of relaxation times, but there may not be any physical significance to the distribution. 

In the sign control method, as in several approaches, the first step is to select the relaxation times in a reasonable manner based on the time scale of the data. In such a process, the relaxation times are not chosen based on any known polymer structure or derived timescales, but are chosen for mathematical convenience. As real polymers contain a continuous distribution of relaxation times, in this approach a sufficient discrete subset of these relaxations are chosen in order to provide a mathematical function that will fit the material data. For a typical data set, such as that in Fig. 7.3, choosing the relaxation times evenly spaced in log time over the data range is reasonable. The number of relaxation times required varies depending on the smoothness of the data, but 10-20 relaxation times over 10 decades of time is a good rule of thumb. To facilitate fitting non-constant values at... 

For a continuous distribution of relaxation times H(x), equivalent to an infinite parallel array of Maxwell elements, the total stress is summed by the integral to give... 

Commercial melts are neither monodisperse nor bidisperse, but usually have a broad, continuous, distribution of relaxation times with polydispersity ratios, M /M, greater than or equal to 2.0. We will start our discussion of the effect of polydispersity by replacing the exponential in Eq. 6.36 with a dimensionless relaxation function P t,M) to obtain a general expression for the relaxation modulus (reptation only) of a monodisperse polymer ... 

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