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Anomalous Transport Bad Honnef, 12th - 16th July,
2006
METHODS OF MEASURING SUBDIFFUSION PARAMETERS
Tadeusz Kosztolowicz
Institute of Physics, Swietokrzyska Academy,
Kielce, Poland
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T. Kosztolowicz, Measuring subdiffusion parameters

• Introduction.
• 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
• Biological application
• Transport of organic acids and salts in the tooth
  enamel
• Final remarks

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Subdiffusion
- subdiffusion parameter
- subdiffusion coefficient
Subdiffusion equation
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Measuring subdiffusion parameters
T. Kosztolowicz, K. Dworecki, S. Mrówczynski, PRL
94, 170602 (2005)
Schematic view of the membrane system
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Near-membrane layer (0,?)
Initial condition
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Boundary conditions at the thin membrane
1.
2.
?
or
?
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In the long time approximation
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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.
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Transport of glucose and sucrose in agarose gel
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
? 0.90, D0.90 (9.8 1.0) ? 104 mm2/s0.90
For sucrose
A 0.064 0.003 for ? 0.08, ? 0.45
? 0.90, D0.90 (6.3 0.9) ? 104 mm2/s0.90
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?P ?/A . The line represents the function
t0.45.
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MEASUREMENT IN NON-TRANSPARENT MEDIUM
theory
T. Kosztolowicz, AIP 800 (2005)
experiment
K. Dworecki, Physica A 359, 24 (2006)
PEG2000 in polyprophylene membrane, 180A pore
size, 9x109 pores/cm2
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Subdiffusion-reaction system
CA(x,0) C0AH(-x)
CB(x,0) C0BH(x)
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The subdiffusion-reaction equations
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Subdiffusion-reaction system
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Time evolution of reaction front in subdiffusive
system
1. D?A D?B
S.B. Yuste, L. Acedo, K. Lindenberg, PRE 69,
036126 (2004)
T. Kosztolowicz, K. Lewandowska cond-mat/0603139
(2006) Phys. Rev. E (submitted)
2. D?A? D?B , D?A, D?B gt 0
T. Kosztolowicz, K. Lewandowska Acta Phys. Pol.
37, 1571 (2006)
3. D?A gt D?B 0
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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.
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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 D?f t ?/2
, we obtain ? 0.64.
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DIFFUSION IMPEDANCE
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A. Compte, R. Metzler, J. Phys. A 30, 7277 (1997)
generalized Cattaneo equation
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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.
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EXPERIMENTAL RESULT
? 0.30 0.06
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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 xfD?ft ?/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.

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

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Fractional derivative
..................................................
..............
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Fractional integral
..................................................
..............
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Fractional derivatives and integrals
The Riemann-Liouville (RL) definition
K.B. Oldham, J. Spanier, The fractional calculus,
AP 1974
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Examples
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Properties of fractional derivatives
Linearity
Chain rule
Leibnizs formula
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Scaling approach
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Scaling approach for subdiffusion ?
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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
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Measuring subdiffusion parameters Short 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. Kosztolowicz, K. Dworecki, S. Mrówczynski, PRL
  94, 170602 (2005)
• Anomalous time evolution of reaction front S.B.
  Yuste, L. Acedo, K. Lindenberg, PRE 69, 036126
  (2004), T. Kosztolowicz, K. Lewandowska
  (submitted)

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Subdiffusion equation
Attention!
so
is not equivalent to
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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.