METHODS OF MEASURING SUBDIFFUSION PARAMETERS Tadeusz 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 - subdiffusion parameter - subdiffusion coefficient Subdiffusion equation

130 mm membrane aqueous solution of 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, (2005)

Near-membrane layer (0,  ) Initial condition

Boundary conditions at the thin membrane or ? ?

In the long time approximation

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 At 0.45.

Transport of glucose and sucrose in agarose gel For glucose: A = ± for  = 0.05,  = 0.45 A = ± for  = 0.08,  = 0.45 A = ± for  = 0.12,  = 0.45 For sucrose: A = ± for  = 0.08,  = 0.45  = 0.90, D 0.90 = (9.8 ± 1.0)  10 –4 mm 2 /s 0.90  = 0.90, D 0.90 = (6.3 ± 0.9)  10 –4 mm 2 /s 0.90

 P =  /A. The line represents the function t 0.45.

MEASUREMENT IN NON-TRANSPARENT MEDIUM theory experiment T. Kosztołowicz, AIP 800 (2005) K. Dworecki, Physica A 359, 24 (2006) PEG2000 in polyprophylene membrane, 180A pore size, 9x10 9 pores/cm 2

Subdiffusion-reaction system C A (x,0) = C 0A H(-x)C B (x,0) = C 0B H(x)

The subdiffusion-reaction equations

Subdiffusion-reaction system

Time evolution of reaction front in subdiffusive system 1. D  A = D  B S.B. Yuste, L. Acedo, K. Lindenberg, PRE 69, (2004) 2. D  A  D  B, D  A, D  B > 0 T. Kosztołowicz, K. Lewandowska cond-mat/ (2006) Phys. Rev. E (submitted) 3. D  A > D  B = 0 T. Kosztołowicz, K. Lewandowska Acta Phys. Pol. 37, 1571 (2006)

The schematic view of the tooth enamel The dotted line represents the concentration of static hydroxyapatite Ca 5 (PO 4 ) 3, the dashed one – the concentration of organic acid HB.

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 x f = 0.39 t Since x f = D  f t  /2, we obtain  = 0.64.

DIFFUSION IMPEDANCE

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

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.

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 x f ~D  f t  /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

Fractional integral

Fractional derivatives and integrals The Riemann-Liouville (RL) definition K.B. Oldham, J. Spanier, The fractional calculus, AP 1974

Examples

Properties of fractional derivatives Leibniz’s formula Linearity Chain rule

Scaling approach

Scaling approach for subdiffusion ?

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

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, (2002) Observing concentration profiles NMR microscopy A. Klemm et al., PRE 65, (2002) Anomalous time evolution of near membrane layer T. Kosztołowicz, K. Dworecki, S. Mrówczyński, PRL 94, (2005) Anomalous time evolution of reaction front S.B. Yuste, L. Acedo, K. Lindenberg, PRE 69, (2004), T. Kosztołowicz, K. Lewandowska (submitted)

Subdiffusion equation Attention! so is not equivalent to

The same experimental data as in previous fig. on log-log scale. The solid lines represent the power function At 0.45, the dotted lines correspond to the function At 0.50.