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

Juan BisquertDepartament de Ciències ExperimentalsUniversitat Jaume I, Spain

València, 2 february 2005

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

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

2

2

0),(

),(x

txfCtxfDt

tD0

t

t tdtt

tf

dt

dtfD

00

)(

)(

)1(

1)(

)0()()(0 fufutfDL t

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-> 0 [R. Hilfer, J. Phys. Chem. B 104 (2000) 3914 ].

)()0,( ,01

0 xfxfI t

x

axa dyyfyxxfI )()(

)(

1)( 1

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.

20

1,0),(

qKu

fquf

ufuf 1,0)0,(

11

,0

)()(

tf

tf

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.

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

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

),(),(2

210 tx

x

fDKtxf

t

20

1

1,0),(

qKu

ufquf

2

2

0),(

),(x

txfCtxfDt

20

1,0),(

qKu

fquf

)()0,( 0 xfxf

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.

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.

Multiple trapping

Propagation of a gaussian packet by drift transport and the result current

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.

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.

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

cBccB

L TkEETk

NEg /)(exp)(

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).

TkEEE Bc /exp)( 0

Transient photocurrent spectroscopy

A fast laser pulse excites electrons to the conduction bandThe carrier density is homogeneous and will be detected by measuring the photocurrent.Initially the electrons will be trappedThen equilibrium will be established at states above a demarcation energy level that depends on time:

TktEEt Bdc /)(exp10

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

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

110)( ttn

1/ cTT

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.

11

,0

)()(

tf

tf

Interpretation of the decaying probability density

It has been remarked in the literaure the divergence of the solution of the FDE as t -> 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.

11

,0

)()(

tf

tf

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

Equivalent circuit of physical processes

Charge storage is represented by capacitorsIn general capacitor is a reversible energy storage

Interfacial charge-transfer, transport, recombination is represented by resistancesIn general resistance is an irreversible process

Diffusion impedance

Equivalent circuit

Boundary conditionsBlocked diffusion

J. Bisquert, J. Phys. Chem. B 106, 325-333 (2002)

Impedance of diffusion is obtained by solving the diffusion equation in Laplace domain

Diffusion impedance

2/ LDd

chCiZ

1

2/1)( iAZ

Impedance of diffusion is obtained by solving the diffusion equation in Laplace domain

Two state model

Impedance of nanostructured TiO2

Z' ()

0 1000 2000 3000 4000

Z'' (

)

0

1000

2000

Z' ()0 200 400

Z'' (

)

0

200

400

substra te so lu tion

TiO 2

W E R E F

lifetime

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. AurbachJ. Electroanal. Chem. 569, 211-223 (2004)

Anomalous diffusion model

Fractional time diffusion

2/)( iAZ

)(

1

iQZ

J. Bisquert and A. CompteJ. Electroanal. Chem. 499, 112-120 (2001).

Anomalous diffusion impedance

J. Bisquert, G. Garcia-Belmonte, A. PitarchChemPhysChem, 4, 287-292 (2003).

Model of 8 traps following an exponential distribution in energy

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 LiClO4–PC 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)

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)

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.htmE-mail: bisquert@uji.es