Fractional diffusion equations in time and frequency domain: models and applications

1
Fractional diffusion equations in time and
frequency domain models and applications

• Juan Bisquert
• Departament de Ciències Experimentals
• Universitat Jaume I, Spain

València, 2 february 2005
2
Outline

• Fractional diffusion equation with decaying
  probability density
• Interpretation of the FDE in the time domain
  meaning of the decay law in multiple trapping
  transport
• Transient diffusion photocurrent
• The frequency domain anomalous diffusion

3
The fractional diffusion equation FDE
We discuss the physical interpretation of a FDE
based on the replacement of the time derivative
in the diffusion equation with a derivative of
noninteger order (1) is the
fractional Riemann-Liouville derivative operator.
The fractional time derivative can be
written and its Laplace transform is

4
Initial condition of the FDE
Eq. (1) was discussed as a possible generalized
diffusion equation describing anomalous diffusion
process, but the feasibility of this application
was doubted because the f is not a normalized
function. Indeed, Eq. (1) requires an initial
condition for the Green function of the form
(2) Where is the Riemann-Liouville
fractional integral operator. This is a nonlocal
initial condition. It implies the divergence of f
as t-gt 0 R. Hilfer, J. Phys. Chem. B 104 (2000)
3914 .

5
Decay of the probability
The expression of the FDE in Laplace-Fourier
space (u, q) is Note that for q 0 Hence the
time decay of the probability in spatially
homogeneous conditions (without diffusion) is
given by Here we aim to provide a physical
interpretation of this decay law and the
implications for an anomalous diffusion process.

6
Background
The mathematical properties of Eq. (1) have been
amply studied in the literature the solutions in
different dimensions of space, the behaviour of
the initial conditions the moments of the
distribution W.R. Schneider and W. Wyss,
Journal of Mathematical Physics 30 (1989)
134. M.M. Meerschaert, D.A. Benson, H.-P.
Scheffler and B. Baeumer, Physical Review E 65
(2002) 041103. M.M. Meerschaert, D.A. Benson,
H.-P. Scheffler and P. Becker-Kern, Physical
Review E 66 (2002) 060102(R). J.-S. Duan,
Journal of Mathematical Physics 46 (2005)
013504. E.K. Lenzi, R.S. Mendes, K.S. Fa,
L.R. da Silva and L.S. Lucena, Journal of
Mathematical Physics 45 (2004) 3444. A.A.
Kilbas, T. Pierantozzi, J.J. Trujillo and L.
Vázquez, J. Phys. A Math. Gen. 37 (2004) 3271.

7
Background
Hilfer described Eq. (2) as a fractional
stationarity condition related to a dissipative
dynamics. Feldman et al. considered Eq. (1) as
the fractional generalization of the Liouville
equation for dissipative systems. Tarasov
provided a fractional analog for the
normalization conditions for distribution
functions. Eq. (1) was used by Nigmatullin to
define a non-exponential relaxation process R.
Hilfer, Fractals 3 (1995) 549 Y. Feldman, A.
Puzenko and Y. Ryabov, Chem. Phys. 284 (2002) 139
V.E. Tarasov, Chaos 14 (2004) 123 R.R.
Nigmatullin and Y.E. Ryabov, Phys. Solid State 39
(1997) 87

8
CTRW model
Another FDE has been amply studied in the
literature of anomalous diffusion
(3) Space-time Fourier-Laplace Eq. (3) is
rigurously related to the Continuous Time Random
Walk formalism of Scher and Montroll.
Compare (1) Eq. (3) has the
initial condition and the
probability is conserved Eqs. (1) and (3) are
not equivalent

9
The time domain

• Recently it was shown
• J. Bisquert, Physical Review Letters 91 (2003)
  010602
• that the FDE in Eq. (1) describes the diffusion
  of free carriers in multiple trapping (MT) with
  an exponential distribution of gap states.
• The present work develops the previous one with
  the goals
• To present a concrete instance of a physical
  system with an intrinsic dissipative dynamics
  leading to a shrinking phase space.
• To study the physical behaviour in experimental
  techniques Transient photocurrents, frequency
  techniques.

10
Multiple trapping
Multiple trapping model has been widely used to
describe carrier transport in amorphous
semiconductors. In 1970s MT model explained long
tails of the electrical current observed in
time-of-flight experiments and the time
dependence of transient photocurrents.
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Multiple trapping
Propagation of a gaussian packet by drift
transport and the result current
12
Multiple trapping
Propagation of a packet by drift transport in
CTRW, the predicted current and the current
measured in amorphous semiconductor
H. Scher and E.W. Montroll, Phys. Rev. B 12
(1975) 2455.
13
Multiple trapping
More recently, starting in 1995, MT has become
very popular for explaining different kinds of
experimental results in nanostructured
semiconductors such as networks of TiO2
nanoparticles used in dye-sensitized solar cells.
14
Multiple trapping
MT applies to a semiconductor with a band of
extended states where transport is fast and a
tail of localized states, which usually takes the
exponential form
15
Multiple trapping
The probability of trapping is similar for all
the bandgap states, but the probability of
release of a trapped electron is exponentially
decreasing with the depth of the trap (detailed
balance).
16
Transient photocurrent spectroscopy
A fast laser pulse excites electrons to the
conduction band The carrier density is
homogeneous and will be detected by measuring the
photocurrent. Initially the electrons will be
trapped Then equilibrium will be established at
states above a demarcation energy level that
depends on time
17
Multiple trapping
Evolution of trapped electrons Thermal
distribution above Ed Increasing occupancy, with
the same shape as the exponential distribution
below Ed. The carrier concentration sinks deeper
in the bandgap. The free carrier concentration is
given by the Boltzmann tails of the peak, and
decreases with time

18
Experimental results
transient photocurrent in a-As2Se3 Only free
electron density is measured The decay law is
given by The fractional exponent depends on
temperature as

19
Interpretation of the decaying probability
density
The measurements of transient photocurrent showed
the decay law that is predicted by the FDE
The decay of the probability is associated to
the removal of some degrees of freedom (trap
levels) in the model. The disappearance of the
probability takes on a perfectly valid physical
meaning, in correspondendence with the
requirements of the experimental technique.

20
Interpretation of the decaying probability
density
It has been remarked in the literaure the
divergence of the solution of the FDE as t -gt
0 In the system described above the decay law
cannot be extrapolated to t 0, because the
decay makes no sense without a minimal time for
initial thermalisation. This time is typically
in the ps range. In normal experiments there is
an initial time associated with the injection or
photogeneration pulse, which is much longer than
this, in the ns range. Transport experiments
resolve the evolution of a large ensemble of
electrons and not individual electronic
transitions. Therefore the divergence of the
initial condition is not an impediment for the
application of the FDE of eq. (1) in the
description of experiments.

21
The frequency domain
Impedance spectroscopy is a common experimental
technique that measures the impedance in a
material system, i.e. the relation between ac
voltage and ac current

22
Equivalent circuit of physical processes
Charge storage is represented by capacitors In
general capacitor is a reversible energy
storage Interfacial charge-transfer,
transport, recombination is represented by
resistances In general resistance is an
irreversible process

23
Diffusion impedance
Impedance of diffusion is obtained by solving the
diffusion equation in Laplace domain
Equivalent circuit
J. Bisquert, J. Phys. Chem. B 106, 325-333 (2002)
24
Diffusion impedance
Impedance of diffusion is obtained by solving the
diffusion equation in Laplace domain

25
Two state model
26
Impedance of nanostructured TiO2

27
Mg2 and Li ion insertions into Mo6S8
Nyquist plots for Mg-ion insertion into the
Chevrel phase, covering the whole frequency
domain

M. D. Levi, H. Gizbar, E. Lancry, Y. Gofer, E.
Levi and D. Aurbach J. Electroanal. Chem. 569,
211-223 (2004)
28
Anomalous diffusion model
Fractional time diffusion
J. Bisquert and A. Compte J. Electroanal. Chem.
499, 112-120 (2001).
29
Anomalous diffusion impedance
Model of 8 traps following an exponential
distribution in energy
J. Bisquert, G. Garcia-Belmonte, A.
Pitarch ChemPhysChem, 4, 287-292 (2003).
30
lithium transport through vanadium pentoxide film

The Nyquist plots of the ac-impedance spectra
measured on the V2O5 xerogel film electrode in a
1 M LiClO4PC solution at the lithium content, ,
0.75, 1.25 and 1.55 which corresponds to the
electrode potential, E, 2.8, 2.5 and 2.2 VLi/Li,
respectively.
Kyu-Nam Jung, Su-Il Pyun and Jong-Won Lee,
Electrochim. Acta 49, 4371-4378 (2004)
31
lithium transport through vanadium pentoxide film

The anodic current transients theoretically
calculated for lithium transport by using random
walk simulation in consideration of the residence
time distribution with 1.3, 1.5 and 1.8. For
comparison, the current transient simulated
without considering the residence time
distribution ( 0) is also presented.
Kyu-Nam Jung, Su-Il Pyun and Jong-Won Lee,
Electrochim. Acta 49, 4371-4378 (2004)
32
Conclusions

• The FDE equation with decaying probability
  density has been interpreted in terms of multiple
  trapping transport.
• The solutions of the FDE appear as a new tool for
  the analysis of important experimental problems,
  both in time and frequency domain
• Homepage www.elp.uji.es/jb.htm
• E-mail bisquert_at_uji.es