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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.
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