A fractional approach is used to describe data from dielectric spectroscopy for several glassy materials. Using composite fractional time evolution propagators a modified law for relaxation in glasses is found that describes the experimental data for broadband dielectric spectroscopy. Properties and solutions of some particular fractional differential equations (fDEQs) are investigated both for rational and irrational order. The laws of Debye, Kohlrausch, Cole-Cole, ColeDavidson and Havriliak-Negam

1. Fractional approaches to dielectric broadband
spectroscopy
Simon Candelaresi, Rudolf Hilfer
ICP, Uni Stuttgart
2008-02-26
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

2. Overview
Dielectric Broadband Spectroscopy
Use composite fractional time evolution to describe dielectric
spectroscopy data of glasses.
Modified Debye relaxation equation with fractional derivatives
Fitting with only 3/4 parameters
Fractional Calculus
Solution to modifed Debye relaxation equation
Fractional differential equation (FDE) is solved directly.
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

3. Dielectric Broadband Spectroscopy
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

4. Dielectric Relaxation in Glasses
(a) 5-methyl-2-hexanol (b) propylene carbonate
(c) glycerol
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

5. Example from Experiment
data: Loidl, Phys. Rev. E 59 6924, (1999)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

6. Fit Models
normalised relaxation function e.g. polarisation
f (t) = Φ(t)/Φ(0)
Debye:
τ
d
dt
f (t) + f (t) = 0, f (0) = 1 f (t) = e−t/τ
L {f (t)} (u) =
∞
0
e−ut
f (t) dt
normalised complex dielectric susceptibility:
χ(u) = 1 − uL {f (t)} (u)
χ(u) =
1
1 + uτ
=
ˆχ(u) − ˆχ∞
ˆχ(0) − ˆχ∞
u = 2πiν
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

7. Fit Models
Debye: χ(u) = 1
1+uτ nparam = 1
Cole-Cole: χ(u) = 1
1+(uτ)α nparam = 2
Cole-Davidson: χ(u) = 1
(1+uτ)γ nparam = 2
Havriliak-Negami [1]: χ(u) = 1
[1+(uτ)α]γ nparam = 3
[1] Havriliak, Negami, Polymer 8 (4) 161, (1967)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

8. Problem
data: Loidl, Phys. Rev. E 59 6924, (1999)
Excess wing cannot be fitted with one function.
superposition (Havriliak-Negami + Cole-Davidson ⇒ 6 fit
parameters)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

9. Fractional Relaxation Model
Apply the idea of composite fractional time evolution [Hilfer, Chem.
Phys. 284 399-408, (2002)]
Modified relaxation equations:
τ1Df (t) + τα
2 Dα
f (t) + f (t) = 0
τ1Df (t) + τα1
1 Dα1
f (t) + τα2
2 Dα2
f (t) + f (t) = 0
corresponding susceptibilities:
3params: χ(u) = 1+(τ2u)α
1+(τ2u)α+τ1u nparam = 3
4params: χ(u) = 1+(τ1u)α1 +(τ2u)α2
1+τ1u+(τ1u)α1 +(τ2u)α2
nparam = 4
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

10. Fits
fractional model
data: Kalinovskaya, Vij, J. Chem. Phys., 112 7, (1999)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

11. Fits
fractional model
data: Kalinovskaya, Vij, J. Chem. Phys., 112 7, (1999)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

12. Fits
fractional model
data: Loidl, Phys. Rev. E 59 6924, (1999)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

13. Fits
fractional model
data: Loidl, Phys. Rev. E 59 6924, (1999)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

14. Fits
fractional model
data: Loidl, Phys. Rev. E 59 6924, (1999)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

15. Fits
fractional model
data: Loidl, Phys. Rev. E 59 6924, (1999)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

16. Fits
fractional model
data: Loidl, Phys. Rev. E 59 6924, (1999)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

17. Fits
fractional model
data: Feldman, J. Phys. Chem. B., 109 (18), (2005)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

18. Fits
fractional model
data: Feldman, J. Phys. Chem. B., 109 (18), (2005)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

19. Fits
fractional model
data: Feldman, J. Phys. Chem. B., 109 (18), (2005)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

20. Fractional Calculus
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

21. Problem
solve the modified relaxation equations
τ1Df (t) + τα
2 Dα
f (t) + f (t) = 0
τ1Df (t) + τα1
1 Dα1
f (t) + τα2
2 Dα2
f (t) + f (t) = 0
f (0) = 1
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

22. Fractional Calculus
Definition
fractional integral:
Iµ
0 f (t) =
1
Γ(µ)
t
0
(t − ξ)µ−1
f (ξ) dξ
Riemann-Liouville:
Dν
0 f (t) = D ν
I
( ν −ν)
0 f (t)
D3.2
0 = D4
I0.8
0
Caputo-Liouville:
Dν
0 f (t) = I
( ν −ν)
0 D ν
f (t)
Hilfer:
Dν,β
0 f (t) = I
β( ν −ν)
0 D ν
I
(1−β)( ν −ν)
0 f (t)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

23. Fractional Calculus
fractiogene e-function
Iµ
0 eat
=
eat
aµΓ(µ)
γ(ν, at)
= E(µ, a; t) fractiogene e-function Et(µ, a)
= tµ
E1,µ+1(at) generalised Mittag-Leffler function
Properties:
Dν
E(µ, a; t) = E(µ − ν, a; t)
E(µ, a; t) = aE(µ + 1, a; t) +
tµ
Γ(µ + 1)
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

24. Fractional Calculus
solution of the FDE
τ1Df (t) + τα1
1 Dα1
f (t) + τα2
2 Dα2
f (t) + f (t) = 0
more general N
i=0 ai Di/Nf (t) = 0
indicial polynomial N
i=0 ai ci
j = 0 j=1,..N
solution f (t) = N
j=1
N−1
k=0 cN−k−1
j E(−k/N, cN
j ; t) Bj
with N
j=1 Bj
k−1
i=0 ci
j aN−k+i+1 = 0 1 ≤ k ≤ N − 1
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

25. Fractional Calculus
solution of the FDE
modified relaxation equation:
τ1Df (t) + τα1
1 Dα1
f (t) + τα2
2 Dα2
f (t) + f (t) = 0
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

26. Summary
Modified relaxation equation describes spectroscopy data
better than Havriliak-Negami and Cole-Cole
Less parameters (3/4) than Havriliak-Negami + Cole-Cole (6)
4 parameters to fit over 9 decades
Fractiogene e-function to solve FDE independently of the type
of derivative
Solution of modified relaxation equation slower than Debye
relaxation
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy

27. Contact
iomsn@icp.uni-stuttgart.de
Simon Candelaresi, Rudolf Hilfer ICP, Uni Stuttgart Fractional approaches to dielectric broadband spectroscopy