Dec 6, 2006 - from Eq. 3.326 of Gradshteyn and Ryzhik. This allows us to define a common decay parameter for exponential and stretched exponential decay ...
stretched exponential function LF 12/06/2006
1
stretched exponential function
The “stretched exponential function”, also known in the field of dielectric relaxation as the Kohlrausch-Williams-Watts (KWW) function, is a frequently used empirical description of the relaxation rates of many physical properties of complex systems such as polymers and glasses. It also appears to accurately describe the removal rates of small, stray bodies in the solar system. The function was introduced by the Germans|German physics|physicist Rudolf Kohlrausch in 1854 to describe the discharge of a capacitor. Although not widely known, it was next used by A. Werner in 1907 to describe complex luminescence decays. It arises in the fluorescence decay law of electronic energy donors, as derived by Theodor F¨orster in 1949. The stretched exponential was again used by G. Williams and D.C. Watts in 1970 to characterize the dielectric relaxation rates in polymers. It is identical to the Cumulative distribution function. The function is a simple extension of the exponential function with one additional parameter φ(t) = e−(t/τKW W )
β
where τKW W , is the characteristic relaxation time of the function and β, is a parameter that can range between 0 and 1 and is referred to as the stretching parameter (or shape parameter).
1
1.1
Distribution function
A wide variety of 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 ρKW W , then the following equation is correct R∞ β e−(t/τKW W ) = 0 e−t/τ ρKW W (τ )dτ Lindsey and Patterson reported a formula for computing ρKW W βk ∞ 1 P (−1)k τ ρKW W (τ ) = − πτ sin(πβk)Γ(βk + 1) k! τKW W k=0
For a more recent and general discussion, see Berberan-Santos et al., ref. given below. The distribution function GKW W plotted in Figure 2 is related to ρKW W via the characteristic time constant τKW W GKW W = τ ρKW W (τ ) Figure 2 shows the same results plotted in both a linear and a Logarithm|log representation. The curves converge to a Kronecker delta|delta function at t/τKW W = 1 as the stretching parameter β approaches one, corresponding to the simple exponential function.
1.2
Average and higher moments
In order to make valid comparisons between exponential decays and stretched exponential decays it is necessary to determine the meaning of the corresponding decay parameters. 2
The decay parameter τ of the exponential decay is the time necessary for decay amplitude to drop by a factor of E (mathematical constant)|”e”. However, this is not the case for stretched exponential decay parameter τKW W and so the decay parameters can not be compared directly. A more meaningful approach can be reached by noticing that the area under the curve of an exponential decay is proportional to the decay parameter. R ∞ −t/τ dt = τ 0 e Thus, the two decay functions might be compared on the basis of their integrals. The integral of the stretched exponential is slightly more complex R ∞ −(t/τ β KW W ) dt = τKW W Γ( 1 ) β β 0 e where G(”x”) is the gamma function, or generalized factorial. This follows immediately from Eq. 3.326 of Gradshteyn and Ryzhik. This allows us to define a common decay parameter for exponential and stretched exponential decay laws τ, exponential τ= τKW W Γ( 1 ), stretched exponential β
β
The moment of the relaxation time can be found without explicit knowledge of the stretched exponential distribution function. It is necessary to show that R ∞ n−1 1 hτ n iKW W = Γ(n) φ(t)dt 0 t is the nth moment of τKW W . Combining this equation with the expression for the stretched exponential in terms of the distribution function leads to R ∞ n−1 −t/τ R ∞ n−1 R ∞ −t/τ R∞ 1 1 ρ(τ )dτ dt = Γ(n) e dt dτ hτ n iKW W = Γ(n) 0 e 0 t 0 t 0 ρ(τ ) where the order of integration has been changed to generate the last equallity. R R∞ 1 n ) dτ = ∞ τ n ρ(τ )dτ hτ n iKW W = Γ(n) ρ(τ ) (Γ(n)τ 0 0 This last integral is the definition of the nth moment. The subscript has been omitted from ρ(τ ) to emphasize that it is not necessary to know the distribution function in this derivation. The final result is then R ∞ n−1 −(t/τ β 1 KW W ) dt Which can be converted to the final form using hτ n iKW W = Γ(n) e 0 t Gradshteyn and Ryzhik integral Eq. 3.478
3
. hτ n iKW W =
2
n τKW W Γ(n/β) β Γ(n)
References F. Alvarez, A. Alegr´ıa, and J. Colmenero. Relationship between the time-domain
kohlrausch-williams-watts and frequency-domain havriliak-negami relaxation functions. Physical Review B, 44:7306–7312, 1991 S.A. Baeurle, A. Hotta, and A.A. Gusev. A new semi-phenomenological approach to
predict the stress relaxation behavior of thermoplastic elastomers. Polymer (journal), 46:4344–4354, 2005 M.N. Berberan-Santos, E.N. Bodunov, and B. Valeur. Mathematical functions for
the analysis of luminescence decays with underlying distributions 1. kohlrausch decay function (stretched exponential). Chemical Physics, 315:171–182, 2005 A. Dobrovolskis, Alvarellos, J., and J. Lissauer. Lifetimes of small bodies in planeto-
centric (or heliocentric) orbits. Icarus (journal), 188:481– 505, 2007 R. Kohlrausch. Theorie des elektrischen r¨ uckstandes in der leidner flasche. Annalen
der Physik und Chemie (Poggendorff ), 91:56–82, 179–213, 1854 C. P. Lindsey and G. D. Patterson. Detailed comparison of the williams-watts and
cole-davidson functions. Journal of Chemical Physics, 73:3348 – 3357, 1980 G. Williams and D. C. Watts. Non-symmetrical dielectric relaxation behavior arising
from a simple empirical decay function. Transactions of the Faraday Society, 66:80 – 85, 1970
2.1
figures
4
6 5
GKWW
4 3 2 1 0 0
2
4
6
8
10
t/τKWW 101
GKWW
100
10-1
10-2
10-3
10-4 -6 10
10-5
10-4
10-3
10-2 t/τKWW
10-1
100
101
102
Figure 1: Lin-Lin and Log-log plot of the stretched exponential distribution function 5 GKW W vs t / τKW W
2
N o r m a l i z e d t i m e d i s t r i b u t i o n ( τ/ τ0 )
1 0
m a x (1 .3 ) m o s t lik e ly m in ( 1 .3 ) n o rm 1 0
1
1 0
0
0 .0
0 .2
0 .4 W W
0 .8
1 .0
p a ra m e te r
1 0
4
1 0
3
1 0
2
1 0
1
2 .0
1 .5
1 .0
W id th o f th e d is tr ib u tio n ( a t Im a x
0 .5
/1 .3 ) [% ]
M o s t l i k e l y d i m e n s i o n l e s s r e l a x a t i o n t i m e ( τ/ τ0 )
βK
0 .6
R e la x tim e S td [% ] 0 .0 0 .0
0 .1
0 .2
0 .3
0 .4 βK
0 .5 W W
0 .6
0 .7
0 .8
0 .9
1 .0
p 6a r a m e t e r
Figure 2: Width of the distribution in time as a function of the βKW W parameter (the width was taken at Ymax /1.3)
