Williams-Watts relaxation

Many relaxation phenomena in polymeric glasses can be described in an empirical way using the Williams-Watts relaxation function t) in which a parameter p accounts for the width of the relaxation spectrum ... 

A particular characteristic feature of dynamic processes in the vicinity of the glass transition is the ubiquity of the Kohlrausch-Williams-Watts (KWW) stretched exponential relaxation 1,7-9... 

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. 

Without using any motional model, the temperature positions of T and Tip minima can be assigned an appropriate frequency 90 MHz at 120 °C from Ti and 43 kHz at - 34 °C from T r These two results fit quite well on the relaxation map of BPA-PC obtained from dynamic mechanical and dielectric relaxation. They support the fact that phenyl ring motions are involved in the /3 relaxation of BPA-PC. Furthermore, the Ti and T f> data can be simulated by considering the Williams-Watts fractional correlation function [33] ... 

Broadness of the relaxation spectrum, as attested, for example, by the low value of the exponent p in the Kolrausch-Williams-Watt expression. 

The considered model of a straight line of M nanoparticles illustrates only general features of dielectric losses caused by an M nanoparticle cluster in polymer matrix. Actually such cluster is a complex fractal system. Analysis of dielectric relaxation parameters of this process allowed the determination of fractal properties of the percolation cluster [104], The dielectric response for this process in the time domain can be described by the Kohlrausch-Williams-Watts (KWW) expression... 

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] ... 

Figure 14 compares the Cole-Cole diagrams for a single relaxation time (P = 1), the Cole-Cole expression with p = 0.5, and the Williams-Watts expression with... 

The Kohlrausch Williams-Watts and Havriliak Negami formalisms are equally capable of representing real experimental data, and this is their main value, rather than an ability to explain the underlying relaxation processes. They are rooted in the time and frequency domains, respectively, and there is no analytical way of transforming from one to the other, but their effective equivalence has been convincingly demonstrated by numerical methods (Alvarez, Alegria and Colmenero, 1991). 

On the other hand, some phenomenological distributions of relaxation times, such as the well known Williams-Watts distribution (see Table 1, WW) provided a rather good description of dielectric relaxation experiments in polymer melts, but they are not of considerable help in understanding molecular phenomena since they are not associated with a molecular model. In the same way, the glass transition theories account well for macroscopic properties such as viscosity, but they are based on general thermodynamic concepts as the free volume or the configurational entropy and they completely ignore the nature of molecular motions. 

One of the features observed in many glass-forming liquids is the non-linear nature of any relaxation processes that occur around and below Tg. The relaxation rate is found to depend on the sign of initial departure of actual sample from the equilibrium state. The relaxation rate is described well by the Kohlrausch-Williams-Watts (KWl O empirical equation. ... 

As mentioned in Chapter 3, glassy relaxation processes are often associated with a fairly broad spectrum of relaxation times. A simple expression that describes this spectrum reasonably well over a wide range of time is the stretched exponential, or Kolrausch-Williams-Watts (KWW) expression (Kolrausch 1847 Williams and Watts 1970 Shlesinger and Montroll 1984) ... 

It is an experimentally demonstrated fact that the a relaxation in the time domain fits the stretch exponential decay function (0 or the Kohlrausch-Williams-Watts (KWW) equation (7,8)... 

The Mittag-Leffler function has interesting properties in both the short-time and the long-time limits. In the short-time limit it yields the Kohlrausch-Williams-Watts Law from stress relaxation in rheology given by... 

Figure 12. The solid curve is the Mittag-Leffler function, the solution to the fractional relaxation equation. The dashed curve is the stretched exponential (Kohlrausch-Williams-Watts Law), and the dotted curve is the inverse power law (Nutting Law).

A model having predictions that are consistent with the aforementioned experimental facts is the Coupling Model (CM) [21-26]. Complex many-body relaxation is necessitated by intermolecular interactions and constraints. The effects of the latter on structural relaxation are the main thrust of the model. The dispersion of structural relaxation times is a consequence of this cooperative dynamics, a conclusion that follows from the presence of fast and slow molecules (or chain segments) interchanging their roles at times on the order of the structural relaxation time Ta [27-29]. The dispersion of the structural relaxation can usually be described by the Kohlrausch-William-Watts (KWW) [30,31] stretched exponential function,... 

Another important characteristic of viscous liquids close to Tg is nonexponential relaxation. Consider the response of a system to a perturbation, such as the polarization in response to an applied electric field, the strain (deformation) resulting from an applied stress, the stress in response to an imposed deformation, the volume response to applied pressure, or the temperature response to a heat flux. It is found experimentally that the temporal behavior of the response function 0(t), following an initial instantaneous response, can often be described by the stretched exponential, or Kohlrausch-Williams-Watts (KWW) function (Kohlrausch, 1854 Williams and Watts, 1970),... 

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

In chapter 7, several aspects of conductivity and dielectric relaxation were discussed. Various other properties such as shear modulus, viscosity, refractive index, volume, enthalpy etc. also exhibit relaxational behaviour particularly in the glass transition region. In this chapter, few further aspects of relaxation are discussed. Relaxation of generalized stress or perturbation whether electrical, mechanical or any other form is typically non-exponential in nature. The associated property is a function of time. A variety of empirical functions, (/) t), have been used to describe the relaxation. Some of them have already been discussed in chapters 6 and 7. The most widely used function is the Kohlraush-Williams-Watts (KWW) function (Kohlraush, 1847 Williams and Watts, 1970 Williams et al., 1971). It is more commonly referred to as the stretched exponential function. The decay or relaxation of the quantity is given by,... 

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