Stretched exponential function

Figure 1. Illustration of the streched exponential fit. The data are rotational anisotropy of anthracene in polyisobutylene of several molecular weights. The plots have been made to overlap by dividing time (t) by the respective characteristic time constant.

The stretched exponential function, also widely known as the Kohlrausch-Williams-Watts (KWW) function is a frequenly used empirical description of the relaxation rates of many physical properties of complex systems such as polymers and and glasses. The function was invented by the German physicist Friedrich Kohlrausch in 1863 to characterize the dielectric relaxation rates in polymers. The stretched exponential was reintroduced by Graham Williams and David C. Watts in 1970 to describe the mechanical creep in glassy fibers.

The function is a simple extension of the exponential function with one additional parameter:

$\phi (t)=e^{-\left({t/\tau _{WW}}\right)^{\beta }}$

where $\tau _{WW}\,$ is the charateristic relaxation time of the function and $\beta \,$ is a parameter that can range between 0 and 1 and is refered to as the stretching parameter. Figure 1 shows the stretched exponential with the parameter of $\beta \,$ equal to 0.52. For comparison, a least squares single and a double exponential fit are also shown. For another example see Figure 2 in the Lindsey and Patterson reference below.

Distribution Function

A wide variety of relaxation relaxation behavior can be fit with the stretched exponential function, however, in most cases the fit is considered purely empirical, that is, it is used because it fits the data with a minimum number of parameters. It is possible, however, to ascribe some physical significance to the stretched exponential fit. In complex systems it may be reasonable to believe that the relaxation is intrinsically exponential, but that there is a large distribution of environments within the sample, each with different characteristics. The differences in the local environment leads to variations in the relaxation time, and when the experiment simultaneously measures a large ensemble of local relaxation times the result looks like a stretched exponential.

If the stretched exponential is the result of a distribution of relaxation times it is worthwhile to describe that distribution. If the distribution function of the stretched exponential is $\rho _{WW}\,$, then the following equation is correct:

$e^{-\left({t/\tau _{WW}}\right)^{\beta }}=\int _{0}^{\infty }{e^{-t/\tau }\rho _{WW}(\tau )}\,d\tau$

Lindsey and Patterson have derrived a formula for computing $\rho _{WW}\,$:

$\rho _{WW}(\tau )=-{{\tau _{S}} \over {\pi \tau ^{2}}}\sum \limits _{k=0}^{\infty }{{{(-1)^{k}} \over {k!}}\sin(\pi \beta k)\Gamma (\beta k+1)\left({\tau \over {\tau _{S}}}\right)}^{\beta k+1}$

The distribution funtion $G_{WW}\,$ ploted in Figure 2 is related to $\rho _{WW}\,$ via the characterisic time constant $\tau _{WW}\,$:

$G_{WW}=\tau _{WW}\rho _{WW}(\tau _{WW})\,$

Figure 2 shows the same results plotted in both a linear and a log representation. The curves converge to a delta function at $t/\tau _{WW}=1$ as the stretching parameter $\beta \,$ approaches one, corresponding to the simple exponetial function.

Figure 2. Linear and log-log plots of the stretched exponential distribution function $G_{WW}$ vs $t/\tau _{WW}$ for values of the stretching parameter β between 0.1 and 0.9.

References

Journal links may require subscription (DOI-Digital object identifier)

• Kohlrausch, F. (1863). "Ueber die elastische Nachwirkung bei der Torsion". Poggendorff's Annalen der Physik. 119: 337–368.Link
• Williams, G. and Watts, D. C. (1970). "Non-Symmetrical Dielectric Relaxation Behavior Arising from a Simple Empirical Decay Function". Transactions of the Faraday Society. 66: 80–85.{{cite journal}}: CS1 maint: multiple names: authors list (link) DOI Link
• Lindsey, C. P. and Patterson, G. D. (1980). "Detailed comparison of the Williams-Watts and Cole-Davidson functions". Journal of Chemical Physics. 73: 3348–3357.{{cite journal}}: CS1 maint: multiple names: authors list (link) DOI Link
• Alvarez, F., Alegría, A. and Colemenero, J. (1991). "Relationship between the time-domain Kohlrausch-Williams-Watts and frequency-domain Havriliak-Negami relaxation functions". Physical Review B. 44: 7306–7312.{{cite journal}}: CS1 maint: multiple names: authors list (link) DOI Link