METHODS OF MEASURING SUBDIFFUSION PARAMETERS

Tadeusz KosztołowiczTadeusz Kosztołowicz

Institute of Physics, Świętokrzyska Academy,

Kielce, Poland

Anomalous Transport

Bad Honnef, 12th - 16th July, 2006

1. Introduction.

2. Measuring subdiffusion parameters:

a) In the system with pure subdiffusion:

Anomalous time evolution of near-membrane layers

b) In the subdiffusive system with chemical reactions:

Anomalous time evolution of reaction front

c) In electrochemical system:

Anomalous impedance

3. Biological application:

Transport of organic acids and salts in the tooth enamel

4. Final remarks

T. Kosztołowicz, Measuring subdiffusion parameters

Subdiffusion

t

Dx

1

22 )10(

- subdiffusion parameter

- subdiffusion coefficient

D

Subdiffusion equation

2

2

1

1 ,,

x

txC

tD

t

txC

130

mmmembrane

aqueous solutionof agarose

aqueous solution of agarose and glucose

glass cuvette

laser beam

Measuring subdiffusion parameters

Schematic view of the membrane system

T. Kosztołowicz, K. Dworecki, S. Mrówczyński, PRL 94, 170602 (2005)

Near-membrane layer (0,)

t,0Ct,C

Initial condition

0,0

0,0, 0

x

xCxC

Boundary conditions at the thin membrane

t,0Jt,0J 1.

2. t,0Ct,0Ct,0J or

t,0Ct,0C ?

?

0t,0Jbt,0Cbt,0Cb 321

20

11,

2

0111

21

10

tD

xH

bb

bCtxC

211

21

3

bb

bDtIn the long time approximation

)0( x

2t,D,At

2

10111 20

11

2,,

HDDA

The experimentally measured thickness of near-membrane layer as a function of time t for glucose with =0.05 (), =0.08 (), and =0.12 () and for sucrose with =0.08 (). The solid lines represent the power function At0.45.

Transport of glucose and sucrose in agarose gel

tAt For glucose: A = 0.091 ± 0.004 for = 0.05, = 0.45

A = 0.081 ± 0.004 for = 0.08, = 0.45

A = 0.071 ± 0.004 for = 0.12, = 0.45

For sucrose: A = 0.064 ± 0.003 for = 0.08, = 0.45

= 0.90, D0.90 = (9.8 ± 1.0) 10–4 mm2/s0.90

= 0.90, D0.90 = (6.3 ± 0.9) 10–4 mm2/s0.90

P = /A . The line represents the function t0.45.

MEASUREMENT IN NON-TRANSPARENT MEDIUM

21~, ttxC d

txxFttxC d~,

theory

experiment

T. Kosztołowicz, AIP 800 (2005)

K. Dworecki, Physica A 359, 24 (2006)

dxx

PEG2000 in polyprophylene membrane, 180A pore size, 9x109 pores/cm2 01.017.0 01.067.0

Subdiffusion-reaction system

)('' inertCBnAm

CA(x,0) = C0AH(-x) CB(x,0) = C0BH(x)

t,xRx

t,xC

tD

t

t,xC2

A2

1

1

AA

t,xRx

t,xC

tD

t

t,xC2

B2

1

1

BB

t,xCt,xkCt

t,xR nB

mA1

1

The subdiffusion-reaction equations

Subdiffusion-reaction system

max),( ttxR f

Time evolution of reaction front in subdiffusive system

1. DA= DB S.B. Yuste, L. Acedo, K. Lindenberg, PRE 69, 036126 (2004)

2 tDtx ff

BA

Bf CC

CDH

00

00111

2

10

21

2. DA DB , DA, DB > 0 T. Kosztołowicz, K. Lewandowskacond-mat/0603139 (2006)Phys. Rev. E (submitted)

2 tDtx ff

B

f

BBA

f

AA D

D

DCD

D

DC

00

11

21

11

20

11 20111

20111 zHzHz

3. DA > DB = 0

2 tDtx ff

B

f

BD

BA

f

AA D

D

DCD

D

DC B

1lim

110

00

T. Kosztołowicz, K. LewandowskaActa Phys. Pol. 37, 1571 (2006)

The schematic view of the tooth enamel

The dotted line represents the concentration of static hydroxyapatite Ca5(PO4)3, the dashed one – the concentration of organic acid HB.

OHBPOHCaHBOHPOCa 2422

345 7357

0

0,5

1

1,5

2

2,5

3

0 50 100 150 200 250 300 350 400

t [h]

x f *

102 [m

m]

Lesion depth versus time

The squares represent experimental data (J. Featherstone et al., Arch. Oral Biol. 24, 101 (1979) ), solid line is the plot of the power function xf = 0.39 t 0.32 . Since xf = Df t /2 , we obtain = 0.64.

sI

sˆsZ

is

s,0J

s,0CRsZ W

DIFFUSION IMPEDANCE

x

t,xC

tD

t

t,xJt,xJ

1

1

x

t,xJD

t

t,xC

2

2

1

1

x

t,xCD

t

t,xC

t

t,xC

2Dv

A. Compte, R. Metzler, J. Phys. A 30, 7277 (1997)

generalized Cattaneo equation

tsinEt,0C 0t,LC

0)0,( xC

i

LitanhRiZ W

s1D

ss

2

s1

Dss 21

2

cotZRe

ZImlim

0

= 0.6 = 0.8 = 1

=

1

=

0.0

1

=

0

THE EXPERIMENTAL SETUP

Impedance is measured using Solartron Frequency Response Analyzer 1360 and Biological Interface Unit 1293 in the frequency range 0.1 Hz to 100 kHz. Amplitude of signal was selected for 1000 mV.

0,00E+00

2,00E+07

4,00E+07

6,00E+07

8,00E+07

1,00E+08

1,20E+08

1,40E+08

1,60E+08

1,80E+08

2,00E+08

0,00E+00 2,00E+07 4,00E+07 6,00E+07 8,00E+07 1,00E+08 1,20E+08 1,40E+08 1,60E+08 1,80E+08 2,00E+08

EXPERIMENTAL RESULT

= 0.30 ± 0.06

Final remarks

•We have developed a method to extract the subdiffusion parameters from experimental data. The method uses the membrane system, where the transported substance diffuses from one vessel to another, and it relies on a fully analytic solution of the fractional subdiffusion equation. We have applied the method to the experimental data on glucose and sucrose subdiffusion in a gel solvent.

•We show that the reaction front evolves in time as xf~Dft /2 with 1. The relation can be used to identify the subdiffusion and to evaluate the subdiffusion parameter in a porous medium such as a tooth enamel.

Final remarks

• Our first method to determine the subdiffusion parameters relies on the time evolution of near-membrane layer =At/2. Why the parameters are not extracted directly from concentration proflies? There are some reasons to choice the near-membrane layers:

1. The near-membrane layer is free of the dependence on the boundary condition at the membrane

2. When the concentration profile is fitted by a solution of subdiffusion equation, there are three free parameters. When the temporal evolution of is discussed, is controlled by time dependence of (t) while D is provided by the coefficient A.

Fractional derivative

xxfxfxx

xfx

1

01

1

lim

xxfxxfxfxx

xfx

22lim 2

02

2

xjxf

jnj

nx

x

xf n

j

jn

xn

n

00 )1()1(

11lim

................................................................

xjxf

jjx

x

xfN

N

j

jN

N

xN

1

0

)(

0 )1()1(

11lim

N

xxx N n

)0(

Fractional integral

1

0)(0

0

001

1

limN

jNN

Nx

x

xjxfxdxxfx

xfN

xjxfnj

jnx

x

xfN

N

j

nN

Nxn

n

N

1

0)(0 )()1(

lim

................................................................

xjxfj

jx

x

xfN

N

jN

N

xN

1

0

)(

0 )()1(lim

1

0

2

)(0

0

00

0

12

2

1lim

1

N

jNN

Nx

xx

xjxfjx

dxxfdxx

xf

N

Fractional derivatives and integrals

dx

xfdxjxf

j

jx

dyyfyxdx

xfd

N

N

jN

Nx

x

RL

N

1

0)(0

0

1

)()1(lim

1

The Riemann-Liouville (RL) definition

)0(

dx

xfd

dx

xfd

dx

d

dx

xfd

RL

n

n

n

n

RL

)0( n

K.B. Oldham, J. Spanier, The fractional calculus, AP 1974

1,

1

1

pxp

p

dx

xd pp

1, pdx

xd p

x

dx

d

1

11

Examples

...

22sin

sin 31

xxx

dx

xd

...

312cos

cos 42

xxx

dx

xd

Properties of fractional derivatives

k

k

k

k

k dx

xgd

dx

xfd

kdx

xgxfd

0

Leibniz’s formula

Linearity

dx

xgdb

dx

xfda

dx

xbgxafd

Chain rule

jPjk

m

k

j j

mk

k j

g

Pf

k

kx

kdx

xgfd

!!

1

1

1

1 10

Scaling approach

attxCA ,

bttxCB ,

m

rttxR ,

,

t

xx f , tx f

Scaling approach for subdiffusion ?

BA DD

,,,

,

0i

i

ii

iA Nt

d

ad

t

txC

,,,

, '

0i

i

ii

iB Nt

d

bd

t

txC

Quasistationary approximation(for normal diffusion-reaction system

Z. Koza, Physica A 240, 622 (1997), J. Stat. Phys. 85, 179 (1996))

Inside the depletion zone:

In the region where R(x,t) ≈ 0

0,0

tC

tC BA

,

20

11/,

/201110

tDxHCtxC AAAA

,

20

11/,

/201110

tDxHCtxC BBBB

0x

0x

Measuring subdiffusion parametersShort history

Observing single particle

• Single particle tracking D.M. Martin et al. Biophys. J. 83, 2109 (2002), P.R. Smith et al., ibid. 76, 3331 (1999)

• Fluorescence correlation spectroscopy P. Schwille et al., Cytometry 36, 176 (1999)

• Magnetic tweezers F. Amblard et al., PRL 77, 4470 (1996)

• Optical tweezers A. Caspi, PRE 66, 011916 (2002)

Observing concentration profiles

• NMR microscopy A. Klemm et al., PRE 65, 021112 (2002)

• Anomalous time evolution of near membrane layer T. Kosztołowicz, K. Dworecki, S. Mrówczyński, PRL 94, 170602 (2005)

• Anomalous time evolution of reaction front S.B. Yuste, L. Acedo, K. Lindenberg, PRE 69, 036126 (2004), T. Kosztołowicz, K. Lewandowska (submitted)

2

2

1

1 ,,

x

txC

tD

t

txC

Subdiffusion equation

Attention!

t

C

t

C

tso

2

2

1

1 ,,

x

txC

tD

t

txC

is not equivalent to

2

2 ,,

x

txCD

t

txC

The same experimental data as in previous fig. on log-log scale. The solid lines represent the power function At0.45, the dotted lines correspond to the function At0.50.