Kohlrausch function

In the previous subsection, we have provided conceptually the rationale and experimentally some data to justify the expectation that the primitive relaxation time To of the CM should correspond to the characteristic relaxation time of the Johari-Go Id stein (JG) secondary relaxation Xjg- Furthermore, it is clear from the CM relation, Ta = ( "to)1 1- , given before by Eq. 6 that To mimics Ta in behavior or vice versa. Thus, the same is expected to hold between Xjg and Ta. This expectation is confirmed in Section V from the properties of tjg- The JG relaxation exists in many glass-formers and hence there are plenty of experimental data to test the prediction, xjG T,P) xo(T,P). Broadband dielectric relaxation data collected over many decades of frequencies are best for carrying out the test. The fit of the a-loss peak by the one-sided Fourier transform of a Kohlrausch function [Eq. (1)] determines n and Ta, and together with tc 2 ps, To is calculated from Eq. 6... 

Although glass transition is conventionally defined by the thermodynamics and kinetic properties of the structural a-relaxation, a fundamental role is played by its precursor, the Johari-Goldstein (JG) secondary relaxation. The JG relaxation time, xjg, like the dispersion of the a-relaxation, is invariant to changes in the temperature and pressure combinations while keeping xa constant in the equilibrium liquid state of a glass-former. For any fixed xa, the ratio, T/G/Ta, is exclusively determined by the dispersion of the a-relaxation or by the fractional exponent, 1 — n, of the Kohlrausch function that fits the dispersion. There is remarkable similarity in properties between the JG relaxation time and the a-relaxation time. Conventional theories and models of glass transition do not account for these nontrivial connections between the JG relaxation and the a-relaxation. For completeness, these theories and models have to be extended to address the JG relaxation and its remarkable properties. 

Figure 7.12. Correlation of nonexponentiality parameter P of Kohlrausch function with decoupling index R, for a variety of ionic glasses (After Angcll, 1990).

Not only does the magnitude of Ta uniquely defrne the dispersion, as shown herein, but also many properties of are governed by or correlated with the width of the dispersion of the structural relaxation or the fractional exponent n of the Kohlrausch function, (t) = exp[-(t/r ) "], frequently used to fit the... 

Figure 6. T-P superposition of loss spectra for 10% QN in tristyrene measured for different T and P combinations but the same Ta= 0.67 s. The line is a Fourier transformed of the Kohlrausch function with Pkww = (1-n) = 0.5. The results demonstrate the co-invariance of three quantities, To, n, and Tjq, to widely different combinations ofT and P.

This equation has been used as the basis to explain the T-P superposition of the O -relaxation of a component in mixtures of van der Waals glass-formers and polymer blends as discussed in Capaccioli and Ngai (2005). Concentration fluctuations in the mixture or blend create a distribution of environments /. Each environment, i, has its own coupling parameter, tii, primitive relaxation time, Toi, and the corresponding Kohlrausch function with stretch exponent, (1 — ,), which determines the relaxation time t, by the CM equation rai =. In the same manner as shown earlier for neat glass-... 

To, as in experiments. (4.) In addition, the relaxation of J(t) at times before and after the j8-relaxation window can be fitted by the Rouse-theory and by a Kohlrausch function with a temperature independent exponent (time-temperature superposition principle), respectively (39 40). 

The response function is chosen, according to Moynihan et at. [50], in a manner of a Kohlrausch function [51], which is also called the stretched exponential function and is often used as a phenomenological description of relaxation in disordered systems. The value of p, which is called the stretching exponent and describes the nonexponential characteristic of the relaxation process, is defined as... 

The Kohlrausch function, which is given at the leading/terminating ion boundary, in fact gives the conditions at any boundary between two adjacent ions A, B. with one common counterion when the boundary migrates in the electric field. 

The GM function has explicit forms for the real and imaginary parts / and but the Kohlrausch function does not In tius case the integral of Eq. (13), which b essentially a Laplace transformation of the derivative of the response function, b calculated numerically. First the current response to a step voltage b measured and the derivative of the Kohlrauscb function b fitted to it, then the Laplace transformation b performed. 

The parameters of the response fuoctioa are dependent on temperature and pressure. The temperature dependence will be discussed here for the Kohlrausch function only. In this fun km. parameter r determines the angular frequency where the dielectric loss and the relaxation lime distribution calculated from Eq. (19) are near to maximum. The temperature dependence of this parameter for such dipolar groups, the thermal motion of which do not influence the main structure appreciably, is... 

Fig. 3.3 Long-term creep predictions by KAHR-Ote model, combined with the isothermal and nonisothermal effective time theory and original Kohlrausch function from short-term response. Thermal histories 97 °C - 21 °C—> 73 °C 27 °C 73 °C. KoMiausch function parameters Do = 0.460 GPa r = 1593 s, = 0.417. The inset is the nonisothtamal temperature history prior to long-term creep tests. Data reproduced with permission from Ref. [73]...

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