Williams-Watts function

Where p defines the shape of the hole energy spectrum. The relaxation time x in Equation 3 is treated as a function of temperature, nonequilibrium glassy state (5), crosslink density and applied stresses instead of as an experimental constant in the Kohlrausch-Williams-Watts function. The macroscopic (global) relaxation time x is related to that of the local state (A) by x = x = i a which results in (11)... 

The misnormalized data of Lee et al.16) was interpreted in terms of two discrete relaxation processes. It was proposed that the relaxation function should be represented as the sum of two Williams-Watts functions. The slope at short times was claimed to be equal to the / for the faster of the two processes. Numerical calculations and graphical representations of exact relaxation functions with parameters equal to those reported by Lee et al.16) were carried out. They did not look even qualitatively similar to their reported data. The slope at the shortest times must be related to a weighted sum of both of the /3 values for the sum of two WW functions. If it was desired to fit the data to a sum of two WW functions, then this could easily be carried out with a nonlinear least squares routine. In most cases it would not be possible to obtain statistically independent values of all six parameters, but at least no further errors would be introduced by faulty manipulations of the data. The graphical procedure of Lee et al.16) cannot be recommended as of any worth in this problem. 

The molecular relaxation process has been studied by the autocorrelation function of normal modes for a linear polymer chain [177]. The relaxation spectrum can be analyzed by the Kohlrausch-Williams-Watts function [177,178] ... 

P = 0.5. The Cole-Cole function forms a symmetric arc, which approaches the intercepts with finite slope and has a maximum e" value less than (er — eu)/2. The Williams-Watts function also forms a flattened arc, but is asymmetric. The shape of the Davidson-Cole function is very similar to the Williams-Watts function, as discussed by Lindsey and Patterson33). The evaluation of the Williams-Watts function requires numerical methods 33,34). Computer programs implementing this function from published tabular values are readily available35). 

Helfand, E., On inversion of the Williams-Watts function for large relaxation times. 7. Chem. Phys. 78,1931 (1983). 

If the first scenario were real, the slower secondary relaxation should express its presence as an excess wing on high frequency side of the a- relaxation peak. To check this we superimposed dielectric loss spectra of octa-O-acetyl-lactose measured above and below Tg to that obtained at T=353 K. Next we fitted a master curve constructed in this way to the Kohlrausch-Williams-Watts function... 

The correlation function for a particular relaxation process can be well described by the Kohlrausch-Williams-Watts function C(x) = 6xp[—(t/tk) ] with tk = 20 ms and p = 0.32. Show by plotting suitable graphs, using a spreadsheet or otherwise, that, in the region X = 0.25 100 ms, the correlation function can be well simulated by the following sum of exponential functions ... 

J. R. Macdonald and R. L. Hurt [1986] Analysis of Dielectric or Conductive System Frequency Response Data Using the Williams-Watts Function, J. Chem. Phys. 84, 496-502. 

Such distribution has been shown to span several orders of magnitude for instance in the case of local motions in the glassy state. Instead of the distribution function, other fitting functions for correlation fimction C (t) are considered. One of these functions is the stretched exponential Kohlraush-Williams-Watts function exp —(f/tkww) that has been found to fit the data quite universally (2). 

This kind of behavior found in an increasing number and variety of systems can often be described remarkably well by either of two empirical relaxation functions the skewed arc function of Davidson and Cole (52) and the Williams-Watts function (53). The former is simpler in the frequency domain with the form... 

A further feature of dielectric relaxations in amorphous polymers is that the a (or ctfi) relaxations are well-described by the stretched-exponential or Kohl-rausch-Williams-Watts function [21-23]... 

The lineshape of the VII spectrum s dominant mode was obtained by Koch, et al. (4) for 23 and 48 kDa polystyrenes in cyclohexane at concentrations 690-810 g/l. Their field correlation functions were accurately described by a Williams-Watts function exp(—(t/r) ). For a 700 g/l solution, = 0.4 was independent of temperature for 21 < T < 52°C. The mean relaxation time (tv//) = for depolarized scattering had the Vogel-Fulcher-Tamman temperature dependence... 

Sun and Wang report a series of studies of polystyrene polymethylmethacrylate mixtures (in benzene, dioxane, and toluene, respectively) using light scattering spectroscopy as the major experimental technique(78-80). Both polymers were in general nondilute. Neither polymer is isorefiractive with any of the solvents. The objective was to study the bimodal spectra that arise under these conditions and to show that the two relaxation times and the mode ampUtude ratio can be used to infer diffusion and cross-diffusion coefiBcients of the two components. Experimental series varied both the total polymer concentration and the concentration ratio of the two components. The theoretical model predicts a biexponential spectrum. The experimental data were fitted by a bimodal distribution of relaxation rates or by a sum of two Williams-Watts functions. The inferred self-diffusion coefiBcients of both species fall with increasing polymer concentration. 

One treatment makes use of the well known and very convenient Williams-Watts function. (6,7,8-12) This function is of the form... 

The dwell portion of the force versus time curves was fit to the Kohlrausch-Williams-Watts function (stretched exponential function) ... 

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